2 dimensions.\n\nAlthough an exact solution at D > 2 may be challenging, the long distance behavior of the Ising model is not so complex. 2 There is an emergent simplicity in the macroscopic physics.For example, the scaling behavior around the critical point is characterized by some critical exponents, which can be determined to good precision by the Monte Carlo simulations of finite size systems (see e.g.[5,6] and references therein).To study the macroscopic properties, one can further consider the continuum limit, whose effective description is expected to be given by the ϕ 4 field theory.The long stance behavior is determined by an infrared fixed point of the renormalization group (RG) flow, corresponding to a scale invariant quantum field theory.Different physical systems can exhibit similar macroscopic properties if they are governed by the same fixed point, i.e., in the same universality class.In D > 4 dimensions, the Ising critical exponents can be derived from the simple mean field theory due to the averaging effects of a large number of nearby sites, associated with the Gaussian fixed point of the RG flow.For 1 < D < 4, the Ising critical behavior is governed 1 The partition function is given by a sum of 4 g Pfaffians [3], where g is the topological genus and 4 g is the number of spin structures. 2 It is interesting to study the D-dimensional Ising model using the ϵ = D − 2 expansion.See [4] for a conformal bootstrap study of the D = 2 + ϵ Ising model at criticality.by a nontrivial fixed point of the RG flow and the critical exponents become nontrivial functions of D. Furthermore, there is strong numerical evidence that scale invariance of the 3D Ising fixed point is enhanced to conformal invariance [7][8][9][10][11][12][13][14][15][16][17][18].It should be easier to solve the 3D Ising model at criticality.Since the revival of the D > 2 conformal bootstrap program [19], one of the most impressive achievements is the high precision determinations of the low-lying data of the 3D Ising conformal field theory (CFT) [7][8][9][10][11]. 3The precise scaling dimensions and operatorproduct-expansion (OPE) coefficients of the two relevant operators are [11] ∆ σ = 0.5181489 (10) , ∆ ϵ = 1.412625 (10) , λ σσϵ = 1.0518537 (41) .\n\n(1.1)\n\nThe determinations of critical exponents η = 0.0362978 (20), ν = 0.629971(4) are consistent with and more precise than the Monte-Carlo determinations η MC = 0.036284 (40), ν MC = 0.62998 (5) in [6].\n\nIn (1.1), the error bars from the numerical bootstrap bounds are rigorous.The tiny allowed region is deduced from the assumptions of unitarity, two relevant operators, and OPE associativity.The unitarity assumption implies that the OPE coefficients are real numbers, leading to positivity constraints.The number of relevant operators and the unitarity bounds give some lower bounds for the scaling dimensions of the irrelevant operators. 4For 4-point functions, the associativity of OPE leads to the crossing equations, i.e., the convergent summations of conformal blocks in different channels should correspond to the same correlator.The crossing equations can be discretized by the Taylor expansion around a well-chosen point in the rapidly convergent regime, where the dominant contributions are associated with operators of low scaling dimension [23].These nonperturbative constraints are labelled by the numbers of derivatives.Together with the positivity constraints and spectral assumptions, one can use the crossing constraints to rule out the inconsistent parameter space.Using a system of mixed correlators, an isolated region of the parameter space for the 3D Ising CFT was found in [9].\n\nIf a correlator admits a rapidly convergent expansion in terms of conformal blocks, the high-lying contributions are suppressed and thus less constrained by the crossing equation.On the other hand, it is mathematically more subtle to study the implications of the crossing equation in a singular limit, as the conformal block summation may cease to be manifestly convergent. 5Nevertheless, a singular limit of the crossing equation can provide nontrivial information about the high-lying states, which is complementary to the rapidly convergent case.The reason is that the emergence of a singularity requires a collective behavior of infinitely many high-lying states, so one may obtain simple asymptotic formulae for the averaged behavior at large quantum numbers.For example, in the lightcone limit, the dominant contribution of the cross-channel vacuum state implies the double-twist behavior at large spin [25,26]. 6The leading asymptotic behavior coincides with that of the generalized free fields.At low twist, the higher spin results from the numerical bootstrap bounds and the extremal functional method [8,[31][32][33] beautifully organize into Regge trajectories, in accordance with the analytic formulae from the lightcone bootstrap [33][34][35].\n\nThe high precision numerical bootstrap bounds (1.1) are obtained from expensive computations.Suppose that we make use of a global understanding of the high-lying operators.Can we determine the low-lying properties of the 3D Ising CFT reasonably well with less computational efforts?Recently, we performed a bootstrap study of the quartic anharmonic oscillator [36] using the asymptotic behavior and nonperturbative self-consistency constraints [37], which can be viewed as a bootstrap study of the 1D version of the massive ϕ 4 field theory. 7As the 1D bootstrap results exhibit rapid convergence, it is tempting to apply this strategy to the 3D field theory.\n\nIn [33], Simmons-Duffin suggested a hybrid bootstrap approach that combines the lightcone approach [25,26] with the truncation approach [42]. 8In the truncation method initiated by Gliozzi, the contributions of the high-lying operators are simply set to zero.As the high-lying contributions are suppressed near z = z = 1/2, it seems reasonable to omit them in a preliminary study.However, if one wants to systematically improve the results, some small but non-negligible high-lying contributions should be taken into account.To determine the additional free parameters associated with the high-lying contributions, one needs to consider more nonperturbative crossing constraints.Without the help of positivity 9 , the computational complexity of a large system of nonlinear equations grows rapidly. 10In a hybrid approach, the high-lying scaling dimensions and OPE coefficients are approximated by simple analytic formulae from the lightcone bootstrap.As the number of free parameters is reduced, less crossing constraints are needed.In this way, one should be able to improve the results more easily.In [33], the original proposal involves the crossing constraints for mixed correlators.Below, we will show that the hybrid analytical/numerical approach can be realized in a simpler way, i.e., by studying only an identical correlator. 11 As discussed later, we are led to an error minimization approach [47] in the hybrid method.\n\nIn this work, we revisit the conformal bootstrap approach to the 3D Ising model using the two complementary types of crossing constraints: 6 We assume that the identity operator has the lowest twist.In perturbation theory, the twist accumulation phenomenon was discussed already in 1973 [27,28].See also [29] for an earlier nonperturbative argument based on a 2D massive theory perspective.Recently, a more rigorous approach was presented in [30] based on positivity constraints. 7This study was inspired by the recent work on the 0D ϕ 4 theory [38,39] and a bootstrap formulation of the quantum anharmonic oscillator [40].See [41] for more details about a novel type of analytic continuation.\n\n8 See [43] for a study of the Ising CFT at finite temperature using this hybrid approach. 9See [44] for a different hybrid approach that combines the positivity constraints with the large spin perturbation theory. 10Using the rational approximations [45], the transcendental equations can be well approximated by polynomial equations, which can be studied by numerical algebraic geometry methods, such as the efficient package HomotopyContinuation.jl [46].However, the computational complexity still grows rather rapidly. 11The reason is that the mixing of subleading Regge trajectories does not have dominant effects on the identical correlator around z = z = 1/2.A CFT with large mixing effects may not be captured by the crossing equation for an identical correlator.\n\n• The singular z → 1 limit leads to a global understanding of the high-lying data.\n\n• The expansion around z = z = 1/2 gives strong constraints on the low-lying data.\n\nWe assume that the 3D Ising CFT has only two relevant operators.We will consider the 4-point correlator of the lowest Z 2 -odd scalar σ with ∆ σ < 3. The other relevant operator ϵ is a Z 2 -even scalar and appears in the σ × σ OPE.It turns out that reasonably accurate results can be derived easily from a small number of nonperturbative crossing constraints.\n\nIn our main results, the total number of derivatives in the expansion around z = z = 1/2 is at most 7.This paper is organized as follows.In Sec. 2, we give a brief overview of the crossing constraints.In Sec. 3, we discuss how to obtain an approximate solution of the 3D Ising CFT using simple analytic approximations and a few nonperturbative crossing constraints.In Sec. 4, we take into account the leading corrections to the analytic approximations for the high-lying contributions.We use the error minimization approach [47] to obtain more accurate results of the 3D Ising CFT data.In Sec. 5, we summarize our results and discuss some directions for further investigations.\n\n\nCrossing constraints\n\nWe will consider the 4-point correlator of the lowest Z 2 -odd scalar σ:\n⟨σ(x 1 )σ(x 2 )σ(x 3 )σ(x 4 )⟩ = G(z, z) x 2∆σ 12 x 2∆σ 34 ,(2.1)\nwhere x i denote the positions of the external operators, and x ij = |x i −x j | are the distances between two operators.The variables (z, z) are defined by\nz z = x 2 12 x 2 34 x 2 13 x 2 24 , (1 − z)(1 − z) = x 2 14 x 2 32\nx 2 13 x 2 24 .\n\n(2.\n\n2)\n\nThe correlator is invariant under the exchange of σ(x 1 ) and σ(x 3 ), so we have the crossing equation\n(1 − z) ∆σ (1 − z) ∆σ G(z, z) = (z z) ∆σ G(1 − z, 1 − z) ,(2.3)\nwhich gives a relation between the direct-channel and cross-channel OPEs and the corresponding conformal block expansions.We can discretize the crossing equation (2.3) by the Taylor expansion around z = z = 1/2:\ni P i F (m,n) ∆ i ,ℓ i = 0 ,(2.4)\nwhere P i = λ 2 σσO i are the squared OPE coefficients, and\n(∆ i , ℓ i ) = (∆ O i , ℓ O i )\nare the scaling dimension and spin of the primary operator O i .The definition of\nF (m,n) ∆ i ,ℓ i is 12 F (m,n) ∆ i ,ℓ i = ∂ m z ∂ n z (1 − z) ∆σ (1 − z) ∆σ F ∆ i ,ℓ i (z, z) − (z ↔ 1 − z) z=z= 1 2 . (2.5)\nBy construction, the nonperturbative crossing constraints (2.4) vanish automatically if m+n is an even integer.Since the crossing equation (2.3) is symmetric in (z, z), we assume m < n.\n\nIn this work, the normalization of a conformal block is fixed by\nF ∆,ℓ (z, z) = z (∆−ℓ)/2 z(∆+ℓ)/2 (1 + O(z)) + O(z (∆−ℓ)/2+1 ) ,(2.6)\nwhere ∆ is the scaling dimension and ℓ is the spin.\n\nAs explained in the introduction, it is also useful to consider the crossing equation in the singular lightcone limit z → 1.After rewriting the crossing equation (2.3)\nas i P i F ∆ i ,ℓ i (z, z) = (z z) ∆σ (1 − z) ∆σ (1 − z) ∆σ i P i F ∆ i ,ℓ i (1 − z, 1 − z) ,(2.7)\none can show that the leading asymptotic behavior at large spin is associated with the lowest twist operator on the right hand side.The asymptotic behavior can be derived from the relation\nh−h 0 =0,2,4,... 2S p (h) k h (z) ∼ 1 − z z p (z → 1) ,(2.8)\nwhere the coefficient function S p (h) is\nS p (h) ≡ Γ(h − p − 1) Γ(−p) 2 Γ(h + p + 1) Γ(h) 2 Γ(2h − 1) ,(2.9)\nthe\nSL(2, R) block k h (z) is k h (z) ≡ zh 2 F 1 (h, h, 2h, z) ,(2.10)\nand the conformal spin h is related to the twist τ and spin ℓ by\nh ≡ τ 2 + ℓ .(2.11)\nAn exact identity version of (2.8) can be found in [33].\n\n\nAsymptotic behavior from the vacuum state\n\nThe leading asymptotic behavior is associated with the vacuum state in the cross-channel, as the identity operator has the lowest twist in the 3D Ising CFT.Using (2.8) and the small z expansion of the conformal block F ∆,ℓ (z, z), one can derive the large-spin asymptotic behavior of the squared OPE coefficients\nP (k=0) {I} (h) ∼ 2S −∆σ (h) ,(3.1)P (k=1) {I} (h) ∼ (∆ σ − D/2 + 1)(h − ∆ σ )(h + ∆ σ − 1) (h − ∆ σ + D/2 − 1)(h + ∆ σ − D/2) S −∆σ (h) ,(3.2)\nwhere the superscript (k) denotes the trajectory [σσ] k,ℓ with twist τ ∼ 2∆ σ + 2k.The subscript {I} indicates that the asymptotic behavior is associated with the contribution of the cross-channel vacuum state.The analytic formulae (3.1),(3.2) coincide with the OPE coefficients of the generalized free fields (GFF) [49,50].They provide a simple approximation for the averaged behavior of the high-lying states.In fact, the scaling dimensions deviate from the double-twist values ∆ [σσ] k,ℓ = 2∆ σ + 2k + ℓ at finite spin, but the effects of their corrections are subleading.Some of these corrections will be taken into account to improve the accuracy in Sec. 4. The low-lying properties deviate more significantly from the GFF solution, so we should separate the low-lying contributions from the high-lying ones.The approximate G(z, z) contains a low-lying part and a GFF part\nG(z, z) ≈ 1 + P ϵ F ∆ϵ,0 (z, z) + P T F 3,2 (z, z) + (GFF) ,(3.3)\nwhere the GFF part does not include the cases of τ = 2∆ σ with ℓ = 0, 2.Here (∆ ϵ , P ϵ , P T ) are free parameters.Note that ϵ is the Z 2 -even relevant operator with 0 < ∆ ϵ < D. We assume that the lowest spin-two operator corresponds to the stress tensor T , so its scaling dimension is given by ∆ T = D = 3.As the generalized free fields correspond to an exact solution to the crossing equation, we can sum over the GFF contributions and obtain a more explicit form of (3.3):\nG(z, z) ≈ 1 + (z z) ∆σ + (z z) ∆σ [(1 − z)(1 − z)] ∆σ +P ϵ F ∆ϵ,0 (z, z) + P T F 3,2 (z, z) − ℓ=0,2 2S −∆σ (∆ σ + ℓ)F 2∆σ+ℓ,ℓ (z, z) . (3.4)\nThe first line contains the leading terms in the lightcone limit, which is manifestly crossing symmetric and thus reminiscent of the inverse bootstrap approach [51]. 13The second line is subleading in both the direct-channel and cross-channel lightcone limits, but becomes important around z = z = 1/2.To satisfy the crossing constraints (2.4), the contributions of the low-lying operator {ϵ, T } should satisfy\nP ϵ F (m,n) ∆ϵ,0 + P T F (m,n) 3,2 − l=0,2 2S −∆σ (∆ σ + ℓ) F (m,n) 2∆σ+ℓ,ℓ ≈ 0 ,(3.5)\nwhich leads to nontrivial constraints on the free parameters (∆ ϵ , P ϵ , P T ) and ∆ σ .For comparison, we also examine the naive truncation without any high-lying contribution.\n\nThe corresponding crossing constraints read\n(naive truncation) F (m,n) 0,0 + P ϵ F (m,n) ∆ϵ,0 + P T F (m,n) 3,2 ≈ 0 ,(3.6)\nwhich contain no contributions from primary operators with ∆ > 3. The low derivative equations are more sensitive to the low-lying contributions due to the rapid convergence in the Euclidean regime [23], so they are expected to be more useful for constraining the low-lying CFT data. 14According to the total number of derivatives, the leading nonperturbative equations correspond to (m, n) = (0, 1), (0, 3), (1, 2) .\n\n(3.7)\n\nFor a given ∆ σ , we can determine (∆ ϵ , P ϵ , P T ) by these three equations.In Fig. 1, we present the solution for ∆ ϵ at various ∆ σ .The blue dots denote the solutions with GFF approximations for the high-lying contributions, while the orange dots 14 For the rapid OPE convergence, it is natural to consider the crossing symmetric point z = z = 1/2 as the reference point because we want to minimize the radial distance [23] in both the direct-and crosschannels.The region around z = z = 1/2 is still expected to be rapidly convergent in both channels and they can be well approximated by the leading terms of the Taylor series.The low derivative crossing constraints can be viewed as certain linear combinations of those around z = z = 1/2, so they are expected to rapidly convergent as well.On the other hand, we need to use many higher derivative terms in the Taylor expansion to approach the singular lightcone limit and many higher spin operators to reconstruct the singularity structure.Therefore, we expect that the high derivative crossing constraints are more sensitive to the high-lying contributions than the low derivative ones.\n\n\nIsing\n\nare associated with the naive truncation without any ∆ > 3 contribution.We can see that the two types of solutions intersect around the free theory solution, but the results with GFF approximations are much closer to the 3D Ising solution (1.1). 15o determine the external scaling dimension ∆ σ , we should introduce one more equation.If we choose (m, n) = (1, 4), the solutions for the free parameters are\n∆ σ ≈ 0.5177 , ∆ ϵ ≈ 1.399 , P ϵ ≈ 1.113 , P T ≈ 0.1065 ,(3.8)\nwhere we only write 4 significant digits.Despite the crude approximation scheme, the solutions of the crossing constraints are unexpectedly close to the correct values in (1.1) and [33].The relative errors are (−0.08%,−0.9%, 0.6%, 0.1%). 16These results are more accurate than those in the previous 3D Ising studies [42,55] by Gliozzi's truncation method, which considered more scaling dimensions as free parameters and relied on some external input from the Monte Carlo method.The only exception is the prediction for ∆ ϵ in the most complicated example in [55] with 10 free parameters for the scaling dimensions and 1 input scaling dimension from the Monte Carlo method.Let us emphasize that we do not use any external input for the scaling dimensions in this work.If all the scaling dimensions are fixed, it is trivial to determine the OPE coefficients by the error minimization method [47].\n\nWe can also estimate the errors without using the results from the positivity bounds.However, it is not clear how to perform a rigorous error estimation in the truncation-like method, as inequality-like positivity constraints are not used.Here we follow a preliminary procedure that does not allow for correlated movement in the variables.Let us consider the remaining 5-derivative crossing constraints with (m, n) = (0, 5), (2,3).We substitute the free parameters with the crossing solutions in (3.8), except for the one under error estimation.Since we use two constraints, we obtain two different values for the parameter under error estimation.The error is set by the largest deviation of this parameter from the crossing constraints.The results are |δ∆ σ | = 0.0084 , |δ∆ ϵ | = 0.027 , |δP ϵ | = 0.029 , |δP T | = 0.0019, which are greater than the error estimates based on the positivity bound results.The error estimation method is similar to the one in [47].Here we can also introduce the η function and solve the chosen constraints by the η minimization.Since we can solve all the constraints, the corresponding local minimum of the η function is zero and we are not able to use the four constraints to estimate the errors.Therefore, we use the two 5-derivative equations to compute the errors.As the two constraints are not encoded in the η function, the resulting errors are much greater than the estimates based on the positivity bounds.\n\nfurther applies this approach to the five-point bootstrap.In [53], the scaling dimensions are input parameters from the positivity bounds and the GFF approximations are used to improve the determinations of the OPE coefficients.Here we use the GFF improved truncation method to determine both the scaling dimensions and OPE coefficients without relying on any input parameters from the positivity bounds.It is more nontrivial to determine the scaling dimensions than only the OPE coefficients.In Sec. 4, we further consider the leading corrections to the GFF approximations, which are not taken into account in [53].Another difference is that the best known value of ∆σ is used to select the set of nonperturbative crossing constraints by viewing certain ∆σ in the crossing constraints as a free parameter in [53]. 16For spinning operators of low twist, it may be more appropriate to consider a percentage based error in terms of twist.\n\nThe critical exponents are related to the scaling dimensions by ∆ σ = 1/2 + η/2, ∆ ϵ = 3 − 1/ν, so we have η ≈ 0.0355 , ν ≈ 0.6247 , (3.9)\n\nwhere the relative errors of the crossing constraint solutions are also small, i.e. (−2%, 0.8%).\n\nThe error estimates from the crossing constraints with (m, n) = (0, 5), (2, 3) are |δη| = 0.0101 and |δν| = 0.0168.\n\nIf we use a different 5-derivative equation, the results would not be so encouraging.For the (0, 5) and (2, 3) constraints, the solution in the upper blue branch in Fig. 1 becomes ∆ σ ≈ 0.504 and ∆ σ ≈ 0.494 respectively. 17One may suspect that the nice results in (3.8) are a pure coincidence.To address this concern, we will take into account the leading corrections to the GFF behavior in Sec. 4, and show that the results can be further improved.Therefore, the encouraging results (3.8) indeed suggest a cheaper avenue towards an accurate solution of the 3D Ising CFT.\n\n\nAsymptotic behavior from the low-lying states\n\nTo obtain more accurate results, we should improve the approximation scheme for the highlying contributions.For the lowest Z 2 -even Regge trajectory with τ ≈ 2∆ σ , the leading corrections to the GFF behavior can be easily deduced from the cross-channel operators {ϵ, T }, which will be discussed later.For the higher Regge trajectories, the GFF asymptotic behavior is an approximation for the averaged contributions. 18We need to study a mixed system of correlators to extract the unmixed trajectories.Above the lowest trajectory [σσ] 0,ℓ , there are two dominant trajectories with τ [σσ] 1 ∼ 2∆ σ + 2 and τ [ϵϵ] 0 ∼ 2∆ ϵ .Their twist spectra are nearly symmetric about τ = ∆ σ + ∆ ϵ + 1 and have large repulsion at low conformal spin.As the twists of the ℓ = 0, 2 operators have significant deviations from the GFF value τ = 2∆ σ + 2, we will consider their twists as free parameters.Nevertheless, their squared OPE coefficients can be approximated by the GFF asymptotic formula (3.2).This is because the ℓ = 0, 2 operators on the lower trajectory [ϵϵ] 0,ℓ decouple 19 and only one dominant operator contributes at each spin ℓ = 0, 2. At higher spin or higher twist, we will again use the simple GFF approximation as the corrections are highly suppressed by the small OPE coefficients.\n\nAs mentioned above, we should take into account the leading corrections to the GFF approximation for the lowest trajectory [σσ] 0,ℓ , which can be deduced from the cross-channel contributions of the low-lying operators {ϵ, T }.In the lightcone limit z → 1, the leading terms of the cross-channel conformal blocks are\nF ∆,ℓ (1 − z, 1 − z) = (1 − z) τ /2 f h (1 − z) + O[(1 − z) τ /2+1 ] = − Γ(2h) Γ(h) 2 (1 − z) τ /2 2H h−1 + log z + . . . ,(4.1)\nwhere H x = Γ ′ (x + 1)/Γ(x + 1) − Γ ′ (1)/Γ(1) denotes the harmonic number and . . .indicates subleading terms in 1 − z or z.The log z terms are related to the anomalous dimensions by z τ /2 = z ∆σ (1 + τ −2∆σ 2 log z + . . .).The crossing equation (2.7) implies that the leading behavior of the anomalous dimensions and the OPE coefficients are\nγ (k=0) {I,ϵ,T } (h) ∼ −4 i=ϵ,T Pi S −∆σ+τ i /2 (h) P (k=0) {I,ϵ,T } (h) ,(4.2)P (k=0) {I,ϵ,T } (h) ∼ 2S ∆σ (h) − 4 i=ϵ,T H h i −1 Pi S −∆σ+τ i /2 (h) ,(4.3)\nwhere Pi = Γ(2h i ) Γ(h i ) 2 P i and the anomalous dimension is defined as γ i ≡ τ i −2∆ σ .The subscript {I, ϵ, T } indicates the asymptotic behavior is associated with the cross-channel contributions of these three low-lying operators.As before, we can resum the GFF contributions.The nonperturbative crossing constraints can be expressed as the difference between the contributions involving the free parameters and their GFF counterparts:\nG (m,n) {I,ϵ,T } ≡ i=ϵ,T P i F (m,n) ∆ i ,ℓ i + ℓ=4,...,ℓ * P (k=0) {I,ϵ,T } (h) F (m,n) 2h−ℓ,ℓ + ℓ=0,2 P (k=1) {I} ∆ + ℓ 2 F (m,n) ∆,ℓ ∆=∆ (k=1) ℓ − ℓ=0,2,...,ℓ * P (k=0) {I} (∆ σ + ℓ) F (m,n) 2∆σ+ℓ,ℓ − ℓ=0,2 P (k=1) {I} ∆ + ℓ 2 F (m,n) ∆,ℓ ∆=2∆σ+2+ℓ ≈ 0 ,(4.4)\nwhere the conformal spin h in the second summation is approximated by\nh = ∆ σ + ℓ + 1 2 γ (k=0) {I,ϵ,T } (∆ σ + ℓ) . (4.5)\nWhen ℓ * is sufficiently large, the results are not sensitive to this parameter. 20Note that the free parameters ∆ (k=1) ℓ=0,2 are the scaling dimensions of the second lowest Z 2 -even operators with spin ℓ = 0, 2. As explained above, their OPE coefficients are still approximated by the GFF formula (3.2).\n\nSince there are 6 free parameters, we consider the following set of nonperturbative crossing constraints (m, n) = (0, 1), (0, 3), (1, 2), (0, 5), (1,4), (2, 3) , (\n\nwhere the total number of derivatives is at most 5.As the number of free parameters and constraints match, one may want to impose G (m,n) {I,ϵ,T } = 0 with (4.6).However, this will rule out the Ising solution, and we only find the free theory solution. 21As a result, we should not impose that G (m,n) {I,ϵ,T } vanish exactly.To obtain the 3D Ising solution, we need to consider a weaker formulation of the crossing constraints (4.4).\n\nA natural modification is to search for the approximate solutions with small errors in the crossing constraints (4.4).To measure the errors of the crossing constraints, we introduce the η function:\nη = m,n 1 m!n! G (m,n) {I,ϵ,T } 2 ,(4.7)\nwhere we use (m!n!) −2 to suppress the high derivative constraints as they converge slower and are relatively less sensitive to the low-lying parameters.Instead of imposing the exact constraints G (m,n) {I,ϵ,T } = 0, we now search for the local minima of the η function.In [47], the error minimization approach was proposed as a potentially more systematic formulation of the truncation approach initiated by Gliozzi [42]. 22In fact, the minimization formulation of [47] was partly motivated by absence of solutions to some crossing symmetric ansatz in the inverse bootstrap approach [51], which is similar to the situation here.The error minimization approach has been adopted in a number of bootstrap studies [4,21,43,[69][70][71][72][73][74].\n\n\n23\n\nAs expected for the 3D Ising CFT, there exists a local minimum of the η function around (∆ σ , ∆ ϵ ) ≈ (0.518, 1.41), which is indicated by the red point in Fig. 2. The resulting low-lying parameters are24\n∆ σ ≈ 0.51810 , ∆ ϵ ≈ 1.4123 , P ϵ ≈ 1.1053 , P T ≈ 0.10650 ,(4.8)\nwhere only 5 significant digits are presented.For comparison, the accuracy of (4.8) is better than the first positive bootstrap results for the 3D Ising model in [7], even though the nonperturbative constraints in [7] involve 23 derivatives and the number of constraints was significantly greater than our case with at most 5 derivatives.Using the positivity bound results as reference values, the relative errors of the η minimization results are (−0.009%,−0.02%, −0.1%, 0.1%).\n\nWe can also estimate the errors without using the positivity bound results.Let us substitute the free parameters with the η minimization results in (4.8), except for the one under error estimation, and then solve for this parameter using one of the crossing constraints in (4.6).The largest deviation from the η minimization result in (4.8) gives an error estimate of this parameter [47] The corresponding critical exponents are\nη ≈ 0.03620 , ν ≈ 0.62984 ,(4.9)\nwhere the relative errors of the η minimization results are (−0.3%,−0.02%).These results are more accurate than those in Sec. 3. The estimated errors without using the positivity bound results are |δη| = 0.00290 , |δν| = 0.00030.We also obtain the scaling dimensions of the low spin operators in the higher trajectory (∆\n(k=1) ℓ=0 , ∆(k=1)\nℓ=2 ) ≈ (3.68, 6.29), which are less accurate.For comparison, the estimates for (∆ [33] are (3.82968(23),5.50915 (44)).\n(k=1) ℓ=0 , ∆ (k=1) ℓ=2 ) in\nUsing the precise bootstrap bound results as the input parameters for the analytic formulae, one can compare the analytic asymptotic formulae with the numerical estimates [33][34][35], whose agreement is surprisingly good and motivated the present work.For completeness, we also present the comparisons in Fig. 3 and Fig. 4, whose agreement with the numerical estimates [33] is excellent for ℓ ≥ 4. The main difference from the earlier studies [33][34][35] is that the input parameters (∆ σ , ∆ ϵ , P ϵ , P T ) in (4.8) are not extracted from the bootstrap bound results.As a step forward, both the low-lying scaling dimensions\n\n{I,ϵ,T } (∆ σ + ℓ).The red dots are the numerical estimates from [33]. and OPE coefficients are determined by the hybrid numerical/analytic approach and the η minimization, without relying on the results from other approaches. 25 After taking into account the leading corrections to the GFF behavior, our results also become much more stable under a change in the selection of derivatives, in comparison to those in Sec. 3. If we substitute the (m, n) = (0, 5) constraint with a 7 derivative constraint, the results are less accurate than the results (4.8) with at most 5 derivatives, but still better than the results (3.8) 25 Since the conformal blocks are highly nonlinear functions in the scaling dimensions, it is more difficult to determine the low-lying scaling dimensions than the coefficients of the conformal blocks.In [43,53,74], the low-lying scaling dimensions are input parameters from the positivity bounds or integrability method and the improved truncation methods are mainly used to determine the coefficients of conformal blocks.\n\n{I,ϵ,T } (h) in (4.3),where h is approximated by (4.5).The red dots are the numerical estimates from [33], which have noticeable errors at ℓ > 26.\n\nwhere the local minimum of the η function is zero expect for the first case.If we replace the (1,4) or (2, 3) constraint with a 7-derivative one, we still find a local minimum of the η function around the 3D Ising solution, but the accuracy is lower than (3.8).It seems that the crossing constraints with m ∼ n are more sensitive to the low-lying contributions than those with m ≪ n, so the former can lead to more accurate determinations for the low-lying parameters.\n\nWe can also add more constraints to the η functions, leading to an overdetermined system with η min > 0. If we use all the crossing constraints with m + n ≤ 7, the local minimum of the η function around (∆ σ , ∆ ϵ ) = (0.52, 1.41) gives ∆ σ ≈ 5.1974 , ∆ ϵ ≈ 1.4192 , P ϵ ≈ 1.1019 , P T ≈ 0.10715 and (∆ 75, 5.38).Although less accurate in comparison to (4.8), the η minimization results for the relevant operators are still around the 3D Ising solution.On the other hand, the irrelevant scaling dimensions (∆\n(k=1) ℓ=0 , ∆ (k=1) ℓ=2 ) ≈ (3.(k=1) ℓ=0 , ∆(k=1)\nℓ=2 ) become more accurate. 26This is consistent with our expectation that the higher derivative equations are more sensitive to the high-lying contributions, so the predic- 26 We also notice this improvement for the irrelevant scaling dimensions by replacing a 5-derivative constraint with a 7-derivative constraint.For example, the cases for (4.10), (4.11), (4.12), (4.13) are (∆ tions for the irrelevant operators are improved, at the price of lowering the accuracy of the low-lying parameters, which are contaminated by the less accurate high-lying contributions.\n\nSince we have 10 constraints with m + n ≤ 7, we can introduce more free parameters.It is natural to consider the lowest spin-4 operator, as its scaling dimension ∆ (k=0) ℓ=4 ≈ 5 is close to the subleading spin-2 scaling dimension, which is viewed as a free parameter in the above.Surprisingly, we arrive at even more accurate results by relaxing the scaling dimension and OPE coefficient of the lowest spin-4 operator: ∆ σ ≈ 0.518152 , ∆ ϵ ≈ 1.41278 , P ϵ ≈ 1.10640 , P T ≈ 0.106226 .are almost as accurate as the latest Monte-Carlo determinations [6].\n\nAs the system is still overdetermined, one may try to introduce more free parameters at higher spin.However, if we do not approximate the lowest spin-6 parameters by the analytic formulae and set them as free parameters, we are not able to find a local minimum around the 3D Ising solution.It seems that the η minimization becomes unstable due to the fact that the squared OPE coefficient of the spin-6 operator is allowed to be negative.Besides reducing computational efforts, stabilization is another important merit of analytic approximations for the high-lying contributions.This is different from the hybrid approach in [44], in which the use of positivity constraints stabilizes the hybrid bootstrap method and thus the effects of the choice for the gluing spin can be studied more systematically.\n\n\nDiscussion\n\nIn this work, we revisited the conformal bootstrap approach to the 3D Ising model using the global asymptotic behavior and the rapidly convergent crossing constraints.We derived surprisingly accurate results (4.14) from simple analytic asymptotic formulae and a small number of nonperturbative crossing constraints, using the error minimization method.In some sense, this furnishes a minimal example for the nonperturbative conformal bootstrap in three dimensions, with the help of an analytic understanding from the lightcone singularity structure.\n\nTo improve the accuracy of the hybrid approach, it is crucial to go beyond the leading corrections to the GFF behavior. 27The asymptotic formulae can be computed more systematically using the methods in [33,75,76].However, one should be more careful as a higher order asymptotic expansion may give worse results at low spin, even if the approximations at large spin are improved.Then the corresponding low spin parameters should be viewed as free parameters and determined by the nonperturbative crossing constraints.To avoid this subtlety, one may use the convergent Lorentzian inversion formula [77][78][79], which includes the nonperturbative contributions [80][81][82][83][84][85][86][87]. 28 29However, the price to pay is a higher computational cost.In practice, it may be useful to combine the asymptotic methods with the convergent methods.\n\nWe also need to unmix the higher trajectories. 30In the present work, we used the decoupling phenomenon to circumvent the mixing problem, and carried out the bootstrap study using only the identical correlator ⟨σ(x 1 )σ(x 2 )σ(x 3 )σ(x 4 )⟩.To properly resolve the mixing problem, one needs to consider a mixed system of correlators [33,[85][86][87].The large repulsion of the twist spectra at low spin is particularly interesting.This can lead to more significant deviations from the leading GFF behavior.We also believe that this is closely related to the decoupling phenomenon in [7,8].\n\nWe would like to emphasize that the goal of the present work is not to surpass the remarkably precise results from the positivity bound approach. 31Instead, we want to show that a hybrid analytical/numerical approach can have the great potential in reducing the computational cost with the help of a global understanding of the high-lying data.The application to the 3D Ising model is just the first example.This strategy can be applied to other well constrained bootstrap solutions, such as the O(N ) models [11,[93][94][95] and the supersymmetric Ising model [87,96,97].For a more complete list of the conformal bootstrap targets, we refer to the comprehensive review [98] and the more recent one [99].As we are not using any positivity constraint, a natural and important direction is to consider the less explored non-positive bootstrap problems that involve D > 2 conformal field theories with nonunitarity [42], defects [100,101], and on nontrivial manifolds [57,102].\n\nFreeFigure 1 :\n1\nFigure 1:The approximate solution for ∆ ϵ at various ∆ σ .The red dots represent the free theory and the Ising solutions in three dimensions.The blue dots are obtained from solving (3.5) with GFF approximations for the high-lying contributions.The orange dots are derived from solving (3.6) with the naive truncation that omits all the high-lying contributions.The nonperturbative crossing constraints are associated with the derivative orders in(3.7).The results with the GFF approximations (blue) are much closer to the 3D Ising solution (1.1).\n\n\nFigure 2 :\n2\nFigure 2: The minimized η function (4.7) at different (∆ σ , ∆ ϵ ).The local minimum (red dot) with η min ≈ 0.00043 determines the 3D Ising results in (4.8).\n\n\nFigure 3 :\n3\nFigure 3: The twist τ = ∆ − ℓ of the lowest Z 2 -even trajectory at various spin ℓ.The curve is based on the analytic expression τ (ℓ) = 2∆ σ + γ\n\n\nFigure 4 :\n4\nFigure 4: The squared OPE coefficients of the leading Z 2 -even trajectory at different spin ℓ, divided by the generalized free case P GFF (ℓ) = 2S −∆σ (∆ σ + ℓ).The curve is based on the analytic expression P (ℓ) = P\n\n\n\n\nℓ=2 ) ≈ (3.75, 6.87), (3.74, 5.36), (3.77, 5.22), (3.72, 5.44).\n\n\n\n\nℓ=2 ) ≈ (3.67, 5.46).These are the most accurate results in this work.In particular, the OPE coefficient λ σσϵ = P 1/2 ϵ ≈ 1.05186 is significantly improved.The local minimization is carried out by Mathematica's FindMinimum, starting at (∆ σ , ∆ ϵ ) = (1, 3/2), (∆ (k=1) ℓ=0 , ∆ (k=1) ℓ=2 , ∆ (k=0) ℓ=4 ) = (3, 5, 5) and (P ϵ , P T , P (k=0) ℓ=4 ) = (1, 1/10, 1/200), which are some crude numbers from the GFF approximation.The corresponding critical exponents η ≈ 0.036304 , ν ≈ 0.63003 (4.15)\n\n\n\n\nbased on the GFF approximations.The corresponding η minimization results are (0, 7) : ∆ σ ≈ 0.51781 , ∆ ϵ ≈ 1.4122 , P ϵ ≈ 1.1042 , P T ≈ 0.10642 ,\n(4.10)(1, 6) : ∆ σ ≈ 0.51867 , ∆ ϵ ≈ 1.4133 , P ϵ ≈ 1.1069 , P T ≈ 0.10658 ,(4.11)(2, 5) : ∆ σ ≈ 0.51876 , ∆ ϵ ≈ 1.4137 , P ϵ ≈ 1.1069 , P T ≈ 0.10659 ,(4.12)(3, 4) : ∆ σ ≈ 0.51862 , ∆ ϵ ≈ 1.4131 , P ϵ ≈ 1.1068 , P T ≈ 0.10657 ,(4.13)\nIn D = 2 dimensions, the Ising conformal field theory is exactly solvable by the conformal bootstrap method and is associated with the simplest unitary minimal model[20].\nIn fact, spectral boundedness is a strong assumption[21,22], which can play a similar central role in the nonpositive bootstrap as the positivity constraints in the bootstrap bound approach.\nSee the section 2.4 of[24] for some examples of dangerous limits.\nIn this work, the conformal blocks are computed numerically using the simpleboot package[48].\nSee[52] for a recent application of the inverse bootstrap approach to the D > 2 Anderson transitions.\nNote added: The improvements by GFF or mean field approximations are also reported in[53], which\nFor the naive truncation (3.6), we do not find a reasonable solution for the Ising scaling dimension ∆σ using a five-derivative equation.\nThe higher trajectories are expected to exhibit more complex behaviors due to the mixing and exponentiation effects.\nWe find that ℓ * = 10 is large enough in our discussion.\nWe do not find a solution satisfying all the nonperturbative constraints around (∆σ, ∆ϵ) = (0.52, 1.4).\nSee[55][56][57][58][59][60][61][62][63][64][65][66][67][68] for the use of Gliozzi's determinant method or the singular value method in the conformal bootstrap studies.\nThe minimization can also be implemented with stochastic algorithms[69][70][71][72]74]. In the recent work[74], the determinations of the OPE coefficients from the truncation method were improved by using 10 exact scaling dimensions from the integrability method and introducing 52 effective operators to approximate the high-lying contributions, where the scaling dimensions of the effective operators are free parameters and are determined by the stochastic minimization under some positivity constraints. In this work, we do not use any external input scaling dimensions nor positivity constraints.\nWe use Mathematica's FindMinimum. The working precision is set to 200.\nSome examples of the subleading corrections are the Jacobian factors in the OPE coefficients and the higher spin contributions from the lowest twist trajectory.\nThe choice of z may be determined by the nonperturbative crossing constraints as the other free parameters.\nA more rigorous alternative approach is to use the CFT dispersive sum rules[88]. See also[24,[89][90][91][92].\nThe high-lying spectrum of the 2D Ising CFT is highly degenerate. One can derive the exact solutions without unmixing the high-lying contributions because the degenerate scaling dimensions are exactly the same. In this way, the averaged behavior encodes the exact information about the total contributions.\nWe need to take into account the subleading effects in a systematic and stable way, which is still not completely clear at this point, but we will try to achieve this in the future.\n19We believe that the absence of these two operators is closely related to the leading kink behavior in[7,8].They may violate the positivity constraints if present[54].AcknowledgmentsI would like to thank Junchen Rong and Ning Su for discussions.I also thank the referee for many constructive suggestions.This work was supported by the Natural Science Foundation of China (Grant No. 12205386) and the Guangzhou Municipal Science and Technology Project (Grant No. 2023A04J0006).\nCrystal statistics. 1. A Two-dimensional model with an order disorder transition. L Onsager, 10.1103/PhysRev.65.117Phys. Rev. 651944\n\nCombinatorial and topological approach to the 3-D Ising model. T Regge, R Zecchina, 10.1088/0305-4470/33/4/308arXiv:cond-mat/9909168J. Phys. A. 332000cond-mat\n\nDimer statistics and phase transitions. P Kasteleyn, 10.1063/1.1703953J. Math. Phys. 41963\n\nIsing model close to d=2. W Li, 10.1103/PhysRevD.105.L091902arXiv:2107.13679Phys. Rev. D. 1059L0919022022hep-th\n\nFinite size scaling study of lattice models in the three-dimensional Ising universality class. M Hasenbusch, 10.1103/PhysRevB.82.174433arXiv:1004.4486Phys. Rev. B. 821744332010cond-mat.stat-mech\n\nRestoring isotropy in a three-dimensional lattice model: The Ising universality class. M Hasenbusch, 10.1103/PhysRevB.104.014426arXiv:2105.09781Phys. Rev. B. 1041144262021cond-mat.stat-mech\n\nSolving the 3D Ising Model with the Conformal Bootstrap. S El-Showk, M F Paulos, D Poland, S Rychkov, D Simmons-Duffin, A Vichi, 10.1103/PhysRevD.86.025022arXiv:1203.6064Phys. Rev. D. 86250222012hep-th\n\nSolving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents. S El-Showk, M F Paulos, D Poland, S Rychkov, D Simmons-Duffin, A Vichi, 10.1007/s10955-014-1042-7arXiv:1403.4545J. Stat. Phys. 1572014hep-th\n\nBootstrapping Mixed Correlators in the 3D Ising Model. F Kos, D Poland, D Simmons-Duffin, 10.1007/JHEP11(2014)109arXiv:1406.4858JHEP. 112014hep-th\n\nA Semidefinite Program Solver for the Conformal Bootstrap. D Simmons-Duffin, 10.1007/JHEP06(2015)174arXiv:1502.02033JHEP. 061742015hep-th\n\nPrecision Islands in the Ising and O(N ) Models. F Kos, D Poland, D Simmons-Duffin, A Vichi, 10.1007/JHEP08(2016)036arXiv:1603.04436JHEP. 08362016hep-th\n\nLine defects in the 3d Ising model. M Billó, M Caselle, D Gaiotto, F Gliozzi, M Meineri, R Pellegrini, 10.1007/JHEP07(2013)055arXiv:1304.4110JHEP. 07552013hep-th\n\nConformal symmetry of the critical 3D Ising model inside a sphere. C Cosme, J M V P Lopes, J Penedones, 10.1007/JHEP08(2015)022arXiv:1503.02011JHEP. 08222015hep-th\n\nUncovering Conformal Symmetry in the 3D Ising Transition: State-Operator Correspondence from a Quantum Fuzzy Sphere Regularization. W Zhu, C Han, E Huffman, J S Hofmann, Y C He, 10.1103/PhysRevX.13.021009arXiv:2210.13482Phys. Rev. X. 132210092023cond-mat.stat-mech\n\nOperator Product Expansion Coefficients of the 3D Ising Criticality via Quantum Fuzzy Spheres. L Hu, Y C He, W Zhu, 10.1103/PhysRevLett.131.031601arXiv:2303.08844Phys. Rev. Lett. 1313316012023cond-mat.stat-mech\n\nConformal four-point correlators of the three-dimensional Ising transition via the quantum fuzzy sphere. C Han, L Hu, W Zhu, Y C He, 10.1103/PhysRevB.108.235123arXiv:2306.04681Phys. Rev. B. 108232351232023cond-mat.stat-mech\n\n3D Ising CFT and exact diagonalization on icosahedron: The power of conformal perturbation theory. B X Lao, S Rychkov, 10.21468/SciPostPhys.15.6.243arXiv:2307.02540SciPost Phys. 1562432023hep-th\n\nSolving Conformal Defects in 3D Conformal Field Theory using Fuzzy Sphere Regularization. L Hu, Y C He, W Zhu, arXiv:2308.01903cond-mat.stat-mech\n\nBounding scalar operator dimensions in 4D CFT. R Rattazzi, V S Rychkov, E Tonni, A Vichi, 10.1088/1126-6708/2008/12/031arXiv:0807.0004JHEP. 12312008hep-th\n\nInfinite Conformal Symmetry in Two-Dimensional Quantum Field Theory. A A Belavin, A M Polyakov, A B Zamolodchikov, 10.1016/0550-3213(84)90052-XNucl. Phys. B. 2411984\n\nNull bootstrap for non-Hermitian Hamiltonians. W Li, 10.1103/PhysRevD.106.125021arXiv:2202.04334Phys. Rev. D. 106121250212022hep-th\n\nTaming Dyson-Schwinger Equations with Null States. W Li, 10.1103/PhysRevLett.131.031603arXiv:2303.10978Phys. Rev. Lett. 1313316032023hep-th\n\nOPE Convergence in Conformal Field Theory. D Pappadopulo, S Rychkov, J Espin, R Rattazzi, 10.1103/PhysRevD.86.105043arXiv:1208.6449Phys. Rev. D. 861050432012hep-th\n\nNonperturbative Mellin Amplitudes: Existence, Properties, Applications. J Penedones, J A Silva, A Zhiboedov, 10.1007/JHEP08(2020)031arXiv:1912.11100JHEP. 08312020hep-th\n\nThe Analytic Bootstrap and AdS Superhorizon Locality. A L Fitzpatrick, J Kaplan, D Poland, D Simmons-Duffin, 10.1007/JHEP12(2013)004arXiv:1212.3616JHEP. 1242013hep-th\n\nConvexity and Liberation at Large Spin. Z Komargodski, A Zhiboedov, 10.1007/JHEP11(2013)140arXiv:1212.4103JHEP. 111402013hep-th\n\nHow to measure the dimension of the parton field. G Parisi, 10.1016/0550-3213(73)90666-4Nucl. Phys. B. 591973\n\nBjorken scaling in quantum field theory. C G CallanJr, D J Gross, 10.1103/PhysRevD.8.4383Phys. Rev. D. 81973\n\nComments on operators with large spin. L F Alday, J M Maldacena, 10.1088/1126-6708/2007/11/019arXiv:0708.0672JHEP. 11192007hep-th\n\nTwist Accumulation in Conformal Field Theory: A Rigorous Approach to the Lightcone Bootstrap. S Pal, J Qiao, S Rychkov, 10.1007/s00220-023-04767-warXiv:2212.04893Commun. Math. Phys. 40232023hep-th\n\nD Poland, D Simmons-Duffin, 10.1007/JHEP05(2011)017arXiv:1009.2087Bounds on 4D Conformal and Superconformal Field Theories. 20110517hep-th\n\nBootstrapping Conformal Field Theories with the Extremal Functional Method. S El-Showk, M F Paulos, 10.1103/PhysRevLett.111.241601arXiv:1211.2810Phys. Rev. Lett. 111242416012013hep-th\n\nThe Lightcone Bootstrap and the Spectrum of the 3d Ising CFT. D Simmons-Duffin, 10.1007/JHEP03(2017)086arXiv:1612.08471JHEP. 03862017hep-th\n\nConformal Bootstrap With Slightly Broken Higher Spin Symmetry. L F Alday, A Zhiboedov, 10.1007/JHEP06(2016)091arXiv:1506.04659JHEP. 06912016hep-th\n\nAn Algebraic Approach to the Analytic Bootstrap. L F Alday, A Zhiboedov, 10.1007/JHEP04(2017)157arXiv:1510.08091JHEP. 041572017hep-th\n\nAnharmonic oscillator. C M Bender, T T Wu, 10.1103/PhysRev.184.1231Phys. Rev. 1841969\n\nPrinciple of minimal singularity for Green's functions. W Li, 10.1103/PhysRevD.109.045012arXiv:2309.02201Phys. Rev. D. 1094450122024hep-th\n\nUnderdetermined Dyson-Schwinger Equations. C M Bender, C Karapoulitidis, S P Klevansky, 10.1103/PhysRevLett.130.101602arXiv:2211.13026Phys. Rev. Lett. 130101016022023math-ph\n\nDyson-Schwinger equations in zero dimensions and polynomial approximations. C M Bender, C Karapoulitidis, S P Klevansky, 10.1103/PhysRevD.108.056002arXiv:2307.01008Phys. Rev. D. 1085560022023math-ph\n\nBootstrapping Matrix Quantum Mechanics. X Han, S A Hartnoll, J Kruthoff, 10.1103/PhysRevLett.125.041601arXiv:2004.10212Phys. Rev. Lett. 1254416012020hep-th\n\nW Li, arXiv:2402.05778The ϕ n trajectory bootstrap. hep-th\n\nMore constraining conformal bootstrap. F Gliozzi, 10.1103/PhysRevLett.111.161602arXiv:1307.3111Phys. Rev. Lett. 1111616022013hep-th\n\nBootstrapping the 3d Ising model at finite temperature. L Iliesiu, M Kolouglu, D Simmons-Duffin, 10.1007/JHEP12(2019)072arXiv:1811.05451JHEP. 12722019hep-th\n\nThe Hybrid Bootstrap. N Su, arXiv:2202.07607hep-th\n\nBootstrapping the O(N ) vector models. F Kos, D Poland, D Simmons-Duffin, 10.1007/JHEP06(2014)091arXiv:1307.6856JHEP. 06912014hep-th\n\nHomotopyContinuation.jl: A Package for Homotopy Continuation in Julia. P Breiding, S Timme, Mathematical Software -ICMS 2018, J.H. Davenport, M. Kauers, G. Labahn and J. Urban2018Springer International Publishing\n\nNew method for the conformal bootstrap with OPE truncations. W Li, arXiv:1711.09075hep-th\n\nsimpleboot: A mathematica framework for bootstrap calculations. N Su, \n\nI Heemskerk, J Penedones, J Polchinski, J Sully, 10.1088/1126-6708/2009/10/079arXiv:0907.0151Holography from Conformal Field Theory. 20091079hep-th\n\nUnitarity and the Holographic S-Matrix. A L Fitzpatrick, J Kaplan, 10.1007/JHEP10(2012)032arXiv:1112.4845JHEP. 10322012hep-th\n\nW Li, 10.1007/JHEP01arXiv:1706.04054Inverse Bootstrapping Conformal Field Theories. 2018. 20180177hep-th\n\nConformal Invariance and Multifractality at Anderson Transitions in Arbitrary Dimensions. J Padayasi, I A Gruzberg, 10.1103/PhysRevLett.131.266401arXiv:2306.07340Phys. Rev. Lett. 131262664012023cond-mat.dis-nn\n\nImproving the five-point bootstrap. D Poland, V Prilepina, P Tadić, arXiv:2312.13344hep-th\n\nFactorized lightcone expansion of conformal blocks. W Li, 10.1007/JHEP05(2021)128arXiv:2012.09710JHEP. 128052021hep-th\n\nCritical exponents of the 3d Ising and related models from Conformal Bootstrap. F Gliozzi, A Rago, 10.1007/JHEP10(2014)042arXiv:1403.6003JHEP. 10422014hep-th\n\nBoundary and Interface CFTs from the Conformal Bootstrap. F Gliozzi, P Liendo, M Meineri, A Rago, 10.1007/JHEP05(2015)036arXiv:1502.07217JHEP. 05362015hep-th\n\nBootstrapping critical Ising model on three-dimensional real projective space. Y Nakayama, 10.1103/PhysRevLett.116.141602arXiv:1601.06851Phys. Rev. Lett. 116141416022016hep-th\n\nTruncatable bootstrap equations in algebraic form and critical surface exponents. F Gliozzi, 10.1007/JHEP10(2016)037arXiv:1605.04175JHEP. 10372016hep-th\n\nClosure of the Operator Product Expansion in the Non-Unitary Bootstrap. I Esterlis, A L Fitzpatrick, D Ramirez, 10.1007/JHEP11(2016)030arXiv:1606.07458JHEP. 11302016hep-th\n\nC Hasegawa, Y Nakayama, 10.1142/S0217732317500456arXiv:1611.06373Expansion in Critical ϕ 3 -Theory on Real Projective Space from Conformal Field Theory. 2017321750045hep-th\n\nConformal bootstrap analysis for the Yang-Lee edge singularity. S Hikami, 10.1093/ptep/pty054arXiv:1707.04813PTEP. 201852018hep-th\n\nConformal Bootstrap Analysis for Single and Branched Polymers. S Hikami, 10.1093/ptep/pty132arXiv:1708.03072PTEP. 2018122018hep-th\n\nDimensional Reduction by Conformal Bootstrap. S Hikami, 10.1093/ptep/ptz081arXiv:1801.09052PTEP. 201982019cond-mat.dis-nn\n\nConformal bootstrap for percolation and polymers. A Leclair, J Squires, 10.1088/1742-5468/aaf10aarXiv:1802.08911J. Stat. Mech. 1812121231052018hep-th\n\nS Hikami, arXiv:1811.05918Conformal Bootstrap Analysis for Localization: Symplectic Case. cond-mat.dis-nn\n\nOn the ϕ 3 theory above six dimensions. J Rong, J Zhu, 10.1007/JHEP04(2020)151arXiv:2001.10864JHEP. 041512020hep-th\n\nIs there supersymmetric Lee-Yang fixed point in three dimensions?. Y Nakayama, 10.1142/S0217751X21501761arXiv:2104.13570Int. J. Mod. Phys. A. 362421501762021hep-th\n\nThe extraordinary boundary transition in the 3d O(N) model via conformal bootstrap. J Padayasi, A Krishnan, M A Metlitski, I A Gruzberg, M Meineri, 10.21468/SciPostPhys.12.6.190arXiv:2111.03071SciPost Phys. 1262022cond-mat.stat-mech\n\nSolving Conformal Field Theories with Artificial Intelligence. G Kántor, V Niarchos, C Papageorgakis, 10.1103/PhysRevLett.128.041601arXiv:2108.08859Phys. Rev. Lett. 1284416012022hep-th\n\nConformal bootstrap with reinforcement learning. G Kántor, V Niarchos, C Papageorgakis, 10.1103/PhysRevD.105.025018arXiv:2108.09330Phys. Rev. D. 1052250182022hep-th\n\nMonte Carlo approach to the conformal bootstrap. A Laio, U L Valenzuela, M Serone, 10.1103/PhysRevD.106.025019arXiv:2206.05193Phys. Rev. D. 1062250192022hep-th\n\n6D (2,0) bootstrap with the soft-actor-critic algorithm. G Kántor, V Niarchos, C Papageorgakis, P Richmond, 10.1103/PhysRevD.107.025005arXiv:2209.02801Phys. Rev. D. 1072250052023hep-th\n\nThe five-point bootstrap. D Poland, V Prilepina, P Tadić, 10.1007/JHEP10(2023)153arXiv:2305.08914JHEP. 101532023hep-th\n\nBootstrability in line-defect CFTs with improved truncation methods. V Niarchos, C Papageorgakis, P Richmond, A G Stapleton, M Woolley, 10.1103/PhysRevD.108.105027arXiv:2306.15730Phys. Rev. D. 108101050272023hep-th\n\nLarge Spin Perturbation Theory for Conformal Field Theories. L F Alday, 10.1103/PhysRevLett.119.111601arXiv:1611.01500Phys. Rev. Lett. 119111116012017hep-th\n\nSolving CFTs with Weakly Broken Higher Spin Symmetry. L F Alday, 10.1007/JHEP10(2017)161arXiv:1612.00696JHEP. 101612017hep-th\n\nAnalyticity in Spin in Conformal Theories. S Caron-Huot, 10.1007/JHEP09(2017)078arXiv:1703.00278JHEP. 1709782017hep-th\n\nA spacetime derivation of the Lorentzian OPE inversion formula. D Simmons-Duffin, D Stanford, E Witten, 10.1007/JHEP07arXiv:1711.03816JHEP. 1807852018. 2018hep-th\n\nLight-ray operators in conformal field theory. P Kravchuk, D Simmons-Duffin, 10.1007/JHEP11arXiv:1805.00098JHEP. 18111022018. 2018hep-th\n\nJ Liu, E Perlmutter, V Rosenhaus, D Simmons-Duffin, 10.1007/JHEP03(2019)052arXiv:1808.00612d-dimensional SYK, AdS Loops, and 6j Symbols. 20190352hep-th\n\nResummation at finite conformal spin. C Cardona, S Guha, S K Kanumilli, K Sen, 10.1007/JHEP01(2019)077arXiv:1811.00213JHEP. 1901772019hep-th\n\nMore Analytic Bootstrap: Nonperturbative Effects and Fermions. S Albayrak, D Meltzer, D Poland, 10.1007/JHEP08(2019)040arXiv:1904.00032JHEP. 1908402019hep-th\n\nClosed-form expression for cross-channel conformal blocks near the lightcone. W Li, 10.1007/JHEP01(2020)055arXiv:1906.00707JHEP. 552001. 2020hep-th\n\nThe Inversion Formula and 6j Symbol for 3d Fermions. S Albayrak, D Meltzer, D Poland, 10.1007/JHEP09(2020)148arXiv:2006.07374JHEP. 091482020hep-th\n\nThe Lorentzian inversion formula and the spectrum of the 3d O(2) CFT. J Liu, D Meltzer, D Poland, D Simmons-Duffin, 10.1007/JHEP09(2020)115arXiv:2007.07914JHEP. 091152020erratum: JHEP 01, 206 (2021). hep-th\n\nThe leading trajectory in the 2+1D Ising CFT. S Caron-Huot, Y Gobeil, Z Zahraee, 10.1007/JHEP02(2023)190arXiv:2007.11647JHEP. 021902023hep-th\n\nPrecision bootstrap for the N = 1 super-Ising model. A Atanasov, A Hillman, D Poland, J Rong, N Su, 10.1007/JHEP08(2022)136arXiv:2201.02206JHEP. 081362022hep-th\n\nDispersive CFT Sum Rules. S Caron-Huot, D Mazac, L Rastelli, D Simmons-Duffin, 10.1007/JHEP05(2021)243arXiv:2008.04931JHEP. 052432021hep-th\n\nA Conformal Dispersion Relation: Correlations from Absorption. D Carmi, S Caron-Huot, 10.1007/JHEP09(2020)009arXiv:1910.12123JHEP. 0992020hep-th\n\nA basis of analytic functionals for CFTs in general dimension. D Mazáč, L Rastelli, X Zhou, 10.1007/JHEP08(2021)140arXiv:1910.12855JHEP. 081402021hep-th\n\nApplications of dispersive sum rules: epsilon-expansion and holography. D Carmi, J Penedones, J A Silva, A Zhiboedov, 10.21468/SciPostPhys.10.6.145arXiv:2009.13506SciPost Phys. 1062021hep-th\n\nMixed correlator dispersive CFT sum rules. A K Trinh, 10.1007/JHEP03(2022)032arXiv:2111.14731JHEP. 03322022hep-th\n\nBootstrapping the O(N) Archipelago. F Kos, D Poland, D Simmons-Duffin, A Vichi, 10.1007/JHEP11(2015)106arXiv:1504.07997JHEP. 111062015hep-th\n\nCarving out OPE space and precise O(2) model critical exponents. S M Chester, W Landry, J Liu, D Poland, D Simmons-Duffin, N Su, A Vichi, 10.1007/JHEP06(2020)142arXiv:1912.03324JHEP. 061422020hep-th\n\nBootstrapping Heisenberg magnets and their cubic instability. S M Chester, W Landry, J Liu, D Poland, D Simmons-Duffin, N Su, A Vichi, 10.1103/PhysRevD.104.105013arXiv:2011.14647Phys. Rev. D. 104101050132021hep-th\n\nBootstrapping the minimal N = 1 superconformal field theory in three dimensions. J Rong, N Su, 10.1007/JHEP06(2021)154arXiv:1807.04434JHEP. 061542021hep-th\n\nBootstrapping the Minimal 3D SCFT. A Atanasov, A Hillman, D Poland, 10.1007/JHEP11arXiv:1807.05702JHEP. 111402018. 2018hep-th\n\nThe Conformal Bootstrap: Theory, Numerical Techniques, and Applications. D Poland, S Rychkov, A Vichi, 10.1103/RevModPhys.91.015002arXiv:1805.04405Rev. Mod. Phys. 91150022019hep-th\n\nS Rychkov, N Su, arXiv:2311.15844New Developments in the Numerical Conformal Bootstrap. hep-th\n\nThe Bootstrap Program for Boundary CFT d. P Liendo, L Rastelli, B C Van Rees, 10.1007/JHEP07(2013)113arXiv:1210.4258JHEP. 071132013hep-th\n\nDefects in conformal field theory. M Billò, V Goncalves, E Lauria, M Meineri, 10.1007/JHEP04(2016)091arXiv:1601.02883JHEP. 04912016hep-th\n\nThe Conformal Bootstrap at Finite Temperature. L Iliesiu, M Kolouglu, R Mahajan, E Perlmutter, D Simmons-Duffin, 10.1007/JHEP10arXiv:1802.10266JHEP. 10702018. 2018hep-th\n"},"corpus_id":{"kind":"number","value":266191513,"string":"266,191,513"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2312.07866\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1007/jhep07(2024)047\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/2312.07866v2.pdf\"\n ],\n \"pdf_sha\": \"156f66a13d5b43567ff18d6321ce649c5c464af3\",\n \"open_access\": {\n \"license\": \"CCBY\",\n \"url\": \"https://doi.org/10.1007/jhep07(2024)047\",\n \"status\": \"GOLD\"\n }\n}"},"title":{"kind":"string","value":"Easy bootstrap for the 3D Ising model: a hybrid approach of the lightcone bootstrap and error minimization methods"},"authors":{"kind":"list like","value":["Wenliang Li liwliang3@mail.sysu.edu.cn \nSchool of Physics\nSun Yat-Sen University\n510275GuangzhouChina"],"string":"[\n \"Wenliang Li liwliang3@mail.sysu.edu.cn \\nSchool of Physics\\nSun Yat-Sen University\\n510275GuangzhouChina\"\n]","isConversational":false},"abstract":{"kind":"string","value":"As a simple lattice model that exhibits a phase transition, the Ising model plays a fundamental role in statistical and condensed matter physics.The Ising transition is realized by physical systems, such as the liquid-vapor transition.Its continuum limit also furnishes a basic example of interacting quantum field theories and universality classes.Motivated by a recent hybrid bootstrap study of the quantum quartic oscillator, we revisit the conformal bootstrap approach to the 3D Ising model at criticality, without resorting to positivity constraints.We use at most 10 nonperturbative crossing constraints at low derivatives from the Taylor expansion around a crossing symmetric point.The high-lying contributions are approximated by simple analytic formulae deduced from the lightcone singularity structure.Surprisingly, the low-lying properties are determined to good accuracy by this computationally very cheap approach.For instance, the results for the two relevant scaling dimensions (∆ σ , ∆ ϵ ) ≈ (0.518153, 1.41278) are close to the most precise rigorous bounds obtained at a much higher computational cost."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Onsager's exact solution for the partition function of the 2D square-lattice Ising model is a landmark in theoretical physics [1].However, an exact solution for the 3D cubic-lattice Ising partition function has remained elusive.Since the 2D Ising model is exactly solvable, one may try to embed the simple cubic lattice in a non-planar surface.This is indeed possible [2], but the price to pay is that the genus is proportional to the number of sites, which becomes infinite in the thermodynamic limit.Then the partition function is given by an infinite sum of complicated terms. 1 The 2D embedding does not seem to be particularly useful due to the interplay between the dimensional complexity and topological complexity.This also explains why it should be significantly more difficult to solve the Ising model in D > 2 dimensions.\nAlthough an exact solution at D > 2 may be challenging, the long distance behavior of the Ising model is not so complex. 2 There is an emergent simplicity in the macroscopic physics.For example, the scaling behavior around the critical point is characterized by some critical exponents, which can be determined to good precision by the Monte Carlo simulations of finite size systems (see e.g.[5,6] and references therein).To study the macroscopic properties, one can further consider the continuum limit, whose effective description is expected to be given by the ϕ 4 field theory.The long stance behavior is determined by an infrared fixed point of the renormalization group (RG) flow, corresponding to a scale invariant quantum field theory.Different physical systems can exhibit similar macroscopic properties if they are governed by the same fixed point, i.e., in the same universality class.In D > 4 dimensions, the Ising critical exponents can be derived from the simple mean field theory due to the averaging effects of a large number of nearby sites, associated with the Gaussian fixed point of the RG flow.For 1 < D < 4, the Ising critical behavior is governed 1 The partition function is given by a sum of 4 g Pfaffians [3], where g is the topological genus and 4 g is the number of spin structures. 2 It is interesting to study the D-dimensional Ising model using the ϵ = D − 2 expansion.See [4] for a conformal bootstrap study of the D = 2 + ϵ Ising model at criticality.by a nontrivial fixed point of the RG flow and the critical exponents become nontrivial functions of D. Furthermore, there is strong numerical evidence that scale invariance of the 3D Ising fixed point is enhanced to conformal invariance [7][8][9][10][11][12][13][14][15][16][17][18].It should be easier to solve the 3D Ising model at criticality.Since the revival of the D > 2 conformal bootstrap program [19], one of the most impressive achievements is the high precision determinations of the low-lying data of the 3D Ising conformal field theory (CFT) [7][8][9][10][11]. 3The precise scaling dimensions and operatorproduct-expansion (OPE) coefficients of the two relevant operators are [11] ∆ σ = 0.5181489 (10) , ∆ ϵ = 1.412625 (10) , λ σσϵ = 1.0518537 (41) .\n(1.1)\nThe determinations of critical exponents η = 0.0362978 (20), ν = 0.629971(4) are consistent with and more precise than the Monte-Carlo determinations η MC = 0.036284 (40), ν MC = 0.62998 (5) in [6].\nIn (1.1), the error bars from the numerical bootstrap bounds are rigorous.The tiny allowed region is deduced from the assumptions of unitarity, two relevant operators, and OPE associativity.The unitarity assumption implies that the OPE coefficients are real numbers, leading to positivity constraints.The number of relevant operators and the unitarity bounds give some lower bounds for the scaling dimensions of the irrelevant operators. 4For 4-point functions, the associativity of OPE leads to the crossing equations, i.e., the convergent summations of conformal blocks in different channels should correspond to the same correlator.The crossing equations can be discretized by the Taylor expansion around a well-chosen point in the rapidly convergent regime, where the dominant contributions are associated with operators of low scaling dimension [23].These nonperturbative constraints are labelled by the numbers of derivatives.Together with the positivity constraints and spectral assumptions, one can use the crossing constraints to rule out the inconsistent parameter space.Using a system of mixed correlators, an isolated region of the parameter space for the 3D Ising CFT was found in [9].\nIf a correlator admits a rapidly convergent expansion in terms of conformal blocks, the high-lying contributions are suppressed and thus less constrained by the crossing equation.On the other hand, it is mathematically more subtle to study the implications of the crossing equation in a singular limit, as the conformal block summation may cease to be manifestly convergent. 5Nevertheless, a singular limit of the crossing equation can provide nontrivial information about the high-lying states, which is complementary to the rapidly convergent case.The reason is that the emergence of a singularity requires a collective behavior of infinitely many high-lying states, so one may obtain simple asymptotic formulae for the averaged behavior at large quantum numbers.For example, in the lightcone limit, the dominant contribution of the cross-channel vacuum state implies the double-twist behavior at large spin [25,26]. 6The leading asymptotic behavior coincides with that of the generalized free fields.At low twist, the higher spin results from the numerical bootstrap bounds and the extremal functional method [8,[31][32][33] beautifully organize into Regge trajectories, in accordance with the analytic formulae from the lightcone bootstrap [33][34][35].\nThe high precision numerical bootstrap bounds (1.1) are obtained from expensive computations.Suppose that we make use of a global understanding of the high-lying operators.Can we determine the low-lying properties of the 3D Ising CFT reasonably well with less computational efforts?Recently, we performed a bootstrap study of the quartic anharmonic oscillator [36] using the asymptotic behavior and nonperturbative self-consistency constraints [37], which can be viewed as a bootstrap study of the 1D version of the massive ϕ 4 field theory. 7As the 1D bootstrap results exhibit rapid convergence, it is tempting to apply this strategy to the 3D field theory.\nIn [33], Simmons-Duffin suggested a hybrid bootstrap approach that combines the lightcone approach [25,26] with the truncation approach [42]. 8In the truncation method initiated by Gliozzi, the contributions of the high-lying operators are simply set to zero.As the high-lying contributions are suppressed near z = z = 1/2, it seems reasonable to omit them in a preliminary study.However, if one wants to systematically improve the results, some small but non-negligible high-lying contributions should be taken into account.To determine the additional free parameters associated with the high-lying contributions, one needs to consider more nonperturbative crossing constraints.Without the help of positivity 9 , the computational complexity of a large system of nonlinear equations grows rapidly. 10In a hybrid approach, the high-lying scaling dimensions and OPE coefficients are approximated by simple analytic formulae from the lightcone bootstrap.As the number of free parameters is reduced, less crossing constraints are needed.In this way, one should be able to improve the results more easily.In [33], the original proposal involves the crossing constraints for mixed correlators.Below, we will show that the hybrid analytical/numerical approach can be realized in a simpler way, i.e., by studying only an identical correlator. 11 As discussed later, we are led to an error minimization approach [47] in the hybrid method.\nIn this work, we revisit the conformal bootstrap approach to the 3D Ising model using the two complementary types of crossing constraints: 6 We assume that the identity operator has the lowest twist.In perturbation theory, the twist accumulation phenomenon was discussed already in 1973 [27,28].See also [29] for an earlier nonperturbative argument based on a 2D massive theory perspective.Recently, a more rigorous approach was presented in [30] based on positivity constraints. 7This study was inspired by the recent work on the 0D ϕ 4 theory [38,39] and a bootstrap formulation of the quantum anharmonic oscillator [40].See [41] for more details about a novel type of analytic continuation.\n8 See [43] for a study of the Ising CFT at finite temperature using this hybrid approach. 9See [44] for a different hybrid approach that combines the positivity constraints with the large spin perturbation theory. 10Using the rational approximations [45], the transcendental equations can be well approximated by polynomial equations, which can be studied by numerical algebraic geometry methods, such as the efficient package HomotopyContinuation.jl [46].However, the computational complexity still grows rather rapidly. 11The reason is that the mixing of subleading Regge trajectories does not have dominant effects on the identical correlator around z = z = 1/2.A CFT with large mixing effects may not be captured by the crossing equation for an identical correlator.\n• The singular z → 1 limit leads to a global understanding of the high-lying data.\n• The expansion around z = z = 1/2 gives strong constraints on the low-lying data.\nWe assume that the 3D Ising CFT has only two relevant operators.We will consider the 4-point correlator of the lowest Z 2 -odd scalar σ with ∆ σ < 3. The other relevant operator ϵ is a Z 2 -even scalar and appears in the σ × σ OPE.It turns out that reasonably accurate results can be derived easily from a small number of nonperturbative crossing constraints.\nIn our main results, the total number of derivatives in the expansion around z = z = 1/2 is at most 7.This paper is organized as follows.In Sec. 2, we give a brief overview of the crossing constraints.In Sec. 3, we discuss how to obtain an approximate solution of the 3D Ising CFT using simple analytic approximations and a few nonperturbative crossing constraints.In Sec. 4, we take into account the leading corrections to the analytic approximations for the high-lying contributions.We use the error minimization approach [47] to obtain more accurate results of the 3D Ising CFT data.In Sec. 5, we summarize our results and discuss some directions for further investigations."},{"title":"Crossing constraints","content":"We will consider the 4-point correlator of the lowest Z 2 -odd scalar σ:\nwhere x i denote the positions of the external operators, and x ij = |x i −x j | are the distances between two operators.The variables (z, z) are defined by\nx 2 13 x 2 24 .\n(2.\n2)\nThe correlator is invariant under the exchange of σ(x 1 ) and σ(x 3 ), so we have the crossing equation\nwhich gives a relation between the direct-channel and cross-channel OPEs and the corresponding conformal block expansions.We can discretize the crossing equation (2.3) by the Taylor expansion around z = z = 1/2:\nwhere P i = λ 2 σσO i are the squared OPE coefficients, and\nare the scaling dimension and spin of the primary operator O i .The definition of\nBy construction, the nonperturbative crossing constraints (2.4) vanish automatically if m+n is an even integer.Since the crossing equation (2.3) is symmetric in (z, z), we assume m < n.\nIn this work, the normalization of a conformal block is fixed by\nwhere ∆ is the scaling dimension and ℓ is the spin.\nAs explained in the introduction, it is also useful to consider the crossing equation in the singular lightcone limit z → 1.After rewriting the crossing equation (2.3)\none can show that the leading asymptotic behavior at large spin is associated with the lowest twist operator on the right hand side.The asymptotic behavior can be derived from the relation\nwhere the coefficient function S p (h) is\nthe\nand the conformal spin h is related to the twist τ and spin ℓ by\nAn exact identity version of (2.8) can be found in [33]."},{"title":"Asymptotic behavior from the vacuum state","content":"The leading asymptotic behavior is associated with the vacuum state in the cross-channel, as the identity operator has the lowest twist in the 3D Ising CFT.Using (2.8) and the small z expansion of the conformal block F ∆,ℓ (z, z), one can derive the large-spin asymptotic behavior of the squared OPE coefficients\nwhere the superscript (k) denotes the trajectory [σσ] k,ℓ with twist τ ∼ 2∆ σ + 2k.The subscript {I} indicates that the asymptotic behavior is associated with the contribution of the cross-channel vacuum state.The analytic formulae (3.1),(3.2) coincide with the OPE coefficients of the generalized free fields (GFF) [49,50].They provide a simple approximation for the averaged behavior of the high-lying states.In fact, the scaling dimensions deviate from the double-twist values ∆ [σσ] k,ℓ = 2∆ σ + 2k + ℓ at finite spin, but the effects of their corrections are subleading.Some of these corrections will be taken into account to improve the accuracy in Sec. 4. The low-lying properties deviate more significantly from the GFF solution, so we should separate the low-lying contributions from the high-lying ones.The approximate G(z, z) contains a low-lying part and a GFF part\nwhere the GFF part does not include the cases of τ = 2∆ σ with ℓ = 0, 2.Here (∆ ϵ , P ϵ , P T ) are free parameters.Note that ϵ is the Z 2 -even relevant operator with 0 < ∆ ϵ < D. We assume that the lowest spin-two operator corresponds to the stress tensor T , so its scaling dimension is given by ∆ T = D = 3.As the generalized free fields correspond to an exact solution to the crossing equation, we can sum over the GFF contributions and obtain a more explicit form of (3.3):\nThe first line contains the leading terms in the lightcone limit, which is manifestly crossing symmetric and thus reminiscent of the inverse bootstrap approach [51]. 13The second line is subleading in both the direct-channel and cross-channel lightcone limits, but becomes important around z = z = 1/2.To satisfy the crossing constraints (2.4), the contributions of the low-lying operator {ϵ, T } should satisfy\nwhich leads to nontrivial constraints on the free parameters (∆ ϵ , P ϵ , P T ) and ∆ σ .For comparison, we also examine the naive truncation without any high-lying contribution.\nThe corresponding crossing constraints read\nwhich contain no contributions from primary operators with ∆ > 3. The low derivative equations are more sensitive to the low-lying contributions due to the rapid convergence in the Euclidean regime [23], so they are expected to be more useful for constraining the low-lying CFT data. 14According to the total number of derivatives, the leading nonperturbative equations correspond to (m, n) = (0, 1), (0, 3), (1, 2) .\n(3.7)\nFor a given ∆ σ , we can determine (∆ ϵ , P ϵ , P T ) by these three equations.In Fig. 1, we present the solution for ∆ ϵ at various ∆ σ .The blue dots denote the solutions with GFF approximations for the high-lying contributions, while the orange dots 14 For the rapid OPE convergence, it is natural to consider the crossing symmetric point z = z = 1/2 as the reference point because we want to minimize the radial distance [23] in both the direct-and crosschannels.The region around z = z = 1/2 is still expected to be rapidly convergent in both channels and they can be well approximated by the leading terms of the Taylor series.The low derivative crossing constraints can be viewed as certain linear combinations of those around z = z = 1/2, so they are expected to rapidly convergent as well.On the other hand, we need to use many higher derivative terms in the Taylor expansion to approach the singular lightcone limit and many higher spin operators to reconstruct the singularity structure.Therefore, we expect that the high derivative crossing constraints are more sensitive to the high-lying contributions than the low derivative ones."},{"title":"Ising","content":"are associated with the naive truncation without any ∆ > 3 contribution.We can see that the two types of solutions intersect around the free theory solution, but the results with GFF approximations are much closer to the 3D Ising solution (1.1). 15o determine the external scaling dimension ∆ σ , we should introduce one more equation.If we choose (m, n) = (1, 4), the solutions for the free parameters are\nwhere we only write 4 significant digits.Despite the crude approximation scheme, the solutions of the crossing constraints are unexpectedly close to the correct values in (1.1) and [33].The relative errors are (−0.08%,−0.9%, 0.6%, 0.1%). 16These results are more accurate than those in the previous 3D Ising studies [42,55] by Gliozzi's truncation method, which considered more scaling dimensions as free parameters and relied on some external input from the Monte Carlo method.The only exception is the prediction for ∆ ϵ in the most complicated example in [55] with 10 free parameters for the scaling dimensions and 1 input scaling dimension from the Monte Carlo method.Let us emphasize that we do not use any external input for the scaling dimensions in this work.If all the scaling dimensions are fixed, it is trivial to determine the OPE coefficients by the error minimization method [47].\nWe can also estimate the errors without using the results from the positivity bounds.However, it is not clear how to perform a rigorous error estimation in the truncation-like method, as inequality-like positivity constraints are not used.Here we follow a preliminary procedure that does not allow for correlated movement in the variables.Let us consider the remaining 5-derivative crossing constraints with (m, n) = (0, 5), (2,3).We substitute the free parameters with the crossing solutions in (3.8), except for the one under error estimation.Since we use two constraints, we obtain two different values for the parameter under error estimation.The error is set by the largest deviation of this parameter from the crossing constraints.The results are |δ∆ σ | = 0.0084 , |δ∆ ϵ | = 0.027 , |δP ϵ | = 0.029 , |δP T | = 0.0019, which are greater than the error estimates based on the positivity bound results.The error estimation method is similar to the one in [47].Here we can also introduce the η function and solve the chosen constraints by the η minimization.Since we can solve all the constraints, the corresponding local minimum of the η function is zero and we are not able to use the four constraints to estimate the errors.Therefore, we use the two 5-derivative equations to compute the errors.As the two constraints are not encoded in the η function, the resulting errors are much greater than the estimates based on the positivity bounds.\nfurther applies this approach to the five-point bootstrap.In [53], the scaling dimensions are input parameters from the positivity bounds and the GFF approximations are used to improve the determinations of the OPE coefficients.Here we use the GFF improved truncation method to determine both the scaling dimensions and OPE coefficients without relying on any input parameters from the positivity bounds.It is more nontrivial to determine the scaling dimensions than only the OPE coefficients.In Sec. 4, we further consider the leading corrections to the GFF approximations, which are not taken into account in [53].Another difference is that the best known value of ∆σ is used to select the set of nonperturbative crossing constraints by viewing certain ∆σ in the crossing constraints as a free parameter in [53]. 16For spinning operators of low twist, it may be more appropriate to consider a percentage based error in terms of twist.\nThe critical exponents are related to the scaling dimensions by ∆ σ = 1/2 + η/2, ∆ ϵ = 3 − 1/ν, so we have η ≈ 0.0355 , ν ≈ 0.6247 , (3.9)\nwhere the relative errors of the crossing constraint solutions are also small, i.e. (−2%, 0.8%).\nThe error estimates from the crossing constraints with (m, n) = (0, 5), (2, 3) are |δη| = 0.0101 and |δν| = 0.0168.\nIf we use a different 5-derivative equation, the results would not be so encouraging.For the (0, 5) and (2, 3) constraints, the solution in the upper blue branch in Fig. 1 becomes ∆ σ ≈ 0.504 and ∆ σ ≈ 0.494 respectively. 17One may suspect that the nice results in (3.8) are a pure coincidence.To address this concern, we will take into account the leading corrections to the GFF behavior in Sec. 4, and show that the results can be further improved.Therefore, the encouraging results (3.8) indeed suggest a cheaper avenue towards an accurate solution of the 3D Ising CFT."},{"title":"Asymptotic behavior from the low-lying states","content":"To obtain more accurate results, we should improve the approximation scheme for the highlying contributions.For the lowest Z 2 -even Regge trajectory with τ ≈ 2∆ σ , the leading corrections to the GFF behavior can be easily deduced from the cross-channel operators {ϵ, T }, which will be discussed later.For the higher Regge trajectories, the GFF asymptotic behavior is an approximation for the averaged contributions. 18We need to study a mixed system of correlators to extract the unmixed trajectories.Above the lowest trajectory [σσ] 0,ℓ , there are two dominant trajectories with τ [σσ] 1 ∼ 2∆ σ + 2 and τ [ϵϵ] 0 ∼ 2∆ ϵ .Their twist spectra are nearly symmetric about τ = ∆ σ + ∆ ϵ + 1 and have large repulsion at low conformal spin.As the twists of the ℓ = 0, 2 operators have significant deviations from the GFF value τ = 2∆ σ + 2, we will consider their twists as free parameters.Nevertheless, their squared OPE coefficients can be approximated by the GFF asymptotic formula (3.2).This is because the ℓ = 0, 2 operators on the lower trajectory [ϵϵ] 0,ℓ decouple 19 and only one dominant operator contributes at each spin ℓ = 0, 2. At higher spin or higher twist, we will again use the simple GFF approximation as the corrections are highly suppressed by the small OPE coefficients.\nAs mentioned above, we should take into account the leading corrections to the GFF approximation for the lowest trajectory [σσ] 0,ℓ , which can be deduced from the cross-channel contributions of the low-lying operators {ϵ, T }.In the lightcone limit z → 1, the leading terms of the cross-channel conformal blocks are\nwhere H x = Γ ′ (x + 1)/Γ(x + 1) − Γ ′ (1)/Γ(1) denotes the harmonic number and . . .indicates subleading terms in 1 − z or z.The log z terms are related to the anomalous dimensions by z τ /2 = z ∆σ (1 + τ −2∆σ 2 log z + . . .).The crossing equation (2.7) implies that the leading behavior of the anomalous dimensions and the OPE coefficients are\nwhere Pi = Γ(2h i ) Γ(h i ) 2 P i and the anomalous dimension is defined as γ i ≡ τ i −2∆ σ .The subscript {I, ϵ, T } indicates the asymptotic behavior is associated with the cross-channel contributions of these three low-lying operators.As before, we can resum the GFF contributions.The nonperturbative crossing constraints can be expressed as the difference between the contributions involving the free parameters and their GFF counterparts:\nwhere the conformal spin h in the second summation is approximated by\nWhen ℓ * is sufficiently large, the results are not sensitive to this parameter. 20Note that the free parameters ∆ (k=1) ℓ=0,2 are the scaling dimensions of the second lowest Z 2 -even operators with spin ℓ = 0, 2. As explained above, their OPE coefficients are still approximated by the GFF formula (3.2).\nSince there are 6 free parameters, we consider the following set of nonperturbative crossing constraints (m, n) = (0, 1), (0, 3), (1, 2), (0, 5), (1,4), (2, 3) , (\nwhere the total number of derivatives is at most 5.As the number of free parameters and constraints match, one may want to impose G (m,n) {I,ϵ,T } = 0 with (4.6).However, this will rule out the Ising solution, and we only find the free theory solution. 21As a result, we should not impose that G (m,n) {I,ϵ,T } vanish exactly.To obtain the 3D Ising solution, we need to consider a weaker formulation of the crossing constraints (4.4).\nA natural modification is to search for the approximate solutions with small errors in the crossing constraints (4.4).To measure the errors of the crossing constraints, we introduce the η function:\nwhere we use (m!n!) −2 to suppress the high derivative constraints as they converge slower and are relatively less sensitive to the low-lying parameters.Instead of imposing the exact constraints G (m,n) {I,ϵ,T } = 0, we now search for the local minima of the η function.In [47], the error minimization approach was proposed as a potentially more systematic formulation of the truncation approach initiated by Gliozzi [42]. 22In fact, the minimization formulation of [47] was partly motivated by absence of solutions to some crossing symmetric ansatz in the inverse bootstrap approach [51], which is similar to the situation here.The error minimization approach has been adopted in a number of bootstrap studies [4,21,43,[69][70][71][72][73][74]."},{"title":"23","content":"As expected for the 3D Ising CFT, there exists a local minimum of the η function around (∆ σ , ∆ ϵ ) ≈ (0.518, 1.41), which is indicated by the red point in Fig. 2. The resulting low-lying parameters are24\nwhere only 5 significant digits are presented.For comparison, the accuracy of (4.8) is better than the first positive bootstrap results for the 3D Ising model in [7], even though the nonperturbative constraints in [7] involve 23 derivatives and the number of constraints was significantly greater than our case with at most 5 derivatives.Using the positivity bound results as reference values, the relative errors of the η minimization results are (−0.009%,−0.02%, −0.1%, 0.1%).\nWe can also estimate the errors without using the positivity bound results.Let us substitute the free parameters with the η minimization results in (4.8), except for the one under error estimation, and then solve for this parameter using one of the crossing constraints in (4.6).The largest deviation from the η minimization result in (4.8) gives an error estimate of this parameter [47] The corresponding critical exponents are\nwhere the relative errors of the η minimization results are (−0.3%,−0.02%).These results are more accurate than those in Sec. 3. The estimated errors without using the positivity bound results are |δη| = 0.00290 , |δν| = 0.00030.We also obtain the scaling dimensions of the low spin operators in the higher trajectory (∆\nℓ=2 ) ≈ (3.68, 6.29), which are less accurate.For comparison, the estimates for (∆ [33] are (3.82968(23),5.50915 (44)).\nUsing the precise bootstrap bound results as the input parameters for the analytic formulae, one can compare the analytic asymptotic formulae with the numerical estimates [33][34][35], whose agreement is surprisingly good and motivated the present work.For completeness, we also present the comparisons in Fig. 3 and Fig. 4, whose agreement with the numerical estimates [33] is excellent for ℓ ≥ 4. The main difference from the earlier studies [33][34][35] is that the input parameters (∆ σ , ∆ ϵ , P ϵ , P T ) in (4.8) are not extracted from the bootstrap bound results.As a step forward, both the low-lying scaling dimensions\n{I,ϵ,T } (∆ σ + ℓ).The red dots are the numerical estimates from [33]. and OPE coefficients are determined by the hybrid numerical/analytic approach and the η minimization, without relying on the results from other approaches. 25 After taking into account the leading corrections to the GFF behavior, our results also become much more stable under a change in the selection of derivatives, in comparison to those in Sec. 3. If we substitute the (m, n) = (0, 5) constraint with a 7 derivative constraint, the results are less accurate than the results (4.8) with at most 5 derivatives, but still better than the results (3.8) 25 Since the conformal blocks are highly nonlinear functions in the scaling dimensions, it is more difficult to determine the low-lying scaling dimensions than the coefficients of the conformal blocks.In [43,53,74], the low-lying scaling dimensions are input parameters from the positivity bounds or integrability method and the improved truncation methods are mainly used to determine the coefficients of conformal blocks.\n{I,ϵ,T } (h) in (4.3),where h is approximated by (4.5).The red dots are the numerical estimates from [33], which have noticeable errors at ℓ > 26.\nwhere the local minimum of the η function is zero expect for the first case.If we replace the (1,4) or (2, 3) constraint with a 7-derivative one, we still find a local minimum of the η function around the 3D Ising solution, but the accuracy is lower than (3.8).It seems that the crossing constraints with m ∼ n are more sensitive to the low-lying contributions than those with m ≪ n, so the former can lead to more accurate determinations for the low-lying parameters.\nWe can also add more constraints to the η functions, leading to an overdetermined system with η min > 0. If we use all the crossing constraints with m + n ≤ 7, the local minimum of the η function around (∆ σ , ∆ ϵ ) = (0.52, 1.41) gives ∆ σ ≈ 5.1974 , ∆ ϵ ≈ 1.4192 , P ϵ ≈ 1.1019 , P T ≈ 0.10715 and (∆ 75, 5.38).Although less accurate in comparison to (4.8), the η minimization results for the relevant operators are still around the 3D Ising solution.On the other hand, the irrelevant scaling dimensions (∆\nℓ=2 ) become more accurate. 26This is consistent with our expectation that the higher derivative equations are more sensitive to the high-lying contributions, so the predic- 26 We also notice this improvement for the irrelevant scaling dimensions by replacing a 5-derivative constraint with a 7-derivative constraint.For example, the cases for (4.10), (4.11), (4.12), (4.13) are (∆ tions for the irrelevant operators are improved, at the price of lowering the accuracy of the low-lying parameters, which are contaminated by the less accurate high-lying contributions.\nSince we have 10 constraints with m + n ≤ 7, we can introduce more free parameters.It is natural to consider the lowest spin-4 operator, as its scaling dimension ∆ (k=0) ℓ=4 ≈ 5 is close to the subleading spin-2 scaling dimension, which is viewed as a free parameter in the above.Surprisingly, we arrive at even more accurate results by relaxing the scaling dimension and OPE coefficient of the lowest spin-4 operator: ∆ σ ≈ 0.518152 , ∆ ϵ ≈ 1.41278 , P ϵ ≈ 1.10640 , P T ≈ 0.106226 .are almost as accurate as the latest Monte-Carlo determinations [6].\nAs the system is still overdetermined, one may try to introduce more free parameters at higher spin.However, if we do not approximate the lowest spin-6 parameters by the analytic formulae and set them as free parameters, we are not able to find a local minimum around the 3D Ising solution.It seems that the η minimization becomes unstable due to the fact that the squared OPE coefficient of the spin-6 operator is allowed to be negative.Besides reducing computational efforts, stabilization is another important merit of analytic approximations for the high-lying contributions.This is different from the hybrid approach in [44], in which the use of positivity constraints stabilizes the hybrid bootstrap method and thus the effects of the choice for the gluing spin can be studied more systematically."},{"title":"Discussion","content":"In this work, we revisited the conformal bootstrap approach to the 3D Ising model using the global asymptotic behavior and the rapidly convergent crossing constraints.We derived surprisingly accurate results (4.14) from simple analytic asymptotic formulae and a small number of nonperturbative crossing constraints, using the error minimization method.In some sense, this furnishes a minimal example for the nonperturbative conformal bootstrap in three dimensions, with the help of an analytic understanding from the lightcone singularity structure.\nTo improve the accuracy of the hybrid approach, it is crucial to go beyond the leading corrections to the GFF behavior. 27The asymptotic formulae can be computed more systematically using the methods in [33,75,76].However, one should be more careful as a higher order asymptotic expansion may give worse results at low spin, even if the approximations at large spin are improved.Then the corresponding low spin parameters should be viewed as free parameters and determined by the nonperturbative crossing constraints.To avoid this subtlety, one may use the convergent Lorentzian inversion formula [77][78][79], which includes the nonperturbative contributions [80][81][82][83][84][85][86][87]. 28 29However, the price to pay is a higher computational cost.In practice, it may be useful to combine the asymptotic methods with the convergent methods.\nWe also need to unmix the higher trajectories. 30In the present work, we used the decoupling phenomenon to circumvent the mixing problem, and carried out the bootstrap study using only the identical correlator ⟨σ(x 1 )σ(x 2 )σ(x 3 )σ(x 4 )⟩.To properly resolve the mixing problem, one needs to consider a mixed system of correlators [33,[85][86][87].The large repulsion of the twist spectra at low spin is particularly interesting.This can lead to more significant deviations from the leading GFF behavior.We also believe that this is closely related to the decoupling phenomenon in [7,8].\nWe would like to emphasize that the goal of the present work is not to surpass the remarkably precise results from the positivity bound approach. 31Instead, we want to show that a hybrid analytical/numerical approach can have the great potential in reducing the computational cost with the help of a global understanding of the high-lying data.The application to the 3D Ising model is just the first example.This strategy can be applied to other well constrained bootstrap solutions, such as the O(N ) models [11,[93][94][95] and the supersymmetric Ising model [87,96,97].For a more complete list of the conformal bootstrap targets, we refer to the comprehensive review [98] and the more recent one [99].As we are not using any positivity constraint, a natural and important direction is to consider the less explored non-positive bootstrap problems that involve D > 2 conformal field theories with nonunitarity [42], defects [100,101], and on nontrivial manifolds [57,102]."},{"title":"FreeFigure 1 :","content":"Figure 1:The approximate solution for ∆ ϵ at various ∆ σ .The red dots represent the free theory and the Ising solutions in three dimensions.The blue dots are obtained from solving (3.5) with GFF approximations for the high-lying contributions.The orange dots are derived from solving (3.6) with the naive truncation that omits all the high-lying contributions.The nonperturbative crossing constraints are associated with the derivative orders in(3.7).The results with the GFF approximations (blue) are much closer to the 3D Ising solution (1.1)."},{"title":"Figure 2 :","content":"Figure 2: The minimized η function (4.7) at different (∆ σ , ∆ ϵ ).The local minimum (red dot) with η min ≈ 0.00043 determines the 3D Ising results in (4.8)."},{"title":"Figure 3 :","content":"Figure 3: The twist τ = ∆ − ℓ of the lowest Z 2 -even trajectory at various spin ℓ.The curve is based on the analytic expression τ (ℓ) = 2∆ σ + γ"},{"title":"Figure 4 :","content":"Figure 4: The squared OPE coefficients of the leading Z 2 -even trajectory at different spin ℓ, divided by the generalized free case P GFF (ℓ) = 2S −∆σ (∆ σ + ℓ).The curve is based on the analytic expression P (ℓ) = P\nℓ=2 ) ≈ (3.75, 6.87), (3.74, 5.36), (3.77, 5.22), (3.72, 5.44).\nℓ=2 ) ≈ (3.67, 5.46).These are the most accurate results in this work.In particular, the OPE coefficient λ σσϵ = P 1/2 ϵ ≈ 1.05186 is significantly improved.The local minimization is carried out by Mathematica's FindMinimum, starting at (∆ σ , ∆ ϵ ) = (1, 3/2), (∆ (k=1) ℓ=0 , ∆ (k=1) ℓ=2 , ∆ (k=0) ℓ=4 ) = (3, 5, 5) and (P ϵ , P T , P (k=0) ℓ=4 ) = (1, 1/10, 1/200), which are some crude numbers from the GFF approximation.The corresponding critical exponents η ≈ 0.036304 , ν ≈ 0.63003 (4.15)\nbased on the GFF approximations.The corresponding η minimization results are (0, 7) : ∆ σ ≈ 0.51781 , ∆ ϵ ≈ 1.4122 , P ϵ ≈ 1.1042 , P T ≈ 0.10642 ,"}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Onsager's exact solution for the partition function of the 2D square-lattice Ising model is a landmark in theoretical physics [1].However, an exact solution for the 3D cubic-lattice Ising partition function has remained elusive.Since the 2D Ising model is exactly solvable, one may try to embed the simple cubic lattice in a non-planar surface.This is indeed possible [2], but the price to pay is that the genus is proportional to the number of sites, which becomes infinite in the thermodynamic limit.Then the partition function is given by an infinite sum of complicated terms. 1 The 2D embedding does not seem to be particularly useful due to the interplay between the dimensional complexity and topological complexity.This also explains why it should be significantly more difficult to solve the Ising model in D > 2 dimensions.\\nAlthough an exact solution at D > 2 may be challenging, the long distance behavior of the Ising model is not so complex. 2 There is an emergent simplicity in the macroscopic physics.For example, the scaling behavior around the critical point is characterized by some critical exponents, which can be determined to good precision by the Monte Carlo simulations of finite size systems (see e.g.[5,6] and references therein).To study the macroscopic properties, one can further consider the continuum limit, whose effective description is expected to be given by the ϕ 4 field theory.The long stance behavior is determined by an infrared fixed point of the renormalization group (RG) flow, corresponding to a scale invariant quantum field theory.Different physical systems can exhibit similar macroscopic properties if they are governed by the same fixed point, i.e., in the same universality class.In D > 4 dimensions, the Ising critical exponents can be derived from the simple mean field theory due to the averaging effects of a large number of nearby sites, associated with the Gaussian fixed point of the RG flow.For 1 < D < 4, the Ising critical behavior is governed 1 The partition function is given by a sum of 4 g Pfaffians [3], where g is the topological genus and 4 g is the number of spin structures. 2 It is interesting to study the D-dimensional Ising model using the ϵ = D − 2 expansion.See [4] for a conformal bootstrap study of the D = 2 + ϵ Ising model at criticality.by a nontrivial fixed point of the RG flow and the critical exponents become nontrivial functions of D. Furthermore, there is strong numerical evidence that scale invariance of the 3D Ising fixed point is enhanced to conformal invariance [7][8][9][10][11][12][13][14][15][16][17][18].It should be easier to solve the 3D Ising model at criticality.Since the revival of the D > 2 conformal bootstrap program [19], one of the most impressive achievements is the high precision determinations of the low-lying data of the 3D Ising conformal field theory (CFT) [7][8][9][10][11]. 3The precise scaling dimensions and operatorproduct-expansion (OPE) coefficients of the two relevant operators are [11] ∆ σ = 0.5181489 (10) , ∆ ϵ = 1.412625 (10) , λ σσϵ = 1.0518537 (41) .\\n(1.1)\\nThe determinations of critical exponents η = 0.0362978 (20), ν = 0.629971(4) are consistent with and more precise than the Monte-Carlo determinations η MC = 0.036284 (40), ν MC = 0.62998 (5) in [6].\\nIn (1.1), the error bars from the numerical bootstrap bounds are rigorous.The tiny allowed region is deduced from the assumptions of unitarity, two relevant operators, and OPE associativity.The unitarity assumption implies that the OPE coefficients are real numbers, leading to positivity constraints.The number of relevant operators and the unitarity bounds give some lower bounds for the scaling dimensions of the irrelevant operators. 4For 4-point functions, the associativity of OPE leads to the crossing equations, i.e., the convergent summations of conformal blocks in different channels should correspond to the same correlator.The crossing equations can be discretized by the Taylor expansion around a well-chosen point in the rapidly convergent regime, where the dominant contributions are associated with operators of low scaling dimension [23].These nonperturbative constraints are labelled by the numbers of derivatives.Together with the positivity constraints and spectral assumptions, one can use the crossing constraints to rule out the inconsistent parameter space.Using a system of mixed correlators, an isolated region of the parameter space for the 3D Ising CFT was found in [9].\\nIf a correlator admits a rapidly convergent expansion in terms of conformal blocks, the high-lying contributions are suppressed and thus less constrained by the crossing equation.On the other hand, it is mathematically more subtle to study the implications of the crossing equation in a singular limit, as the conformal block summation may cease to be manifestly convergent. 5Nevertheless, a singular limit of the crossing equation can provide nontrivial information about the high-lying states, which is complementary to the rapidly convergent case.The reason is that the emergence of a singularity requires a collective behavior of infinitely many high-lying states, so one may obtain simple asymptotic formulae for the averaged behavior at large quantum numbers.For example, in the lightcone limit, the dominant contribution of the cross-channel vacuum state implies the double-twist behavior at large spin [25,26]. 6The leading asymptotic behavior coincides with that of the generalized free fields.At low twist, the higher spin results from the numerical bootstrap bounds and the extremal functional method [8,[31][32][33] beautifully organize into Regge trajectories, in accordance with the analytic formulae from the lightcone bootstrap [33][34][35].\\nThe high precision numerical bootstrap bounds (1.1) are obtained from expensive computations.Suppose that we make use of a global understanding of the high-lying operators.Can we determine the low-lying properties of the 3D Ising CFT reasonably well with less computational efforts?Recently, we performed a bootstrap study of the quartic anharmonic oscillator [36] using the asymptotic behavior and nonperturbative self-consistency constraints [37], which can be viewed as a bootstrap study of the 1D version of the massive ϕ 4 field theory. 7As the 1D bootstrap results exhibit rapid convergence, it is tempting to apply this strategy to the 3D field theory.\\nIn [33], Simmons-Duffin suggested a hybrid bootstrap approach that combines the lightcone approach [25,26] with the truncation approach [42]. 8In the truncation method initiated by Gliozzi, the contributions of the high-lying operators are simply set to zero.As the high-lying contributions are suppressed near z = z = 1/2, it seems reasonable to omit them in a preliminary study.However, if one wants to systematically improve the results, some small but non-negligible high-lying contributions should be taken into account.To determine the additional free parameters associated with the high-lying contributions, one needs to consider more nonperturbative crossing constraints.Without the help of positivity 9 , the computational complexity of a large system of nonlinear equations grows rapidly. 10In a hybrid approach, the high-lying scaling dimensions and OPE coefficients are approximated by simple analytic formulae from the lightcone bootstrap.As the number of free parameters is reduced, less crossing constraints are needed.In this way, one should be able to improve the results more easily.In [33], the original proposal involves the crossing constraints for mixed correlators.Below, we will show that the hybrid analytical/numerical approach can be realized in a simpler way, i.e., by studying only an identical correlator. 11 As discussed later, we are led to an error minimization approach [47] in the hybrid method.\\nIn this work, we revisit the conformal bootstrap approach to the 3D Ising model using the two complementary types of crossing constraints: 6 We assume that the identity operator has the lowest twist.In perturbation theory, the twist accumulation phenomenon was discussed already in 1973 [27,28].See also [29] for an earlier nonperturbative argument based on a 2D massive theory perspective.Recently, a more rigorous approach was presented in [30] based on positivity constraints. 7This study was inspired by the recent work on the 0D ϕ 4 theory [38,39] and a bootstrap formulation of the quantum anharmonic oscillator [40].See [41] for more details about a novel type of analytic continuation.\\n8 See [43] for a study of the Ising CFT at finite temperature using this hybrid approach. 9See [44] for a different hybrid approach that combines the positivity constraints with the large spin perturbation theory. 10Using the rational approximations [45], the transcendental equations can be well approximated by polynomial equations, which can be studied by numerical algebraic geometry methods, such as the efficient package HomotopyContinuation.jl [46].However, the computational complexity still grows rather rapidly. 11The reason is that the mixing of subleading Regge trajectories does not have dominant effects on the identical correlator around z = z = 1/2.A CFT with large mixing effects may not be captured by the crossing equation for an identical correlator.\\n• The singular z → 1 limit leads to a global understanding of the high-lying data.\\n• The expansion around z = z = 1/2 gives strong constraints on the low-lying data.\\nWe assume that the 3D Ising CFT has only two relevant operators.We will consider the 4-point correlator of the lowest Z 2 -odd scalar σ with ∆ σ < 3. The other relevant operator ϵ is a Z 2 -even scalar and appears in the σ × σ OPE.It turns out that reasonably accurate results can be derived easily from a small number of nonperturbative crossing constraints.\\nIn our main results, the total number of derivatives in the expansion around z = z = 1/2 is at most 7.This paper is organized as follows.In Sec. 2, we give a brief overview of the crossing constraints.In Sec. 3, we discuss how to obtain an approximate solution of the 3D Ising CFT using simple analytic approximations and a few nonperturbative crossing constraints.In Sec. 4, we take into account the leading corrections to the analytic approximations for the high-lying contributions.We use the error minimization approach [47] to obtain more accurate results of the 3D Ising CFT data.In Sec. 5, we summarize our results and discuss some directions for further investigations.\"\n },\n {\n \"title\": \"Crossing constraints\",\n \"content\": \"We will consider the 4-point correlator of the lowest Z 2 -odd scalar σ:\\nwhere x i denote the positions of the external operators, and x ij = |x i −x j | are the distances between two operators.The variables (z, z) are defined by\\nx 2 13 x 2 24 .\\n(2.\\n2)\\nThe correlator is invariant under the exchange of σ(x 1 ) and σ(x 3 ), so we have the crossing equation\\nwhich gives a relation between the direct-channel and cross-channel OPEs and the corresponding conformal block expansions.We can discretize the crossing equation (2.3) by the Taylor expansion around z = z = 1/2:\\nwhere P i = λ 2 σσO i are the squared OPE coefficients, and\\nare the scaling dimension and spin of the primary operator O i .The definition of\\nBy construction, the nonperturbative crossing constraints (2.4) vanish automatically if m+n is an even integer.Since the crossing equation (2.3) is symmetric in (z, z), we assume m < n.\\nIn this work, the normalization of a conformal block is fixed by\\nwhere ∆ is the scaling dimension and ℓ is the spin.\\nAs explained in the introduction, it is also useful to consider the crossing equation in the singular lightcone limit z → 1.After rewriting the crossing equation (2.3)\\none can show that the leading asymptotic behavior at large spin is associated with the lowest twist operator on the right hand side.The asymptotic behavior can be derived from the relation\\nwhere the coefficient function S p (h) is\\nthe\\nand the conformal spin h is related to the twist τ and spin ℓ by\\nAn exact identity version of (2.8) can be found in [33].\"\n },\n {\n \"title\": \"Asymptotic behavior from the vacuum state\",\n \"content\": \"The leading asymptotic behavior is associated with the vacuum state in the cross-channel, as the identity operator has the lowest twist in the 3D Ising CFT.Using (2.8) and the small z expansion of the conformal block F ∆,ℓ (z, z), one can derive the large-spin asymptotic behavior of the squared OPE coefficients\\nwhere the superscript (k) denotes the trajectory [σσ] k,ℓ with twist τ ∼ 2∆ σ + 2k.The subscript {I} indicates that the asymptotic behavior is associated with the contribution of the cross-channel vacuum state.The analytic formulae (3.1),(3.2) coincide with the OPE coefficients of the generalized free fields (GFF) [49,50].They provide a simple approximation for the averaged behavior of the high-lying states.In fact, the scaling dimensions deviate from the double-twist values ∆ [σσ] k,ℓ = 2∆ σ + 2k + ℓ at finite spin, but the effects of their corrections are subleading.Some of these corrections will be taken into account to improve the accuracy in Sec. 4. The low-lying properties deviate more significantly from the GFF solution, so we should separate the low-lying contributions from the high-lying ones.The approximate G(z, z) contains a low-lying part and a GFF part\\nwhere the GFF part does not include the cases of τ = 2∆ σ with ℓ = 0, 2.Here (∆ ϵ , P ϵ , P T ) are free parameters.Note that ϵ is the Z 2 -even relevant operator with 0 < ∆ ϵ < D. We assume that the lowest spin-two operator corresponds to the stress tensor T , so its scaling dimension is given by ∆ T = D = 3.As the generalized free fields correspond to an exact solution to the crossing equation, we can sum over the GFF contributions and obtain a more explicit form of (3.3):\\nThe first line contains the leading terms in the lightcone limit, which is manifestly crossing symmetric and thus reminiscent of the inverse bootstrap approach [51]. 13The second line is subleading in both the direct-channel and cross-channel lightcone limits, but becomes important around z = z = 1/2.To satisfy the crossing constraints (2.4), the contributions of the low-lying operator {ϵ, T } should satisfy\\nwhich leads to nontrivial constraints on the free parameters (∆ ϵ , P ϵ , P T ) and ∆ σ .For comparison, we also examine the naive truncation without any high-lying contribution.\\nThe corresponding crossing constraints read\\nwhich contain no contributions from primary operators with ∆ > 3. The low derivative equations are more sensitive to the low-lying contributions due to the rapid convergence in the Euclidean regime [23], so they are expected to be more useful for constraining the low-lying CFT data. 14According to the total number of derivatives, the leading nonperturbative equations correspond to (m, n) = (0, 1), (0, 3), (1, 2) .\\n(3.7)\\nFor a given ∆ σ , we can determine (∆ ϵ , P ϵ , P T ) by these three equations.In Fig. 1, we present the solution for ∆ ϵ at various ∆ σ .The blue dots denote the solutions with GFF approximations for the high-lying contributions, while the orange dots 14 For the rapid OPE convergence, it is natural to consider the crossing symmetric point z = z = 1/2 as the reference point because we want to minimize the radial distance [23] in both the direct-and crosschannels.The region around z = z = 1/2 is still expected to be rapidly convergent in both channels and they can be well approximated by the leading terms of the Taylor series.The low derivative crossing constraints can be viewed as certain linear combinations of those around z = z = 1/2, so they are expected to rapidly convergent as well.On the other hand, we need to use many higher derivative terms in the Taylor expansion to approach the singular lightcone limit and many higher spin operators to reconstruct the singularity structure.Therefore, we expect that the high derivative crossing constraints are more sensitive to the high-lying contributions than the low derivative ones.\"\n },\n {\n \"title\": \"Ising\",\n \"content\": \"are associated with the naive truncation without any ∆ > 3 contribution.We can see that the two types of solutions intersect around the free theory solution, but the results with GFF approximations are much closer to the 3D Ising solution (1.1). 15o determine the external scaling dimension ∆ σ , we should introduce one more equation.If we choose (m, n) = (1, 4), the solutions for the free parameters are\\nwhere we only write 4 significant digits.Despite the crude approximation scheme, the solutions of the crossing constraints are unexpectedly close to the correct values in (1.1) and [33].The relative errors are (−0.08%,−0.9%, 0.6%, 0.1%). 16These results are more accurate than those in the previous 3D Ising studies [42,55] by Gliozzi's truncation method, which considered more scaling dimensions as free parameters and relied on some external input from the Monte Carlo method.The only exception is the prediction for ∆ ϵ in the most complicated example in [55] with 10 free parameters for the scaling dimensions and 1 input scaling dimension from the Monte Carlo method.Let us emphasize that we do not use any external input for the scaling dimensions in this work.If all the scaling dimensions are fixed, it is trivial to determine the OPE coefficients by the error minimization method [47].\\nWe can also estimate the errors without using the results from the positivity bounds.However, it is not clear how to perform a rigorous error estimation in the truncation-like method, as inequality-like positivity constraints are not used.Here we follow a preliminary procedure that does not allow for correlated movement in the variables.Let us consider the remaining 5-derivative crossing constraints with (m, n) = (0, 5), (2,3).We substitute the free parameters with the crossing solutions in (3.8), except for the one under error estimation.Since we use two constraints, we obtain two different values for the parameter under error estimation.The error is set by the largest deviation of this parameter from the crossing constraints.The results are |δ∆ σ | = 0.0084 , |δ∆ ϵ | = 0.027 , |δP ϵ | = 0.029 , |δP T | = 0.0019, which are greater than the error estimates based on the positivity bound results.The error estimation method is similar to the one in [47].Here we can also introduce the η function and solve the chosen constraints by the η minimization.Since we can solve all the constraints, the corresponding local minimum of the η function is zero and we are not able to use the four constraints to estimate the errors.Therefore, we use the two 5-derivative equations to compute the errors.As the two constraints are not encoded in the η function, the resulting errors are much greater than the estimates based on the positivity bounds.\\nfurther applies this approach to the five-point bootstrap.In [53], the scaling dimensions are input parameters from the positivity bounds and the GFF approximations are used to improve the determinations of the OPE coefficients.Here we use the GFF improved truncation method to determine both the scaling dimensions and OPE coefficients without relying on any input parameters from the positivity bounds.It is more nontrivial to determine the scaling dimensions than only the OPE coefficients.In Sec. 4, we further consider the leading corrections to the GFF approximations, which are not taken into account in [53].Another difference is that the best known value of ∆σ is used to select the set of nonperturbative crossing constraints by viewing certain ∆σ in the crossing constraints as a free parameter in [53]. 16For spinning operators of low twist, it may be more appropriate to consider a percentage based error in terms of twist.\\nThe critical exponents are related to the scaling dimensions by ∆ σ = 1/2 + η/2, ∆ ϵ = 3 − 1/ν, so we have η ≈ 0.0355 , ν ≈ 0.6247 , (3.9)\\nwhere the relative errors of the crossing constraint solutions are also small, i.e. (−2%, 0.8%).\\nThe error estimates from the crossing constraints with (m, n) = (0, 5), (2, 3) are |δη| = 0.0101 and |δν| = 0.0168.\\nIf we use a different 5-derivative equation, the results would not be so encouraging.For the (0, 5) and (2, 3) constraints, the solution in the upper blue branch in Fig. 1 becomes ∆ σ ≈ 0.504 and ∆ σ ≈ 0.494 respectively. 17One may suspect that the nice results in (3.8) are a pure coincidence.To address this concern, we will take into account the leading corrections to the GFF behavior in Sec. 4, and show that the results can be further improved.Therefore, the encouraging results (3.8) indeed suggest a cheaper avenue towards an accurate solution of the 3D Ising CFT.\"\n },\n {\n \"title\": \"Asymptotic behavior from the low-lying states\",\n \"content\": \"To obtain more accurate results, we should improve the approximation scheme for the highlying contributions.For the lowest Z 2 -even Regge trajectory with τ ≈ 2∆ σ , the leading corrections to the GFF behavior can be easily deduced from the cross-channel operators {ϵ, T }, which will be discussed later.For the higher Regge trajectories, the GFF asymptotic behavior is an approximation for the averaged contributions. 18We need to study a mixed system of correlators to extract the unmixed trajectories.Above the lowest trajectory [σσ] 0,ℓ , there are two dominant trajectories with τ [σσ] 1 ∼ 2∆ σ + 2 and τ [ϵϵ] 0 ∼ 2∆ ϵ .Their twist spectra are nearly symmetric about τ = ∆ σ + ∆ ϵ + 1 and have large repulsion at low conformal spin.As the twists of the ℓ = 0, 2 operators have significant deviations from the GFF value τ = 2∆ σ + 2, we will consider their twists as free parameters.Nevertheless, their squared OPE coefficients can be approximated by the GFF asymptotic formula (3.2).This is because the ℓ = 0, 2 operators on the lower trajectory [ϵϵ] 0,ℓ decouple 19 and only one dominant operator contributes at each spin ℓ = 0, 2. At higher spin or higher twist, we will again use the simple GFF approximation as the corrections are highly suppressed by the small OPE coefficients.\\nAs mentioned above, we should take into account the leading corrections to the GFF approximation for the lowest trajectory [σσ] 0,ℓ , which can be deduced from the cross-channel contributions of the low-lying operators {ϵ, T }.In the lightcone limit z → 1, the leading terms of the cross-channel conformal blocks are\\nwhere H x = Γ ′ (x + 1)/Γ(x + 1) − Γ ′ (1)/Γ(1) denotes the harmonic number and . . .indicates subleading terms in 1 − z or z.The log z terms are related to the anomalous dimensions by z τ /2 = z ∆σ (1 + τ −2∆σ 2 log z + . . .).The crossing equation (2.7) implies that the leading behavior of the anomalous dimensions and the OPE coefficients are\\nwhere Pi = Γ(2h i ) Γ(h i ) 2 P i and the anomalous dimension is defined as γ i ≡ τ i −2∆ σ .The subscript {I, ϵ, T } indicates the asymptotic behavior is associated with the cross-channel contributions of these three low-lying operators.As before, we can resum the GFF contributions.The nonperturbative crossing constraints can be expressed as the difference between the contributions involving the free parameters and their GFF counterparts:\\nwhere the conformal spin h in the second summation is approximated by\\nWhen ℓ * is sufficiently large, the results are not sensitive to this parameter. 20Note that the free parameters ∆ (k=1) ℓ=0,2 are the scaling dimensions of the second lowest Z 2 -even operators with spin ℓ = 0, 2. As explained above, their OPE coefficients are still approximated by the GFF formula (3.2).\\nSince there are 6 free parameters, we consider the following set of nonperturbative crossing constraints (m, n) = (0, 1), (0, 3), (1, 2), (0, 5), (1,4), (2, 3) , (\\nwhere the total number of derivatives is at most 5.As the number of free parameters and constraints match, one may want to impose G (m,n) {I,ϵ,T } = 0 with (4.6).However, this will rule out the Ising solution, and we only find the free theory solution. 21As a result, we should not impose that G (m,n) {I,ϵ,T } vanish exactly.To obtain the 3D Ising solution, we need to consider a weaker formulation of the crossing constraints (4.4).\\nA natural modification is to search for the approximate solutions with small errors in the crossing constraints (4.4).To measure the errors of the crossing constraints, we introduce the η function:\\nwhere we use (m!n!) −2 to suppress the high derivative constraints as they converge slower and are relatively less sensitive to the low-lying parameters.Instead of imposing the exact constraints G (m,n) {I,ϵ,T } = 0, we now search for the local minima of the η function.In [47], the error minimization approach was proposed as a potentially more systematic formulation of the truncation approach initiated by Gliozzi [42]. 22In fact, the minimization formulation of [47] was partly motivated by absence of solutions to some crossing symmetric ansatz in the inverse bootstrap approach [51], which is similar to the situation here.The error minimization approach has been adopted in a number of bootstrap studies [4,21,43,[69][70][71][72][73][74].\"\n },\n {\n \"title\": \"23\",\n \"content\": \"As expected for the 3D Ising CFT, there exists a local minimum of the η function around (∆ σ , ∆ ϵ ) ≈ (0.518, 1.41), which is indicated by the red point in Fig. 2. The resulting low-lying parameters are24\\nwhere only 5 significant digits are presented.For comparison, the accuracy of (4.8) is better than the first positive bootstrap results for the 3D Ising model in [7], even though the nonperturbative constraints in [7] involve 23 derivatives and the number of constraints was significantly greater than our case with at most 5 derivatives.Using the positivity bound results as reference values, the relative errors of the η minimization results are (−0.009%,−0.02%, −0.1%, 0.1%).\\nWe can also estimate the errors without using the positivity bound results.Let us substitute the free parameters with the η minimization results in (4.8), except for the one under error estimation, and then solve for this parameter using one of the crossing constraints in (4.6).The largest deviation from the η minimization result in (4.8) gives an error estimate of this parameter [47] The corresponding critical exponents are\\nwhere the relative errors of the η minimization results are (−0.3%,−0.02%).These results are more accurate than those in Sec. 3. The estimated errors without using the positivity bound results are |δη| = 0.00290 , |δν| = 0.00030.We also obtain the scaling dimensions of the low spin operators in the higher trajectory (∆\\nℓ=2 ) ≈ (3.68, 6.29), which are less accurate.For comparison, the estimates for (∆ [33] are (3.82968(23),5.50915 (44)).\\nUsing the precise bootstrap bound results as the input parameters for the analytic formulae, one can compare the analytic asymptotic formulae with the numerical estimates [33][34][35], whose agreement is surprisingly good and motivated the present work.For completeness, we also present the comparisons in Fig. 3 and Fig. 4, whose agreement with the numerical estimates [33] is excellent for ℓ ≥ 4. The main difference from the earlier studies [33][34][35] is that the input parameters (∆ σ , ∆ ϵ , P ϵ , P T ) in (4.8) are not extracted from the bootstrap bound results.As a step forward, both the low-lying scaling dimensions\\n{I,ϵ,T } (∆ σ + ℓ).The red dots are the numerical estimates from [33]. and OPE coefficients are determined by the hybrid numerical/analytic approach and the η minimization, without relying on the results from other approaches. 25 After taking into account the leading corrections to the GFF behavior, our results also become much more stable under a change in the selection of derivatives, in comparison to those in Sec. 3. If we substitute the (m, n) = (0, 5) constraint with a 7 derivative constraint, the results are less accurate than the results (4.8) with at most 5 derivatives, but still better than the results (3.8) 25 Since the conformal blocks are highly nonlinear functions in the scaling dimensions, it is more difficult to determine the low-lying scaling dimensions than the coefficients of the conformal blocks.In [43,53,74], the low-lying scaling dimensions are input parameters from the positivity bounds or integrability method and the improved truncation methods are mainly used to determine the coefficients of conformal blocks.\\n{I,ϵ,T } (h) in (4.3),where h is approximated by (4.5).The red dots are the numerical estimates from [33], which have noticeable errors at ℓ > 26.\\nwhere the local minimum of the η function is zero expect for the first case.If we replace the (1,4) or (2, 3) constraint with a 7-derivative one, we still find a local minimum of the η function around the 3D Ising solution, but the accuracy is lower than (3.8).It seems that the crossing constraints with m ∼ n are more sensitive to the low-lying contributions than those with m ≪ n, so the former can lead to more accurate determinations for the low-lying parameters.\\nWe can also add more constraints to the η functions, leading to an overdetermined system with η min > 0. If we use all the crossing constraints with m + n ≤ 7, the local minimum of the η function around (∆ σ , ∆ ϵ ) = (0.52, 1.41) gives ∆ σ ≈ 5.1974 , ∆ ϵ ≈ 1.4192 , P ϵ ≈ 1.1019 , P T ≈ 0.10715 and (∆ 75, 5.38).Although less accurate in comparison to (4.8), the η minimization results for the relevant operators are still around the 3D Ising solution.On the other hand, the irrelevant scaling dimensions (∆\\nℓ=2 ) become more accurate. 26This is consistent with our expectation that the higher derivative equations are more sensitive to the high-lying contributions, so the predic- 26 We also notice this improvement for the irrelevant scaling dimensions by replacing a 5-derivative constraint with a 7-derivative constraint.For example, the cases for (4.10), (4.11), (4.12), (4.13) are (∆ tions for the irrelevant operators are improved, at the price of lowering the accuracy of the low-lying parameters, which are contaminated by the less accurate high-lying contributions.\\nSince we have 10 constraints with m + n ≤ 7, we can introduce more free parameters.It is natural to consider the lowest spin-4 operator, as its scaling dimension ∆ (k=0) ℓ=4 ≈ 5 is close to the subleading spin-2 scaling dimension, which is viewed as a free parameter in the above.Surprisingly, we arrive at even more accurate results by relaxing the scaling dimension and OPE coefficient of the lowest spin-4 operator: ∆ σ ≈ 0.518152 , ∆ ϵ ≈ 1.41278 , P ϵ ≈ 1.10640 , P T ≈ 0.106226 .are almost as accurate as the latest Monte-Carlo determinations [6].\\nAs the system is still overdetermined, one may try to introduce more free parameters at higher spin.However, if we do not approximate the lowest spin-6 parameters by the analytic formulae and set them as free parameters, we are not able to find a local minimum around the 3D Ising solution.It seems that the η minimization becomes unstable due to the fact that the squared OPE coefficient of the spin-6 operator is allowed to be negative.Besides reducing computational efforts, stabilization is another important merit of analytic approximations for the high-lying contributions.This is different from the hybrid approach in [44], in which the use of positivity constraints stabilizes the hybrid bootstrap method and thus the effects of the choice for the gluing spin can be studied more systematically.\"\n },\n {\n \"title\": \"Discussion\",\n \"content\": \"In this work, we revisited the conformal bootstrap approach to the 3D Ising model using the global asymptotic behavior and the rapidly convergent crossing constraints.We derived surprisingly accurate results (4.14) from simple analytic asymptotic formulae and a small number of nonperturbative crossing constraints, using the error minimization method.In some sense, this furnishes a minimal example for the nonperturbative conformal bootstrap in three dimensions, with the help of an analytic understanding from the lightcone singularity structure.\\nTo improve the accuracy of the hybrid approach, it is crucial to go beyond the leading corrections to the GFF behavior. 27The asymptotic formulae can be computed more systematically using the methods in [33,75,76].However, one should be more careful as a higher order asymptotic expansion may give worse results at low spin, even if the approximations at large spin are improved.Then the corresponding low spin parameters should be viewed as free parameters and determined by the nonperturbative crossing constraints.To avoid this subtlety, one may use the convergent Lorentzian inversion formula [77][78][79], which includes the nonperturbative contributions [80][81][82][83][84][85][86][87]. 28 29However, the price to pay is a higher computational cost.In practice, it may be useful to combine the asymptotic methods with the convergent methods.\\nWe also need to unmix the higher trajectories. 30In the present work, we used the decoupling phenomenon to circumvent the mixing problem, and carried out the bootstrap study using only the identical correlator ⟨σ(x 1 )σ(x 2 )σ(x 3 )σ(x 4 )⟩.To properly resolve the mixing problem, one needs to consider a mixed system of correlators [33,[85][86][87].The large repulsion of the twist spectra at low spin is particularly interesting.This can lead to more significant deviations from the leading GFF behavior.We also believe that this is closely related to the decoupling phenomenon in [7,8].\\nWe would like to emphasize that the goal of the present work is not to surpass the remarkably precise results from the positivity bound approach. 31Instead, we want to show that a hybrid analytical/numerical approach can have the great potential in reducing the computational cost with the help of a global understanding of the high-lying data.The application to the 3D Ising model is just the first example.This strategy can be applied to other well constrained bootstrap solutions, such as the O(N ) models [11,[93][94][95] and the supersymmetric Ising model [87,96,97].For a more complete list of the conformal bootstrap targets, we refer to the comprehensive review [98] and the more recent one [99].As we are not using any positivity constraint, a natural and important direction is to consider the less explored non-positive bootstrap problems that involve D > 2 conformal field theories with nonunitarity [42], defects [100,101], and on nontrivial manifolds [57,102].\"\n },\n {\n \"title\": \"FreeFigure 1 :\",\n \"content\": \"Figure 1:The approximate solution for ∆ ϵ at various ∆ σ .The red dots represent the free theory and the Ising solutions in three dimensions.The blue dots are obtained from solving (3.5) with GFF approximations for the high-lying contributions.The orange dots are derived from solving (3.6) with the naive truncation that omits all the high-lying contributions.The nonperturbative crossing constraints are associated with the derivative orders in(3.7).The results with the GFF approximations (blue) are much closer to the 3D Ising solution (1.1).\"\n },\n {\n \"title\": \"Figure 2 :\",\n \"content\": \"Figure 2: The minimized η function (4.7) at different (∆ σ , ∆ ϵ ).The local minimum (red dot) with η min ≈ 0.00043 determines the 3D Ising results in (4.8).\"\n },\n {\n \"title\": \"Figure 3 :\",\n \"content\": \"Figure 3: The twist τ = ∆ − ℓ of the lowest Z 2 -even trajectory at various spin ℓ.The curve is based on the analytic expression τ (ℓ) = 2∆ σ + γ\"\n },\n {\n \"title\": \"Figure 4 :\",\n \"content\": \"Figure 4: The squared OPE coefficients of the leading Z 2 -even trajectory at different spin ℓ, divided by the generalized free case P GFF (ℓ) = 2S −∆σ (∆ σ + ℓ).The curve is based on the analytic expression P (ℓ) = P\\nℓ=2 ) ≈ (3.75, 6.87), (3.74, 5.36), (3.77, 5.22), (3.72, 5.44).\\nℓ=2 ) ≈ (3.67, 5.46).These are the most accurate results in this work.In particular, the OPE coefficient λ σσϵ = P 1/2 ϵ ≈ 1.05186 is significantly improved.The local minimization is carried out by Mathematica's FindMinimum, starting at (∆ σ , ∆ ϵ ) = (1, 3/2), (∆ (k=1) ℓ=0 , ∆ (k=1) ℓ=2 , ∆ (k=0) ℓ=4 ) = (3, 5, 5) and (P ϵ , P T , P (k=0) ℓ=4 ) = (1, 1/10, 1/200), which are some crude numbers from the GFF approximation.The corresponding critical exponents η ≈ 0.036304 , ν ≈ 0.63003 (4.15)\\nbased on the GFF approximations.The corresponding η minimization results are (0, 7) : ∆ σ ≈ 0.51781 , ∆ ϵ ≈ 1.4122 , P ϵ ≈ 1.1042 , P T ≈ 0.10642 ,\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Figure 1:The approximate solution for ∆ ϵ at various ∆ σ .The red dots represent the free theory and the Ising solutions in three dimensions.The blue dots are obtained from solving (3.5) with GFF approximations for the high-lying contributions.The orange dots are derived from solving (3.6) with the naive truncation that omits all the high-lying contributions.The nonperturbative crossing constraints are associated with the derivative orders in(3.7).The results with the GFF approximations (blue) are much closer to the 3D Ising solution (1.1).","content":"FreeFigure 1 :\n1\nFigure 1:The approximate solution for ∆ ϵ at various ∆ σ .The red dots represent the free theory and the Ising solutions in three dimensions.The blue dots are obtained from solving (3.5) with GFF approximations for the high-lying contributions.The orange dots are derived from solving (3.6) with the naive truncation that omits all the high-lying contributions.The nonperturbative crossing constraints are associated with the derivative orders in(3.7).The results with the GFF approximations (blue) are much closer to the 3D Ising solution (1.1)."},{"id":"","caption":"Figure 2: The minimized η function (4.7) at different (∆ σ , ∆ ϵ ).The local minimum (red dot) with η min ≈ 0.00043 determines the 3D Ising results in (4.8).","content":"Figure 2 :\n2\nFigure 2: The minimized η function (4.7) at different (∆ σ , ∆ ϵ ).The local minimum (red dot) with η min ≈ 0.00043 determines the 3D Ising results in (4.8)."},{"id":"","caption":"Figure 3: The twist τ = ∆ − ℓ of the lowest Z 2 -even trajectory at various spin ℓ.The curve is based on the analytic expression τ (ℓ) = 2∆ σ + γ","content":"Figure 3 :\n3\nFigure 3: The twist τ = ∆ − ℓ of the lowest Z 2 -even trajectory at various spin ℓ.The curve is based on the analytic expression τ (ℓ) = 2∆ σ + γ"},{"id":"","caption":"Figure 4: The squared OPE coefficients of the leading Z 2 -even trajectory at different spin ℓ, divided by the generalized free case P GFF (ℓ) = 2S −∆σ (∆ σ + ℓ).The curve is based on the analytic expression P (ℓ) = P","content":"Figure 4 :\n4\nFigure 4: The squared OPE coefficients of the leading Z 2 -even trajectory at different spin ℓ, divided by the generalized free case P GFF (ℓ) = 2S −∆σ (∆ σ + ℓ).The curve is based on the analytic expression P (ℓ) = P"},{"id":"","caption":"ℓ=2 ) ≈ (3.75, 6.87), (3.74, 5.36), (3.77, 5.22), (3.72, 5.44).","content":"ℓ=2 ) ≈ (3.75, 6.87), (3.74, 5.36), (3.77, 5.22), (3.72, 5.44)."},{"id":"","caption":"ℓ=2 ) ≈ (3.67, 5.46).These are the most accurate results in this work.In particular, the OPE coefficient λ σσϵ = P 1/2 ϵ ≈ 1.05186 is significantly improved.The local minimization is carried out by Mathematica's FindMinimum, starting at (∆ σ , ∆ ϵ ) = (1, 3/2), (∆ (k=1) ℓ=0 , ∆ (k=1) ℓ=2 , ∆ (k=0) ℓ=4 ) = (3, 5, 5) and (P ϵ , P T , P (k=0) ℓ=4 ) = (1, 1/10, 1/200), which are some crude numbers from the GFF approximation.The corresponding critical exponents η ≈ 0.036304 , ν ≈ 0.63003 (4.15)","content":"ℓ=2 ) ≈ (3.67, 5.46).These are the most accurate results in this work.In particular, the OPE coefficient λ σσϵ = P 1/2 ϵ ≈ 1.05186 is significantly improved.The local minimization is carried out by Mathematica's FindMinimum, starting at (∆ σ , ∆ ϵ ) = (1, 3/2), (∆ (k=1) ℓ=0 , ∆ (k=1) ℓ=2 , ∆ (k=0) ℓ=4 ) = (3, 5, 5) and (P ϵ , P T , P (k=0) ℓ=4 ) = (1, 1/10, 1/200), which are some crude numbers from the GFF approximation.The corresponding critical exponents η ≈ 0.036304 , ν ≈ 0.63003 (4.15)"},{"id":"","caption":"based on the GFF approximations.The corresponding η minimization results are (0, 7) : ∆ σ ≈ 0.51781 , ∆ ϵ ≈ 1.4122 , P ϵ ≈ 1.1042 , P T ≈ 0.10642 ,","content":"based on the GFF approximations.The corresponding η minimization results are (0, 7) : ∆ σ ≈ 0.51781 , ∆ ϵ ≈ 1.4122 , P ϵ ≈ 1.1042 , P T ≈ 0.10642 ,\n(4.10)(1, 6) : ∆ σ ≈ 0.51867 , ∆ ϵ ≈ 1.4133 , P ϵ ≈ 1.1069 , P T ≈ 0.10658 ,(4.11)(2, 5) : ∆ σ ≈ 0.51876 , ∆ ϵ ≈ 1.4137 , P ϵ ≈ 1.1069 , P T ≈ 0.10659 ,(4.12)(3, 4) : ∆ σ ≈ 0.51862 , ∆ ϵ ≈ 1.4131 , P ϵ ≈ 1.1068 , P T ≈ 0.10657 ,(4.13)"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Figure 1:The approximate solution for ∆ ϵ at various ∆ σ .The red dots represent the free theory and the Ising solutions in three dimensions.The blue dots are obtained from solving (3.5) with GFF approximations for the high-lying contributions.The orange dots are derived from solving (3.6) with the naive truncation that omits all the high-lying contributions.The nonperturbative crossing constraints are associated with the derivative orders in(3.7).The results with the GFF approximations (blue) are much closer to the 3D Ising solution (1.1).\",\n \"content\": \"FreeFigure 1 :\\n1\\nFigure 1:The approximate solution for ∆ ϵ at various ∆ σ .The red dots represent the free theory and the Ising solutions in three dimensions.The blue dots are obtained from solving (3.5) with GFF approximations for the high-lying contributions.The orange dots are derived from solving (3.6) with the naive truncation that omits all the high-lying contributions.The nonperturbative crossing constraints are associated with the derivative orders in(3.7).The results with the GFF approximations (blue) are much closer to the 3D Ising solution (1.1).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 2: The minimized η function (4.7) at different (∆ σ , ∆ ϵ ).The local minimum (red dot) with η min ≈ 0.00043 determines the 3D Ising results in (4.8).\",\n \"content\": \"Figure 2 :\\n2\\nFigure 2: The minimized η function (4.7) at different (∆ σ , ∆ ϵ ).The local minimum (red dot) with η min ≈ 0.00043 determines the 3D Ising results in (4.8).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 3: The twist τ = ∆ − ℓ of the lowest Z 2 -even trajectory at various spin ℓ.The curve is based on the analytic expression τ (ℓ) = 2∆ σ + γ\",\n \"content\": \"Figure 3 :\\n3\\nFigure 3: The twist τ = ∆ − ℓ of the lowest Z 2 -even trajectory at various spin ℓ.The curve is based on the analytic expression τ (ℓ) = 2∆ σ + γ\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 4: The squared OPE coefficients of the leading Z 2 -even trajectory at different spin ℓ, divided by the generalized free case P GFF (ℓ) = 2S −∆σ (∆ σ + ℓ).The curve is based on the analytic expression P (ℓ) = P\",\n \"content\": \"Figure 4 :\\n4\\nFigure 4: The squared OPE coefficients of the leading Z 2 -even trajectory at different spin ℓ, divided by the generalized free case P GFF (ℓ) = 2S −∆σ (∆ σ + ℓ).The curve is based on the analytic expression P (ℓ) = P\"\n },\n {\n \"id\": \"\",\n \"caption\": \"ℓ=2 ) ≈ (3.75, 6.87), (3.74, 5.36), (3.77, 5.22), (3.72, 5.44).\",\n \"content\": \"ℓ=2 ) ≈ (3.75, 6.87), (3.74, 5.36), (3.77, 5.22), (3.72, 5.44).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"ℓ=2 ) ≈ (3.67, 5.46).These are the most accurate results in this work.In particular, the OPE coefficient λ σσϵ = P 1/2 ϵ ≈ 1.05186 is significantly improved.The local minimization is carried out by Mathematica's FindMinimum, starting at (∆ σ , ∆ ϵ ) = (1, 3/2), (∆ (k=1) ℓ=0 , ∆ (k=1) ℓ=2 , ∆ (k=0) ℓ=4 ) = (3, 5, 5) and (P ϵ , P T , P (k=0) ℓ=4 ) = (1, 1/10, 1/200), which are some crude numbers from the GFF approximation.The corresponding critical exponents η ≈ 0.036304 , ν ≈ 0.63003 (4.15)\",\n \"content\": \"ℓ=2 ) ≈ (3.67, 5.46).These are the most accurate results in this work.In particular, the OPE coefficient λ σσϵ = P 1/2 ϵ ≈ 1.05186 is significantly improved.The local minimization is carried out by Mathematica's FindMinimum, starting at (∆ σ , ∆ ϵ ) = (1, 3/2), (∆ (k=1) ℓ=0 , ∆ (k=1) ℓ=2 , ∆ (k=0) ℓ=4 ) = (3, 5, 5) and (P ϵ , P T , P (k=0) ℓ=4 ) = (1, 1/10, 1/200), which are some crude numbers from the GFF approximation.The corresponding critical exponents η ≈ 0.036304 , ν ≈ 0.63003 (4.15)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"based on the GFF approximations.The corresponding η minimization results are (0, 7) : ∆ σ ≈ 0.51781 , ∆ ϵ ≈ 1.4122 , P ϵ ≈ 1.1042 , P T ≈ 0.10642 ,\",\n \"content\": \"based on the GFF approximations.The corresponding η minimization results are (0, 7) : ∆ σ ≈ 0.51781 , ∆ ϵ ≈ 1.4122 , P ϵ ≈ 1.1042 , P T ≈ 0.10642 ,\\n(4.10)(1, 6) : ∆ σ ≈ 0.51867 , ∆ ϵ ≈ 1.4133 , P ϵ ≈ 1.1069 , P T ≈ 0.10658 ,(4.11)(2, 5) : ∆ σ ≈ 0.51876 , ∆ ϵ ≈ 1.4137 , P ϵ ≈ 1.1069 , P T ≈ 0.10659 ,(4.12)(3, 4) : ∆ σ ≈ 0.51862 , ∆ ϵ ≈ 1.4131 , P ϵ ≈ 1.1068 , P T ≈ 0.10657 ,(4.13)\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Crystal statistics. 1. A Two-dimensional model with an order disorder transition. L Onsager, 10.1103/PhysRev.65.117Phys. Rev. 651944","doi":""},{"text":"Combinatorial and topological approach to the 3-D Ising model. T Regge, R Zecchina, 10.1088/0305-4470/33/4/308arXiv:cond-mat/9909168J. Phys. A. 332000cond-mat","doi":""},{"text":"Dimer statistics and phase transitions. P Kasteleyn, 10.1063/1.1703953J. Math. Phys. 41963","doi":""},{"text":"Ising model close to d=2. W Li, 10.1103/PhysRevD.105.L091902arXiv:2107.13679Phys. Rev. D. 1059L0919022022hep-th","doi":""},{"text":"Finite size scaling study of lattice models in the three-dimensional Ising universality class. M Hasenbusch, 10.1103/PhysRevB.82.174433arXiv:1004.4486Phys. Rev. B. 821744332010cond-mat.stat-mech","doi":""},{"text":"Restoring isotropy in a three-dimensional lattice model: The Ising universality class. M Hasenbusch, 10.1103/PhysRevB.104.014426arXiv:2105.09781Phys. Rev. B. 1041144262021cond-mat.stat-mech","doi":""},{"text":"Solving the 3D Ising Model with the Conformal Bootstrap. S El-Showk, M F Paulos, D Poland, S Rychkov, D Simmons-Duffin, A Vichi, 10.1103/PhysRevD.86.025022arXiv:1203.6064Phys. Rev. D. 86250222012hep-th","doi":""},{"text":"Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents. S El-Showk, M F Paulos, D Poland, S Rychkov, D Simmons-Duffin, A Vichi, 10.1007/s10955-014-1042-7arXiv:1403.4545J. Stat. Phys. 1572014hep-th","doi":""},{"text":"Bootstrapping Mixed Correlators in the 3D Ising Model. F Kos, D Poland, D Simmons-Duffin, 10.1007/JHEP11(2014)109arXiv:1406.4858JHEP. 112014hep-th","doi":""},{"text":"A Semidefinite Program Solver for the Conformal Bootstrap. D Simmons-Duffin, 10.1007/JHEP06(2015)174arXiv:1502.02033JHEP. 061742015hep-th","doi":""},{"text":"Precision Islands in the Ising and O(N ) Models. F Kos, D Poland, D Simmons-Duffin, A Vichi, 10.1007/JHEP08(2016)036arXiv:1603.04436JHEP. 08362016hep-th","doi":""},{"text":"Line defects in the 3d Ising model. M Billó, M Caselle, D Gaiotto, F Gliozzi, M Meineri, R Pellegrini, 10.1007/JHEP07(2013)055arXiv:1304.4110JHEP. 07552013hep-th","doi":""},{"text":"Conformal symmetry of the critical 3D Ising model inside a sphere. C Cosme, J M V P Lopes, J Penedones, 10.1007/JHEP08(2015)022arXiv:1503.02011JHEP. 08222015hep-th","doi":""},{"text":"Uncovering Conformal Symmetry in the 3D Ising Transition: State-Operator Correspondence from a Quantum Fuzzy Sphere Regularization. W Zhu, C Han, E Huffman, J S Hofmann, Y C He, 10.1103/PhysRevX.13.021009arXiv:2210.13482Phys. Rev. X. 132210092023cond-mat.stat-mech","doi":""},{"text":"Operator Product Expansion Coefficients of the 3D Ising Criticality via Quantum Fuzzy Spheres. L Hu, Y C He, W Zhu, 10.1103/PhysRevLett.131.031601arXiv:2303.08844Phys. Rev. Lett. 1313316012023cond-mat.stat-mech","doi":""},{"text":"Conformal four-point correlators of the three-dimensional Ising transition via the quantum fuzzy sphere. C Han, L Hu, W Zhu, Y C He, 10.1103/PhysRevB.108.235123arXiv:2306.04681Phys. Rev. B. 108232351232023cond-mat.stat-mech","doi":""},{"text":"3D Ising CFT and exact diagonalization on icosahedron: The power of conformal perturbation theory. B X Lao, S Rychkov, 10.21468/SciPostPhys.15.6.243arXiv:2307.02540SciPost Phys. 1562432023hep-th","doi":""},{"text":"Solving Conformal Defects in 3D Conformal Field Theory using Fuzzy Sphere Regularization. L Hu, Y C He, W Zhu, arXiv:2308.01903cond-mat.stat-mech","doi":""},{"text":"Bounding scalar operator dimensions in 4D CFT. R Rattazzi, V S Rychkov, E Tonni, A Vichi, 10.1088/1126-6708/2008/12/031arXiv:0807.0004JHEP. 12312008hep-th","doi":""},{"text":"Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory. A A Belavin, A M Polyakov, A B Zamolodchikov, 10.1016/0550-3213(84)90052-XNucl. Phys. B. 2411984","doi":""},{"text":"Null bootstrap for non-Hermitian Hamiltonians. W Li, 10.1103/PhysRevD.106.125021arXiv:2202.04334Phys. Rev. D. 106121250212022hep-th","doi":""},{"text":"Taming Dyson-Schwinger Equations with Null States. W Li, 10.1103/PhysRevLett.131.031603arXiv:2303.10978Phys. Rev. Lett. 1313316032023hep-th","doi":""},{"text":"OPE Convergence in Conformal Field Theory. D Pappadopulo, S Rychkov, J Espin, R Rattazzi, 10.1103/PhysRevD.86.105043arXiv:1208.6449Phys. Rev. D. 861050432012hep-th","doi":""},{"text":"Nonperturbative Mellin Amplitudes: Existence, Properties, Applications. J Penedones, J A Silva, A Zhiboedov, 10.1007/JHEP08(2020)031arXiv:1912.11100JHEP. 08312020hep-th","doi":""},{"text":"The Analytic Bootstrap and AdS Superhorizon Locality. A L Fitzpatrick, J Kaplan, D Poland, D Simmons-Duffin, 10.1007/JHEP12(2013)004arXiv:1212.3616JHEP. 1242013hep-th","doi":""},{"text":"Convexity and Liberation at Large Spin. Z Komargodski, A Zhiboedov, 10.1007/JHEP11(2013)140arXiv:1212.4103JHEP. 111402013hep-th","doi":""},{"text":"How to measure the dimension of the parton field. G Parisi, 10.1016/0550-3213(73)90666-4Nucl. Phys. B. 591973","doi":""},{"text":"Bjorken scaling in quantum field theory. C G CallanJr, D J Gross, 10.1103/PhysRevD.8.4383Phys. Rev. D. 81973","doi":""},{"text":"Comments on operators with large spin. L F Alday, J M Maldacena, 10.1088/1126-6708/2007/11/019arXiv:0708.0672JHEP. 11192007hep-th","doi":""},{"text":"Twist Accumulation in Conformal Field Theory: A Rigorous Approach to the Lightcone Bootstrap. S Pal, J Qiao, S Rychkov, 10.1007/s00220-023-04767-warXiv:2212.04893Commun. Math. Phys. 40232023hep-th","doi":""},{"text":"D Poland, D Simmons-Duffin, 10.1007/JHEP05(2011)017arXiv:1009.2087Bounds on 4D Conformal and Superconformal Field Theories. 20110517hep-th","doi":""},{"text":"Bootstrapping Conformal Field Theories with the Extremal Functional Method. S El-Showk, M F Paulos, 10.1103/PhysRevLett.111.241601arXiv:1211.2810Phys. Rev. Lett. 111242416012013hep-th","doi":""},{"text":"The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT. D Simmons-Duffin, 10.1007/JHEP03(2017)086arXiv:1612.08471JHEP. 03862017hep-th","doi":""},{"text":"Conformal Bootstrap With Slightly Broken Higher Spin Symmetry. L F Alday, A Zhiboedov, 10.1007/JHEP06(2016)091arXiv:1506.04659JHEP. 06912016hep-th","doi":""},{"text":"An Algebraic Approach to the Analytic Bootstrap. L F Alday, A Zhiboedov, 10.1007/JHEP04(2017)157arXiv:1510.08091JHEP. 041572017hep-th","doi":""},{"text":"Anharmonic oscillator. C M Bender, T T Wu, 10.1103/PhysRev.184.1231Phys. Rev. 1841969","doi":""},{"text":"Principle of minimal singularity for Green's functions. W Li, 10.1103/PhysRevD.109.045012arXiv:2309.02201Phys. Rev. D. 1094450122024hep-th","doi":""},{"text":"Underdetermined Dyson-Schwinger Equations. C M Bender, C Karapoulitidis, S P Klevansky, 10.1103/PhysRevLett.130.101602arXiv:2211.13026Phys. Rev. Lett. 130101016022023math-ph","doi":""},{"text":"Dyson-Schwinger equations in zero dimensions and polynomial approximations. C M Bender, C Karapoulitidis, S P Klevansky, 10.1103/PhysRevD.108.056002arXiv:2307.01008Phys. Rev. D. 1085560022023math-ph","doi":""},{"text":"Bootstrapping Matrix Quantum Mechanics. X Han, S A Hartnoll, J Kruthoff, 10.1103/PhysRevLett.125.041601arXiv:2004.10212Phys. Rev. Lett. 1254416012020hep-th","doi":""},{"text":"W Li, arXiv:2402.05778The ϕ n trajectory bootstrap. hep-th","doi":""},{"text":"More constraining conformal bootstrap. F Gliozzi, 10.1103/PhysRevLett.111.161602arXiv:1307.3111Phys. Rev. Lett. 1111616022013hep-th","doi":""},{"text":"Bootstrapping the 3d Ising model at finite temperature. L Iliesiu, M Kolouglu, D Simmons-Duffin, 10.1007/JHEP12(2019)072arXiv:1811.05451JHEP. 12722019hep-th","doi":""},{"text":"The Hybrid Bootstrap. N Su, arXiv:2202.07607hep-th","doi":""},{"text":"Bootstrapping the O(N ) vector models. F Kos, D Poland, D Simmons-Duffin, 10.1007/JHEP06(2014)091arXiv:1307.6856JHEP. 06912014hep-th","doi":""},{"text":"HomotopyContinuation.jl: A Package for Homotopy Continuation in Julia. P Breiding, S Timme, Mathematical Software -ICMS 2018, J.H. Davenport, M. Kauers, G. Labahn and J. Urban2018Springer International Publishing","doi":""},{"text":"New method for the conformal bootstrap with OPE truncations. W Li, arXiv:1711.09075hep-th","doi":""},{"text":"simpleboot: A mathematica framework for bootstrap calculations. N Su,","doi":""},{"text":"I Heemskerk, J Penedones, J Polchinski, J Sully, 10.1088/1126-6708/2009/10/079arXiv:0907.0151Holography from Conformal Field Theory. 20091079hep-th","doi":""},{"text":"Unitarity and the Holographic S-Matrix. A L Fitzpatrick, J Kaplan, 10.1007/JHEP10(2012)032arXiv:1112.4845JHEP. 10322012hep-th","doi":""},{"text":"W Li, 10.1007/JHEP01arXiv:1706.04054Inverse Bootstrapping Conformal Field Theories. 2018. 20180177hep-th","doi":""},{"text":"Conformal Invariance and Multifractality at Anderson Transitions in Arbitrary Dimensions. J Padayasi, I A Gruzberg, 10.1103/PhysRevLett.131.266401arXiv:2306.07340Phys. Rev. Lett. 131262664012023cond-mat.dis-nn","doi":""},{"text":"Improving the five-point bootstrap. D Poland, V Prilepina, P Tadić, arXiv:2312.13344hep-th","doi":""},{"text":"Factorized lightcone expansion of conformal blocks. W Li, 10.1007/JHEP05(2021)128arXiv:2012.09710JHEP. 128052021hep-th","doi":""},{"text":"Critical exponents of the 3d Ising and related models from Conformal Bootstrap. F Gliozzi, A Rago, 10.1007/JHEP10(2014)042arXiv:1403.6003JHEP. 10422014hep-th","doi":""},{"text":"Boundary and Interface CFTs from the Conformal Bootstrap. F Gliozzi, P Liendo, M Meineri, A Rago, 10.1007/JHEP05(2015)036arXiv:1502.07217JHEP. 05362015hep-th","doi":""},{"text":"Bootstrapping critical Ising model on three-dimensional real projective space. Y Nakayama, 10.1103/PhysRevLett.116.141602arXiv:1601.06851Phys. Rev. Lett. 116141416022016hep-th","doi":""},{"text":"Truncatable bootstrap equations in algebraic form and critical surface exponents. F Gliozzi, 10.1007/JHEP10(2016)037arXiv:1605.04175JHEP. 10372016hep-th","doi":""},{"text":"Closure of the Operator Product Expansion in the Non-Unitary Bootstrap. I Esterlis, A L Fitzpatrick, D Ramirez, 10.1007/JHEP11(2016)030arXiv:1606.07458JHEP. 11302016hep-th","doi":""},{"text":"C Hasegawa, Y Nakayama, 10.1142/S0217732317500456arXiv:1611.06373Expansion in Critical ϕ 3 -Theory on Real Projective Space from Conformal Field Theory. 2017321750045hep-th","doi":""},{"text":"Conformal bootstrap analysis for the Yang-Lee edge singularity. S Hikami, 10.1093/ptep/pty054arXiv:1707.04813PTEP. 201852018hep-th","doi":""},{"text":"Conformal Bootstrap Analysis for Single and Branched Polymers. S Hikami, 10.1093/ptep/pty132arXiv:1708.03072PTEP. 2018122018hep-th","doi":""},{"text":"Dimensional Reduction by Conformal Bootstrap. S Hikami, 10.1093/ptep/ptz081arXiv:1801.09052PTEP. 201982019cond-mat.dis-nn","doi":""},{"text":"Conformal bootstrap for percolation and polymers. A Leclair, J Squires, 10.1088/1742-5468/aaf10aarXiv:1802.08911J. Stat. Mech. 1812121231052018hep-th","doi":""},{"text":"S Hikami, arXiv:1811.05918Conformal Bootstrap Analysis for Localization: Symplectic Case. cond-mat.dis-nn","doi":""},{"text":"On the ϕ 3 theory above six dimensions. J Rong, J Zhu, 10.1007/JHEP04(2020)151arXiv:2001.10864JHEP. 041512020hep-th","doi":""},{"text":"Is there supersymmetric Lee-Yang fixed point in three dimensions?. Y Nakayama, 10.1142/S0217751X21501761arXiv:2104.13570Int. J. Mod. Phys. A. 362421501762021hep-th","doi":""},{"text":"The extraordinary boundary transition in the 3d O(N) model via conformal bootstrap. J Padayasi, A Krishnan, M A Metlitski, I A Gruzberg, M Meineri, 10.21468/SciPostPhys.12.6.190arXiv:2111.03071SciPost Phys. 1262022cond-mat.stat-mech","doi":""},{"text":"Solving Conformal Field Theories with Artificial Intelligence. G Kántor, V Niarchos, C Papageorgakis, 10.1103/PhysRevLett.128.041601arXiv:2108.08859Phys. Rev. Lett. 1284416012022hep-th","doi":""},{"text":"Conformal bootstrap with reinforcement learning. G Kántor, V Niarchos, C Papageorgakis, 10.1103/PhysRevD.105.025018arXiv:2108.09330Phys. Rev. D. 1052250182022hep-th","doi":""},{"text":"Monte Carlo approach to the conformal bootstrap. A Laio, U L Valenzuela, M Serone, 10.1103/PhysRevD.106.025019arXiv:2206.05193Phys. Rev. D. 1062250192022hep-th","doi":""},{"text":"6D (2,0) bootstrap with the soft-actor-critic algorithm. G Kántor, V Niarchos, C Papageorgakis, P Richmond, 10.1103/PhysRevD.107.025005arXiv:2209.02801Phys. Rev. D. 1072250052023hep-th","doi":""},{"text":"The five-point bootstrap. D Poland, V Prilepina, P Tadić, 10.1007/JHEP10(2023)153arXiv:2305.08914JHEP. 101532023hep-th","doi":""},{"text":"Bootstrability in line-defect CFTs with improved truncation methods. V Niarchos, C Papageorgakis, P Richmond, A G Stapleton, M Woolley, 10.1103/PhysRevD.108.105027arXiv:2306.15730Phys. Rev. D. 108101050272023hep-th","doi":""},{"text":"Large Spin Perturbation Theory for Conformal Field Theories. L F Alday, 10.1103/PhysRevLett.119.111601arXiv:1611.01500Phys. Rev. Lett. 119111116012017hep-th","doi":""},{"text":"Solving CFTs with Weakly Broken Higher Spin Symmetry. L F Alday, 10.1007/JHEP10(2017)161arXiv:1612.00696JHEP. 101612017hep-th","doi":""},{"text":"Analyticity in Spin in Conformal Theories. S Caron-Huot, 10.1007/JHEP09(2017)078arXiv:1703.00278JHEP. 1709782017hep-th","doi":""},{"text":"A spacetime derivation of the Lorentzian OPE inversion formula. D Simmons-Duffin, D Stanford, E Witten, 10.1007/JHEP07arXiv:1711.03816JHEP. 1807852018. 2018hep-th","doi":""},{"text":"Light-ray operators in conformal field theory. P Kravchuk, D Simmons-Duffin, 10.1007/JHEP11arXiv:1805.00098JHEP. 18111022018. 2018hep-th","doi":""},{"text":"J Liu, E Perlmutter, V Rosenhaus, D Simmons-Duffin, 10.1007/JHEP03(2019)052arXiv:1808.00612d-dimensional SYK, AdS Loops, and 6j Symbols. 20190352hep-th","doi":""},{"text":"Resummation at finite conformal spin. C Cardona, S Guha, S K Kanumilli, K Sen, 10.1007/JHEP01(2019)077arXiv:1811.00213JHEP. 1901772019hep-th","doi":""},{"text":"More Analytic Bootstrap: Nonperturbative Effects and Fermions. S Albayrak, D Meltzer, D Poland, 10.1007/JHEP08(2019)040arXiv:1904.00032JHEP. 1908402019hep-th","doi":""},{"text":"Closed-form expression for cross-channel conformal blocks near the lightcone. W Li, 10.1007/JHEP01(2020)055arXiv:1906.00707JHEP. 552001. 2020hep-th","doi":""},{"text":"The Inversion Formula and 6j Symbol for 3d Fermions. S Albayrak, D Meltzer, D Poland, 10.1007/JHEP09(2020)148arXiv:2006.07374JHEP. 091482020hep-th","doi":""},{"text":"The Lorentzian inversion formula and the spectrum of the 3d O(2) CFT. J Liu, D Meltzer, D Poland, D Simmons-Duffin, 10.1007/JHEP09(2020)115arXiv:2007.07914JHEP. 091152020erratum: JHEP 01, 206 (2021). hep-th","doi":""},{"text":"The leading trajectory in the 2+1D Ising CFT. S Caron-Huot, Y Gobeil, Z Zahraee, 10.1007/JHEP02(2023)190arXiv:2007.11647JHEP. 021902023hep-th","doi":""},{"text":"Precision bootstrap for the N = 1 super-Ising model. A Atanasov, A Hillman, D Poland, J Rong, N Su, 10.1007/JHEP08(2022)136arXiv:2201.02206JHEP. 081362022hep-th","doi":""},{"text":"Dispersive CFT Sum Rules. S Caron-Huot, D Mazac, L Rastelli, D Simmons-Duffin, 10.1007/JHEP05(2021)243arXiv:2008.04931JHEP. 052432021hep-th","doi":""},{"text":"A Conformal Dispersion Relation: Correlations from Absorption. D Carmi, S Caron-Huot, 10.1007/JHEP09(2020)009arXiv:1910.12123JHEP. 0992020hep-th","doi":""},{"text":"A basis of analytic functionals for CFTs in general dimension. D Mazáč, L Rastelli, X Zhou, 10.1007/JHEP08(2021)140arXiv:1910.12855JHEP. 081402021hep-th","doi":""},{"text":"Applications of dispersive sum rules: epsilon-expansion and holography. D Carmi, J Penedones, J A Silva, A Zhiboedov, 10.21468/SciPostPhys.10.6.145arXiv:2009.13506SciPost Phys. 1062021hep-th","doi":""},{"text":"Mixed correlator dispersive CFT sum rules. A K Trinh, 10.1007/JHEP03(2022)032arXiv:2111.14731JHEP. 03322022hep-th","doi":""},{"text":"Bootstrapping the O(N) Archipelago. F Kos, D Poland, D Simmons-Duffin, A Vichi, 10.1007/JHEP11(2015)106arXiv:1504.07997JHEP. 111062015hep-th","doi":""},{"text":"Carving out OPE space and precise O(2) model critical exponents. S M Chester, W Landry, J Liu, D Poland, D Simmons-Duffin, N Su, A Vichi, 10.1007/JHEP06(2020)142arXiv:1912.03324JHEP. 061422020hep-th","doi":""},{"text":"Bootstrapping Heisenberg magnets and their cubic instability. S M Chester, W Landry, J Liu, D Poland, D Simmons-Duffin, N Su, A Vichi, 10.1103/PhysRevD.104.105013arXiv:2011.14647Phys. Rev. D. 104101050132021hep-th","doi":""},{"text":"Bootstrapping the minimal N = 1 superconformal field theory in three dimensions. J Rong, N Su, 10.1007/JHEP06(2021)154arXiv:1807.04434JHEP. 061542021hep-th","doi":""},{"text":"Bootstrapping the Minimal 3D SCFT. A Atanasov, A Hillman, D Poland, 10.1007/JHEP11arXiv:1807.05702JHEP. 111402018. 2018hep-th","doi":""},{"text":"The Conformal Bootstrap: Theory, Numerical Techniques, and Applications. D Poland, S Rychkov, A Vichi, 10.1103/RevModPhys.91.015002arXiv:1805.04405Rev. Mod. Phys. 91150022019hep-th","doi":""},{"text":"S Rychkov, N Su, arXiv:2311.15844New Developments in the Numerical Conformal Bootstrap. hep-th","doi":""},{"text":"The Bootstrap Program for Boundary CFT d. P Liendo, L Rastelli, B C Van Rees, 10.1007/JHEP07(2013)113arXiv:1210.4258JHEP. 071132013hep-th","doi":""},{"text":"Defects in conformal field theory. M Billò, V Goncalves, E Lauria, M Meineri, 10.1007/JHEP04(2016)091arXiv:1601.02883JHEP. 04912016hep-th","doi":""},{"text":"The Conformal Bootstrap at Finite Temperature. L Iliesiu, M Kolouglu, R Mahajan, E Perlmutter, D Simmons-Duffin, 10.1007/JHEP10arXiv:1802.10266JHEP. 10702018. 2018hep-th","doi":""}],"string":"[\n {\n \"text\": \"Crystal statistics. 1. A Two-dimensional model with an order disorder transition. L Onsager, 10.1103/PhysRev.65.117Phys. Rev. 651944\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Combinatorial and topological approach to the 3-D Ising model. T Regge, R Zecchina, 10.1088/0305-4470/33/4/308arXiv:cond-mat/9909168J. Phys. A. 332000cond-mat\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dimer statistics and phase transitions. P Kasteleyn, 10.1063/1.1703953J. Math. Phys. 41963\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Ising model close to d=2. W Li, 10.1103/PhysRevD.105.L091902arXiv:2107.13679Phys. Rev. D. 1059L0919022022hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Finite size scaling study of lattice models in the three-dimensional Ising universality class. M Hasenbusch, 10.1103/PhysRevB.82.174433arXiv:1004.4486Phys. Rev. B. 821744332010cond-mat.stat-mech\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Restoring isotropy in a three-dimensional lattice model: The Ising universality class. M Hasenbusch, 10.1103/PhysRevB.104.014426arXiv:2105.09781Phys. Rev. B. 1041144262021cond-mat.stat-mech\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Solving the 3D Ising Model with the Conformal Bootstrap. S El-Showk, M F Paulos, D Poland, S Rychkov, D Simmons-Duffin, A Vichi, 10.1103/PhysRevD.86.025022arXiv:1203.6064Phys. Rev. D. 86250222012hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents. S El-Showk, M F Paulos, D Poland, S Rychkov, D Simmons-Duffin, A Vichi, 10.1007/s10955-014-1042-7arXiv:1403.4545J. Stat. Phys. 1572014hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bootstrapping Mixed Correlators in the 3D Ising Model. F Kos, D Poland, D Simmons-Duffin, 10.1007/JHEP11(2014)109arXiv:1406.4858JHEP. 112014hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Semidefinite Program Solver for the Conformal Bootstrap. D Simmons-Duffin, 10.1007/JHEP06(2015)174arXiv:1502.02033JHEP. 061742015hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Precision Islands in the Ising and O(N ) Models. F Kos, D Poland, D Simmons-Duffin, A Vichi, 10.1007/JHEP08(2016)036arXiv:1603.04436JHEP. 08362016hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Line defects in the 3d Ising model. M Billó, M Caselle, D Gaiotto, F Gliozzi, M Meineri, R Pellegrini, 10.1007/JHEP07(2013)055arXiv:1304.4110JHEP. 07552013hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Conformal symmetry of the critical 3D Ising model inside a sphere. C Cosme, J M V P Lopes, J Penedones, 10.1007/JHEP08(2015)022arXiv:1503.02011JHEP. 08222015hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Uncovering Conformal Symmetry in the 3D Ising Transition: State-Operator Correspondence from a Quantum Fuzzy Sphere Regularization. W Zhu, C Han, E Huffman, J S Hofmann, Y C He, 10.1103/PhysRevX.13.021009arXiv:2210.13482Phys. Rev. X. 132210092023cond-mat.stat-mech\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Operator Product Expansion Coefficients of the 3D Ising Criticality via Quantum Fuzzy Spheres. L Hu, Y C He, W Zhu, 10.1103/PhysRevLett.131.031601arXiv:2303.08844Phys. Rev. Lett. 1313316012023cond-mat.stat-mech\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Conformal four-point correlators of the three-dimensional Ising transition via the quantum fuzzy sphere. C Han, L Hu, W Zhu, Y C He, 10.1103/PhysRevB.108.235123arXiv:2306.04681Phys. Rev. B. 108232351232023cond-mat.stat-mech\",\n \"doi\": \"\"\n },\n {\n \"text\": \"3D Ising CFT and exact diagonalization on icosahedron: The power of conformal perturbation theory. B X Lao, S Rychkov, 10.21468/SciPostPhys.15.6.243arXiv:2307.02540SciPost Phys. 1562432023hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Solving Conformal Defects in 3D Conformal Field Theory using Fuzzy Sphere Regularization. L Hu, Y C He, W Zhu, arXiv:2308.01903cond-mat.stat-mech\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bounding scalar operator dimensions in 4D CFT. R Rattazzi, V S Rychkov, E Tonni, A Vichi, 10.1088/1126-6708/2008/12/031arXiv:0807.0004JHEP. 12312008hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory. A A Belavin, A M Polyakov, A B Zamolodchikov, 10.1016/0550-3213(84)90052-XNucl. Phys. B. 2411984\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Null bootstrap for non-Hermitian Hamiltonians. W Li, 10.1103/PhysRevD.106.125021arXiv:2202.04334Phys. Rev. D. 106121250212022hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Taming Dyson-Schwinger Equations with Null States. W Li, 10.1103/PhysRevLett.131.031603arXiv:2303.10978Phys. Rev. Lett. 1313316032023hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"OPE Convergence in Conformal Field Theory. D Pappadopulo, S Rychkov, J Espin, R Rattazzi, 10.1103/PhysRevD.86.105043arXiv:1208.6449Phys. Rev. D. 861050432012hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nonperturbative Mellin Amplitudes: Existence, Properties, Applications. J Penedones, J A Silva, A Zhiboedov, 10.1007/JHEP08(2020)031arXiv:1912.11100JHEP. 08312020hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Analytic Bootstrap and AdS Superhorizon Locality. A L Fitzpatrick, J Kaplan, D Poland, D Simmons-Duffin, 10.1007/JHEP12(2013)004arXiv:1212.3616JHEP. 1242013hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Convexity and Liberation at Large Spin. Z Komargodski, A Zhiboedov, 10.1007/JHEP11(2013)140arXiv:1212.4103JHEP. 111402013hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"How to measure the dimension of the parton field. G Parisi, 10.1016/0550-3213(73)90666-4Nucl. Phys. B. 591973\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bjorken scaling in quantum field theory. C G CallanJr, D J Gross, 10.1103/PhysRevD.8.4383Phys. Rev. D. 81973\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Comments on operators with large spin. L F Alday, J M Maldacena, 10.1088/1126-6708/2007/11/019arXiv:0708.0672JHEP. 11192007hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Twist Accumulation in Conformal Field Theory: A Rigorous Approach to the Lightcone Bootstrap. S Pal, J Qiao, S Rychkov, 10.1007/s00220-023-04767-warXiv:2212.04893Commun. Math. Phys. 40232023hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"D Poland, D Simmons-Duffin, 10.1007/JHEP05(2011)017arXiv:1009.2087Bounds on 4D Conformal and Superconformal Field Theories. 20110517hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bootstrapping Conformal Field Theories with the Extremal Functional Method. S El-Showk, M F Paulos, 10.1103/PhysRevLett.111.241601arXiv:1211.2810Phys. Rev. Lett. 111242416012013hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT. D Simmons-Duffin, 10.1007/JHEP03(2017)086arXiv:1612.08471JHEP. 03862017hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Conformal Bootstrap With Slightly Broken Higher Spin Symmetry. L F Alday, A Zhiboedov, 10.1007/JHEP06(2016)091arXiv:1506.04659JHEP. 06912016hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An Algebraic Approach to the Analytic Bootstrap. L F Alday, A Zhiboedov, 10.1007/JHEP04(2017)157arXiv:1510.08091JHEP. 041572017hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Anharmonic oscillator. C M Bender, T T Wu, 10.1103/PhysRev.184.1231Phys. Rev. 1841969\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Principle of minimal singularity for Green's functions. W Li, 10.1103/PhysRevD.109.045012arXiv:2309.02201Phys. Rev. D. 1094450122024hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Underdetermined Dyson-Schwinger Equations. C M Bender, C Karapoulitidis, S P Klevansky, 10.1103/PhysRevLett.130.101602arXiv:2211.13026Phys. Rev. Lett. 130101016022023math-ph\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dyson-Schwinger equations in zero dimensions and polynomial approximations. C M Bender, C Karapoulitidis, S P Klevansky, 10.1103/PhysRevD.108.056002arXiv:2307.01008Phys. Rev. D. 1085560022023math-ph\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bootstrapping Matrix Quantum Mechanics. X Han, S A Hartnoll, J Kruthoff, 10.1103/PhysRevLett.125.041601arXiv:2004.10212Phys. Rev. Lett. 1254416012020hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"W Li, arXiv:2402.05778The ϕ n trajectory bootstrap. hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"More constraining conformal bootstrap. F Gliozzi, 10.1103/PhysRevLett.111.161602arXiv:1307.3111Phys. Rev. Lett. 1111616022013hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bootstrapping the 3d Ising model at finite temperature. L Iliesiu, M Kolouglu, D Simmons-Duffin, 10.1007/JHEP12(2019)072arXiv:1811.05451JHEP. 12722019hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Hybrid Bootstrap. N Su, arXiv:2202.07607hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bootstrapping the O(N ) vector models. F Kos, D Poland, D Simmons-Duffin, 10.1007/JHEP06(2014)091arXiv:1307.6856JHEP. 06912014hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"HomotopyContinuation.jl: A Package for Homotopy Continuation in Julia. P Breiding, S Timme, Mathematical Software -ICMS 2018, J.H. Davenport, M. Kauers, G. Labahn and J. Urban2018Springer International Publishing\",\n \"doi\": \"\"\n },\n {\n \"text\": \"New method for the conformal bootstrap with OPE truncations. W Li, arXiv:1711.09075hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"simpleboot: A mathematica framework for bootstrap calculations. N Su,\",\n \"doi\": \"\"\n },\n {\n \"text\": \"I Heemskerk, J Penedones, J Polchinski, J Sully, 10.1088/1126-6708/2009/10/079arXiv:0907.0151Holography from Conformal Field Theory. 20091079hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Unitarity and the Holographic S-Matrix. A L Fitzpatrick, J Kaplan, 10.1007/JHEP10(2012)032arXiv:1112.4845JHEP. 10322012hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"W Li, 10.1007/JHEP01arXiv:1706.04054Inverse Bootstrapping Conformal Field Theories. 2018. 20180177hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Conformal Invariance and Multifractality at Anderson Transitions in Arbitrary Dimensions. J Padayasi, I A Gruzberg, 10.1103/PhysRevLett.131.266401arXiv:2306.07340Phys. Rev. Lett. 131262664012023cond-mat.dis-nn\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Improving the five-point bootstrap. D Poland, V Prilepina, P Tadić, arXiv:2312.13344hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Factorized lightcone expansion of conformal blocks. W Li, 10.1007/JHEP05(2021)128arXiv:2012.09710JHEP. 128052021hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Critical exponents of the 3d Ising and related models from Conformal Bootstrap. F Gliozzi, A Rago, 10.1007/JHEP10(2014)042arXiv:1403.6003JHEP. 10422014hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Boundary and Interface CFTs from the Conformal Bootstrap. F Gliozzi, P Liendo, M Meineri, A Rago, 10.1007/JHEP05(2015)036arXiv:1502.07217JHEP. 05362015hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bootstrapping critical Ising model on three-dimensional real projective space. Y Nakayama, 10.1103/PhysRevLett.116.141602arXiv:1601.06851Phys. Rev. Lett. 116141416022016hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Truncatable bootstrap equations in algebraic form and critical surface exponents. F Gliozzi, 10.1007/JHEP10(2016)037arXiv:1605.04175JHEP. 10372016hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Closure of the Operator Product Expansion in the Non-Unitary Bootstrap. I Esterlis, A L Fitzpatrick, D Ramirez, 10.1007/JHEP11(2016)030arXiv:1606.07458JHEP. 11302016hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"C Hasegawa, Y Nakayama, 10.1142/S0217732317500456arXiv:1611.06373Expansion in Critical ϕ 3 -Theory on Real Projective Space from Conformal Field Theory. 2017321750045hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Conformal bootstrap analysis for the Yang-Lee edge singularity. S Hikami, 10.1093/ptep/pty054arXiv:1707.04813PTEP. 201852018hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Conformal Bootstrap Analysis for Single and Branched Polymers. S Hikami, 10.1093/ptep/pty132arXiv:1708.03072PTEP. 2018122018hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dimensional Reduction by Conformal Bootstrap. S Hikami, 10.1093/ptep/ptz081arXiv:1801.09052PTEP. 201982019cond-mat.dis-nn\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Conformal bootstrap for percolation and polymers. A Leclair, J Squires, 10.1088/1742-5468/aaf10aarXiv:1802.08911J. Stat. Mech. 1812121231052018hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"S Hikami, arXiv:1811.05918Conformal Bootstrap Analysis for Localization: Symplectic Case. cond-mat.dis-nn\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the ϕ 3 theory above six dimensions. J Rong, J Zhu, 10.1007/JHEP04(2020)151arXiv:2001.10864JHEP. 041512020hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Is there supersymmetric Lee-Yang fixed point in three dimensions?. Y Nakayama, 10.1142/S0217751X21501761arXiv:2104.13570Int. J. Mod. Phys. A. 362421501762021hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The extraordinary boundary transition in the 3d O(N) model via conformal bootstrap. J Padayasi, A Krishnan, M A Metlitski, I A Gruzberg, M Meineri, 10.21468/SciPostPhys.12.6.190arXiv:2111.03071SciPost Phys. 1262022cond-mat.stat-mech\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Solving Conformal Field Theories with Artificial Intelligence. G Kántor, V Niarchos, C Papageorgakis, 10.1103/PhysRevLett.128.041601arXiv:2108.08859Phys. Rev. Lett. 1284416012022hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Conformal bootstrap with reinforcement learning. G Kántor, V Niarchos, C Papageorgakis, 10.1103/PhysRevD.105.025018arXiv:2108.09330Phys. Rev. D. 1052250182022hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Monte Carlo approach to the conformal bootstrap. A Laio, U L Valenzuela, M Serone, 10.1103/PhysRevD.106.025019arXiv:2206.05193Phys. Rev. D. 1062250192022hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"6D (2,0) bootstrap with the soft-actor-critic algorithm. G Kántor, V Niarchos, C Papageorgakis, P Richmond, 10.1103/PhysRevD.107.025005arXiv:2209.02801Phys. Rev. D. 1072250052023hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The five-point bootstrap. D Poland, V Prilepina, P Tadić, 10.1007/JHEP10(2023)153arXiv:2305.08914JHEP. 101532023hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bootstrability in line-defect CFTs with improved truncation methods. V Niarchos, C Papageorgakis, P Richmond, A G Stapleton, M Woolley, 10.1103/PhysRevD.108.105027arXiv:2306.15730Phys. Rev. D. 108101050272023hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Large Spin Perturbation Theory for Conformal Field Theories. L F Alday, 10.1103/PhysRevLett.119.111601arXiv:1611.01500Phys. Rev. Lett. 119111116012017hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Solving CFTs with Weakly Broken Higher Spin Symmetry. L F Alday, 10.1007/JHEP10(2017)161arXiv:1612.00696JHEP. 101612017hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Analyticity in Spin in Conformal Theories. S Caron-Huot, 10.1007/JHEP09(2017)078arXiv:1703.00278JHEP. 1709782017hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A spacetime derivation of the Lorentzian OPE inversion formula. D Simmons-Duffin, D Stanford, E Witten, 10.1007/JHEP07arXiv:1711.03816JHEP. 1807852018. 2018hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Light-ray operators in conformal field theory. P Kravchuk, D Simmons-Duffin, 10.1007/JHEP11arXiv:1805.00098JHEP. 18111022018. 2018hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J Liu, E Perlmutter, V Rosenhaus, D Simmons-Duffin, 10.1007/JHEP03(2019)052arXiv:1808.00612d-dimensional SYK, AdS Loops, and 6j Symbols. 20190352hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Resummation at finite conformal spin. C Cardona, S Guha, S K Kanumilli, K Sen, 10.1007/JHEP01(2019)077arXiv:1811.00213JHEP. 1901772019hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"More Analytic Bootstrap: Nonperturbative Effects and Fermions. S Albayrak, D Meltzer, D Poland, 10.1007/JHEP08(2019)040arXiv:1904.00032JHEP. 1908402019hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Closed-form expression for cross-channel conformal blocks near the lightcone. W Li, 10.1007/JHEP01(2020)055arXiv:1906.00707JHEP. 552001. 2020hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Inversion Formula and 6j Symbol for 3d Fermions. S Albayrak, D Meltzer, D Poland, 10.1007/JHEP09(2020)148arXiv:2006.07374JHEP. 091482020hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Lorentzian inversion formula and the spectrum of the 3d O(2) CFT. J Liu, D Meltzer, D Poland, D Simmons-Duffin, 10.1007/JHEP09(2020)115arXiv:2007.07914JHEP. 091152020erratum: JHEP 01, 206 (2021). hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The leading trajectory in the 2+1D Ising CFT. S Caron-Huot, Y Gobeil, Z Zahraee, 10.1007/JHEP02(2023)190arXiv:2007.11647JHEP. 021902023hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Precision bootstrap for the N = 1 super-Ising model. A Atanasov, A Hillman, D Poland, J Rong, N Su, 10.1007/JHEP08(2022)136arXiv:2201.02206JHEP. 081362022hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dispersive CFT Sum Rules. S Caron-Huot, D Mazac, L Rastelli, D Simmons-Duffin, 10.1007/JHEP05(2021)243arXiv:2008.04931JHEP. 052432021hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Conformal Dispersion Relation: Correlations from Absorption. D Carmi, S Caron-Huot, 10.1007/JHEP09(2020)009arXiv:1910.12123JHEP. 0992020hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A basis of analytic functionals for CFTs in general dimension. D Mazáč, L Rastelli, X Zhou, 10.1007/JHEP08(2021)140arXiv:1910.12855JHEP. 081402021hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Applications of dispersive sum rules: epsilon-expansion and holography. D Carmi, J Penedones, J A Silva, A Zhiboedov, 10.21468/SciPostPhys.10.6.145arXiv:2009.13506SciPost Phys. 1062021hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Mixed correlator dispersive CFT sum rules. A K Trinh, 10.1007/JHEP03(2022)032arXiv:2111.14731JHEP. 03322022hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bootstrapping the O(N) Archipelago. F Kos, D Poland, D Simmons-Duffin, A Vichi, 10.1007/JHEP11(2015)106arXiv:1504.07997JHEP. 111062015hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Carving out OPE space and precise O(2) model critical exponents. S M Chester, W Landry, J Liu, D Poland, D Simmons-Duffin, N Su, A Vichi, 10.1007/JHEP06(2020)142arXiv:1912.03324JHEP. 061422020hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bootstrapping Heisenberg magnets and their cubic instability. S M Chester, W Landry, J Liu, D Poland, D Simmons-Duffin, N Su, A Vichi, 10.1103/PhysRevD.104.105013arXiv:2011.14647Phys. Rev. D. 104101050132021hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bootstrapping the minimal N = 1 superconformal field theory in three dimensions. J Rong, N Su, 10.1007/JHEP06(2021)154arXiv:1807.04434JHEP. 061542021hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bootstrapping the Minimal 3D SCFT. A Atanasov, A Hillman, D Poland, 10.1007/JHEP11arXiv:1807.05702JHEP. 111402018. 2018hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Conformal Bootstrap: Theory, Numerical Techniques, and Applications. D Poland, S Rychkov, A Vichi, 10.1103/RevModPhys.91.015002arXiv:1805.04405Rev. Mod. Phys. 91150022019hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"S Rychkov, N Su, arXiv:2311.15844New Developments in the Numerical Conformal Bootstrap. hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Bootstrap Program for Boundary CFT d. P Liendo, L Rastelli, B C Van Rees, 10.1007/JHEP07(2013)113arXiv:1210.4258JHEP. 071132013hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Defects in conformal field theory. M Billò, V Goncalves, E Lauria, M Meineri, 10.1007/JHEP04(2016)091arXiv:1601.02883JHEP. 04912016hep-th\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Conformal Bootstrap at Finite Temperature. L Iliesiu, M Kolouglu, R Mahajan, E Perlmutter, D Simmons-Duffin, 10.1007/JHEP10arXiv:1802.10266JHEP. 10702018. 2018hep-th\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":470,"cells":{"text":{"kind":"string","value":"\nAn Extended Kalman Filter for Data-enabled Predictive Control\n\n\nDaniele Alpago \nFlorian Dörfler \nJohn Lygeros \nAn Extended Kalman Filter for Data-enabled Predictive Control\n\nThe literature dealing with data-driven analysis and control problems has significantly grown in the recent years. Most of the recent literature deals with linear timeinvariant systems in which the uncertainty (if any) is assumed to be deterministic and bounded; relatively little attention has been devoted to stochastic linear time-invariant systems. As a first step in this direction, we propose to equip the recently introduced Data-enabled Predictive Control algorithm with a data-based Extended Kalman Filter to make use of additional available input-output data for reducing the effect of noise, without increasing the computational load of the optimization procedure.arXiv:2003.08269v2 [math.OC] 10 May 2020\n\nI. INTRODUCTION\n\nWith the increasing complexity of applications in science and engineering, the use of model-based control techniques is becoming more and more challenging as they usually require accurate descriptions of model and uncertainties, often difficult to obtain. The need of data-driven approaches was already perceived in the 1980s with the advent of system identification [1] and adaptive control [2], and received new impetuous in the 1990s with approaches such as Iterative Feedback Tuning, Correlation-based Tuning, Virtual Reference Feedback Tuning [3]- [5] and many others. In recent years, data-driven analysis and control has experienced a renewed interest [6]. One promising approach is due to the rediscovery of a result originally formulated in the context of behavioral system theory [7], [8] by J. C. Willems and coauthors, known as the Fundamental Lemma [9]. The result states that if the input signal is sufficiently rich, all possible trajectories of a deterministic, linear time-invariant (LTI) system can be generated from linear combinations of past trajectories of the same system. This allows one to use a Hankel matrix constructed from input-output data as an implicit representation of the underlying dynamics. This representation has been first exploited for data-driven simulation and linear-quadratic tracking for deterministic LTI systems in a behavioral setting [10], [11]. This has resulted in a growing stream of literature dealing with data-driven analysis [12], [13] and control problems [14]- [17].\n\nIn this context, particular attention has been devoted to the problem of optimal trajectory tracking [18]- [22], widely studied in model-based control. The aim is to compute an optimal control policy based on output feedback that drives the system along an output reference while minimizing a stage cost and satisfying input and output constraints. Model Predictive Control (MPC) has been one of the most effective methods to tackle such problems. MPC requires an accurate model of the system and an accurate description of the uncertainty (if any), which might be challenging and expensive to obtain in many applications [23], [24]. The success of MPC and the difficulties arising from the requirement for models have led to the introduction of a Data-enabled Predictive Control (DeePC) algorithm. The latter does not rely on a particular parametric system representation, but rather on a Hankel-matrix representation of the underlying dynamics, constructed from the system's input-output trajectories [19]- [21]. Unlike classical model-based predictive control methods, DeePC performs the control computation without identifying the system by solving a (parametric) convex optimization problem that includes the Hankel matrix in the constraints. The complexity of such optimization depends on the system dimensions, the MPC horizon and the amount of available data. Therefore, it is not obvious how to efficiently include additional data (for example, past measurements of the output of the closed-loop system) in the procedure, without increasing the computational burden of the optimization that has to be solved online.\n\nUnlike earlier methods, recent data-driven approaches based on the Fundamental Lemma have devoted little attention to stochastic systems. Some approaches adopt a \"robust control\" perspective and treat the uncertainty as a deterministic and bounded sequence [14], [16], [22], sometimes affecting just the output. The focus of the present paper is to extend the DeePC algorithm to stochastic LTI systems and design methods for including more data to improve closed-loop performance, without increasing the computational load of the optimization. This is accomplished through a combination of an offline averaging of Hankel matrix predictors and an online, data-driven Extended Kalman Filter (EKF). The latter is based on an implicit model constructed from the parametric solution of the DeePC optimization program. The combination of off-line averaging with online EKF dramatically improves the closed-loop performance, as evidenced by numerical experiments.\n\nIn Section II we introduce the notation and recall some preliminary results. In Section III the DeePC algorithm is reviewed and the issues that motivate our approach are highlighted. Section IV-B introduces the proposed approach and Section V presents numerical experiments. Finally, in Section VI we draw conclusions and outline future lines of research.\n\n\nII. NOTATION AND PRELIMINARIES\n\nWe recall the definition of persistently exciting signals and a state-space version of a result from behavioral system theory [7], [8], known as the Fundamental Lemma [9], [10].\n\nNotation. Given a matrix A ∈ R m×n , A denotes its transpose and A −1 its inverse (if m = n). The notation A(i : j, :), i ≤ j ≤ m, stands for the sub-matrix of A that goes from the i-th row of A to the j-th row of A, included. If i = j we simply write A(i, :). We denote with I m the identity matrix of dimension m×m. With diag(A 1 , . . . , A n ) we denote the blockdiagonal matrix formed with the matrices A 1 , . . . , A n . Given a sequence of matrices A h , . . . , A h+k in R m×n , with k > 0, we denote by A h,h+k := col(A h , . . . , A h+k ) := [A h · · · A h+k ] ∈ R (k+1)m×n . Given a vector w ∈ R m we introduce the quadratic form w 2 P = w Pw which coincides with the squared Euclidean norm w 2 2 when P = I m . The symbol E[·] denotes the expectation operator.\nPersistency of Excitation. Let w 0 , . . . , w N−1 be N ∈ N samples of a given sequence (w k ) k∈Z taking values in R q . For M ∈ N, M ≤ N, let H M (w 0,N−1 ) := w 0 w 1 · · · w N−M w 1 w 2 · · · w N−M+2 . . . . . . . . . . . . w M−1 w M · · · w N−1 \nbe the block-Hankel matrix associated to the trajectory w 0,N−1 , with M block-rows and the maximal number of columns N − M + 1. We say that the signal w 0,N−1 is persistently exciting of order M if the Hankel matrix H M (w 0,N−1 ) has full row-rank qM. This requires the sequence w 0,N−1 to be rich and long enough, in particular N ≥ (q + 1) M − 1.\n\nFundamental Lemma. Consider a state-space representation of an n-dimensional LTI system\nx k+1 = A x k + B u k y k = C x k + D u k(1)\nwhere (x k ) k∈Z is the n-dimensional state-process, (u k ) k∈Z the m-dimensional input process, and (y k ) k∈Z is the pdimensional output process. Given an initial condition x 0 ∈ R n and a sequence of inputs u 0,k−1 ∈ R mk , the output of the system can be written as\ny k−1 = CA k−1 x 0 + CA k−2 B · · · CB D u 0,k−1 , k ≥ 2,(2)\nand y 0 = C x 0 + D u 0 . Let T ∈ N and collect T -long inputoutput trajectories u 0,T −1 , y 0,T −1 of the system (1). For fixed positive integers N p , N f ∈ N and k ≥ N p , we can associate to the vectors u (k)\np := u k−N p ,k−1 , y (k) p := y k−N p ,k−1 , u (k) f := u k,k+N f −1 , y (k) f := y k,k+N f −1 ,(3)\nthe block-Hankel matrices\nU p U f = u (N p ) p · · · u (T −N f −1) p u (N p ) f · · · u (T −N f −1) f =: H N p +N f (u 0,T −1 ), Y p Y f = y (N p ) p · · · y (T −N f −1) p y (N p ) f · · · y (T −N f −1) f =: H N p +N f (y 0,T −1 ). From (2) we can then construct the (m + p)(N p + N f ) × (T − N p − N f + 1) data matrix H := U p U f Y p Y f = 0 I m(N p +N f ) 0 O p (A,C) T p (B, D) O f (A,C) T f (B, D) X U ,(4)\nwhere U := [u 0 · · · u T −1 ] and X := [x 0 · · · x T −1 ] are the block-Hankel matrices containing the inputs and the corresponding sequence of states, respectively. Here,\nO p (A,C) := col(C, CA, · · · , CA N p −1 ), O f (A,C) := col(CA N p , CA N p +1 , · · · , CA N f −1 ),\nare observability matrices,\nT p (B, D) := D 0 · · · 0 0 · · · 0 CB D · · · 0 0 · · · 0 CAB CB · · · 0 0 . . . 0 . . . . . . . . . . . . . . . · · · 0 CA N p −2 B CA N p −3 B · · · D 0 · · · 0 ,(5)\nand T f (B, D) is defined similarly. The following fundamental result from [9] guarantees that the matrix (4) can be used in place of the parametric representation (1), as long as the input u 0,T −1 is persistently exciting.\n\nLemma 1 (Fundamental Lemma): Assume the system (1) to be controllable and the input trajectory u 0,T −1 to be persistently exciting of order N p + N f + n. Then, a sequence col(u p , u f , y p , y f ) is an input-output trajectory of the system (1) if and only if it is in the range space of H.\n\nRecalling the persistency of excitation condition, T ≥ (m+ 1)(N p + N f + n) + 1 is a necessary condition for u 0,T −1 to be persistently exciting of order N p + N f + n. Lemma 1 has been originally proven using the behavioral language in [9, Theorem 1]. For an equivalent recent state-space proof see [12] and [16, Appendix A].\n\n\nIII. DATA-ENABLED PREDICTIVE CONTROL REVIEW\n\nWe briefly introduce the recently proposed Data-enabled Predictive Control (DeePC) method [19], and highlight some related issues when dealing with stochastic LTI systems. This will serve as a motivation for what follows.\n\nConsider the stochastic version of (1)\nx k+1 = A x k + B u k + E v k y k = C x k + D u k + F v k(6)\nwhere (v k ) k∈Z is p-dimensional white noise (zero mean and unit-variance). We collect sufficiently long input-output\ntrajectories u 0,T −1 , y 0,T −1 , i.e. T ≥ (m + 1)(N p + N f + n) + 1.\nThe same computations leading to (4), lead to the matrix corresponding to the stochastic model (6):\nH := U p U f Y p Y f = 0 I m(N p +N f ) 0 0 0 O p (A,C) T p (B, D) T p (E, F) O f (A,C) T f (B, D) T f (E, F) X U V , (7) where V := [v 0 · · · v T −1 ] is the block-Hankel matrix gath- ering the noise samples and T p (E, F), T f (E, F) are block- Toeplitz matrices similar to T p (B, D), T f (B, D) in (5).\nThe DeePC algorithm proposed in [19] attempts to compute an optimal-control action based on past input-output data coming from the unknown system (6), without previous identification. The control action is selected through an MPC-like optimization problem that allows one to enforce constraints ensuring safety and performance requirements. The previously collected data is directly used on-line in the MPC optimization problem; the predictor is therefore implicit and arises as the outcome of the optimization problem. In particular, at the generic iteration k ≥ N p , the DeePC computes optimal control actions by solving\nmin g,u i ,y i N f ∑ i=1 y i − r k+i 2 Q + u i 2 R + λ y Y p g − y (k) p 2 2 + λ g g 2 2 subject to U p U f Y f g = u (k) p u i y i , u i ∈ U, i = 1, . . . , N f , y i ∈ Y, i = 1, . . . , N f ,(8)\nwhere N f is the prediction horizon, r ∈ R pN f is the output reference signal we want to track, U ⊆ R m , Y ⊆ R p are the input and output constraint sets, respectively, Q ∈ R p×p is the output cost matrix (positive semidefinite), R ∈ R m×m is the control cost matrix (positive definite), λ y ≥ 0 and λ g ≥ 0 are the regularization parameters, and u\n(k) p , y (k)\np are the most recent N p input-output measurements from (6), according to the notation (3). For simplicity we consider input and output box constraints of the form U = [u min , u max ] and Y = [y min , y max ], respectively. Note that the block-Hankel matrices U p ,U f ,Y p ,Y f are fixed throughout the online iterations.\n\nIf we let g k be the optimal solution of problem (8) at iteration k, DeePC provides an implicit predictor y (k) f = Y f g k whose model is never derived explicitly but whose predictions are implicitly used for obtaining the optimal control actions u p to the most recent input-output measurements and set k to k + N c + 1.\n\nEquation (8) is a relaxation of the corresponding problem enforcing the constraint Y p g = y (k) p , which was proven to be equivalent to the classical receding-horizon MPC in the case of deterministic LTI systems [19]. Here, to cope with potential infeasibility due to the disturbances, the constraint is substituted with the least-squares regularization term Y p g − y (k) p 2 2 . The two-norm regularization on g has been introduced to avoid overfitting and it has been shown to relate to distributional robustness of the method with respect to a range of uncertainties [20]. One can also see that, besides infeasibility issues, considering stochastic models makes both the implicit predictor model y (k) f = Y f g k and the control actions u (k) f = U f g k defined by the optimization (8) to be non-linear in the past-data (y\n(k) p , u (k) p ) (Section IV).\nWe conclude this section with a note on the effect of acquiring more data. Classical LTI system identification methods use historical data off-line to compute estimates of the matrices in the system dynamics. The resulting matrices are then used on-line to generate state estimates and perform the prediction [1]. Other prediction methods estimate the linear relation from u\n(k) p , u (k) f , y (k) p to y (k)\nf off-line, and then use it online to carry on the predictions, [25] and references therein. This kind of prediction architectures naturally possesses a data-compression mechanism. For the classical system identification methods only storage of the estimates of the system matrices is required. This depends on input and output dimensions m and p and on the (guessed) state dimension, but not on the amount T of data nor on the prediction horizon N f . Likewise, the storage requirements for linear predictors depends on input and output dimensions m and p, and the horizons N p , N f , but not on the amount T of data. If more data become available one can then use it to improve predictions with no need to store it or use it online. The DeePC approach, on the other hand, requires one to carry all the data at every iteration and the size of the optimization problem (8) increases both with the amount T of available data and with the parameters m, p, N p , and N f . Though additional data would in principle also be beneficial for a DeePC controller, it is not clear how one can incorporate it without increasing the on-line computational burden.\n\n\nIV. METHOD DESCRIPTION\n\nWe introduce a possible way to effectively incorporate more data in the DeePC framework to reduce the effect of noise in the solution of problem (8). We propose to use additional data that may be available off-line to de-noise the block-Hankel matrices Y p and Y f and to equip the DeePC with an EKF based on data to handle the noise in the online measurements y (k) p .\n\n\nA. Averaging Data Matrices\n\nThe output matrices Y p and Y f are constructed off-line from the trajectory y 0,T −1 . Let the corresponding initial conditions. Using those additional data on-line to improve the prediction will lead to an intractable optimization problem. However, we can make use of additional trajectories off-line to construct N different data matrices H (1) , . . . , H (N) defined analogously to (7), and average those matrices to obtain 0,T −1 , respectively. Since (v k ) k∈Z is assumed to be white-noise, the Law of Large Numbers guarantees thatV N → 0 as N → ∞ [26]. The averaging procedure makes use of additional data to mitigate the effect of noise in the data-driven model, hence reducing the risk of overfitting that would be present if the data was used directly in DeePC. Accordingly, the more matrices are involved in the average, the smaller the value of the regularization parameter λ g that gives the optimal closed-loop cost (see Figure 3, Section V).\nH N := 1 N N ∑ i=1 H (i) = 0 I m(N p +N f ) 0 0 0 O p (A,C) T p (B, D) T p (E, F) O f (A,C) T f (B, D) T f (E, F) X N U N V N (9) whereV N := 1 N ∑ N i=1 V (i) is the average of the matrices V (i) = [v (i) 0 · · · v\nWe recognize that such a method heavily exploits the underlying linear structure of the problem. However, this is meant to be a first attempt to exploit additional data for improving the performance of the algorithm when dealing with stochastic systems, without increasing the dimension of the optimization problem (11) to be solved online. Indeed, thanks to the linear structure and superposition, the sub-matricesX N andŪ N in the averaged data matrix (9) still represent valid system trajectories corresponding to the average of the initial states x 0 = 1 N ∑ i x (i) 0 and the average of the input sequences u = 1 N ∑ i u (i) , used in the experiments that generated the data for each H (i) . The only point that requires attention is ensuring that the resulting average input sequence respects the persistence of excitation requirement. The simplest way to ensure this is to assume that the same persistently exciting input sequence is applied in all cases.\n\n\nB. An EKF for DeePC\n\nThe averaging procedure represents a simple way to make use of multiple T -long trajectories to denoise off-line the data matrix representing the dynamics in the constraints of (8). Numerical evidence suggests that additional denoising of the on-line data y (k) p that enters the cost of (8) can lead to a further, significant improvement in performance (Section V). In a model based setting, such on-line denoising could be performed by a Kalman filter. In our data-driven setting, however, the classical Kalman filter cannot be applied as it requires a model of the system dynamics. Here we show how the non-linear one-step predictor implicit in (8) can be used to derive an EKF for integrating past measurements into an implicit \"state estimate\"; this can in turn be used to improve the asymptotic performance of the algorithm.\n\nWe introduce a fictitious state vector z k := col(y k−N p+1 , · · · , y k ) ∈ R pN p and keep track of a filtered state-estimateẑ k|k and the corresponding error covariance P k|k . At the generic iteration k ≥ N p , the EKF-DeePC algorithm uses the estimateẑ k|k by solving the following optimization problem\nmin g∈R d N f ∑ i=1 (Y f g) i − r k+i 2 Q + (U f g) i 2 R + λ y Y p g −ẑ k|k 2 2 + λ g g 2 2 subject to U p g = u (k) p , u min ≤ (U f g) i ≤ u max , i = 1, . . . , N f , y min ≤ (Y f g) i ≤ y max , i = 1, . . . , N f .(10)\nThe formulation (10) is obtained from (8) by substituting the constraints u i = U f g and y i = Y f g in the cost, leaving g as the only decision variable. The crucial difference with respect to (8) is that the past data Y p g used in the implicit predictor is now required to fit the state estimateẑ k|k instead of the N p most-recent measurements y (k) p . Rewriting (10) as a multi-parametric quadratic program (mp-QP) in the parameter θ k := col(ẑ k|k , u (k) p ) leads to an explicit relation between of the optimum g k and the parameter θ k ; we exploit this relation to build the EKF. The mp-QP form of (10) is\nmin g∈R d 1 2 g Pg + (Gθ k + q k ) g + θ k Hθ k + 1 2 r k Qr k subject to U p g = B eq θ k , A in g ≤ b in ,(11)\nwhere the inequality constraint defined by\nb in := col(1 mN f u max , − 1 mN f u min , 1 pN f y max , − 1 pN f y min ), A in := col(U f , −U f ,Y f , −Y f ),\nhas to be understood component-wise, i.e. (A in g) i ≤ s i for i = 1, . . . , 2(m + p)N f . The cost is then defined by the reference signal r k := col(r k+1 , · · · , r k+N f ) and the matrices\nP := Y f QY f +U f RU f + λ y Y p Y p + λ g I d , q k := −Y f Qr k G := −λ y Y p 0 B eq := 0 I mN p\n, H := (λ y /2) · I pN p 0 0 0 .\n\nHere 1 N := col(1, 1, · · · , 1) ∈ R N , Q = diag(Q, . . . , Q) and R = diag(R, . . . , R). Assuming P = P > 0 and the KKTmatrix for problem (11) to be positive semi-definite (which is always the case for an MPC problem with input weighting matrix R > 0 [27]), the optimizer g k is a piecewise affine function of the parameters, and can be written as g k =\nA kẑk|k +B k u (k)\np +h k . In particular, the equality constraints in (11) and the noise in the model are responsible of the affine structure while the inequality constraints implies this affine relation to hold just locally, i.e. in a neighborhood of the parameter θ k , known as the critical region. The coefficients A k ,B k andh k coming from the KKT conditions for problem (11), are therefore region-dependent themselves [27]: we consider the affine expansion of g k pertaining to a specific value for the parameter θ k , fixed by the previous iteration.\n\nThe fact that, under suitable assumptions, the piecewise affine relation between the optimizer g k of (11) and the parameter θ k can be derived from the KKT conditions for problem (11), might suggest that the predictor implicit in (10) could be made explicit and be constructed off-line. Doing this would require one to construct all the regions on which the affine expansion is defined, for all the possible values of the parameters. As the number of these regions scales exponentially with the QP size (parameters plus constraints), hence with the amount T of available data and the horizons N p and N f , this approach is likely to be computationally intractable. This is the main reason for keeping the predictor implicit.\n\nLet M := col(Y p (2 : pN p , :), Y f (1, :)) be the prediction map. We can exploit the piece-wise affine form of the optimizer g k to incorporate the implicit predictor provided by (11) in an EKF-like architecture. To fix ideas, suppose we start running the algorithm at k = N p . From an initial guess of the meanẑ N p |N p = E[z N p ] and the covariance matrix\nP N p |N p = E[(z N p −ẑ N p |N p )(z N p −ẑ N p |N p ) ]\n, we then compute recursively (at every point in time) the standard Kalman filter update steps in Figure 1. We note that, because of its dependence on g k , the implicit predictor is piecewise affine (the matrices A k , B k , h k at the current iteration, depend on the critical region) making this a data-driven analogue to an Extended Kalman Filter.\n\n\nPrediction\n\nUpdate The matrices Q k ∈ R pN p ×pN p and R k ∈ R p×p are the variances of the process and measurement noise implicitly affecting the state and output dynamics z k+1 = A k z k + B k u (k) p + h k and y k = C z k , respectively. In practice, to implement the filter in Figure 1 one needs to address the same issues as for any Kalman filter, i.e. choosing the initial conditionsẑ N p |N p and P N p |N p , and obtaining an estimate (or guess) of the variances Q k and R k . For a model-based Kalman filter, one can use the residuals of the system identification to obtain estimates for these quantities; we speculate that something similar would be possible using the residuals between the average and individual data matrices in Section IV-A, though we do not pursue this direction here, due to space limitations. The steps outlined above are summarized in Figure 2.\nP k+1|k = A k P k|k A k + Q k P k+1|k+1 = (I − K k+1 C)P k+1|k z k+1|k = M g * k = A kẑk|k + B k u (k) p + h kẑk+1|k+1 =ẑ k+1|k + K k+1 (y k+1 − Cẑ k+1|k ) y k+1 = Cẑ k+1|k\n\nV. NUMERICAL VALIDATION\n\nWe present numerical experiments assessing the effects of the proposed solution in tackling the noise acting in the optimization (8). Consider the stochastic system\nx k+1 = Ax k + Bu k + w k y k = Cx k + v k , A = 0.8 1 0 0.8 , B = [0 1] , C = [1 1].(12)\nwhere (w k ) k∈Z , (v k ) k∈Z are independent zero-mean Gaussian white noises with covariance Σ w := EE and Σ v := FF respectively; below we consider Σ w = σ 2 w I n and Σ v = σ 2 v I p and report results for different values of σ 2 w and σ 2 v . Model (12) is reachable and observable. The performance metric we will consider throughout is the closed-loop cost J(u, y) = ∑ N sim k=1 y k − r k 2 Q + u k 2 R , where r k = 5 sin(0.3k) is the reference signal. To isolate the effects of averaging and the EKF, for each numerical experiment we tuned the regularization parameters λ y and λ g through exhaustive search to minimize the closed-loop cost for the standard DeePC, the averaged DeePC and the averaged DeePC with EKF. All numerical evidence comes from 100 repetitions for different data-sets. Unless otherwise stated, we set T = 100, N sim = 100, N p = 3, N f = 5, Q = I p , R = I m , σ 2 w = σ 2 v = 0.5 and N = 40 data matrices in the average (9).\n\nFirst we show the effect of the averaging introduced in Section IV-A. To isolate the averaging effect, we consider noisy Y p , Y f but y (k) p generated from the model with neither process noise, not measurement noise. The result, shown in Figure 3, is as expected: the cost decreases towards the ideal cost of MPC with the same horizon but perfect model and full state measurement while on the right, the optimal (numerically found) value of the regularization parameter λ g decreases to zero, as we expect from standard results in optimization. Figure 4 shows a comparison between the closed-loop costs of the standard DeePC and the averaged DeePC with EKF algorithms when varying the noise variances σ 2 w and σ 2 v . Figure 4 shows that introducing averaging and the EKF substantially improves the performance of the DeePC algorithm. Further experiments (data not shown) with averaged DeePC without EKF and EKF without averaging confirmed that DeePC with a combination of averaging and EKF substantially outperformed all other combinations; indeed this was the case across a broad range of values of the regularizes λ y and λ g , suggesting that averaged DeePC with EKF is easier to tune. Figure cost of the two algorithms behaves with respect to the past horizon N p . The improvement in the performance is due to the introduction of the averaging plus the EKF, highlighting the difference between fixed horizon and recursive filtering.\n\n\nVI. CONCLUSION AND OUTLOOK\n\nWe presented an extension of the data-enabled predictive control (DeePC) algorithm introduced in [19] to tackle control problems on unknown stochastic LTI systems, by making use of additional data without increasing the dimension of the on-line optimization problem. The procedure features an on-line data-driven EKF that filters out the noise in the measurements, and an (off-line) averaging of multiple data matrices to get a cleaner data-driven model. The performance of the proposed method has been validated experimentally on an LTI stochastic system showing improvements with respect to standard DeePC. Future work includes testing the proposed method on a real-case scenario and comparing it to algorithms such as [3]- [5].\n\n\nResearch supported by the ERC under the project OCAL, the SEMP and the Gini Foundation. D. Alpago is with the Department of Information Engineering at University of Padova, Italy, dalpago@dei.unipd.it; F. Dörfler and J. Lygeros are with the Department of Information Technology and Electrical Engineering at ETH Zürich, Switzerland, dorfler@ethz.ch, jlygeros@ethz.ch.\n\n1\n1U f g k . Problem(8) is solved in a receding-horizon fashion: of the N f -long optimal control sequence u (k) f = col(u k , . . . , u k+N f −1 ) solution of (8), we apply a sub-sequence u k , . . . , u k+N c , for some N c ≤ N f −\n\n\nT −1 ) be available T -long input-output trajectories and denote with x (i)0 , i = 1, . . . N,\n\nT\n−1 ], i = 1, . . . , N, constructed from the different noise realizations v similarlyX N andŪ N contain the averaged state trajectories corresponding to the different initial conditions x\n\nFig. 1 :Fig. 2 :\n12EKF equations. Here, C := [0 · · · 0 I p ] by construction, A k := MÃ k , B k := MB k , h k := Mh k and S k+1 = CP k+1|k C + R k , K k+1 = P k+1|k C S −1 k+1 , are the variance of the innovation process and the filter gain, respectively. EKF-DeePC Algorithm.\n\nFig. 3 :\n3Closed-loop cost (left) and optimal λ g (right) when averaging N data matrices. The dashed-green line shows the MPC cost with the same horizon, but perfect model and full state measurement.\n\nFig. 4 :Fig. 5 :\n45Closed-loop cost of standard DeePC and averaged DeePC with EKF for different σ 2 v while keeping σ 2 w = 0.1 (left), and different σ 2 w while keeping σ 2 v = 0.Closed-loop costs while varying the past horizon N p .\nACKNOWLEDGMENTSThe authors are grateful to Colin Jones and Francesco Micheli for detailed discussions and the anonymous reviewers for insightful suggestions.\nSystem Identification: Theory for the User. Prentice Hall information and system sciences series. L Ljung, Prentice Hall PTRL. Ljung, System Identification: Theory for the User. Prentice Hall information and system sciences series, Prentice Hall PTR, 1999.\n\nAdaptive Filtering Prediction and Control. G C Goodwin, K S Sin, Dover Publications, IncUSAG. C. Goodwin and K. S. Sin, Adaptive Filtering Prediction and Control. USA: Dover Publications, Inc., 2009.\n\nIterative feedback tuning: theory and applications. H Hjalmarsson, M Gevers, S Gunnarsson, O Lequin, IEEE Control Systems Magazine. 184H. Hjalmarsson, M. Gevers, S. Gunnarsson, and O. Lequin, \"Iterative feedback tuning: theory and applications,\" IEEE Control Systems Magazine, vol. 18, no. 4, pp. 26-41, 1998.\n\nConvergence analysis of an iterative correlation-based controller tuning method. A Karimi, L Mikovi, D Bonvin, IFAC Proceedings Volumes. 3515th IFAC World CongressA. Karimi, L. Mikovi, and D. Bonvin, \"Convergence analysis of an iterative correlation-based controller tuning method,\" IFAC Proceed- ings Volumes, vol. 35, no. 1, pp. 413 -418, 2002. 15th IFAC World Congress.\n\nVirtual reference feedback tuning: a direct method for the design of feedback controllers. M Campi, A Lecchini, S Savaresi, Automatica. 388M. Campi, A. Lecchini, and S. Savaresi, \"Virtual reference feedback tuning: a direct method for the design of feedback controllers,\" Automatica, vol. 38, no. 8, pp. 1337 -1346, 2002.\n\nFrom model-based control to data-driven control: Survey, classification and perspective. Z.-S Hou, Z Wang, Data-based Control, Decision, Scheduling and Fault Diagnostics. 235Z.-S. Hou and Z. Wang, \"From model-based control to data-driven control: Survey, classification and perspective,\" Information Sciences, vol. 235, pp. 3 -35, 2013. Data-based Control, Decision, Scheduling and Fault Diagnostics.\n\nParadigms and puzzles in the theory of dynamical systems. J C Willems, IEEE Transactions on Automatic Control. 36J. C. Willems, \"Paradigms and puzzles in the theory of dynamical systems,\" IEEE Transactions on Automatic Control, vol. 36, pp. 259- 294, March 1991.\n\nJ C Willems, J W Polderman, Introduction to mathematical systems theory: a behavioral approach. Springer Science & Business Media26J. C. Willems and J. W. Polderman, Introduction to mathematical systems theory: a behavioral approach, vol. 26. Springer Science & Business Media, 2013.\n\nA note on persistency of excitation. J C Willems, P Rapisarda, I Markovsky, B L D Moor, Systems & Control Letters. 544J. C. Willems, P. Rapisarda, I. Markovsky, and B. L. D. Moor, \"A note on persistency of excitation,\" Systems & Control Letters, vol. 54, no. 4, pp. 325 -329, 2005.\n\nData-driven simulation and control. I Markovsky, P Rapisarda, International Journal of Control. 8112I. Markovsky and P. Rapisarda, \"Data-driven simulation and control,\" International Journal of Control, vol. 81, no. 12, pp. 1946-1959, 2008.\n\nOn the linear quadratic data-driven control. I Markovsky, P Rapisarda, 2007 European Control Conference (ECC). IEEEI. Markovsky and P. Rapisarda, \"On the linear quadratic data-driven control,\" in 2007 European Control Conference (ECC), pp. 5313- 5318, IEEE, 2007.\n\nWillems' fundamental lemma for state-space systems and its extension to multiple datasets. H J Van Waarde, C D Persis, M K Camlibel, P Tesi, H. J. van Waarde, C. D. Persis, M. K. Camlibel, and P. Tesi, \"Willems' fundamental lemma for state-space systems and its extension to multiple datasets,\" 2020.\n\nOne-shot verification of dissipativity properties from inputoutput data. A Romer, J Berberich, J Khler, F Allgwer, IEEE Control Systems Letters. 3A. Romer, J. Berberich, J. Khler, and F. Allgwer, \"One-shot verifica- tion of dissipativity properties from inputoutput data,\" IEEE Control Systems Letters, vol. 3, pp. 709-714, July 2019.\n\nRobust datadriven state-feedback design. J Berberich, A Romer, C W Scherer, F Allgöwer, arXiv:1909.04314arXiv preprintJ. Berberich, A. Romer, C. W. Scherer, and F. Allgöwer, \"Robust data- driven state-feedback design,\" arXiv preprint arXiv:1909.04314, 2019.\n\nA trajectory-based framework for datadriven system analysis and control. J Berberich, F Allgöwer, arXiv:1903.10723arXiv preprintJ. Berberich and F. Allgöwer, \"A trajectory-based framework for data- driven system analysis and control,\" arXiv preprint arXiv:1903.10723, 2019.\n\nFormulas for data-driven control: Stabilization, optimality, and robustness. C De Persis, P Tesi, IEEE Transactions on Automatic Control. 65C. De Persis and P. Tesi, \"Formulas for data-driven control: Stabi- lization, optimality, and robustness,\" IEEE Transactions on Automatic Control, vol. 65, pp. 909-924, March 2020.\n\nData informativity: a new perspective on data-driven analysis and control. H J Van Waarde, J Eising, H L Trentelman, M K Camlibel, H. J. van Waarde, J. Eising, H. L. Trentelman, and M. K. Camlibel, \"Data informativity: a new perspective on data-driven analysis and control,\" 2019.\n\nSpc: Subspace predictive control. W Favoreel, B D Moor, M Gevers, IFAC Proceedings Volumes. Beijing, Chia3214th IFAC World CongressW. Favoreel, B. D. Moor, and M. Gevers, \"Spc: Subspace predictive control,\" IFAC Proceedings Volumes, vol. 32, no. 2, pp. 4004 -4009, 1999. 14th IFAC World Congress 1999, Beijing, Chia, 5-9 July.\n\nData-enabled predictive control: In the shallows of the deepc. J Coulson, J Lygeros, F Dörfler, 2019 18th European Control Conference (ECC). IEEEJ. Coulson, J. Lygeros, and F. Dörfler, \"Data-enabled predictive control: In the shallows of the deepc,\" in 2019 18th European Control Conference (ECC), pp. 307-312, IEEE, 2019.\n\nRegularized and distributionally robust data-enabled predictive control. J Coulson, J Lygeros, F Dörfler, arXiv:1903.06804arXiv preprintJ. Coulson, J. Lygeros, and F. Dörfler, \"Regularized and distri- butionally robust data-enabled predictive control,\" arXiv preprint arXiv:1903.06804, 2019.\n\nData-enabled predictive control for grid-connected power converters. L Huang, J Coulson, J Lygeros, F Dorfler, arXiv:1903.07339arXiv preprintL. Huang, J. Coulson, J. Lygeros, and F. Dorfler, \"Data-enabled predictive control for grid-connected power converters,\" arXiv preprint arXiv:1903.07339, 2019.\n\nData-driven model predictive control with stability and robustness guarantees. J Berberich, J Köhler, M A Müller, F Allgöwer, arXiv:1906.04679arXiv preprintJ. Berberich, J. Köhler, M. A. Müller, and F. Allgöwer, \"Data-driven model predictive control with stability and robustness guarantees,\" arXiv preprint arXiv:1906.04679, 2019.\n\nData-Based Modelling and System Identification. H Hjalmarsson, Automatica. 413From experiment design to closed-loop controlH. Hjalmarsson, \"From experiment design to closed-loop control,\" Automatica, vol. 41, no. 3, pp. 393 -438, 2005. Data-Based Modelling and System Identification.\n\nIndustrial application of model based predictive control as a supervisory system. B Bulut, M R Katebi, M A Johnson, Proceedings of the 2000 American Control Conference. the 2000 American Control Conference6B. Bulut, M. R. Katebi, and M. A. Johnson, \"Industrial application of model based predictive control as a supervisory system,\" in Proceed- ings of the 2000 American Control Conference, vol. 6, pp. 3763-3767 vol.6, June 2000.\n\nSubspace identification for linear systems: TheoryImplementationApplications. P Van Overschee, B. De Moor, Springer Science & Business MediaP. Van Overschee and B. De Moor, Subspace identification for linear systems: TheoryImplementationApplications. Springer Science & Business Media, 2012.\n\n. P Billingsley, Probability and Measure. Wiley Series in Probability and Statistics. WileyP. Billingsley, Probability and Measure. Wiley Series in Probability and Statistics, Wiley, 1995.\n\nThe explicit linear quadratic regulator for constrained systems. A Bemporad, M Morari, V Dua, E N Pistikopoulos, Automatica. 381A. Bemporad, M. Morari, V. Dua, and E. N. Pistikopoulos, \"The ex- plicit linear quadratic regulator for constrained systems,\" Automatica, vol. 38, no. 1, pp. 3 -20, 2002.\n"},"corpus_id":{"kind":"number","value":212747784,"string":"212,747,784"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2003.08269\",\n \"mag\": 3029097646,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"journals/csysl/AlpagoDL20\",\n \"doi\": \"10.1109/lcsys.2020.2998296\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/2003.08269v2.pdf\"\n ],\n \"pdf_sha\": \"d1a421a5e3a9388a27dd4c7edfbbed27c4a51c90\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"https://www.research-collection.ethz.ch/bitstream/20.500.11850/425063/3/EKF_DeePC_AAM.pdf\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"An Extended Kalman Filter for Data-enabled Predictive Control"},"authors":{"kind":"list like","value":["Daniele Alpago","Florian Dörfler","John Lygeros"],"string":"[\n \"Daniele Alpago\",\n \"Florian Dörfler\",\n \"John Lygeros\"\n]","isConversational":false},"abstract":{"kind":"string","value":"The literature dealing with data-driven analysis and control problems has significantly grown in the recent years. Most of the recent literature deals with linear timeinvariant systems in which the uncertainty (if any) is assumed to be deterministic and bounded; relatively little attention has been devoted to stochastic linear time-invariant systems. As a first step in this direction, we propose to equip the recently introduced Data-enabled Predictive Control algorithm with a data-based Extended Kalman Filter to make use of additional available input-output data for reducing the effect of noise, without increasing the computational load of the optimization procedure.arXiv:2003.08269v2 [math.OC] 10 May 2020"},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":"With the increasing complexity of applications in science and engineering, the use of model-based control techniques is becoming more and more challenging as they usually require accurate descriptions of model and uncertainties, often difficult to obtain. The need of data-driven approaches was already perceived in the 1980s with the advent of system identification [1] and adaptive control [2], and received new impetuous in the 1990s with approaches such as Iterative Feedback Tuning, Correlation-based Tuning, Virtual Reference Feedback Tuning [3]- [5] and many others. In recent years, data-driven analysis and control has experienced a renewed interest [6]. One promising approach is due to the rediscovery of a result originally formulated in the context of behavioral system theory [7], [8] by J. C. Willems and coauthors, known as the Fundamental Lemma [9]. The result states that if the input signal is sufficiently rich, all possible trajectories of a deterministic, linear time-invariant (LTI) system can be generated from linear combinations of past trajectories of the same system. This allows one to use a Hankel matrix constructed from input-output data as an implicit representation of the underlying dynamics. This representation has been first exploited for data-driven simulation and linear-quadratic tracking for deterministic LTI systems in a behavioral setting [10], [11]. This has resulted in a growing stream of literature dealing with data-driven analysis [12], [13] and control problems [14]- [17].\nIn this context, particular attention has been devoted to the problem of optimal trajectory tracking [18]- [22], widely studied in model-based control. The aim is to compute an optimal control policy based on output feedback that drives the system along an output reference while minimizing a stage cost and satisfying input and output constraints. Model Predictive Control (MPC) has been one of the most effective methods to tackle such problems. MPC requires an accurate model of the system and an accurate description of the uncertainty (if any), which might be challenging and expensive to obtain in many applications [23], [24]. The success of MPC and the difficulties arising from the requirement for models have led to the introduction of a Data-enabled Predictive Control (DeePC) algorithm. The latter does not rely on a particular parametric system representation, but rather on a Hankel-matrix representation of the underlying dynamics, constructed from the system's input-output trajectories [19]- [21]. Unlike classical model-based predictive control methods, DeePC performs the control computation without identifying the system by solving a (parametric) convex optimization problem that includes the Hankel matrix in the constraints. The complexity of such optimization depends on the system dimensions, the MPC horizon and the amount of available data. Therefore, it is not obvious how to efficiently include additional data (for example, past measurements of the output of the closed-loop system) in the procedure, without increasing the computational burden of the optimization that has to be solved online.\nUnlike earlier methods, recent data-driven approaches based on the Fundamental Lemma have devoted little attention to stochastic systems. Some approaches adopt a \"robust control\" perspective and treat the uncertainty as a deterministic and bounded sequence [14], [16], [22], sometimes affecting just the output. The focus of the present paper is to extend the DeePC algorithm to stochastic LTI systems and design methods for including more data to improve closed-loop performance, without increasing the computational load of the optimization. This is accomplished through a combination of an offline averaging of Hankel matrix predictors and an online, data-driven Extended Kalman Filter (EKF). The latter is based on an implicit model constructed from the parametric solution of the DeePC optimization program. The combination of off-line averaging with online EKF dramatically improves the closed-loop performance, as evidenced by numerical experiments.\nIn Section II we introduce the notation and recall some preliminary results. In Section III the DeePC algorithm is reviewed and the issues that motivate our approach are highlighted. Section IV-B introduces the proposed approach and Section V presents numerical experiments. Finally, in Section VI we draw conclusions and outline future lines of research."},{"title":"II. NOTATION AND PRELIMINARIES","content":"We recall the definition of persistently exciting signals and a state-space version of a result from behavioral system theory [7], [8], known as the Fundamental Lemma [9], [10].\nNotation. Given a matrix A ∈ R m×n , A denotes its transpose and A −1 its inverse (if m = n). The notation A(i : j, :), i ≤ j ≤ m, stands for the sub-matrix of A that goes from the i-th row of A to the j-th row of A, included. If i = j we simply write A(i, :). We denote with I m the identity matrix of dimension m×m. With diag(A 1 , . . . , A n ) we denote the blockdiagonal matrix formed with the matrices A 1 , . . . , A n . Given a sequence of matrices A h , . . . , A h+k in R m×n , with k > 0, we denote by A h,h+k := col(A h , . . . , A h+k ) := [A h · · · A h+k ] ∈ R (k+1)m×n . Given a vector w ∈ R m we introduce the quadratic form w 2 P = w Pw which coincides with the squared Euclidean norm w 2 2 when P = I m . The symbol E[·] denotes the expectation operator.\nbe the block-Hankel matrix associated to the trajectory w 0,N−1 , with M block-rows and the maximal number of columns N − M + 1. We say that the signal w 0,N−1 is persistently exciting of order M if the Hankel matrix H M (w 0,N−1 ) has full row-rank qM. This requires the sequence w 0,N−1 to be rich and long enough, in particular N ≥ (q + 1) M − 1.\nFundamental Lemma. Consider a state-space representation of an n-dimensional LTI system\nwhere (x k ) k∈Z is the n-dimensional state-process, (u k ) k∈Z the m-dimensional input process, and (y k ) k∈Z is the pdimensional output process. Given an initial condition x 0 ∈ R n and a sequence of inputs u 0,k−1 ∈ R mk , the output of the system can be written as\nand y 0 = C x 0 + D u 0 . Let T ∈ N and collect T -long inputoutput trajectories u 0,T −1 , y 0,T −1 of the system (1). For fixed positive integers N p , N f ∈ N and k ≥ N p , we can associate to the vectors u (k)\nthe block-Hankel matrices\nwhere U := [u 0 · · · u T −1 ] and X := [x 0 · · · x T −1 ] are the block-Hankel matrices containing the inputs and the corresponding sequence of states, respectively. Here,\nare observability matrices,\nand T f (B, D) is defined similarly. The following fundamental result from [9] guarantees that the matrix (4) can be used in place of the parametric representation (1), as long as the input u 0,T −1 is persistently exciting.\nLemma 1 (Fundamental Lemma): Assume the system (1) to be controllable and the input trajectory u 0,T −1 to be persistently exciting of order N p + N f + n. Then, a sequence col(u p , u f , y p , y f ) is an input-output trajectory of the system (1) if and only if it is in the range space of H.\nRecalling the persistency of excitation condition, T ≥ (m+ 1)(N p + N f + n) + 1 is a necessary condition for u 0,T −1 to be persistently exciting of order N p + N f + n. Lemma 1 has been originally proven using the behavioral language in [9, Theorem 1]. For an equivalent recent state-space proof see [12] and [16, Appendix A]."},{"title":"III. DATA-ENABLED PREDICTIVE CONTROL REVIEW","content":"We briefly introduce the recently proposed Data-enabled Predictive Control (DeePC) method [19], and highlight some related issues when dealing with stochastic LTI systems. This will serve as a motivation for what follows.\nConsider the stochastic version of (1)\nwhere (v k ) k∈Z is p-dimensional white noise (zero mean and unit-variance). We collect sufficiently long input-output\nThe same computations leading to (4), lead to the matrix corresponding to the stochastic model (6):\nThe DeePC algorithm proposed in [19] attempts to compute an optimal-control action based on past input-output data coming from the unknown system (6), without previous identification. The control action is selected through an MPC-like optimization problem that allows one to enforce constraints ensuring safety and performance requirements. The previously collected data is directly used on-line in the MPC optimization problem; the predictor is therefore implicit and arises as the outcome of the optimization problem. In particular, at the generic iteration k ≥ N p , the DeePC computes optimal control actions by solving\nwhere N f is the prediction horizon, r ∈ R pN f is the output reference signal we want to track, U ⊆ R m , Y ⊆ R p are the input and output constraint sets, respectively, Q ∈ R p×p is the output cost matrix (positive semidefinite), R ∈ R m×m is the control cost matrix (positive definite), λ y ≥ 0 and λ g ≥ 0 are the regularization parameters, and u\np are the most recent N p input-output measurements from (6), according to the notation (3). For simplicity we consider input and output box constraints of the form U = [u min , u max ] and Y = [y min , y max ], respectively. Note that the block-Hankel matrices U p ,U f ,Y p ,Y f are fixed throughout the online iterations.\nIf we let g k be the optimal solution of problem (8) at iteration k, DeePC provides an implicit predictor y (k) f = Y f g k whose model is never derived explicitly but whose predictions are implicitly used for obtaining the optimal control actions u p to the most recent input-output measurements and set k to k + N c + 1.\nEquation (8) is a relaxation of the corresponding problem enforcing the constraint Y p g = y (k) p , which was proven to be equivalent to the classical receding-horizon MPC in the case of deterministic LTI systems [19]. Here, to cope with potential infeasibility due to the disturbances, the constraint is substituted with the least-squares regularization term Y p g − y (k) p 2 2 . The two-norm regularization on g has been introduced to avoid overfitting and it has been shown to relate to distributional robustness of the method with respect to a range of uncertainties [20]. One can also see that, besides infeasibility issues, considering stochastic models makes both the implicit predictor model y (k) f = Y f g k and the control actions u (k) f = U f g k defined by the optimization (8) to be non-linear in the past-data (y\nWe conclude this section with a note on the effect of acquiring more data. Classical LTI system identification methods use historical data off-line to compute estimates of the matrices in the system dynamics. The resulting matrices are then used on-line to generate state estimates and perform the prediction [1]. Other prediction methods estimate the linear relation from u\nf off-line, and then use it online to carry on the predictions, [25] and references therein. This kind of prediction architectures naturally possesses a data-compression mechanism. For the classical system identification methods only storage of the estimates of the system matrices is required. This depends on input and output dimensions m and p and on the (guessed) state dimension, but not on the amount T of data nor on the prediction horizon N f . Likewise, the storage requirements for linear predictors depends on input and output dimensions m and p, and the horizons N p , N f , but not on the amount T of data. If more data become available one can then use it to improve predictions with no need to store it or use it online. The DeePC approach, on the other hand, requires one to carry all the data at every iteration and the size of the optimization problem (8) increases both with the amount T of available data and with the parameters m, p, N p , and N f . Though additional data would in principle also be beneficial for a DeePC controller, it is not clear how one can incorporate it without increasing the on-line computational burden."},{"title":"IV. METHOD DESCRIPTION","content":"We introduce a possible way to effectively incorporate more data in the DeePC framework to reduce the effect of noise in the solution of problem (8). We propose to use additional data that may be available off-line to de-noise the block-Hankel matrices Y p and Y f and to equip the DeePC with an EKF based on data to handle the noise in the online measurements y (k) p ."},{"title":"A. Averaging Data Matrices","content":"The output matrices Y p and Y f are constructed off-line from the trajectory y 0,T −1 . Let the corresponding initial conditions. Using those additional data on-line to improve the prediction will lead to an intractable optimization problem. However, we can make use of additional trajectories off-line to construct N different data matrices H (1) , . . . , H (N) defined analogously to (7), and average those matrices to obtain 0,T −1 , respectively. Since (v k ) k∈Z is assumed to be white-noise, the Law of Large Numbers guarantees thatV N → 0 as N → ∞ [26]. The averaging procedure makes use of additional data to mitigate the effect of noise in the data-driven model, hence reducing the risk of overfitting that would be present if the data was used directly in DeePC. Accordingly, the more matrices are involved in the average, the smaller the value of the regularization parameter λ g that gives the optimal closed-loop cost (see Figure 3, Section V).\nWe recognize that such a method heavily exploits the underlying linear structure of the problem. However, this is meant to be a first attempt to exploit additional data for improving the performance of the algorithm when dealing with stochastic systems, without increasing the dimension of the optimization problem (11) to be solved online. Indeed, thanks to the linear structure and superposition, the sub-matricesX N andŪ N in the averaged data matrix (9) still represent valid system trajectories corresponding to the average of the initial states x 0 = 1 N ∑ i x (i) 0 and the average of the input sequences u = 1 N ∑ i u (i) , used in the experiments that generated the data for each H (i) . The only point that requires attention is ensuring that the resulting average input sequence respects the persistence of excitation requirement. The simplest way to ensure this is to assume that the same persistently exciting input sequence is applied in all cases."},{"title":"B. An EKF for DeePC","content":"The averaging procedure represents a simple way to make use of multiple T -long trajectories to denoise off-line the data matrix representing the dynamics in the constraints of (8). Numerical evidence suggests that additional denoising of the on-line data y (k) p that enters the cost of (8) can lead to a further, significant improvement in performance (Section V). In a model based setting, such on-line denoising could be performed by a Kalman filter. In our data-driven setting, however, the classical Kalman filter cannot be applied as it requires a model of the system dynamics. Here we show how the non-linear one-step predictor implicit in (8) can be used to derive an EKF for integrating past measurements into an implicit \"state estimate\"; this can in turn be used to improve the asymptotic performance of the algorithm.\nWe introduce a fictitious state vector z k := col(y k−N p+1 , · · · , y k ) ∈ R pN p and keep track of a filtered state-estimateẑ k|k and the corresponding error covariance P k|k . At the generic iteration k ≥ N p , the EKF-DeePC algorithm uses the estimateẑ k|k by solving the following optimization problem\nThe formulation (10) is obtained from (8) by substituting the constraints u i = U f g and y i = Y f g in the cost, leaving g as the only decision variable. The crucial difference with respect to (8) is that the past data Y p g used in the implicit predictor is now required to fit the state estimateẑ k|k instead of the N p most-recent measurements y (k) p . Rewriting (10) as a multi-parametric quadratic program (mp-QP) in the parameter θ k := col(ẑ k|k , u (k) p ) leads to an explicit relation between of the optimum g k and the parameter θ k ; we exploit this relation to build the EKF. The mp-QP form of (10) is\nwhere the inequality constraint defined by\nhas to be understood component-wise, i.e. (A in g) i ≤ s i for i = 1, . . . , 2(m + p)N f . The cost is then defined by the reference signal r k := col(r k+1 , · · · , r k+N f ) and the matrices\n, H := (λ y /2) · I pN p 0 0 0 .\nHere 1 N := col(1, 1, · · · , 1) ∈ R N , Q = diag(Q, . . . , Q) and R = diag(R, . . . , R). Assuming P = P > 0 and the KKTmatrix for problem (11) to be positive semi-definite (which is always the case for an MPC problem with input weighting matrix R > 0 [27]), the optimizer g k is a piecewise affine function of the parameters, and can be written as g k =\np +h k . In particular, the equality constraints in (11) and the noise in the model are responsible of the affine structure while the inequality constraints implies this affine relation to hold just locally, i.e. in a neighborhood of the parameter θ k , known as the critical region. The coefficients A k ,B k andh k coming from the KKT conditions for problem (11), are therefore region-dependent themselves [27]: we consider the affine expansion of g k pertaining to a specific value for the parameter θ k , fixed by the previous iteration.\nThe fact that, under suitable assumptions, the piecewise affine relation between the optimizer g k of (11) and the parameter θ k can be derived from the KKT conditions for problem (11), might suggest that the predictor implicit in (10) could be made explicit and be constructed off-line. Doing this would require one to construct all the regions on which the affine expansion is defined, for all the possible values of the parameters. As the number of these regions scales exponentially with the QP size (parameters plus constraints), hence with the amount T of available data and the horizons N p and N f , this approach is likely to be computationally intractable. This is the main reason for keeping the predictor implicit.\nLet M := col(Y p (2 : pN p , :), Y f (1, :)) be the prediction map. We can exploit the piece-wise affine form of the optimizer g k to incorporate the implicit predictor provided by (11) in an EKF-like architecture. To fix ideas, suppose we start running the algorithm at k = N p . From an initial guess of the meanẑ N p |N p = E[z N p ] and the covariance matrix\n, we then compute recursively (at every point in time) the standard Kalman filter update steps in Figure 1. We note that, because of its dependence on g k , the implicit predictor is piecewise affine (the matrices A k , B k , h k at the current iteration, depend on the critical region) making this a data-driven analogue to an Extended Kalman Filter."},{"title":"Prediction","content":"Update The matrices Q k ∈ R pN p ×pN p and R k ∈ R p×p are the variances of the process and measurement noise implicitly affecting the state and output dynamics z k+1 = A k z k + B k u (k) p + h k and y k = C z k , respectively. In practice, to implement the filter in Figure 1 one needs to address the same issues as for any Kalman filter, i.e. choosing the initial conditionsẑ N p |N p and P N p |N p , and obtaining an estimate (or guess) of the variances Q k and R k . For a model-based Kalman filter, one can use the residuals of the system identification to obtain estimates for these quantities; we speculate that something similar would be possible using the residuals between the average and individual data matrices in Section IV-A, though we do not pursue this direction here, due to space limitations. The steps outlined above are summarized in Figure 2."},{"title":"V. NUMERICAL VALIDATION","content":"We present numerical experiments assessing the effects of the proposed solution in tackling the noise acting in the optimization (8). Consider the stochastic system\nwhere (w k ) k∈Z , (v k ) k∈Z are independent zero-mean Gaussian white noises with covariance Σ w := EE and Σ v := FF respectively; below we consider Σ w = σ 2 w I n and Σ v = σ 2 v I p and report results for different values of σ 2 w and σ 2 v . Model (12) is reachable and observable. The performance metric we will consider throughout is the closed-loop cost J(u, y) = ∑ N sim k=1 y k − r k 2 Q + u k 2 R , where r k = 5 sin(0.3k) is the reference signal. To isolate the effects of averaging and the EKF, for each numerical experiment we tuned the regularization parameters λ y and λ g through exhaustive search to minimize the closed-loop cost for the standard DeePC, the averaged DeePC and the averaged DeePC with EKF. All numerical evidence comes from 100 repetitions for different data-sets. Unless otherwise stated, we set T = 100, N sim = 100, N p = 3, N f = 5, Q = I p , R = I m , σ 2 w = σ 2 v = 0.5 and N = 40 data matrices in the average (9).\nFirst we show the effect of the averaging introduced in Section IV-A. To isolate the averaging effect, we consider noisy Y p , Y f but y (k) p generated from the model with neither process noise, not measurement noise. The result, shown in Figure 3, is as expected: the cost decreases towards the ideal cost of MPC with the same horizon but perfect model and full state measurement while on the right, the optimal (numerically found) value of the regularization parameter λ g decreases to zero, as we expect from standard results in optimization. Figure 4 shows a comparison between the closed-loop costs of the standard DeePC and the averaged DeePC with EKF algorithms when varying the noise variances σ 2 w and σ 2 v . Figure 4 shows that introducing averaging and the EKF substantially improves the performance of the DeePC algorithm. Further experiments (data not shown) with averaged DeePC without EKF and EKF without averaging confirmed that DeePC with a combination of averaging and EKF substantially outperformed all other combinations; indeed this was the case across a broad range of values of the regularizes λ y and λ g , suggesting that averaged DeePC with EKF is easier to tune. Figure cost of the two algorithms behaves with respect to the past horizon N p . The improvement in the performance is due to the introduction of the averaging plus the EKF, highlighting the difference between fixed horizon and recursive filtering."},{"title":"VI. CONCLUSION AND OUTLOOK","content":"We presented an extension of the data-enabled predictive control (DeePC) algorithm introduced in [19] to tackle control problems on unknown stochastic LTI systems, by making use of additional data without increasing the dimension of the on-line optimization problem. The procedure features an on-line data-driven EKF that filters out the noise in the measurements, and an (off-line) averaging of multiple data matrices to get a cleaner data-driven model. The performance of the proposed method has been validated experimentally on an LTI stochastic system showing improvements with respect to standard DeePC. Future work includes testing the proposed method on a real-case scenario and comparing it to algorithms such as [3]- [5]."},{"title":"1","content":""},{"title":"T","content":""},{"title":"Fig. 1 :Fig. 2 :","content":""},{"title":"Fig. 3 :","content":""},{"title":"Fig. 4 :Fig. 5 :","content":""}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"With the increasing complexity of applications in science and engineering, the use of model-based control techniques is becoming more and more challenging as they usually require accurate descriptions of model and uncertainties, often difficult to obtain. The need of data-driven approaches was already perceived in the 1980s with the advent of system identification [1] and adaptive control [2], and received new impetuous in the 1990s with approaches such as Iterative Feedback Tuning, Correlation-based Tuning, Virtual Reference Feedback Tuning [3]- [5] and many others. In recent years, data-driven analysis and control has experienced a renewed interest [6]. One promising approach is due to the rediscovery of a result originally formulated in the context of behavioral system theory [7], [8] by J. C. Willems and coauthors, known as the Fundamental Lemma [9]. The result states that if the input signal is sufficiently rich, all possible trajectories of a deterministic, linear time-invariant (LTI) system can be generated from linear combinations of past trajectories of the same system. This allows one to use a Hankel matrix constructed from input-output data as an implicit representation of the underlying dynamics. This representation has been first exploited for data-driven simulation and linear-quadratic tracking for deterministic LTI systems in a behavioral setting [10], [11]. This has resulted in a growing stream of literature dealing with data-driven analysis [12], [13] and control problems [14]- [17].\\nIn this context, particular attention has been devoted to the problem of optimal trajectory tracking [18]- [22], widely studied in model-based control. The aim is to compute an optimal control policy based on output feedback that drives the system along an output reference while minimizing a stage cost and satisfying input and output constraints. Model Predictive Control (MPC) has been one of the most effective methods to tackle such problems. MPC requires an accurate model of the system and an accurate description of the uncertainty (if any), which might be challenging and expensive to obtain in many applications [23], [24]. The success of MPC and the difficulties arising from the requirement for models have led to the introduction of a Data-enabled Predictive Control (DeePC) algorithm. The latter does not rely on a particular parametric system representation, but rather on a Hankel-matrix representation of the underlying dynamics, constructed from the system's input-output trajectories [19]- [21]. Unlike classical model-based predictive control methods, DeePC performs the control computation without identifying the system by solving a (parametric) convex optimization problem that includes the Hankel matrix in the constraints. The complexity of such optimization depends on the system dimensions, the MPC horizon and the amount of available data. Therefore, it is not obvious how to efficiently include additional data (for example, past measurements of the output of the closed-loop system) in the procedure, without increasing the computational burden of the optimization that has to be solved online.\\nUnlike earlier methods, recent data-driven approaches based on the Fundamental Lemma have devoted little attention to stochastic systems. Some approaches adopt a \\\"robust control\\\" perspective and treat the uncertainty as a deterministic and bounded sequence [14], [16], [22], sometimes affecting just the output. The focus of the present paper is to extend the DeePC algorithm to stochastic LTI systems and design methods for including more data to improve closed-loop performance, without increasing the computational load of the optimization. This is accomplished through a combination of an offline averaging of Hankel matrix predictors and an online, data-driven Extended Kalman Filter (EKF). The latter is based on an implicit model constructed from the parametric solution of the DeePC optimization program. The combination of off-line averaging with online EKF dramatically improves the closed-loop performance, as evidenced by numerical experiments.\\nIn Section II we introduce the notation and recall some preliminary results. In Section III the DeePC algorithm is reviewed and the issues that motivate our approach are highlighted. Section IV-B introduces the proposed approach and Section V presents numerical experiments. Finally, in Section VI we draw conclusions and outline future lines of research.\"\n },\n {\n \"title\": \"II. NOTATION AND PRELIMINARIES\",\n \"content\": \"We recall the definition of persistently exciting signals and a state-space version of a result from behavioral system theory [7], [8], known as the Fundamental Lemma [9], [10].\\nNotation. Given a matrix A ∈ R m×n , A denotes its transpose and A −1 its inverse (if m = n). The notation A(i : j, :), i ≤ j ≤ m, stands for the sub-matrix of A that goes from the i-th row of A to the j-th row of A, included. If i = j we simply write A(i, :). We denote with I m the identity matrix of dimension m×m. With diag(A 1 , . . . , A n ) we denote the blockdiagonal matrix formed with the matrices A 1 , . . . , A n . Given a sequence of matrices A h , . . . , A h+k in R m×n , with k > 0, we denote by A h,h+k := col(A h , . . . , A h+k ) := [A h · · · A h+k ] ∈ R (k+1)m×n . Given a vector w ∈ R m we introduce the quadratic form w 2 P = w Pw which coincides with the squared Euclidean norm w 2 2 when P = I m . The symbol E[·] denotes the expectation operator.\\nbe the block-Hankel matrix associated to the trajectory w 0,N−1 , with M block-rows and the maximal number of columns N − M + 1. We say that the signal w 0,N−1 is persistently exciting of order M if the Hankel matrix H M (w 0,N−1 ) has full row-rank qM. This requires the sequence w 0,N−1 to be rich and long enough, in particular N ≥ (q + 1) M − 1.\\nFundamental Lemma. Consider a state-space representation of an n-dimensional LTI system\\nwhere (x k ) k∈Z is the n-dimensional state-process, (u k ) k∈Z the m-dimensional input process, and (y k ) k∈Z is the pdimensional output process. Given an initial condition x 0 ∈ R n and a sequence of inputs u 0,k−1 ∈ R mk , the output of the system can be written as\\nand y 0 = C x 0 + D u 0 . Let T ∈ N and collect T -long inputoutput trajectories u 0,T −1 , y 0,T −1 of the system (1). For fixed positive integers N p , N f ∈ N and k ≥ N p , we can associate to the vectors u (k)\\nthe block-Hankel matrices\\nwhere U := [u 0 · · · u T −1 ] and X := [x 0 · · · x T −1 ] are the block-Hankel matrices containing the inputs and the corresponding sequence of states, respectively. Here,\\nare observability matrices,\\nand T f (B, D) is defined similarly. The following fundamental result from [9] guarantees that the matrix (4) can be used in place of the parametric representation (1), as long as the input u 0,T −1 is persistently exciting.\\nLemma 1 (Fundamental Lemma): Assume the system (1) to be controllable and the input trajectory u 0,T −1 to be persistently exciting of order N p + N f + n. Then, a sequence col(u p , u f , y p , y f ) is an input-output trajectory of the system (1) if and only if it is in the range space of H.\\nRecalling the persistency of excitation condition, T ≥ (m+ 1)(N p + N f + n) + 1 is a necessary condition for u 0,T −1 to be persistently exciting of order N p + N f + n. Lemma 1 has been originally proven using the behavioral language in [9, Theorem 1]. For an equivalent recent state-space proof see [12] and [16, Appendix A].\"\n },\n {\n \"title\": \"III. DATA-ENABLED PREDICTIVE CONTROL REVIEW\",\n \"content\": \"We briefly introduce the recently proposed Data-enabled Predictive Control (DeePC) method [19], and highlight some related issues when dealing with stochastic LTI systems. This will serve as a motivation for what follows.\\nConsider the stochastic version of (1)\\nwhere (v k ) k∈Z is p-dimensional white noise (zero mean and unit-variance). We collect sufficiently long input-output\\nThe same computations leading to (4), lead to the matrix corresponding to the stochastic model (6):\\nThe DeePC algorithm proposed in [19] attempts to compute an optimal-control action based on past input-output data coming from the unknown system (6), without previous identification. The control action is selected through an MPC-like optimization problem that allows one to enforce constraints ensuring safety and performance requirements. The previously collected data is directly used on-line in the MPC optimization problem; the predictor is therefore implicit and arises as the outcome of the optimization problem. In particular, at the generic iteration k ≥ N p , the DeePC computes optimal control actions by solving\\nwhere N f is the prediction horizon, r ∈ R pN f is the output reference signal we want to track, U ⊆ R m , Y ⊆ R p are the input and output constraint sets, respectively, Q ∈ R p×p is the output cost matrix (positive semidefinite), R ∈ R m×m is the control cost matrix (positive definite), λ y ≥ 0 and λ g ≥ 0 are the regularization parameters, and u\\np are the most recent N p input-output measurements from (6), according to the notation (3). For simplicity we consider input and output box constraints of the form U = [u min , u max ] and Y = [y min , y max ], respectively. Note that the block-Hankel matrices U p ,U f ,Y p ,Y f are fixed throughout the online iterations.\\nIf we let g k be the optimal solution of problem (8) at iteration k, DeePC provides an implicit predictor y (k) f = Y f g k whose model is never derived explicitly but whose predictions are implicitly used for obtaining the optimal control actions u p to the most recent input-output measurements and set k to k + N c + 1.\\nEquation (8) is a relaxation of the corresponding problem enforcing the constraint Y p g = y (k) p , which was proven to be equivalent to the classical receding-horizon MPC in the case of deterministic LTI systems [19]. Here, to cope with potential infeasibility due to the disturbances, the constraint is substituted with the least-squares regularization term Y p g − y (k) p 2 2 . The two-norm regularization on g has been introduced to avoid overfitting and it has been shown to relate to distributional robustness of the method with respect to a range of uncertainties [20]. One can also see that, besides infeasibility issues, considering stochastic models makes both the implicit predictor model y (k) f = Y f g k and the control actions u (k) f = U f g k defined by the optimization (8) to be non-linear in the past-data (y\\nWe conclude this section with a note on the effect of acquiring more data. Classical LTI system identification methods use historical data off-line to compute estimates of the matrices in the system dynamics. The resulting matrices are then used on-line to generate state estimates and perform the prediction [1]. Other prediction methods estimate the linear relation from u\\nf off-line, and then use it online to carry on the predictions, [25] and references therein. This kind of prediction architectures naturally possesses a data-compression mechanism. For the classical system identification methods only storage of the estimates of the system matrices is required. This depends on input and output dimensions m and p and on the (guessed) state dimension, but not on the amount T of data nor on the prediction horizon N f . Likewise, the storage requirements for linear predictors depends on input and output dimensions m and p, and the horizons N p , N f , but not on the amount T of data. If more data become available one can then use it to improve predictions with no need to store it or use it online. The DeePC approach, on the other hand, requires one to carry all the data at every iteration and the size of the optimization problem (8) increases both with the amount T of available data and with the parameters m, p, N p , and N f . Though additional data would in principle also be beneficial for a DeePC controller, it is not clear how one can incorporate it without increasing the on-line computational burden.\"\n },\n {\n \"title\": \"IV. METHOD DESCRIPTION\",\n \"content\": \"We introduce a possible way to effectively incorporate more data in the DeePC framework to reduce the effect of noise in the solution of problem (8). We propose to use additional data that may be available off-line to de-noise the block-Hankel matrices Y p and Y f and to equip the DeePC with an EKF based on data to handle the noise in the online measurements y (k) p .\"\n },\n {\n \"title\": \"A. Averaging Data Matrices\",\n \"content\": \"The output matrices Y p and Y f are constructed off-line from the trajectory y 0,T −1 . Let the corresponding initial conditions. Using those additional data on-line to improve the prediction will lead to an intractable optimization problem. However, we can make use of additional trajectories off-line to construct N different data matrices H (1) , . . . , H (N) defined analogously to (7), and average those matrices to obtain 0,T −1 , respectively. Since (v k ) k∈Z is assumed to be white-noise, the Law of Large Numbers guarantees thatV N → 0 as N → ∞ [26]. The averaging procedure makes use of additional data to mitigate the effect of noise in the data-driven model, hence reducing the risk of overfitting that would be present if the data was used directly in DeePC. Accordingly, the more matrices are involved in the average, the smaller the value of the regularization parameter λ g that gives the optimal closed-loop cost (see Figure 3, Section V).\\nWe recognize that such a method heavily exploits the underlying linear structure of the problem. However, this is meant to be a first attempt to exploit additional data for improving the performance of the algorithm when dealing with stochastic systems, without increasing the dimension of the optimization problem (11) to be solved online. Indeed, thanks to the linear structure and superposition, the sub-matricesX N andŪ N in the averaged data matrix (9) still represent valid system trajectories corresponding to the average of the initial states x 0 = 1 N ∑ i x (i) 0 and the average of the input sequences u = 1 N ∑ i u (i) , used in the experiments that generated the data for each H (i) . The only point that requires attention is ensuring that the resulting average input sequence respects the persistence of excitation requirement. The simplest way to ensure this is to assume that the same persistently exciting input sequence is applied in all cases.\"\n },\n {\n \"title\": \"B. An EKF for DeePC\",\n \"content\": \"The averaging procedure represents a simple way to make use of multiple T -long trajectories to denoise off-line the data matrix representing the dynamics in the constraints of (8). Numerical evidence suggests that additional denoising of the on-line data y (k) p that enters the cost of (8) can lead to a further, significant improvement in performance (Section V). In a model based setting, such on-line denoising could be performed by a Kalman filter. In our data-driven setting, however, the classical Kalman filter cannot be applied as it requires a model of the system dynamics. Here we show how the non-linear one-step predictor implicit in (8) can be used to derive an EKF for integrating past measurements into an implicit \\\"state estimate\\\"; this can in turn be used to improve the asymptotic performance of the algorithm.\\nWe introduce a fictitious state vector z k := col(y k−N p+1 , · · · , y k ) ∈ R pN p and keep track of a filtered state-estimateẑ k|k and the corresponding error covariance P k|k . At the generic iteration k ≥ N p , the EKF-DeePC algorithm uses the estimateẑ k|k by solving the following optimization problem\\nThe formulation (10) is obtained from (8) by substituting the constraints u i = U f g and y i = Y f g in the cost, leaving g as the only decision variable. The crucial difference with respect to (8) is that the past data Y p g used in the implicit predictor is now required to fit the state estimateẑ k|k instead of the N p most-recent measurements y (k) p . Rewriting (10) as a multi-parametric quadratic program (mp-QP) in the parameter θ k := col(ẑ k|k , u (k) p ) leads to an explicit relation between of the optimum g k and the parameter θ k ; we exploit this relation to build the EKF. The mp-QP form of (10) is\\nwhere the inequality constraint defined by\\nhas to be understood component-wise, i.e. (A in g) i ≤ s i for i = 1, . . . , 2(m + p)N f . The cost is then defined by the reference signal r k := col(r k+1 , · · · , r k+N f ) and the matrices\\n, H := (λ y /2) · I pN p 0 0 0 .\\nHere 1 N := col(1, 1, · · · , 1) ∈ R N , Q = diag(Q, . . . , Q) and R = diag(R, . . . , R). Assuming P = P > 0 and the KKTmatrix for problem (11) to be positive semi-definite (which is always the case for an MPC problem with input weighting matrix R > 0 [27]), the optimizer g k is a piecewise affine function of the parameters, and can be written as g k =\\np +h k . In particular, the equality constraints in (11) and the noise in the model are responsible of the affine structure while the inequality constraints implies this affine relation to hold just locally, i.e. in a neighborhood of the parameter θ k , known as the critical region. The coefficients A k ,B k andh k coming from the KKT conditions for problem (11), are therefore region-dependent themselves [27]: we consider the affine expansion of g k pertaining to a specific value for the parameter θ k , fixed by the previous iteration.\\nThe fact that, under suitable assumptions, the piecewise affine relation between the optimizer g k of (11) and the parameter θ k can be derived from the KKT conditions for problem (11), might suggest that the predictor implicit in (10) could be made explicit and be constructed off-line. Doing this would require one to construct all the regions on which the affine expansion is defined, for all the possible values of the parameters. As the number of these regions scales exponentially with the QP size (parameters plus constraints), hence with the amount T of available data and the horizons N p and N f , this approach is likely to be computationally intractable. This is the main reason for keeping the predictor implicit.\\nLet M := col(Y p (2 : pN p , :), Y f (1, :)) be the prediction map. We can exploit the piece-wise affine form of the optimizer g k to incorporate the implicit predictor provided by (11) in an EKF-like architecture. To fix ideas, suppose we start running the algorithm at k = N p . From an initial guess of the meanẑ N p |N p = E[z N p ] and the covariance matrix\\n, we then compute recursively (at every point in time) the standard Kalman filter update steps in Figure 1. We note that, because of its dependence on g k , the implicit predictor is piecewise affine (the matrices A k , B k , h k at the current iteration, depend on the critical region) making this a data-driven analogue to an Extended Kalman Filter.\"\n },\n {\n \"title\": \"Prediction\",\n \"content\": \"Update The matrices Q k ∈ R pN p ×pN p and R k ∈ R p×p are the variances of the process and measurement noise implicitly affecting the state and output dynamics z k+1 = A k z k + B k u (k) p + h k and y k = C z k , respectively. In practice, to implement the filter in Figure 1 one needs to address the same issues as for any Kalman filter, i.e. choosing the initial conditionsẑ N p |N p and P N p |N p , and obtaining an estimate (or guess) of the variances Q k and R k . For a model-based Kalman filter, one can use the residuals of the system identification to obtain estimates for these quantities; we speculate that something similar would be possible using the residuals between the average and individual data matrices in Section IV-A, though we do not pursue this direction here, due to space limitations. The steps outlined above are summarized in Figure 2.\"\n },\n {\n \"title\": \"V. NUMERICAL VALIDATION\",\n \"content\": \"We present numerical experiments assessing the effects of the proposed solution in tackling the noise acting in the optimization (8). Consider the stochastic system\\nwhere (w k ) k∈Z , (v k ) k∈Z are independent zero-mean Gaussian white noises with covariance Σ w := EE and Σ v := FF respectively; below we consider Σ w = σ 2 w I n and Σ v = σ 2 v I p and report results for different values of σ 2 w and σ 2 v . Model (12) is reachable and observable. The performance metric we will consider throughout is the closed-loop cost J(u, y) = ∑ N sim k=1 y k − r k 2 Q + u k 2 R , where r k = 5 sin(0.3k) is the reference signal. To isolate the effects of averaging and the EKF, for each numerical experiment we tuned the regularization parameters λ y and λ g through exhaustive search to minimize the closed-loop cost for the standard DeePC, the averaged DeePC and the averaged DeePC with EKF. All numerical evidence comes from 100 repetitions for different data-sets. Unless otherwise stated, we set T = 100, N sim = 100, N p = 3, N f = 5, Q = I p , R = I m , σ 2 w = σ 2 v = 0.5 and N = 40 data matrices in the average (9).\\nFirst we show the effect of the averaging introduced in Section IV-A. To isolate the averaging effect, we consider noisy Y p , Y f but y (k) p generated from the model with neither process noise, not measurement noise. The result, shown in Figure 3, is as expected: the cost decreases towards the ideal cost of MPC with the same horizon but perfect model and full state measurement while on the right, the optimal (numerically found) value of the regularization parameter λ g decreases to zero, as we expect from standard results in optimization. Figure 4 shows a comparison between the closed-loop costs of the standard DeePC and the averaged DeePC with EKF algorithms when varying the noise variances σ 2 w and σ 2 v . Figure 4 shows that introducing averaging and the EKF substantially improves the performance of the DeePC algorithm. Further experiments (data not shown) with averaged DeePC without EKF and EKF without averaging confirmed that DeePC with a combination of averaging and EKF substantially outperformed all other combinations; indeed this was the case across a broad range of values of the regularizes λ y and λ g , suggesting that averaged DeePC with EKF is easier to tune. Figure cost of the two algorithms behaves with respect to the past horizon N p . The improvement in the performance is due to the introduction of the averaging plus the EKF, highlighting the difference between fixed horizon and recursive filtering.\"\n },\n {\n \"title\": \"VI. CONCLUSION AND OUTLOOK\",\n \"content\": \"We presented an extension of the data-enabled predictive control (DeePC) algorithm introduced in [19] to tackle control problems on unknown stochastic LTI systems, by making use of additional data without increasing the dimension of the on-line optimization problem. The procedure features an on-line data-driven EKF that filters out the noise in the measurements, and an (off-line) averaging of multiple data matrices to get a cleaner data-driven model. The performance of the proposed method has been validated experimentally on an LTI stochastic system showing improvements with respect to standard DeePC. Future work includes testing the proposed method on a real-case scenario and comparing it to algorithms such as [3]- [5].\"\n },\n {\n \"title\": \"1\",\n \"content\": \"\"\n },\n {\n \"title\": \"T\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 1 :Fig. 2 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 3 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 4 :Fig. 5 :\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Research supported by the ERC under the project OCAL, the SEMP and the Gini Foundation. D. Alpago is with the Department of Information Engineering at University of Padova, Italy, dalpago@dei.unipd.it; F. Dörfler and J. Lygeros are with the Department of Information Technology and Electrical Engineering at ETH Zürich, Switzerland, dorfler@ethz.ch, jlygeros@ethz.ch.","content":"Research supported by the ERC under the project OCAL, the SEMP and the Gini Foundation. D. Alpago is with the Department of Information Engineering at University of Padova, Italy, dalpago@dei.unipd.it; F. Dörfler and J. Lygeros are with the Department of Information Technology and Electrical Engineering at ETH Zürich, Switzerland, dorfler@ethz.ch, jlygeros@ethz.ch."},{"id":"","caption":"U f g k . Problem(8) is solved in a receding-horizon fashion: of the N f -long optimal control sequence u (k) f = col(u k , . . . , u k+N f −1 ) solution of (8), we apply a sub-sequence u k , . . . , u k+N c , for some N c ≤ N f −","content":"1\n1U f g k . Problem(8) is solved in a receding-horizon fashion: of the N f -long optimal control sequence u (k) f = col(u k , . . . , u k+N f −1 ) solution of (8), we apply a sub-sequence u k , . . . , u k+N c , for some N c ≤ N f −"},{"id":"","caption":"T −1 ) be available T -long input-output trajectories and denote with x (i)0 , i = 1, . . . N,","content":"T −1 ) be available T -long input-output trajectories and denote with x (i)0 , i = 1, . . . N,"},{"id":"","caption":"−1 ], i = 1, . . . , N, constructed from the different noise realizations v similarlyX N andŪ N contain the averaged state trajectories corresponding to the different initial conditions x","content":"T\n−1 ], i = 1, . . . , N, constructed from the different noise realizations v similarlyX N andŪ N contain the averaged state trajectories corresponding to the different initial conditions x"},{"id":"","caption":"EKF equations. Here, C := [0 · · · 0 I p ] by construction, A k := MÃ k , B k := MB k , h k := Mh k and S k+1 = CP k+1|k C + R k , K k+1 = P k+1|k C S −1 k+1 , are the variance of the innovation process and the filter gain, respectively. EKF-DeePC Algorithm.","content":"Fig. 1 :Fig. 2 :\n12EKF equations. Here, C := [0 · · · 0 I p ] by construction, A k := MÃ k , B k := MB k , h k := Mh k and S k+1 = CP k+1|k C + R k , K k+1 = P k+1|k C S −1 k+1 , are the variance of the innovation process and the filter gain, respectively. EKF-DeePC Algorithm."},{"id":"","caption":"Closed-loop cost (left) and optimal λ g (right) when averaging N data matrices. The dashed-green line shows the MPC cost with the same horizon, but perfect model and full state measurement.","content":"Fig. 3 :\n3Closed-loop cost (left) and optimal λ g (right) when averaging N data matrices. The dashed-green line shows the MPC cost with the same horizon, but perfect model and full state measurement."},{"id":"","caption":"Closed-loop cost of standard DeePC and averaged DeePC with EKF for different σ 2 v while keeping σ 2 w = 0.1 (left), and different σ 2 w while keeping σ 2 v = 0.Closed-loop costs while varying the past horizon N p .","content":"Fig. 4 :Fig. 5 :\n45Closed-loop cost of standard DeePC and averaged DeePC with EKF for different σ 2 v while keeping σ 2 w = 0.1 (left), and different σ 2 w while keeping σ 2 v = 0.Closed-loop costs while varying the past horizon N p ."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Research supported by the ERC under the project OCAL, the SEMP and the Gini Foundation. D. Alpago is with the Department of Information Engineering at University of Padova, Italy, dalpago@dei.unipd.it; F. Dörfler and J. Lygeros are with the Department of Information Technology and Electrical Engineering at ETH Zürich, Switzerland, dorfler@ethz.ch, jlygeros@ethz.ch.\",\n \"content\": \"Research supported by the ERC under the project OCAL, the SEMP and the Gini Foundation. D. Alpago is with the Department of Information Engineering at University of Padova, Italy, dalpago@dei.unipd.it; F. Dörfler and J. Lygeros are with the Department of Information Technology and Electrical Engineering at ETH Zürich, Switzerland, dorfler@ethz.ch, jlygeros@ethz.ch.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"U f g k . Problem(8) is solved in a receding-horizon fashion: of the N f -long optimal control sequence u (k) f = col(u k , . . . , u k+N f −1 ) solution of (8), we apply a sub-sequence u k , . . . , u k+N c , for some N c ≤ N f −\",\n \"content\": \"1\\n1U f g k . Problem(8) is solved in a receding-horizon fashion: of the N f -long optimal control sequence u (k) f = col(u k , . . . , u k+N f −1 ) solution of (8), we apply a sub-sequence u k , . . . , u k+N c , for some N c ≤ N f −\"\n },\n {\n \"id\": \"\",\n \"caption\": \"T −1 ) be available T -long input-output trajectories and denote with x (i)0 , i = 1, . . . N,\",\n \"content\": \"T −1 ) be available T -long input-output trajectories and denote with x (i)0 , i = 1, . . . N,\"\n },\n {\n \"id\": \"\",\n \"caption\": \"−1 ], i = 1, . . . , N, constructed from the different noise realizations v similarlyX N andŪ N contain the averaged state trajectories corresponding to the different initial conditions x\",\n \"content\": \"T\\n−1 ], i = 1, . . . , N, constructed from the different noise realizations v similarlyX N andŪ N contain the averaged state trajectories corresponding to the different initial conditions x\"\n },\n {\n \"id\": \"\",\n \"caption\": \"EKF equations. Here, C := [0 · · · 0 I p ] by construction, A k := MÃ k , B k := MB k , h k := Mh k and S k+1 = CP k+1|k C + R k , K k+1 = P k+1|k C S −1 k+1 , are the variance of the innovation process and the filter gain, respectively. EKF-DeePC Algorithm.\",\n \"content\": \"Fig. 1 :Fig. 2 :\\n12EKF equations. Here, C := [0 · · · 0 I p ] by construction, A k := MÃ k , B k := MB k , h k := Mh k and S k+1 = CP k+1|k C + R k , K k+1 = P k+1|k C S −1 k+1 , are the variance of the innovation process and the filter gain, respectively. EKF-DeePC Algorithm.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Closed-loop cost (left) and optimal λ g (right) when averaging N data matrices. The dashed-green line shows the MPC cost with the same horizon, but perfect model and full state measurement.\",\n \"content\": \"Fig. 3 :\\n3Closed-loop cost (left) and optimal λ g (right) when averaging N data matrices. The dashed-green line shows the MPC cost with the same horizon, but perfect model and full state measurement.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Closed-loop cost of standard DeePC and averaged DeePC with EKF for different σ 2 v while keeping σ 2 w = 0.1 (left), and different σ 2 w while keeping σ 2 v = 0.Closed-loop costs while varying the past horizon N p .\",\n \"content\": \"Fig. 4 :Fig. 5 :\\n45Closed-loop cost of standard DeePC and averaged DeePC with EKF for different σ 2 v while keeping σ 2 w = 0.1 (left), and different σ 2 w while keeping σ 2 v = 0.Closed-loop costs while varying the past horizon N p .\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"System Identification: Theory for the User. Prentice Hall information and system sciences series. L Ljung, Prentice Hall PTRL. Ljung, System Identification: Theory for the User. Prentice Hall information and system sciences series, Prentice Hall PTR, 1999.","doi":""},{"text":"Adaptive Filtering Prediction and Control. G C Goodwin, K S Sin, Dover Publications, IncUSAG. C. Goodwin and K. S. Sin, Adaptive Filtering Prediction and Control. USA: Dover Publications, Inc., 2009.","doi":""},{"text":"Iterative feedback tuning: theory and applications. H Hjalmarsson, M Gevers, S Gunnarsson, O Lequin, IEEE Control Systems Magazine. 184H. Hjalmarsson, M. Gevers, S. Gunnarsson, and O. Lequin, \"Iterative feedback tuning: theory and applications,\" IEEE Control Systems Magazine, vol. 18, no. 4, pp. 26-41, 1998.","doi":""},{"text":"Convergence analysis of an iterative correlation-based controller tuning method. A Karimi, L Mikovi, D Bonvin, IFAC Proceedings Volumes. 3515th IFAC World CongressA. Karimi, L. Mikovi, and D. Bonvin, \"Convergence analysis of an iterative correlation-based controller tuning method,\" IFAC Proceed- ings Volumes, vol. 35, no. 1, pp. 413 -418, 2002. 15th IFAC World Congress.","doi":""},{"text":"Virtual reference feedback tuning: a direct method for the design of feedback controllers. M Campi, A Lecchini, S Savaresi, Automatica. 388M. Campi, A. Lecchini, and S. Savaresi, \"Virtual reference feedback tuning: a direct method for the design of feedback controllers,\" Automatica, vol. 38, no. 8, pp. 1337 -1346, 2002.","doi":""},{"text":"From model-based control to data-driven control: Survey, classification and perspective. Z.-S Hou, Z Wang, Data-based Control, Decision, Scheduling and Fault Diagnostics. 235Z.-S. Hou and Z. Wang, \"From model-based control to data-driven control: Survey, classification and perspective,\" Information Sciences, vol. 235, pp. 3 -35, 2013. Data-based Control, Decision, Scheduling and Fault Diagnostics.","doi":""},{"text":"Paradigms and puzzles in the theory of dynamical systems. J C Willems, IEEE Transactions on Automatic Control. 36J. C. Willems, \"Paradigms and puzzles in the theory of dynamical systems,\" IEEE Transactions on Automatic Control, vol. 36, pp. 259- 294, March 1991.","doi":""},{"text":"J C Willems, J W Polderman, Introduction to mathematical systems theory: a behavioral approach. Springer Science & Business Media26J. C. Willems and J. W. Polderman, Introduction to mathematical systems theory: a behavioral approach, vol. 26. Springer Science & Business Media, 2013.","doi":""},{"text":"A note on persistency of excitation. J C Willems, P Rapisarda, I Markovsky, B L D Moor, Systems & Control Letters. 544J. C. Willems, P. Rapisarda, I. Markovsky, and B. L. D. Moor, \"A note on persistency of excitation,\" Systems & Control Letters, vol. 54, no. 4, pp. 325 -329, 2005.","doi":""},{"text":"Data-driven simulation and control. I Markovsky, P Rapisarda, International Journal of Control. 8112I. Markovsky and P. Rapisarda, \"Data-driven simulation and control,\" International Journal of Control, vol. 81, no. 12, pp. 1946-1959, 2008.","doi":""},{"text":"On the linear quadratic data-driven control. I Markovsky, P Rapisarda, 2007 European Control Conference (ECC). IEEEI. Markovsky and P. Rapisarda, \"On the linear quadratic data-driven control,\" in 2007 European Control Conference (ECC), pp. 5313- 5318, IEEE, 2007.","doi":""},{"text":"Willems' fundamental lemma for state-space systems and its extension to multiple datasets. H J Van Waarde, C D Persis, M K Camlibel, P Tesi, H. J. van Waarde, C. D. Persis, M. K. Camlibel, and P. Tesi, \"Willems' fundamental lemma for state-space systems and its extension to multiple datasets,\" 2020.","doi":""},{"text":"One-shot verification of dissipativity properties from inputoutput data. A Romer, J Berberich, J Khler, F Allgwer, IEEE Control Systems Letters. 3A. Romer, J. Berberich, J. Khler, and F. Allgwer, \"One-shot verifica- tion of dissipativity properties from inputoutput data,\" IEEE Control Systems Letters, vol. 3, pp. 709-714, July 2019.","doi":""},{"text":"Robust datadriven state-feedback design. J Berberich, A Romer, C W Scherer, F Allgöwer, arXiv:1909.04314arXiv preprintJ. Berberich, A. Romer, C. W. Scherer, and F. Allgöwer, \"Robust data- driven state-feedback design,\" arXiv preprint arXiv:1909.04314, 2019.","doi":"arXiv:1909.04314"},{"text":"A trajectory-based framework for datadriven system analysis and control. J Berberich, F Allgöwer, arXiv:1903.10723arXiv preprintJ. Berberich and F. Allgöwer, \"A trajectory-based framework for data- driven system analysis and control,\" arXiv preprint arXiv:1903.10723, 2019.","doi":"arXiv:1903.10723"},{"text":"Formulas for data-driven control: Stabilization, optimality, and robustness. C De Persis, P Tesi, IEEE Transactions on Automatic Control. 65C. De Persis and P. Tesi, \"Formulas for data-driven control: Stabi- lization, optimality, and robustness,\" IEEE Transactions on Automatic Control, vol. 65, pp. 909-924, March 2020.","doi":""},{"text":"Data informativity: a new perspective on data-driven analysis and control. H J Van Waarde, J Eising, H L Trentelman, M K Camlibel, H. J. van Waarde, J. Eising, H. L. Trentelman, and M. K. Camlibel, \"Data informativity: a new perspective on data-driven analysis and control,\" 2019.","doi":""},{"text":"Spc: Subspace predictive control. W Favoreel, B D Moor, M Gevers, IFAC Proceedings Volumes. Beijing, Chia3214th IFAC World CongressW. Favoreel, B. D. Moor, and M. Gevers, \"Spc: Subspace predictive control,\" IFAC Proceedings Volumes, vol. 32, no. 2, pp. 4004 -4009, 1999. 14th IFAC World Congress 1999, Beijing, Chia, 5-9 July.","doi":""},{"text":"Data-enabled predictive control: In the shallows of the deepc. J Coulson, J Lygeros, F Dörfler, 2019 18th European Control Conference (ECC). IEEEJ. Coulson, J. Lygeros, and F. Dörfler, \"Data-enabled predictive control: In the shallows of the deepc,\" in 2019 18th European Control Conference (ECC), pp. 307-312, IEEE, 2019.","doi":""},{"text":"Regularized and distributionally robust data-enabled predictive control. J Coulson, J Lygeros, F Dörfler, arXiv:1903.06804arXiv preprintJ. Coulson, J. Lygeros, and F. Dörfler, \"Regularized and distri- butionally robust data-enabled predictive control,\" arXiv preprint arXiv:1903.06804, 2019.","doi":"arXiv:1903.06804"},{"text":"Data-enabled predictive control for grid-connected power converters. L Huang, J Coulson, J Lygeros, F Dorfler, arXiv:1903.07339arXiv preprintL. Huang, J. Coulson, J. Lygeros, and F. Dorfler, \"Data-enabled predictive control for grid-connected power converters,\" arXiv preprint arXiv:1903.07339, 2019.","doi":"arXiv:1903.07339"},{"text":"Data-driven model predictive control with stability and robustness guarantees. J Berberich, J Köhler, M A Müller, F Allgöwer, arXiv:1906.04679arXiv preprintJ. Berberich, J. Köhler, M. A. Müller, and F. Allgöwer, \"Data-driven model predictive control with stability and robustness guarantees,\" arXiv preprint arXiv:1906.04679, 2019.","doi":"arXiv:1906.04679"},{"text":"Data-Based Modelling and System Identification. H Hjalmarsson, Automatica. 413From experiment design to closed-loop controlH. Hjalmarsson, \"From experiment design to closed-loop control,\" Automatica, vol. 41, no. 3, pp. 393 -438, 2005. Data-Based Modelling and System Identification.","doi":""},{"text":"Industrial application of model based predictive control as a supervisory system. B Bulut, M R Katebi, M A Johnson, Proceedings of the 2000 American Control Conference. the 2000 American Control Conference6B. Bulut, M. R. Katebi, and M. A. Johnson, \"Industrial application of model based predictive control as a supervisory system,\" in Proceed- ings of the 2000 American Control Conference, vol. 6, pp. 3763-3767 vol.6, June 2000.","doi":""},{"text":"Subspace identification for linear systems: TheoryImplementationApplications. P Van Overschee, B. De Moor, Springer Science & Business MediaP. Van Overschee and B. De Moor, Subspace identification for linear systems: TheoryImplementationApplications. Springer Science & Business Media, 2012.","doi":""},{"text":". P Billingsley, Probability and Measure. Wiley Series in Probability and Statistics. WileyP. Billingsley, Probability and Measure. Wiley Series in Probability and Statistics, Wiley, 1995.","doi":""},{"text":"The explicit linear quadratic regulator for constrained systems. A Bemporad, M Morari, V Dua, E N Pistikopoulos, Automatica. 381A. Bemporad, M. Morari, V. Dua, and E. N. Pistikopoulos, \"The ex- plicit linear quadratic regulator for constrained systems,\" Automatica, vol. 38, no. 1, pp. 3 -20, 2002.","doi":""}],"string":"[\n {\n \"text\": \"System Identification: Theory for the User. Prentice Hall information and system sciences series. L Ljung, Prentice Hall PTRL. Ljung, System Identification: Theory for the User. Prentice Hall information and system sciences series, Prentice Hall PTR, 1999.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Adaptive Filtering Prediction and Control. G C Goodwin, K S Sin, Dover Publications, IncUSAG. C. Goodwin and K. S. Sin, Adaptive Filtering Prediction and Control. USA: Dover Publications, Inc., 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Iterative feedback tuning: theory and applications. H Hjalmarsson, M Gevers, S Gunnarsson, O Lequin, IEEE Control Systems Magazine. 184H. Hjalmarsson, M. Gevers, S. Gunnarsson, and O. Lequin, \\\"Iterative feedback tuning: theory and applications,\\\" IEEE Control Systems Magazine, vol. 18, no. 4, pp. 26-41, 1998.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Convergence analysis of an iterative correlation-based controller tuning method. A Karimi, L Mikovi, D Bonvin, IFAC Proceedings Volumes. 3515th IFAC World CongressA. Karimi, L. Mikovi, and D. Bonvin, \\\"Convergence analysis of an iterative correlation-based controller tuning method,\\\" IFAC Proceed- ings Volumes, vol. 35, no. 1, pp. 413 -418, 2002. 15th IFAC World Congress.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Virtual reference feedback tuning: a direct method for the design of feedback controllers. M Campi, A Lecchini, S Savaresi, Automatica. 388M. Campi, A. Lecchini, and S. Savaresi, \\\"Virtual reference feedback tuning: a direct method for the design of feedback controllers,\\\" Automatica, vol. 38, no. 8, pp. 1337 -1346, 2002.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"From model-based control to data-driven control: Survey, classification and perspective. Z.-S Hou, Z Wang, Data-based Control, Decision, Scheduling and Fault Diagnostics. 235Z.-S. Hou and Z. Wang, \\\"From model-based control to data-driven control: Survey, classification and perspective,\\\" Information Sciences, vol. 235, pp. 3 -35, 2013. Data-based Control, Decision, Scheduling and Fault Diagnostics.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Paradigms and puzzles in the theory of dynamical systems. J C Willems, IEEE Transactions on Automatic Control. 36J. C. Willems, \\\"Paradigms and puzzles in the theory of dynamical systems,\\\" IEEE Transactions on Automatic Control, vol. 36, pp. 259- 294, March 1991.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J C Willems, J W Polderman, Introduction to mathematical systems theory: a behavioral approach. Springer Science & Business Media26J. C. Willems and J. W. Polderman, Introduction to mathematical systems theory: a behavioral approach, vol. 26. Springer Science & Business Media, 2013.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A note on persistency of excitation. J C Willems, P Rapisarda, I Markovsky, B L D Moor, Systems & Control Letters. 544J. C. Willems, P. Rapisarda, I. Markovsky, and B. L. D. Moor, \\\"A note on persistency of excitation,\\\" Systems & Control Letters, vol. 54, no. 4, pp. 325 -329, 2005.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Data-driven simulation and control. I Markovsky, P Rapisarda, International Journal of Control. 8112I. Markovsky and P. Rapisarda, \\\"Data-driven simulation and control,\\\" International Journal of Control, vol. 81, no. 12, pp. 1946-1959, 2008.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the linear quadratic data-driven control. I Markovsky, P Rapisarda, 2007 European Control Conference (ECC). IEEEI. Markovsky and P. Rapisarda, \\\"On the linear quadratic data-driven control,\\\" in 2007 European Control Conference (ECC), pp. 5313- 5318, IEEE, 2007.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Willems' fundamental lemma for state-space systems and its extension to multiple datasets. H J Van Waarde, C D Persis, M K Camlibel, P Tesi, H. J. van Waarde, C. D. Persis, M. K. Camlibel, and P. Tesi, \\\"Willems' fundamental lemma for state-space systems and its extension to multiple datasets,\\\" 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"One-shot verification of dissipativity properties from inputoutput data. A Romer, J Berberich, J Khler, F Allgwer, IEEE Control Systems Letters. 3A. Romer, J. Berberich, J. Khler, and F. Allgwer, \\\"One-shot verifica- tion of dissipativity properties from inputoutput data,\\\" IEEE Control Systems Letters, vol. 3, pp. 709-714, July 2019.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Robust datadriven state-feedback design. J Berberich, A Romer, C W Scherer, F Allgöwer, arXiv:1909.04314arXiv preprintJ. Berberich, A. Romer, C. W. Scherer, and F. Allgöwer, \\\"Robust data- driven state-feedback design,\\\" arXiv preprint arXiv:1909.04314, 2019.\",\n \"doi\": \"arXiv:1909.04314\"\n },\n {\n \"text\": \"A trajectory-based framework for datadriven system analysis and control. J Berberich, F Allgöwer, arXiv:1903.10723arXiv preprintJ. Berberich and F. Allgöwer, \\\"A trajectory-based framework for data- driven system analysis and control,\\\" arXiv preprint arXiv:1903.10723, 2019.\",\n \"doi\": \"arXiv:1903.10723\"\n },\n {\n \"text\": \"Formulas for data-driven control: Stabilization, optimality, and robustness. C De Persis, P Tesi, IEEE Transactions on Automatic Control. 65C. De Persis and P. Tesi, \\\"Formulas for data-driven control: Stabi- lization, optimality, and robustness,\\\" IEEE Transactions on Automatic Control, vol. 65, pp. 909-924, March 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Data informativity: a new perspective on data-driven analysis and control. H J Van Waarde, J Eising, H L Trentelman, M K Camlibel, H. J. van Waarde, J. Eising, H. L. Trentelman, and M. K. Camlibel, \\\"Data informativity: a new perspective on data-driven analysis and control,\\\" 2019.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Spc: Subspace predictive control. W Favoreel, B D Moor, M Gevers, IFAC Proceedings Volumes. Beijing, Chia3214th IFAC World CongressW. Favoreel, B. D. Moor, and M. Gevers, \\\"Spc: Subspace predictive control,\\\" IFAC Proceedings Volumes, vol. 32, no. 2, pp. 4004 -4009, 1999. 14th IFAC World Congress 1999, Beijing, Chia, 5-9 July.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Data-enabled predictive control: In the shallows of the deepc. J Coulson, J Lygeros, F Dörfler, 2019 18th European Control Conference (ECC). IEEEJ. Coulson, J. Lygeros, and F. Dörfler, \\\"Data-enabled predictive control: In the shallows of the deepc,\\\" in 2019 18th European Control Conference (ECC), pp. 307-312, IEEE, 2019.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Regularized and distributionally robust data-enabled predictive control. J Coulson, J Lygeros, F Dörfler, arXiv:1903.06804arXiv preprintJ. Coulson, J. Lygeros, and F. Dörfler, \\\"Regularized and distri- butionally robust data-enabled predictive control,\\\" arXiv preprint arXiv:1903.06804, 2019.\",\n \"doi\": \"arXiv:1903.06804\"\n },\n {\n \"text\": \"Data-enabled predictive control for grid-connected power converters. L Huang, J Coulson, J Lygeros, F Dorfler, arXiv:1903.07339arXiv preprintL. Huang, J. Coulson, J. Lygeros, and F. Dorfler, \\\"Data-enabled predictive control for grid-connected power converters,\\\" arXiv preprint arXiv:1903.07339, 2019.\",\n \"doi\": \"arXiv:1903.07339\"\n },\n {\n \"text\": \"Data-driven model predictive control with stability and robustness guarantees. J Berberich, J Köhler, M A Müller, F Allgöwer, arXiv:1906.04679arXiv preprintJ. Berberich, J. Köhler, M. A. Müller, and F. Allgöwer, \\\"Data-driven model predictive control with stability and robustness guarantees,\\\" arXiv preprint arXiv:1906.04679, 2019.\",\n \"doi\": \"arXiv:1906.04679\"\n },\n {\n \"text\": \"Data-Based Modelling and System Identification. H Hjalmarsson, Automatica. 413From experiment design to closed-loop controlH. Hjalmarsson, \\\"From experiment design to closed-loop control,\\\" Automatica, vol. 41, no. 3, pp. 393 -438, 2005. Data-Based Modelling and System Identification.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Industrial application of model based predictive control as a supervisory system. B Bulut, M R Katebi, M A Johnson, Proceedings of the 2000 American Control Conference. the 2000 American Control Conference6B. Bulut, M. R. Katebi, and M. A. Johnson, \\\"Industrial application of model based predictive control as a supervisory system,\\\" in Proceed- ings of the 2000 American Control Conference, vol. 6, pp. 3763-3767 vol.6, June 2000.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Subspace identification for linear systems: TheoryImplementationApplications. P Van Overschee, B. De Moor, Springer Science & Business MediaP. Van Overschee and B. De Moor, Subspace identification for linear systems: TheoryImplementationApplications. Springer Science & Business Media, 2012.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Billingsley, Probability and Measure. Wiley Series in Probability and Statistics. WileyP. Billingsley, Probability and Measure. Wiley Series in Probability and Statistics, Wiley, 1995.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The explicit linear quadratic regulator for constrained systems. A Bemporad, M Morari, V Dua, E N Pistikopoulos, Automatica. 381A. Bemporad, M. Morari, V. Dua, and E. N. Pistikopoulos, \\\"The ex- plicit linear quadratic regulator for constrained systems,\\\" Automatica, vol. 38, no. 1, pp. 3 -20, 2002.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":471,"cells":{"text":{"kind":"string","value":"\nMultiple Object Tracking with Correlation Learning\n\n\nQiang Wang \nMachine Intelligence Technology Lab, Alibaba Group\n\n\nYun Zheng zhengyun.zy@alibaba-inc.com \nMachine Intelligence Technology Lab, Alibaba Group\n\n\nPan Pan \nMachine Intelligence Technology Lab, Alibaba Group\n\n\nYinghui Xu \nMachine Intelligence Technology Lab, Alibaba Group\n\n\nMultiple Object Tracking with Correlation Learning\n\nRecent works have shown that convolutional networks have substantially improved the performance of multiple object tracking by simultaneously learning detection and appearance features. However, due to the local perception of the convolutional network structure itself, the long-range dependencies in both the spatial and temporal cannot be obtained efficiently. To incorporate the spatial layout, we propose to exploit the local correlation module to model the topological relationship between targets and their surrounding environment, which can enhance the discriminative power of our model in crowded scenes. Specifically, we establish dense correspondences of each spatial location and its context, and explicitly constrain the correlation volumes through self-supervised learning. To exploit the temporal context, existing approaches generally utilize two or more adjacent frames to construct an enhanced feature representation, but the dynamic motion scene is inherently difficult to depict via CNNs. Instead, our paper proposes a learnable correlation operator to establish frameto-frame matches over convolutional feature maps in the different layers to align and propagate temporal context. With extensive experimental results on the MOT datasets, our approach demonstrates the effectiveness of correlation learning with the superior performance and obtains stateof-the-art MOTA of 76.5% and IDF1 of 73.6% on MOT17.\n\nIntroduction\n\nMulti-Object Tracking (MOT) is an essential component for computer vision with many applications, such as video surveillance [31] and modern autonomous driving [19,41]. It aims to continuously locate trajectories of multiple targets in video frames. Decades of research efforts have led to impressive performance on challenging benchmarks [24,30,8].\n\nMOT has traditionally adopted the tracking-by-detection paradigm [3,5,1,58], which capitalizes on the natural division of detection and data association tasks for the problem. These algorithms extract appearance features within each detection patches and record object location information for subsequent data association [51,5]. This tracking paradigm makes researchers mainly focus on optimizing detection [56,16], feature representation [27,15], or data association [3,5,17]. With the rapid progress of detection algorithms [13,14,36,35], the detection based tracking has achieved great performance improvement [56,16]. Although tremendous strides have been made in MOT, there still exists tough challenges in determining distractors and frequent occlusions, especially in complex interactive scenes [8]. Additionally, the above cascaded structure is inefficient and prevents the joint optimization between stages. One promising approach is to extend the end-to-end trainable detection framework [36,35,64] to jointly learn detection and appearance feature, which has largely advanced the state-of-the-art in MOT [44,50,29,58]. However as illustrated in Fig. 1, in the case of existing similar distractors, the appearance feature generates undistinctive and inaccurate matching confidences (Fig. 1a), severely affecting the performance of association. These methods are limited in local descriptors, and it is difficult to distinguish similar objects. While as shown in Fig. 1c, the context relation map can help to easily distinguish different targets.\n\nBased on those observations, we propose a correlation network to learn the topological information of the object and context. Specifically, we use a spatial correlation layer to record the relationship between targets and relative spatial positions. While constructing a full correlation (e.g., non-local [48]) for all locations is computationally prohibitive for real-time MOT, this work constructs a local correlation volume by limiting the search range at each feature pyramid. Besides, our correlation learning is not limited for targets of interest category [53,49]. Background contexts, such as vehicles, are also modeled to help target recognition and relational reasoning (Fig. 1c). We establish dense correspondences of each spatial location and its context, and explicitly constrain the correlation volumes through self-supervised learning.\n\nFurther, the detector in MOT usually uses independent frames as input and therefore does not make full use of temporal information. This detection method makes the algorithm suffer from missing detection in crowded scenes, and further increases the difficulty of subsequent data association. Recently, adjacent frames [64,34] or three frames [32] are adopted to enhance the temporal consistence. The performance of the algorithm in occlusion scenes has been improved to a certain extent, but these methods are still limited with fewer frames. CenterTrack [64] attempt to use an aggressive data augmentation to increase the ability of target alignment, but convolution networks itself are inherently limited in local receptive fields. To solve the above problem, we extend the spatial correlation module to the temporal dimension and incorporate the historical information to reduce ambiguities in object detections.\n\nTo summarize, we make the following contributions:\n\n• We propose CorrTracker, a unified correlation tracker to intensively model associations between objects and transmit information through associations. • We propose a local structure-aware network and enhance the discriminability of similar objects with selfsupervised learning. • We extend the local correlation network to model temporal information efficiently. • CorrTracker shows significant improvements over existing state-of-the-art results in four MOT benchmarks.\n\nIn particular, we achieve 76.5% MOTA and IDF1 of 73.6% on MOT17.\n\n\nRelated Work\n\nReal-time Tracking. As MOT has strong practical merit, the tracking speed attracts much attention. The researchers start from the simplest IOUTracker [4], which only uses the intersection-over-union of bounding boxes for tracking, to add the motion model of Kalman Filter [3] to predict the position of the rectangular boxes for matching. Although they have achieved amazing speed, stable tracking cannot be achieved under challenges such as target interleaving. Researchers [27,51] introduce Person Re-Identification (ReID) features as an appearance model to increase the discriminative power of the tracker. However, the individual calculation for patches makes the object classification and ReID feature extraction as a computational bottleneck. MOTDT [27] achieves real-time tracking by using RoI-pooling [14] on a shared feature map. In order to further decrease the computational cost of ReID feature extraction, JDE [50] adds a ReID branch in a single-stage detector YOLOv3 [35] to achieve efficient ReID feature calculation. FairMOT [58] explores the importance of detection and recognition tasks and uses anchor-free method [65] to reduce the ambiguity of anchors. We are mainly based on FairMOT, which achieves the state-of-the-art performance with a more balanced ReID and detection.\n\nOther researchers [1,64,34,32] explore new tracking paradigms to remove ReID recognition. Tracktor [1] uses the the bounding boxes in the previous frame to directly regress the current locations. CenterTrack [64], Chained-Tracker [34], and TubeTK [32] use multiple frames to simultaneously predict the bounding boxes for adjacent frames to achieve short association, thereby merging to long-term tracks. However, these methods usually have many identity switches because they cannot model longterm dependencies. Tracking with Graph Model. MOT has traditionally been approached as a hand-crafted graph optimization problem [61,17], where the cropped targets are treated as nodes. Recently, graph neural network based methods [53,49,5] have been shown as a promising alternative to traditional optimization methods. State-of-the-art approach [53] utilizes graph convolutional network to propagate features in the spatial-temporal space. MPN [5] introduces the message passing network to dissect the information and associate detections through the edge classification. Different from these methods, feature propagation is carried out at the frame feature level, which can absorb the information of both the foreground and background and reduce the loss of contextual information. Tracking with Optical Flow. FlowTrack [60] introduces optical flow to predict the target location. But explicitly using optical flow is not only time-consuming, but also only encodes the pixel-level motion. CenterTrack [64] borrows the method of optical flow to directly predict the movement of the target center between two frames, which is called instance flow. However, directly predicting the offset on the concatenated feature map needs to provide training samples with all displacement, which requires excessive data augmentation. Our correlation method predicts a dense set of matching confidence for each target, which is intrinsically invariant to translation of the paired frames. Our correlation operation is similar to the correlation volume in optical flow [10,40,42] and correlation filter [7]. We both predict dense local correlation, and regard it as a part of the feature description. However, optical flow does not calculate the internal correlation of the image, nor does it have propagated the feature from multiple frames. D&T [12] also utilizes correlation layer to predict candidate motion between pair of consecutive frames. Compared with it, our anchor-free framework is more compact and efficient. Attention Mechanism. Our modeling of local correlation is similar to the self-attention mechanism and Transformer. Transformer has been a huge success in the NLP field [43] and has also been adapted to the computer vision [48,18,28] to capture long-range dependencies. In order to reduce the quadratic complexity of the non-local operation, the researchers propose to shrink the attention span with local region [33], or only along individual axes [47]. Different from these methods, we mainly encode the context identity through local correlation weighting, and use this cues to increase the model robustness. Figure 2 shows the overall pipeline of the proposed Cor-rTracker. Our method can be distilled down to three stages: (1) general feature extraction, (2) simultaneous learning correlation from spatial-temporal dependencies and predicting the detection, and (3) performing data association to assign detections into their most likely trajectories, where stage (1) and stage (2) are differentiable and composed into an end-to-end trainable architecture. We adopt a compact association technique that is similar to the one used by DeepSORT [51] to control the initialization and the termination of tracks. The main contribution is the highly efficient modeling for the correlation between dense location and their context on feature maps, which helps suppressing distractors in complex scenes.\n\n\nMethodology\n\n\nMotivation\n\nFor each input video frame I t ∈ R H×W ×3 , an object detector is applied to find all candidate\ndetections D t = {d i t } N i=1 , d i t = (x i t , y i t , w i t , h i t )\nappearing in this frame and we have existing trajectories\nT t−1 = {T j t−1 } M j=1 , T j t−1 = {d j 1 , ..., d j t−2 , d j t−1 }.\nThen the affinity matrix A ∈ R N ×M is estimated by pair-wise comparisons of cropped patches and existing trajectories. The metric jointly considers both the appearance features f (·) ∈ R d and geometric representations.\nA ij = dist(f (d i t ),f (T j t−1 )) + αIoU(d i t ,d j t ),(1)\nThe discriminative featuref (T j t−1 ) of a trajectory is usually updated with a constant-weighting strategy to follow the appearance changes. Each confidence value for appearance feature is obtained in a distance metric, e.g., the inner product space. However, the sole reliance on person-to-person feature comparisons are often insufficient to disambiguate multiple similar regions in an image. As illustrated in Fig. 1, in the case of similar distractors, the feature extractor usually generates inaccurate and uninformative matching confidences ( Fig. 1a), severely affecting the performance of data association. This is the key limitation of appearance feature matching, since co-occurring similar objects are all pervasive in MOT.\n\nPatch based feature extraction is applied as a prevalent scheme in MOT owing to its intuition. However the correlation information between the cropped image patches is lost directly, and the adjacency spatial relationship is only retained in coordinates d i t . Although the subsequent data association will be globally optimized, directly using ReID features without considering the context tends to introduce more identity switches, lagging the tracking robustness. To deal with this problem, we model the local structure of objects to distinguish it from distractors.\n\nInspired by correlation volume from optical flow [10], we observe that a confidence value in the correlation vol-ume models the geometric structure of each target. We design a novel dense correlation module, aiming to explore the context information for MOT. The relative position is encoded in the correlation volumes, which can be used as an auxiliary discriminant information.\n\n\nSpatial Local Correlation Layers\n\nIn this work, we use Spatial Local Correlation Layers to model the relational structure for associating a target with its neighbour. In our local correlation layer, the feature similarity is only evaluated in the neighbourhood of the target image coordinate. Formally, we let l denote the level in the feature pyramid and the correlation volume C l between the query feature F l q ∈ R H l ×W l ×d l and reference feature F l r ∈ R H l ×W l ×d l is defined as,\nC l (F q , F r , x, d) = F l q (x) T F l r (x + d), d ∞ ≤ R,(2)\nwhere x ∈ Z 2 is a coordinate in the query feature map and d ∈ Z 2 is the displacement from this location. The displacement is constrained to d ∞ ≤ R , i.e. the maximum motion in any direction is R. While most naturally thought of as a 4-dimensional tensor, the two displacement dimensions are usually vectorized into one to simplify further processing in the CNN. The resulting 3-d correlation volume C l thus is of size H l × W l × (2R + 1) 2 . We also introduce the dilation tricks [55], which can increase the receptive field without additional cost. We use element-wise addition to incorporate the correlation feature into a unified appearance representation. This context correlation features are encoded by a feed-forward Multilayer Perceptron (MLP) to match the number of channels d l in appearance features F l t .\nF l C = F l t + MLP l C l F l t , F l t )).(3)\nThe non-local [48] module is to explicitly model all pairwise interactions between elements in a feature maps F l t ∈ R H l ×W l ×d l . The resulting four-dimensional correlation volume N L(F l t ) ∈ R H l ×W l ×H l ×W l captures dense matching confidences between every pair of image locations. They build a full connection volume at a single scale, which is both computationally expensive and memory intensive. By contrast, our work shows that constructing a local correlation volume leads to both effective and efficient models. In comparison with the global correlation method, our local correlation model adds less overhead to the latency (see Table 1).\n\n\nCorrelation at Multiple Pyramid Levels\n\nIn order to achieve long-range correlation, we propose to learn correlation at feature pyramids, as shown in Figure 3. On the one hand, we hope that our correlation module can obtain long-distance dependencies as much as possible, but as the local region size R increases, both calculation and storage increase significantly, which hinders the application. On the other hand, MOT naturally needs to deal with multi-scale targets. The two-stage detection [36] uses RoI pooling [14] to eliminate the difference in target scales, but this type of method usually suffers from high processing latency. In order to solve the above problems, we utilize the general pyramid structure in the convolutional network and learn correlation on the feature pyramids. Our multi-scale pyramid correlation can also be regarded as a comparison of multi-granularity features, covering the spatial context in the range of [0, R × D × 2 l ], where D refers to the dilation rate. And, we pass this correlation from the top layer to the bottom layer,\nF l−1 C = Conv(Upsample(F l C )) + F l−1 C ,(4)\nIn this way, we can obtain an approximate correlation between the target and the entire global context, while keep the compactness and efficiency. Our pyramid correlation leverages the natural spatial-temporal coherence in videos.\n\nMulti-object tracking can be decomposed into multiple independent single-object tracking. Our method can be equivalent to a dense siamese network tracking [2] on the feature pyramid. On the other hand, from the perspective of set matching, global characteristics need to be considered. Our multi-scale correlation takes into account both aspects of information transmission.\n\n\nTemporal Correlation Learning\n\nThe correlation between different frames are usually ignored by the MOT field, and trackers usually overcome occlusion through data association. Single frame detector is difficult to ensure a good temporal consistency [59]. This makes the algorithm's performance drop significantly in occlusion, motion blur and small target scenes, which becomes the bottleneck for MOT. We extend the spatial local correlation from Section 3.2 to the temporal dimension, and establish correlation for the targets in different frames. The correlation between two frames can be viewed as the establishment of motion information learning. We also use this correlation to enhance the feature representation, which can increase the detection accuracy.\n\nSpecifically, we establish multi-scale correlation between different frames, and use reference images as memory to enhance image features. This method helps tracker overcome target occlusion and motion blur, and increases the consistency of detection and identity features.\nF q (x) = ∀ d ∞ <R C l (F q , F r , x, d) (2R + 1) 2 F r (x + d) (5) C l (F q , F r , x, d) = F l q (x) T F l r (x + d), d ∞ ≤ R (6)\nSimilar to the multi-head attention [43], we adopt the embedded features and dot-product similarity. In our case, we set the normalization factor as (2R + 1) 2 and locally aggregate features. This shrinked region design also comes from the motion prior of the MOT scene. For the minimal memory consumption and fastest run-time, we can only save the previous features F t−1 in the memory. For the maximum accuracy, our long-term model saves the latest 5 frame image features by default.\n\n\nSelf-supervised Feature Learning\n\nIn Section 3.2 and Section 3.4, we present how we model the correlation in spatial and temporal dimension. We can simply use the proposed correlation module as a plugin module without explicitly adding constraints, similar to the non-local module, which has shown significant improvement. Here we investigate a multi-task learning approach that imposes a semantic supervision from visual object tracking [2] and self-supervised training from correspondence flow [45] on correlation volumes.\n\nOur correlation module is interpretable, measuring the similarity between different objects. Actually, our method intensively performs M ×N siamese tracking operations [2] to increases the discrimination. In this view, we can explicitly impose a tracking supervision. Specifically, we set up the ground-truth label as\nC l (F q , F r , x, d) = 1 if y q (x) = y r (x + d) 0 if y q (x)! = y r (x + d), −1 if y q (x) < 0(7)\nwhere y is the identity label of the corresponding position in feature maps. We ignore the position without objects y q (x) < 0 and use a class-balanced cross-entropy loss [2]. Inspired by the recent advances of self-supervised tracking [45], we use colorization as a proxy task for training our local correlation.\nI q (x) = ∀ d ∞ <R C l (F q , F r , x, d) (2R + 1) 2 I r (x + d),(8)\nwe use the cross-entropy categorical loss after quantizing the color-space into discrete categories [45].\n\n\nTracking Framework\n\nWe modify the FairMOT [58] backbone by adding correlation module before the iterative deep aggregation module [57]. Our model retains the detection and ReID branches, and adding correlation loss in Sec. 3.5 for multitask learning. For the tracking inference, our tracker first calculates the similarity between the detections of the current frame and the previous trajectories according to Eq. (2), and use the Hungarian algorithm [23] for finding the optimal matching. The unmatched detections are used to initialize new trajectories. In order to reduce false positives, we mark these new trajectories as \"inactive\" until the next frame is matched again and confirmed as \"active\". The unmatched trajectories are set to the \"lost\" state. When the continuous lost time t loss of a trajectory exceeds the threshold τ loss , we put it in the remove set. If there is a success matching before removing, we restore the trajectory to the active state. We use Kalman Filter [20] to model the pedestrian motion and keep the same settings in FairMOT [58].\n\n\nExperiments\n\nTo demonstrate the advantages of the proposed correlation tracker, we first compare the correlation module with other relational reasoning methods [48,46] and evaluate different settings to justify our design choices in Section 4.2. Then we show that our correlation tracker outperforms the state-of-the-art methods on four MOT benchmarks [24,30,8] in Section 4.3. Finally, we visualize the tracking trajectories in Section 4.4 and compare with other motion prediction based trackers [64,34,32].\n\n\nImplementation Details\n\nNetwork Setup.\n\nThe implementation and hyperparameters mostly follow [58], we adopt CenterNet [65] detector with a variant of Deep Layer Aggregation (DLA-34) [57] as backbone and utilize the iterative deep aggregation module (IDA) to recover a high-resolution feature map with stride 4. We also add a 3 × 3 deformable convolution layer [66] before every upsampling stage. The backbone network is initialized with the parameters pre-trained on COCO [26] and then pre-trained on CrowdHuman [39] with self-supervised learning as FairMOT [58]. The proposed correlation module is augmented before IDA module to fuse multi-scale correlation. For the correlation module, we set local region size R = 5 and dilation rate D = 2.\n\nTraining and Validation Datasets. For a fair comparison, we also use the default training datasets as Fair-MOT [58]. There are six training datasets including the ETH [11], CityPerson [62], CalTech [9], CUHK-SYSU [52], PRW [63] and MOT17 [30]. ETH and CityPerson only provide box annotations, so we ignore the ReID losses from these datasets. CalTech, CUHK-SYSU, PRW We train on an input resolution of 1088 × 608, which yields an output resolution of 272 × 152. We use random flip, random scaling (between 0.5 to 1.2), cropping, and color jittering as data augmentation, and use Adam [22] to optimize the overall objective. The learning rate is initialized as 1e −4 and then decayed to 1e −5 in the last 10 epochs. We train with a batch-size of 12 (on 2 GPUs) for 30 epochs. In the training phase, we sample 5 temporally ordered frames within a random interval of less than 3.\n\nTest Datasets and Evaluation Metrics. We evaluate the performance of our correlation tracker on the 2DMOT2015 [24], MOT16 [30], MOT17 [30], and MOT20 [8]. In particular, 2DMOT2015 [24] contains 11 test videos. MOT16 [30] and MOT17 [30] contain same 7 test videos with part different annotations. The MOT20 [8] consists of 4 test videos on extremely crowded scenes, which makes them really challenging.\n\nWe adopt the standard metrics of MOT Benchmarks for evaluation, including Multiple Object Tracking Accuracy (MOTA) [21], ID F1 Score [37], Mostly tracked targets (MT), Mostly lost targets (ML), the number of False Positives (FP), the number of False Negatives (FN), and the number of Identity Switch (ID Sw.) [25]. The run time is also provided and evaluated on a NVIDIA Tesla V100 GPU.\n\n\nAblation Studies\n\nTo elaborate on the effectiveness of the proposed approach, we conduct extensive ablation studies. First, we give detailed correlation analysis with different settings to justify our design choices, as presented in Table. 1. Next, the tracking accuracy and runtime for different region sizes of the correlation module is explored. Different building blocks are compared to illustrate the effectiveness and efficiency of the full correlation tracker. Spatial correlation. In order to evaluate the effectiveness Table 2. Ablation studies on MOT17 validation set. \"LT\" and \"Self\" denote using the proposed long-term memory and selfsupervised loss, respectively. of our spatial local correlation module (SLC), we compared our baseline model [58] and two relation methods, the nonlocal module [48] and CorrNet [46]. Directly use of nonlocal brings performance degradation on MOT since nonlocal module does not record the relative positional relationship between the targets and usually brings performance drops on small object [54]. In addition, non-local method has a huge overhead in memory and computation. In video recognition, a learnable correlation filter network [46] is proposed, and the grouped convolution is used to reduce the amount of calculation. Our method has achieved similar MOTA compared to CorrNet, but has a large improvement on IDF1, which comes from our multi-scale correlation design. Compared with the baseline, our IDF1 has increased by 2.9%, and identity switches have been reduced by 15%, demonstrating the discrimination of our spatial correlation. The Re-ID embeddings cannot easily distinguish similar distractor, our correlation feature models geometric information and is better suited for MOT. Temporal correlation. In the field of video object detection [38], temporal and global information are commonly used to improve performance [6]. The research of temporal detection for MOT is still preliminary. Recently, CenterTrack [64] concatenates the previous frame and the current frame, and Chained-Tracker [34] concatenates the high-level image features for two frames to fuse temporal information. We compare these two methods, dubbed concat-raw and concat-feat, with our temporal local correlation module (TLC). These two methods brought 0.2% and 1.3% MOTA improvements over the single frame baseline. Compared with these two methods, our temporal local correlation module achieves consistent improvements in both MOTA and IDF1. Our temporal correlation module helps for the temporal feature alignment around frames. At the same time, our method only adds a small overhead to feature-level concatenation, which proves the efficiency of our algorithm. Compared with the baseline FairMOT, employing both spatial and temporal local correlation (STLC) yields a IDF1 of 76.1% , which brings 3.2% improvement.\n\nLong-term dependences. We also analyze the performance improvement with long-term correlation. Compared with the method using two frames as the source cues, our long-term method achieves a large improvement of 0.6% MOTA, due to the increased capacity for object detections.\n\nThe improvement of MOTA also means an increase in the upper bound of our tracker. Self-supervised learning. For correlation learning, explicit supervision is usually not imposed [48,46]. We have proposed two supervision methods in section 3.5. It can be seen that the siamese tracking supervision imposed to training has achieved a relatively good improvement in IDF1.\n\nThere is no change in the run time of our algorithm, because the change in training loss does not change the inference processing. Self-supervised losses have also been appropriately improved on both MOTA and IDF1 due to more positive samples employed in correlation volume.\n\nChoice of local region. Figure 4 shows the MOTA and run time of our correlation module for different region size R ∈ {1, 2, ..., 8}. As expected, a larger local size R can cover a larger neighborhood while matching pixels, thus yields a higher accuracy. But the improvements become marginal beyond R = 5, possibly due to the low resolution of the feature maps. Note that non-local module usually doubles the run time of the backbone and the cost of explicitly computing optical flow [10] can be very high as well. This shows that our correlation module is more efficient by learning motion information from features directly. Region size R = 5 yields a good trade-off between speed and accuracy. The computation overhead is relatively small, compared to the complexity of the whole detection networks.\n\n\nExperiments on MOT Challenges\n\nTo extensively evaluate the proposed method, we compare it with 8 state-of-the-art trackers, which cover most of current representative methods. There are 2 joint detection and embedding methods (JDE [50] and FairMOT [58]), 2 multi-frame prediction methods (Tube TK [32] and CTracker [34], 2 graph network based methods (MPN [5] and JDMOTGNN [49]), 2 offset prediction based methods (CenterTracker [64] and Tracktor++v2 [1]). The results are summarized in Table 3.\n\n2DMOT2015 [24]. The evaluation on 2DMOT2015 is performed by the official toolkit. As shown in Table 3, our correlation tracker outperforms the top private method of 2DMOT2015, (i.e., FairMOT [58]), by 1.7% in MOTA and 1.0% in IDF1. It is worth noting that the ID Switches are decreased by 13%, which shows the robustness of our cor- Table 3. Comparisons of our method with state-of-the-arts on MOT benchmarks [24,30,8]. We set new state-of-the-art results by a significant margin in terms of MOTA and IDF1. Our correlation tracker is more accurate while running with high speed. relation module. Moreover, our tracker is superior to the recent end-to-end graph trackers JDMOTGNN [49]. Our feature propagation approach can absorb both foreground and background information, which improves our tracker by 1.6% in terms of MOTA. MOT16 [30] and MOT17 [30]. Table 3 reports the evaluation results with the comparisons to recent prevailing trackers on MOT16. The recent proposed FairMOTv2 [58] achieves the second performance in MOTA and IDF1, while our method ranks first with 76.6% MOTA, outperforming other private approaches by a significant margin. Our CorrTracker achieves the best performance on MOT17, surpassing FairMOTv2 by 2.8% MOTA and 1.3% IDF1. Moreover, the FN of our CorrTracker surpasses FairMOTv2 by 15%, which means nearly 20,000 new bounding boxes are added to the association process. In this case, our algorithm still maintains comparable or even superior ID Switches, which actually proves that our method significantly improves the tracking association. As reported in Table 3, our CorrTracker, CenterTrack [64] and CTrack [34] all use multi-frame cues to predict detections, our FN is largely decreased by 30%.\n\nMOT20 [8]. To further evaluate the proposed models, we report the results on MOT20, which is more challenging than MOT17. The final results is presented in the bottom block of Table 3. Our CorrTracker achieves MOTA score of 65.2%, substantially outperforming FairMOTv2 [58] with MOTA of 61.8%. Although our approach is an order of magnitude faster than JDMOTGNN [49] in speed, our accuracy is slightly worse due to the anchor-free design. MOT20-06 MOT20-08 Figure 6. Qualitative results of our correlation tracker on MOT17 [30] and MOT20 [8]. The color of each bounding box indicates the target identity. The dotted line under each bounding box denotes the recent tracklet of each target. The proposed tracker predicts trajectories with substantially robust and temporally consistent.\n\n\nVisualization\n\nWe visualize the tracking trajectories for prior methods, i.e., the center offset [64] and multi-frame bounding boxes [34,32], in Figure 5. We observe that our correlation map focuses on the entire context, while the regular appearance feature concentrates on the local region of the target. Our correlation module improves the reliability of recognition since it provides a global view of the target. Methods based on offset prediction, e.g., Cen-terTrack [64] and CTracker [34], can easily generate id switches when encountered with complex object interactions. Figure 6 shows qualitative results of our Correlation Tracker on MOT17 and MOT20, the advantage over existing method is most pronounced on the robust to occlusion and tiny objects.\n\n\nConclusion\n\nIn this work, we propose a novel correlation tracking framework based upon the observation that the relational structure helps to distinguish similar objects. Our correlation module densely matches all targets with their local context and learn a discriminative embeddings from the correlation volumes. Furthermore, we show how to extend the correlation module from spatial layout to the adjacent frames for strengthening the temporal modeling ability. We explore that self-supervised learning to impose a discriminative constraint on the correlation volume, which explicitly predicts a instance flow. Extensive experiments on four MOT challenges demonstrate that our CorrTracker achieves state-of-the-art performance and is efficient in inference.\n\nFigure 1 .\n1Visualization of the matching confidences (a)-(b) computed between the indicated target (white cross) in the reference image and all locations of the query image. The appearance feature based tracker[58] (a) generates undistinctive and inaccurate confidences due to the existing of similar distractors. In contrast, our Correlation Tracker (b) predicts a distinct high-confidence value at the correct location, with correlation learning (c).\n\nFigure 2 .\n2Overview. We enhance the appearance features with correlation layer, which densely encoding pairwise relation of object and their spatial-temporal neighbourhoods. The local correlation volumes are optimized in a self-supervised manner.\n\nFigure 3 .\n3Correlation at Multiple Pyramid Levels. For a feature tensor in F l , we take the inner product with local region (R = 1, D = 2) to generate a 3-d correlation volume W l × H l × (2R + 1) 2 , D is the dilation rate.\n\nFigure 4 .\n4Effect of filter size R on speed and MOTA accuracy on MOT17-val.\n\nFigure 5 .\n5Qualitative comparisons against several prior methods[58,64,34,32] in occlusion situations. Frames are sampled from MOT17-03. Our CorrTracker can identify objects via mining the context patterns around targets.\n\nTable 1 .\n1Evaluation of correlation architecture on the MOT17[30] validation set.Method \nTwo \nframes \nMOTA ↑ \nIDF1↑ \nID Sw. ↓ Speed↑ \n\nbaseline [58] \n\n69.1% \n72.9% \n299 \n25.6 \nnon-local [48] \n\n67.7% \n70.4% \n311 \n16.60 \nCorrNet [46] \n\n70.0% \n73.3% \n303 \n22.93 \nSLC (ours) \n\n70.3% \n75.8% \n258 \n20.19 \nconcat-raw [64] \n\n69.3% \n74.1% \n336 \n23.99 \nconcat-feat [34] \n\n70.4% \n74.0% \n308 \n19.77 \nTLC (ours) \n\n70.9% \n74.7% \n326 \n19.26 \nSTLC (ours) \n\n71.5% \n76.1% \n307 \n16.56 \n\nand MOT17 provide both box and identity annotations \nwhich allows us to train both branches. We de-duplicate \nthe overlaps between ETH datasets and MOT16 for fair \ncomparison. For all validation experiments, we use the \nsix datasets mentioned above and the first half frames of \nMOT17 as training, and the second half of MOT17 as the \nvalidation set. \n\n\n\nMethod MOTA ↑ IDF1 ↑ MT ↑ ML ↓ FP ↓ FN ↓ ID Sw. ↓ Hz ↑2D MOT 2015 [24] \n\nCorrTracker (ours) \n62.3 \n65.7 49.0 12.9 6909 \n15728 \n513 \n17.9 \nJDMOTGNN [49] \n60.7 \n64.6 47.0 10.5 7334 \n16358 \n477 \n2.4 \nFairMOTv2 [58] \n60.6 \n64.7 47.6 11.0 7854 \n15785 \n591 \n30.5 \nTube TK POI [32] \n58.4 \n53.1 39.3 18.0 5756 \n18961 \n854 \n5.8 \nMPN [5] \n51.5 \n58.6 31.2 25.9 7620 \n21780 \n375 \n6.5 \nTracktor++v2 [1] \n46.6 \n47.6 18.2 27.9 4624 \n26896 \n1290 \n1.8 \n\nMOT16 [30] \n\nCorrTracker (ours) \n76.6 \n74.3 47.8 13.3 10860 30756 \n979 \n14.8 \nFairMOTv2 [58] \n74.9 \n72.8 44.7 15.9 10163 34484 \n1074 \n25.4 \nCTracker [34] \n67.6 \n57.2 32.9 19.0 8934 \n48305 \n1897 \n6.8 \nTube TK POI [32] \n66.9 \n62.2 39.0 16.1 11544 47502 \n1236 \n1.0 \nPOI [56] \n66.1 \n65.1 34.0 20.8 5061 \n55914 \n805 \n9.9 \nTracktor++v2 [1] \n56.2 \n54.9 20.7 35.8 2394 \n76844 \n617 \n1.8 \n\nMOT17 [30] \n\nCorrTracker (ours) \n76.5 \n73.6 47.6 12.7 29808 99510 \n3369 \n14.8 \nFairMOTv2 [58] \n73.7 \n72.3 43.2 17.3 27507 117477 \n3303 \n25.9 \nCenterTrack [64] \n67.8 \n64.7 34.6 24.6 18498 160,332 3039 \n3.8 \nCTracker [34] \n66.6 \n57.4 32.2 24.2 22284 160491 \n5529 \n6.8 \nTracktor++v2 [1] \n56.3 \n55.1 21.1 35.3 8866 235449 \n1987 \n1.8 \n\nMOT20 [8] \n\nCorrTracker (ours) \n65.2 \n69.1 66.4 8.9 79429 95855 \n5183 \n8.5 \nJDMOTGNN [49] \n67.1 \n67.5 53.1 13.2 31913 135409 \n3131 \n0.9 \nFairMOTv2 [58] \n61.8 \n67.3 68.8 7.6 103440 88901 \n5243 \n13.2 \n\n\n\nTracking without bells and whistles. Philipp Bergmann, Tim Meinhardt, Laura Leal-Taixe, Proceedings of the IEEE international conference on computer vision. the IEEE international conference on computer visionPhilipp Bergmann, Tim Meinhardt, and Laura Leal-Taixe. Tracking without bells and whistles. In Proceedings of the IEEE international conference on computer vision, pages 941-951, 2019.\n\nFully-convolutional siamese networks for object tracking. Luca Bertinetto, Jack Valmadre, Joao F Henriques, Andrea Vedaldi, Philip Hs Torr, European conference on computer vision. SpringerLuca Bertinetto, Jack Valmadre, Joao F Henriques, Andrea Vedaldi, and Philip HS Torr. Fully-convolutional siamese networks for object tracking. In European conference on computer vision, pages 850-865. Springer, 2016.\n\nSimple online and realtime tracking. Alex Bewley, Zongyuan Ge, Lionel Ott, Fabio Ramos, Ben Upcroft, 2016 IEEE International Conference on Image Processing (ICIP). IEEEAlex Bewley, Zongyuan Ge, Lionel Ott, Fabio Ramos, and Ben Upcroft. Simple online and realtime tracking. In 2016 IEEE International Conference on Image Processing (ICIP), pages 3464-3468. IEEE, 2016.\n\nHighspeed tracking-by-detection without using image information. Erik Bochinski, Thomas Volker Eiselein, Sikora, 14th IEEE International Conference on Advanced Video and Signal Based Surveillance (AVSS). IEEEErik Bochinski, Volker Eiselein, and Thomas Sikora. High- speed tracking-by-detection without using image informa- tion. In 2017 14th IEEE International Conference on Ad- vanced Video and Signal Based Surveillance (AVSS), pages 1-6. IEEE.\n\nLearning a neural solver for multiple object tracking. Guillem Brasó, Laura Leal-Taixé, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionGuillem Brasó and Laura Leal-Taixé. Learning a neu- ral solver for multiple object tracking. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 6247-6257, 2020.\n\nMemory enhanced global-local aggregation for video object detection. Yihong Chen, Yue Cao, Han Hu, Liwei Wang, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionYihong Chen, Yue Cao, Han Hu, and Liwei Wang. Memory enhanced global-local aggregation for video object detec- tion. In Proceedings of the IEEE/CVF Conference on Com- puter Vision and Pattern Recognition, pages 10337-10346, 2020.\n\nFamnet: Joint learning of feature, affinity and multi-dimensional assignment for online multiple object tracking. Peng Chu, Haibin Ling, Proceedings of the IEEE International Conference on Computer Vision. the IEEE International Conference on Computer VisionPeng Chu and Haibin Ling. Famnet: Joint learning of fea- ture, affinity and multi-dimensional assignment for online multiple object tracking. In Proceedings of the IEEE Inter- national Conference on Computer Vision, pages 6172-6181, 2019.\n\nP Dendorfer, H Rezatofighi, A Milan, J Shi, D Cremers, I Reid, S Roth, K Schindler, L Leal-Taixé, arXiv:2003.09003arXiv: 2003.09003Mot20: A benchmark for multi object tracking in crowded scenes. P. Dendorfer, H. Rezatofighi, A. Milan, J. Shi, D. Cremers, I. Reid, S. Roth, K. Schindler, and L. Leal-Taixé. Mot20: A benchmark for multi object tracking in crowded scenes. arXiv:2003.09003[cs], Mar. 2020. arXiv: 2003.09003.\n\nBernt Schiele, and Pietro Perona. Pedestrian detection: A benchmark. Piotr Dollár, Christian Wojek, IEEE Conference on Computer Vision and Pattern Recognition. IEEEPiotr Dollár, Christian Wojek, Bernt Schiele, and Pietro Per- ona. Pedestrian detection: A benchmark. In 2009 IEEE Con- ference on Computer Vision and Pattern Recognition, pages 304-311. IEEE, 2009.\n\nFlownet: Learning optical flow with convolutional networks. Alexey Dosovitskiy, Philipp Fischer, Eddy Ilg, Philip Hausser, Caner Hazirbas, Vladimir Golkov, Patrick Van Der, Daniel Smagt, Thomas Cremers, Brox, Proceedings of the IEEE international conference on computer vision. the IEEE international conference on computer visionAlexey Dosovitskiy, Philipp Fischer, Eddy Ilg, Philip Hausser, Caner Hazirbas, Vladimir Golkov, Patrick Van Der Smagt, Daniel Cremers, and Thomas Brox. Flownet: Learning optical flow with convolutional networks. In Pro- ceedings of the IEEE international conference on computer vision, pages 2758-2766, 2015.\n\nA mobile vision system for robust multi-person tracking. Andreas Ess, Bastian Leibe, Konrad Schindler, Luc Van Gool, 2008 IEEE Conference on Computer Vision and Pattern Recognition. IEEEAndreas Ess, Bastian Leibe, Konrad Schindler, and Luc Van Gool. A mobile vision system for robust multi-person tracking. In 2008 IEEE Conference on Computer Vision and Pattern Recognition, pages 1-8. IEEE, 2008.\n\nDetect to track and track to detect. Christoph Feichtenhofer, Axel Pinz, Andrew Zisserman, Proceedings of the IEEE International Conference on Computer Vision. the IEEE International Conference on Computer VisionChristoph Feichtenhofer, Axel Pinz, and Andrew Zisserman. Detect to track and track to detect. In Proceedings of the IEEE International Conference on Computer Vision, pages 3038-3046, 2017.\n\nObject detection with discriminatively trained part-based models. F Pedro, Ross B Felzenszwalb, David Girshick, Deva Mcallester, Ramanan, IEEE transactions on pattern analysis and machine intelligence. 32Pedro F Felzenszwalb, Ross B Girshick, David McAllester, and Deva Ramanan. Object detection with discriminatively trained part-based models. IEEE transactions on pattern analysis and machine intelligence, 32(9):1627-1645, 2009.\n\nFast r-cnn. Ross Girshick, Proceedings of the IEEE international conference on computer vision. the IEEE international conference on computer visionRoss Girshick. Fast r-cnn. In Proceedings of the IEEE inter- national conference on computer vision, pages 1440-1448, 2015.\n\nTracking with multi-level features. Roberto Henschel, Laura Leal-Taixé, Bodo Rosenhahn, Konrad Schindler, arXiv:1607.07304arXiv preprintRoberto Henschel, Laura Leal-Taixé, Bodo Rosenhahn, and Konrad Schindler. Tracking with multi-level features. arXiv preprint arXiv:1607.07304, 2016.\n\nMultiple people tracking using body and joint detections. Roberto Henschel, Yunzhe Zou, Bodo Rosenhahn, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops. the IEEE Conference on Computer Vision and Pattern Recognition WorkshopsRoberto Henschel, Yunzhe Zou, and Bodo Rosenhahn. Mul- tiple people tracking using body and joint detections. In Pro- ceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, pages 0-0, 2019.\n\nLifted disjoint paths with application in multiple object tracking. Andrea Hornakova, Roberto Henschel, Bodo Rosenhahn, Paul Swoboda, The 37th International Conference on Machine Learning (ICML). Andrea Hornakova, Roberto Henschel, Bodo Rosenhahn, and Paul Swoboda. Lifted disjoint paths with application in multiple object tracking. In The 37th International Con- ference on Machine Learning (ICML), July 2020.\n\nRelation networks for object detection. Han Hu, Jiayuan Gu, Zheng Zhang, Jifeng Dai, Yichen Wei, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionHan Hu, Jiayuan Gu, Zheng Zhang, Jifeng Dai, and Yichen Wei. Relation networks for object detection. In Proceed- ings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3588-3597, 2018.\n\nComputer vision for autonomous vehicles: Problems, datasets and state of the art. Joel Janai, Fatma Güney, Aseem Behl, Andreas Geiger, Foundations and Trends® in Computer Graphics and Vision. 121-3Joel Janai, Fatma Güney, Aseem Behl, Andreas Geiger, et al. Computer vision for autonomous vehicles: Problems, datasets and state of the art. Foundations and Trends® in Computer Graphics and Vision, 12(1-3):1-308, 2020.\n\nA new approach to linear filtering and prediction problems. R E Kalman, ASME Journal of Basic Engineering. R. E. Kalman. A new approach to linear filtering and predic- tion problems. ASME Journal of Basic Engineering, 1960.\n\nFramework for performance evaluation of face, text, and vehicle detection and tracking in video: Data, metrics, and protocol. Rangachar Kasturi, Dmitry Goldgof, Padmanabhan Soundararajan, Vasant Manohar, John Garofolo, Rachel Bowers, Matthew Boonstra, Valentina Korzhova, Jing Zhang, IEEE Transactions on Pattern Analysis and Machine Intelligence. 312Rangachar Kasturi, Dmitry Goldgof, Padmanabhan Soundararajan, Vasant Manohar, John Garofolo, Rachel Bowers, Matthew Boonstra, Valentina Korzhova, and Jing Zhang. Framework for performance evaluation of face, text, and vehicle detection and tracking in video: Data, metrics, and protocol. IEEE Transactions on Pattern Analysis and Machine Intelligence, 31(2):319-336, 2008.\n\nAdam: A method for stochastic optimization. P Diederik, Jimmy Kingma, Ba, arXiv:1412.6980arXiv preprintDiederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.\n\nThe hungarian method for the assignment problem. Harold W Kuhn, Naval research logistics quarterly. 2Harold W Kuhn. The hungarian method for the assignment problem. Naval research logistics quarterly, 2(1-2):83-97, 1955.\n\nL Leal-Taixé, A Milan, I Reid, S Roth, K Schindler, Motchallenge, arXiv:1504.01942arXiv: 1504.01942Towards a benchmark for multitarget tracking. L. Leal-Taixé, A. Milan, I. Reid, S. Roth, and K. Schindler. MOTChallenge 2015: Towards a benchmark for multi- target tracking. arXiv:1504.01942 [cs], Apr. 2015. arXiv: 1504.01942.\n\nLearning to associate: Hybridboosted multi-target tracker for crowded scene. Yuan Li, Chang Huang, Ram Nevatia, 2009 IEEE Conference on Computer Vision and Pattern Recognition. IEEEYuan Li, Chang Huang, and Ram Nevatia. Learning to asso- ciate: Hybridboosted multi-target tracker for crowded scene. In 2009 IEEE Conference on Computer Vision and Pattern Recognition, pages 2953-2960. IEEE, 2009.\n\nMicrosoft coco: Common objects in context. Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, C Lawrence Zitnick, European conference on computer vision. SpringerTsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C Lawrence Zitnick. Microsoft coco: Common objects in context. In European conference on computer vision, pages 740-755. Springer, 2014.\n\nReal-time multiple people tracking with deeply learned candidate selection and person re-identification. Chen Long, Ai Haizhou, Zhuang Zijie, Shang Chong, ICME. Chen Long, Ai Haizhou, Zhuang Zijie, and Shang Chong. Real-time multiple people tracking with deeply learned can- didate selection and person re-identification. In ICME, 2018.\n\nSee more, know more: Unsupervised video object segmentation with co-attention siamese networks. Xiankai Lu, Wenguan Wang, Chao Ma, Jianbing Shen, Ling Shao, Fatih Porikli, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionXiankai Lu, Wenguan Wang, Chao Ma, Jianbing Shen, Ling Shao, and Fatih Porikli. See more, know more: Unsuper- vised video object segmentation with co-attention siamese networks. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3623- 3632, 2019.\n\nRetinatrack: Online single stage joint detection and tracking. Zhichao Lu, Vivek Rathod, Ronny Votel, Jonathan Huang, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionZhichao Lu, Vivek Rathod, Ronny Votel, and Jonathan Huang. Retinatrack: Online single stage joint detection and tracking. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 14668- 14678, 2020.\n\nA Milan, L Leal-Taixé, I Reid, S Roth, K Schindler, arXiv:1603.00831arXiv: 1603.00831MOT16: A benchmark for multi-object tracking. A. Milan, L. Leal-Taixé, I. Reid, S. Roth, and K. Schindler. MOT16: A benchmark for multi-object tracking. arXiv:1603.00831 [cs], Mar. 2016. arXiv: 1603.00831.\n\nA largescale benchmark dataset for event recognition in surveillance video. Sangmin Oh, Anthony Hoogs, Amitha Perera, Naresh Cuntoor, Chia-Chih Chen, Jong Taek Lee, Saurajit Mukherjee, Hyungtae Aggarwal, Larry Lee, Davis, CVPR 2011. IEEESangmin Oh, Anthony Hoogs, Amitha Perera, Naresh Cun- toor, Chia-Chih Chen, Jong Taek Lee, Saurajit Mukherjee, JK Aggarwal, Hyungtae Lee, Larry Davis, et al. A large- scale benchmark dataset for event recognition in surveillance video. In CVPR 2011, pages 3153-3160. IEEE, 2011.\n\nTubetk: Adopting tubes to track multi-object in a one-step training model. Bo Pang, Yizhuo Li, Yifan Zhang, Muchen Li, Cewu Lu, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionBo Pang, Yizhuo Li, Yifan Zhang, Muchen Li, and Cewu Lu. Tubetk: Adopting tubes to track multi-object in a one-step training model. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 6308- 6318, 2020.\n\nImage transformer. Niki Parmar, Ashish Vaswani, Jakob Uszkoreit, Lukasz Kaiser, Noam Shazeer, Alexander Ku, Dustin Tran, Proceedings of the 35th International Conference on Machine Learning. the 35th International Conference on Machine LearningPMLRNiki Parmar, Ashish Vaswani, Jakob Uszkoreit, Lukasz Kaiser, Noam Shazeer, Alexander Ku, and Dustin Tran. Im- age transformer. In Proceedings of the 35th International Conference on Machine Learning, pages 4055-4064. PMLR, 2018.\n\nChained-tracker: Chaining paired attentive regression results for end-to-end joint multiple-object detection and tracking. Jinlong Peng, Changan Wang, Fangbin Wan, Yang Wu, Yabiao Wang, Ying Tai, Chengjie Wang, Jilin Li, Feiyue Huang, Yanwei Fu, Proceedings of the European Conference on Computer Vision. the European Conference on Computer VisionJinlong Peng, Changan Wang, Fangbin Wan, Yang Wu, Yabiao Wang, Ying Tai, Chengjie Wang, Jilin Li, Feiyue Huang, and Yanwei Fu. Chained-tracker: Chaining paired at- tentive regression results for end-to-end joint multiple-object detection and tracking. In Proceedings of the European Con- ference on Computer Vision, 2020.\n\nJoseph Redmon, Ali Farhadi, arXiv:1804.02767Yolov3: An incremental improvement. arXiv preprintJoseph Redmon and Ali Farhadi. Yolov3: An incremental improvement. arXiv preprint arXiv:1804.02767, 2018.\n\nFaster r-cnn: Towards real-time object detection with region proposal networks. Kaiming Shaoqing Ren, Ross He, Jian Girshick, Sun, Advances in neural information processing systems. Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. Faster r-cnn: Towards real-time object detection with region proposal networks. In Advances in neural information pro- cessing systems, pages 91-99, 2015.\n\nPerformance measures and a data set for multi-target, multi-camera tracking. Ergys Ristani, Francesco Solera, Roger Zou, Rita Cucchiara, Carlo Tomasi, European Conference on Computer Vision. SpringerErgys Ristani, Francesco Solera, Roger Zou, Rita Cucchiara, and Carlo Tomasi. Performance measures and a data set for multi-target, multi-camera tracking. In European Conference on Computer Vision, pages 17-35. Springer, 2016.\n\nImagenet large scale visual recognition challenge. Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, International journal of computer vision. 1153Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, San- jeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, et al. Imagenet large scale visual recognition challenge. International journal of computer vision, 115(3):211-252, 2015.\n\nCrowdhuman: A benchmark for detecting human in a crowd. Shuai Shao, Zijian Zhao, Boxun Li, Tete Xiao, Gang Yu, Xiangyu Zhang, Jian Sun, arXiv:1805.00123arXiv preprintShuai Shao, Zijian Zhao, Boxun Li, Tete Xiao, Gang Yu, Xiangyu Zhang, and Jian Sun. Crowdhuman: A bench- mark for detecting human in a crowd. arXiv preprint arXiv:1805.00123, 2018.\n\nPwc-net: Cnns for optical flow using pyramid, warping, and cost volume. Deqing Sun, Xiaodong Yang, Ming-Yu Liu, Jan Kautz, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionDeqing Sun, Xiaodong Yang, Ming-Yu Liu, and Jan Kautz. Pwc-net: Cnns for optical flow using pyramid, warping, and cost volume. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 8934-8943, 2018.\n\nJonathon Shlens, Zhifeng Chen, and Dragomir Anguelov. Scalability in perception for autonomous driving: Waymo open dataset. Pei Sun, Henrik Kretzschmar, Xerxes Dotiwalla, Aurelien Chouard, Vijaysai Patnaik, Paul Tsui, James Guo, Yin Zhou, Yuning Chai, Benjamin Caine, Vijay Vasudevan, Wei Han, Jiquan Ngiam, Hang Zhao, Aleksei Timofeev, Scott Ettinger, Maxim Krivokon, Amy Gao, Aditya Joshi, Yu Zhang, IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). Pei Sun, Henrik Kretzschmar, Xerxes Dotiwalla, Aurelien Chouard, Vijaysai Patnaik, Paul Tsui, James Guo, Yin Zhou, Yuning Chai, Benjamin Caine, Vijay Vasudevan, Wei Han, Jiquan Ngiam, Hang Zhao, Aleksei Timofeev, Scott Et- tinger, Maxim Krivokon, Amy Gao, Aditya Joshi, Yu Zhang, Jonathon Shlens, Zhifeng Chen, and Dragomir Anguelov. Scalability in perception for autonomous driving: Waymo open dataset. In IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2020.\n\nRaft: Recurrent allpairs field transforms for optical flow. Zachary Teed, Jia Deng, arXiv:2003.12039arXiv preprintZachary Teed and Jia Deng. Raft: Recurrent all- pairs field transforms for optical flow. arXiv preprint arXiv:2003.12039, 2020.\n\nAttention is all you need. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, Illia Polosukhin, Advances in neural information processing systems. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszko- reit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in neural information processing systems, pages 5998-6008, 2017.\n\nMots: Multi-object tracking and segmentation. Paul Voigtlaender, Michael Krause, Aljosa Osep, Jonathon Luiten, Berin Balachandar Gnana, Andreas Sekar, Bastian Geiger, Leibe, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionPaul Voigtlaender, Michael Krause, Aljosa Osep, Jonathon Luiten, Berin Balachandar Gnana Sekar, Andreas Geiger, and Bastian Leibe. Mots: Multi-object tracking and segmen- tation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7942-7951, 2019.\n\nTracking emerges by colorizing videos. Carl Vondrick, Abhinav Shrivastava, Alireza Fathi, Sergio Guadarrama, Kevin Murphy, Proceedings of the European conference on computer vision (ECCV). the European conference on computer vision (ECCV)Carl Vondrick, Abhinav Shrivastava, Alireza Fathi, Sergio Guadarrama, and Kevin Murphy. Tracking emerges by col- orizing videos. In Proceedings of the European conference on computer vision (ECCV), pages 391-408, 2018.\n\nVideo modeling with correlation networks. Heng Wang, Du Tran, Lorenzo Torresani, Matt Feiszli, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionHeng Wang, Du Tran, Lorenzo Torresani, and Matt Feiszli. Video modeling with correlation networks. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 352-361, 2020.\n\nAxial-deeplab: Standalone axial-attention for panoptic segmentation. Huiyu Wang, Yukun Zhu, Bradley Green, Hartwig Adam, Alan Yuille, Liang-Chieh Chen, arXiv:2003.07853arXiv preprintHuiyu Wang, Yukun Zhu, Bradley Green, Hartwig Adam, Alan Yuille, and Liang-Chieh Chen. Axial-deeplab: Stand- alone axial-attention for panoptic segmentation. arXiv preprint arXiv:2003.07853, 2020.\n\nNon-local neural networks. Xiaolong Wang, Ross Girshick, Abhinav Gupta, Kaiming He, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionXiaolong Wang, Ross Girshick, Abhinav Gupta, and Kaim- ing He. Non-local neural networks. In Proceedings of the IEEE conference on computer vision and pattern recogni- tion, pages 7794-7803, 2018.\n\nJoint detection and multi-object tracking with graph neural networks. Yongxin Wang, Xinshuo Weng, Kris Kitani, arXiv:2006.13164arXiv preprintYongxin Wang, Xinshuo Weng, and Kris Kitani. Joint detec- tion and multi-object tracking with graph neural networks. arXiv preprint arXiv:2006.13164, 2020.\n\nTowards real-time multi-object tracking. Zhongdao Wang, Liang Zheng, Yixuan Liu, Shengjin Wang, European Conference on Computer Vision. Zhongdao Wang, Liang Zheng, Yixuan Liu, and Shengjin Wang. Towards real-time multi-object tracking. In European Conference on Computer Vision, 2020.\n\nSimple online and realtime tracking with a deep association metric. Nicolai Wojke, Alex Bewley, Dietrich Paulus, 2017 IEEE international conference on image processing (ICIP). IEEENicolai Wojke, Alex Bewley, and Dietrich Paulus. Simple online and realtime tracking with a deep association metric. In 2017 IEEE international conference on image processing (ICIP), pages 3645-3649. IEEE, 2017.\n\nJoint detection and identification feature learning for person search. Tong Xiao, Shuang Li, Bochao Wang, Liang Lin, Xiaogang Wang, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionTong Xiao, Shuang Li, Bochao Wang, Liang Lin, and Xiao- gang Wang. Joint detection and identification feature learn- ing for person search. In Proceedings of the IEEE Con- ference on Computer Vision and Pattern Recognition, pages 3415-3424, 2017.\n\nSpatialtemporal relation networks for multi-object tracking. Jiarui Xu, Yue Cao, Zheng Zhang, Han Hu, Proceedings of the IEEE International Conference on Computer Vision. the IEEE International Conference on Computer VisionJiarui Xu, Yue Cao, Zheng Zhang, and Han Hu. Spatial- temporal relation networks for multi-object tracking. In Pro- ceedings of the IEEE International Conference on Computer Vision, pages 3988-3998, 2019.\n\nDisentangled non-local neural networks. Minghao Yin, Zhuliang Yao, Yue Cao, Xiu Li, Zheng Zhang, Stephen Lin, Han Hu, arXiv:2006.06668arXiv preprintMinghao Yin, Zhuliang Yao, Yue Cao, Xiu Li, Zheng Zhang, Stephen Lin, and Han Hu. Disentangled non-local neural networks. arXiv preprint arXiv:2006.06668, 2020.\n\nDilated residual networks. Fisher Yu, Vladlen Koltun, Thomas Funkhouser, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionFisher Yu, Vladlen Koltun, and Thomas Funkhouser. Dilated residual networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 472-480, 2017.\n\nPoi: Multiple object tracking with high performance detection and appearance feature. Fengwei Yu, Wenbo Li, Quanquan Li, Yu Liu, Xiaohua Shi, Junjie Yan, European Conference on Computer Vision. SpringerFengwei Yu, Wenbo Li, Quanquan Li, Yu Liu, Xiaohua Shi, and Junjie Yan. Poi: Multiple object tracking with high per- formance detection and appearance feature. In European Conference on Computer Vision, pages 36-42. Springer, 2016.\n\nDeep layer aggregation. Fisher Yu, Dequan Wang, Evan Shelhamer, Trevor Darrell, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionFisher Yu, Dequan Wang, Evan Shelhamer, and Trevor Darrell. Deep layer aggregation. In Proceedings of the IEEE conference on computer vision and pattern recogni- tion, pages 2403-2412, 2018.\n\nA simple baseline for multi-object tracking. Yifu Zhan, Chunyu Wang, Xinggang Wang, Wenjun Zeng, Wenyu Liu, arXiv:2004.01888arXiv preprintYifu Zhan, Chunyu Wang, Xinggang Wang, Wenjun Zeng, and Wenyu Liu. A simple baseline for multi-object tracking. arXiv preprint arXiv:2004.01888, 2020.\n\nOn the stability of video detection and tracking. Hong Zhang, Naiyan Wang, arXiv:1611.06467arXiv preprintHong Zhang and Naiyan Wang. On the stability of video detection and tracking. arXiv preprint arXiv:1611.06467, 2016.\n\nJimuyang Zhang, Sanping Zhou, Xin Chang, Fangbin Wan, Jinjun Wang, Yang Wu, Dong Huang, arXiv:2001.11180Multiple object tracking by flowing and fusing. arXiv preprintJimuyang Zhang, Sanping Zhou, Xin Chang, Fangbin Wan, Jinjun Wang, Yang Wu, and Dong Huang. Multiple object tracking by flowing and fusing. arXiv preprint arXiv:2001.11180, 2020.\n\nGlobal data association for multi-object tracking using network flows. Li Zhang, Yuan Li, Ramakant Nevatia, 2008 IEEE Conference on Computer Vision and Pattern Recognition. IEEELi Zhang, Yuan Li, and Ramakant Nevatia. Global data association for multi-object tracking using network flows. In 2008 IEEE Conference on Computer Vision and Pattern Recognition, pages 1-8. IEEE, 2008.\n\nCitypersons: A diverse dataset for pedestrian detection. Shanshan Zhang, Rodrigo Benenson, Bernt Schiele, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionShanshan Zhang, Rodrigo Benenson, and Bernt Schiele. Citypersons: A diverse dataset for pedestrian detection. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3213-3221, 2017.\n\nPerson re-identification in the wild. Liang Zheng, Hengheng Zhang, Shaoyan Sun, Manmohan Chandraker, Yi Yang, Qi Tian, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionLiang Zheng, Hengheng Zhang, Shaoyan Sun, Manmohan Chandraker, Yi Yang, and Qi Tian. Person re-identification in the wild. In Proceedings of the IEEE Conference on Com- puter Vision and Pattern Recognition, pages 1367-1376, 2017.\n\nTracking objects as points. Xingyi Zhou, Vladlen Koltun, Philipp Krähenbühl, European Conference on Computer Vision. Xingyi Zhou, Vladlen Koltun, and Philipp Krähenbühl. Tracking objects as points. In European Conference on Com- puter Vision, 2020.\n\nObjects as points. Xingyi Zhou, Dequan Wang, Philipp Krähenbühl, arXiv:1904.07850arXiv preprintXingyi Zhou, Dequan Wang, and Philipp Krähenbühl. Ob- jects as points. In arXiv preprint arXiv:1904.07850, 2019.\n\nDeformable convnets v2: More deformable, better results. Xizhou Zhu, Han Hu, Stephen Lin, Jifeng Dai, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionXizhou Zhu, Han Hu, Stephen Lin, and Jifeng Dai. De- formable convnets v2: More deformable, better results. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 9308-9316, 2019.\n"},"corpus_id":{"kind":"number","value":233182057,"string":"233,182,057"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2104.03541\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"conf/cvpr/WangZPX21\",\n \"doi\": \"10.1109/cvpr46437.2021.00387\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/2104.03541v1.pdf\"\n ],\n \"pdf_sha\": \"cecfb3f6f900a01b8495f5075e3d62390f0a0bdd\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/2104.03541\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Multiple Object Tracking with Correlation Learning"},"authors":{"kind":"list like","value":["Qiang Wang \nMachine Intelligence Technology Lab, Alibaba Group","Yun Zheng zhengyun.zy@alibaba-inc.com \nMachine Intelligence Technology Lab, Alibaba Group","Pan Pan \nMachine Intelligence Technology Lab, Alibaba Group","Yinghui Xu \nMachine Intelligence Technology Lab, Alibaba Group"],"string":"[\n \"Qiang Wang \\nMachine Intelligence Technology Lab, Alibaba Group\",\n \"Yun Zheng zhengyun.zy@alibaba-inc.com \\nMachine Intelligence Technology Lab, Alibaba Group\",\n \"Pan Pan \\nMachine Intelligence Technology Lab, Alibaba Group\",\n \"Yinghui Xu \\nMachine Intelligence Technology Lab, Alibaba Group\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Recent works have shown that convolutional networks have substantially improved the performance of multiple object tracking by simultaneously learning detection and appearance features. However, due to the local perception of the convolutional network structure itself, the long-range dependencies in both the spatial and temporal cannot be obtained efficiently. To incorporate the spatial layout, we propose to exploit the local correlation module to model the topological relationship between targets and their surrounding environment, which can enhance the discriminative power of our model in crowded scenes. Specifically, we establish dense correspondences of each spatial location and its context, and explicitly constrain the correlation volumes through self-supervised learning. To exploit the temporal context, existing approaches generally utilize two or more adjacent frames to construct an enhanced feature representation, but the dynamic motion scene is inherently difficult to depict via CNNs. Instead, our paper proposes a learnable correlation operator to establish frameto-frame matches over convolutional feature maps in the different layers to align and propagate temporal context. With extensive experimental results on the MOT datasets, our approach demonstrates the effectiveness of correlation learning with the superior performance and obtains stateof-the-art MOTA of 76.5% and IDF1 of 73.6% on MOT17."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Multi-Object Tracking (MOT) is an essential component for computer vision with many applications, such as video surveillance [31] and modern autonomous driving [19,41]. It aims to continuously locate trajectories of multiple targets in video frames. Decades of research efforts have led to impressive performance on challenging benchmarks [24,30,8].\nMOT has traditionally adopted the tracking-by-detection paradigm [3,5,1,58], which capitalizes on the natural division of detection and data association tasks for the problem. These algorithms extract appearance features within each detection patches and record object location information for subsequent data association [51,5]. This tracking paradigm makes researchers mainly focus on optimizing detection [56,16], feature representation [27,15], or data association [3,5,17]. With the rapid progress of detection algorithms [13,14,36,35], the detection based tracking has achieved great performance improvement [56,16]. Although tremendous strides have been made in MOT, there still exists tough challenges in determining distractors and frequent occlusions, especially in complex interactive scenes [8]. Additionally, the above cascaded structure is inefficient and prevents the joint optimization between stages. One promising approach is to extend the end-to-end trainable detection framework [36,35,64] to jointly learn detection and appearance feature, which has largely advanced the state-of-the-art in MOT [44,50,29,58]. However as illustrated in Fig. 1, in the case of existing similar distractors, the appearance feature generates undistinctive and inaccurate matching confidences (Fig. 1a), severely affecting the performance of association. These methods are limited in local descriptors, and it is difficult to distinguish similar objects. While as shown in Fig. 1c, the context relation map can help to easily distinguish different targets.\nBased on those observations, we propose a correlation network to learn the topological information of the object and context. Specifically, we use a spatial correlation layer to record the relationship between targets and relative spatial positions. While constructing a full correlation (e.g., non-local [48]) for all locations is computationally prohibitive for real-time MOT, this work constructs a local correlation volume by limiting the search range at each feature pyramid. Besides, our correlation learning is not limited for targets of interest category [53,49]. Background contexts, such as vehicles, are also modeled to help target recognition and relational reasoning (Fig. 1c). We establish dense correspondences of each spatial location and its context, and explicitly constrain the correlation volumes through self-supervised learning.\nFurther, the detector in MOT usually uses independent frames as input and therefore does not make full use of temporal information. This detection method makes the algorithm suffer from missing detection in crowded scenes, and further increases the difficulty of subsequent data association. Recently, adjacent frames [64,34] or three frames [32] are adopted to enhance the temporal consistence. The performance of the algorithm in occlusion scenes has been improved to a certain extent, but these methods are still limited with fewer frames. CenterTrack [64] attempt to use an aggressive data augmentation to increase the ability of target alignment, but convolution networks itself are inherently limited in local receptive fields. To solve the above problem, we extend the spatial correlation module to the temporal dimension and incorporate the historical information to reduce ambiguities in object detections.\nTo summarize, we make the following contributions:\n• We propose CorrTracker, a unified correlation tracker to intensively model associations between objects and transmit information through associations. • We propose a local structure-aware network and enhance the discriminability of similar objects with selfsupervised learning. • We extend the local correlation network to model temporal information efficiently. • CorrTracker shows significant improvements over existing state-of-the-art results in four MOT benchmarks.\nIn particular, we achieve 76.5% MOTA and IDF1 of 73.6% on MOT17."},{"title":"Related Work","content":"Real-time Tracking. As MOT has strong practical merit, the tracking speed attracts much attention. The researchers start from the simplest IOUTracker [4], which only uses the intersection-over-union of bounding boxes for tracking, to add the motion model of Kalman Filter [3] to predict the position of the rectangular boxes for matching. Although they have achieved amazing speed, stable tracking cannot be achieved under challenges such as target interleaving. Researchers [27,51] introduce Person Re-Identification (ReID) features as an appearance model to increase the discriminative power of the tracker. However, the individual calculation for patches makes the object classification and ReID feature extraction as a computational bottleneck. MOTDT [27] achieves real-time tracking by using RoI-pooling [14] on a shared feature map. In order to further decrease the computational cost of ReID feature extraction, JDE [50] adds a ReID branch in a single-stage detector YOLOv3 [35] to achieve efficient ReID feature calculation. FairMOT [58] explores the importance of detection and recognition tasks and uses anchor-free method [65] to reduce the ambiguity of anchors. We are mainly based on FairMOT, which achieves the state-of-the-art performance with a more balanced ReID and detection.\nOther researchers [1,64,34,32] explore new tracking paradigms to remove ReID recognition. Tracktor [1] uses the the bounding boxes in the previous frame to directly regress the current locations. CenterTrack [64], Chained-Tracker [34], and TubeTK [32] use multiple frames to simultaneously predict the bounding boxes for adjacent frames to achieve short association, thereby merging to long-term tracks. However, these methods usually have many identity switches because they cannot model longterm dependencies. Tracking with Graph Model. MOT has traditionally been approached as a hand-crafted graph optimization problem [61,17], where the cropped targets are treated as nodes. Recently, graph neural network based methods [53,49,5] have been shown as a promising alternative to traditional optimization methods. State-of-the-art approach [53] utilizes graph convolutional network to propagate features in the spatial-temporal space. MPN [5] introduces the message passing network to dissect the information and associate detections through the edge classification. Different from these methods, feature propagation is carried out at the frame feature level, which can absorb the information of both the foreground and background and reduce the loss of contextual information. Tracking with Optical Flow. FlowTrack [60] introduces optical flow to predict the target location. But explicitly using optical flow is not only time-consuming, but also only encodes the pixel-level motion. CenterTrack [64] borrows the method of optical flow to directly predict the movement of the target center between two frames, which is called instance flow. However, directly predicting the offset on the concatenated feature map needs to provide training samples with all displacement, which requires excessive data augmentation. Our correlation method predicts a dense set of matching confidence for each target, which is intrinsically invariant to translation of the paired frames. Our correlation operation is similar to the correlation volume in optical flow [10,40,42] and correlation filter [7]. We both predict dense local correlation, and regard it as a part of the feature description. However, optical flow does not calculate the internal correlation of the image, nor does it have propagated the feature from multiple frames. D&T [12] also utilizes correlation layer to predict candidate motion between pair of consecutive frames. Compared with it, our anchor-free framework is more compact and efficient. Attention Mechanism. Our modeling of local correlation is similar to the self-attention mechanism and Transformer. Transformer has been a huge success in the NLP field [43] and has also been adapted to the computer vision [48,18,28] to capture long-range dependencies. In order to reduce the quadratic complexity of the non-local operation, the researchers propose to shrink the attention span with local region [33], or only along individual axes [47]. Different from these methods, we mainly encode the context identity through local correlation weighting, and use this cues to increase the model robustness. Figure 2 shows the overall pipeline of the proposed Cor-rTracker. Our method can be distilled down to three stages: (1) general feature extraction, (2) simultaneous learning correlation from spatial-temporal dependencies and predicting the detection, and (3) performing data association to assign detections into their most likely trajectories, where stage (1) and stage (2) are differentiable and composed into an end-to-end trainable architecture. We adopt a compact association technique that is similar to the one used by DeepSORT [51] to control the initialization and the termination of tracks. The main contribution is the highly efficient modeling for the correlation between dense location and their context on feature maps, which helps suppressing distractors in complex scenes."},{"title":"Methodology","content":""},{"title":"Motivation","content":"For each input video frame I t ∈ R H×W ×3 , an object detector is applied to find all candidate\nappearing in this frame and we have existing trajectories\nThen the affinity matrix A ∈ R N ×M is estimated by pair-wise comparisons of cropped patches and existing trajectories. The metric jointly considers both the appearance features f (·) ∈ R d and geometric representations.\nThe discriminative featuref (T j t−1 ) of a trajectory is usually updated with a constant-weighting strategy to follow the appearance changes. Each confidence value for appearance feature is obtained in a distance metric, e.g., the inner product space. However, the sole reliance on person-to-person feature comparisons are often insufficient to disambiguate multiple similar regions in an image. As illustrated in Fig. 1, in the case of similar distractors, the feature extractor usually generates inaccurate and uninformative matching confidences ( Fig. 1a), severely affecting the performance of data association. This is the key limitation of appearance feature matching, since co-occurring similar objects are all pervasive in MOT.\nPatch based feature extraction is applied as a prevalent scheme in MOT owing to its intuition. However the correlation information between the cropped image patches is lost directly, and the adjacency spatial relationship is only retained in coordinates d i t . Although the subsequent data association will be globally optimized, directly using ReID features without considering the context tends to introduce more identity switches, lagging the tracking robustness. To deal with this problem, we model the local structure of objects to distinguish it from distractors.\nInspired by correlation volume from optical flow [10], we observe that a confidence value in the correlation vol-ume models the geometric structure of each target. We design a novel dense correlation module, aiming to explore the context information for MOT. The relative position is encoded in the correlation volumes, which can be used as an auxiliary discriminant information."},{"title":"Spatial Local Correlation Layers","content":"In this work, we use Spatial Local Correlation Layers to model the relational structure for associating a target with its neighbour. In our local correlation layer, the feature similarity is only evaluated in the neighbourhood of the target image coordinate. Formally, we let l denote the level in the feature pyramid and the correlation volume C l between the query feature F l q ∈ R H l ×W l ×d l and reference feature F l r ∈ R H l ×W l ×d l is defined as,\nwhere x ∈ Z 2 is a coordinate in the query feature map and d ∈ Z 2 is the displacement from this location. The displacement is constrained to d ∞ ≤ R , i.e. the maximum motion in any direction is R. While most naturally thought of as a 4-dimensional tensor, the two displacement dimensions are usually vectorized into one to simplify further processing in the CNN. The resulting 3-d correlation volume C l thus is of size H l × W l × (2R + 1) 2 . We also introduce the dilation tricks [55], which can increase the receptive field without additional cost. We use element-wise addition to incorporate the correlation feature into a unified appearance representation. This context correlation features are encoded by a feed-forward Multilayer Perceptron (MLP) to match the number of channels d l in appearance features F l t .\nThe non-local [48] module is to explicitly model all pairwise interactions between elements in a feature maps F l t ∈ R H l ×W l ×d l . The resulting four-dimensional correlation volume N L(F l t ) ∈ R H l ×W l ×H l ×W l captures dense matching confidences between every pair of image locations. They build a full connection volume at a single scale, which is both computationally expensive and memory intensive. By contrast, our work shows that constructing a local correlation volume leads to both effective and efficient models. In comparison with the global correlation method, our local correlation model adds less overhead to the latency (see Table 1)."},{"title":"Correlation at Multiple Pyramid Levels","content":"In order to achieve long-range correlation, we propose to learn correlation at feature pyramids, as shown in Figure 3. On the one hand, we hope that our correlation module can obtain long-distance dependencies as much as possible, but as the local region size R increases, both calculation and storage increase significantly, which hinders the application. On the other hand, MOT naturally needs to deal with multi-scale targets. The two-stage detection [36] uses RoI pooling [14] to eliminate the difference in target scales, but this type of method usually suffers from high processing latency. In order to solve the above problems, we utilize the general pyramid structure in the convolutional network and learn correlation on the feature pyramids. Our multi-scale pyramid correlation can also be regarded as a comparison of multi-granularity features, covering the spatial context in the range of [0, R × D × 2 l ], where D refers to the dilation rate. And, we pass this correlation from the top layer to the bottom layer,\nIn this way, we can obtain an approximate correlation between the target and the entire global context, while keep the compactness and efficiency. Our pyramid correlation leverages the natural spatial-temporal coherence in videos.\nMulti-object tracking can be decomposed into multiple independent single-object tracking. Our method can be equivalent to a dense siamese network tracking [2] on the feature pyramid. On the other hand, from the perspective of set matching, global characteristics need to be considered. Our multi-scale correlation takes into account both aspects of information transmission."},{"title":"Temporal Correlation Learning","content":"The correlation between different frames are usually ignored by the MOT field, and trackers usually overcome occlusion through data association. Single frame detector is difficult to ensure a good temporal consistency [59]. This makes the algorithm's performance drop significantly in occlusion, motion blur and small target scenes, which becomes the bottleneck for MOT. We extend the spatial local correlation from Section 3.2 to the temporal dimension, and establish correlation for the targets in different frames. The correlation between two frames can be viewed as the establishment of motion information learning. We also use this correlation to enhance the feature representation, which can increase the detection accuracy.\nSpecifically, we establish multi-scale correlation between different frames, and use reference images as memory to enhance image features. This method helps tracker overcome target occlusion and motion blur, and increases the consistency of detection and identity features.\nSimilar to the multi-head attention [43], we adopt the embedded features and dot-product similarity. In our case, we set the normalization factor as (2R + 1) 2 and locally aggregate features. This shrinked region design also comes from the motion prior of the MOT scene. For the minimal memory consumption and fastest run-time, we can only save the previous features F t−1 in the memory. For the maximum accuracy, our long-term model saves the latest 5 frame image features by default."},{"title":"Self-supervised Feature Learning","content":"In Section 3.2 and Section 3.4, we present how we model the correlation in spatial and temporal dimension. We can simply use the proposed correlation module as a plugin module without explicitly adding constraints, similar to the non-local module, which has shown significant improvement. Here we investigate a multi-task learning approach that imposes a semantic supervision from visual object tracking [2] and self-supervised training from correspondence flow [45] on correlation volumes.\nOur correlation module is interpretable, measuring the similarity between different objects. Actually, our method intensively performs M ×N siamese tracking operations [2] to increases the discrimination. In this view, we can explicitly impose a tracking supervision. Specifically, we set up the ground-truth label as\nwhere y is the identity label of the corresponding position in feature maps. We ignore the position without objects y q (x) < 0 and use a class-balanced cross-entropy loss [2]. Inspired by the recent advances of self-supervised tracking [45], we use colorization as a proxy task for training our local correlation.\nwe use the cross-entropy categorical loss after quantizing the color-space into discrete categories [45]."},{"title":"Tracking Framework","content":"We modify the FairMOT [58] backbone by adding correlation module before the iterative deep aggregation module [57]. Our model retains the detection and ReID branches, and adding correlation loss in Sec. 3.5 for multitask learning. For the tracking inference, our tracker first calculates the similarity between the detections of the current frame and the previous trajectories according to Eq. (2), and use the Hungarian algorithm [23] for finding the optimal matching. The unmatched detections are used to initialize new trajectories. In order to reduce false positives, we mark these new trajectories as \"inactive\" until the next frame is matched again and confirmed as \"active\". The unmatched trajectories are set to the \"lost\" state. When the continuous lost time t loss of a trajectory exceeds the threshold τ loss , we put it in the remove set. If there is a success matching before removing, we restore the trajectory to the active state. We use Kalman Filter [20] to model the pedestrian motion and keep the same settings in FairMOT [58]."},{"title":"Experiments","content":"To demonstrate the advantages of the proposed correlation tracker, we first compare the correlation module with other relational reasoning methods [48,46] and evaluate different settings to justify our design choices in Section 4.2. Then we show that our correlation tracker outperforms the state-of-the-art methods on four MOT benchmarks [24,30,8] in Section 4.3. Finally, we visualize the tracking trajectories in Section 4.4 and compare with other motion prediction based trackers [64,34,32]."},{"title":"Implementation Details","content":"Network Setup.\nThe implementation and hyperparameters mostly follow [58], we adopt CenterNet [65] detector with a variant of Deep Layer Aggregation (DLA-34) [57] as backbone and utilize the iterative deep aggregation module (IDA) to recover a high-resolution feature map with stride 4. We also add a 3 × 3 deformable convolution layer [66] before every upsampling stage. The backbone network is initialized with the parameters pre-trained on COCO [26] and then pre-trained on CrowdHuman [39] with self-supervised learning as FairMOT [58]. The proposed correlation module is augmented before IDA module to fuse multi-scale correlation. For the correlation module, we set local region size R = 5 and dilation rate D = 2.\nTraining and Validation Datasets. For a fair comparison, we also use the default training datasets as Fair-MOT [58]. There are six training datasets including the ETH [11], CityPerson [62], CalTech [9], CUHK-SYSU [52], PRW [63] and MOT17 [30]. ETH and CityPerson only provide box annotations, so we ignore the ReID losses from these datasets. CalTech, CUHK-SYSU, PRW We train on an input resolution of 1088 × 608, which yields an output resolution of 272 × 152. We use random flip, random scaling (between 0.5 to 1.2), cropping, and color jittering as data augmentation, and use Adam [22] to optimize the overall objective. The learning rate is initialized as 1e −4 and then decayed to 1e −5 in the last 10 epochs. We train with a batch-size of 12 (on 2 GPUs) for 30 epochs. In the training phase, we sample 5 temporally ordered frames within a random interval of less than 3.\nTest Datasets and Evaluation Metrics. We evaluate the performance of our correlation tracker on the 2DMOT2015 [24], MOT16 [30], MOT17 [30], and MOT20 [8]. In particular, 2DMOT2015 [24] contains 11 test videos. MOT16 [30] and MOT17 [30] contain same 7 test videos with part different annotations. The MOT20 [8] consists of 4 test videos on extremely crowded scenes, which makes them really challenging.\nWe adopt the standard metrics of MOT Benchmarks for evaluation, including Multiple Object Tracking Accuracy (MOTA) [21], ID F1 Score [37], Mostly tracked targets (MT), Mostly lost targets (ML), the number of False Positives (FP), the number of False Negatives (FN), and the number of Identity Switch (ID Sw.) [25]. The run time is also provided and evaluated on a NVIDIA Tesla V100 GPU."},{"title":"Ablation Studies","content":"To elaborate on the effectiveness of the proposed approach, we conduct extensive ablation studies. First, we give detailed correlation analysis with different settings to justify our design choices, as presented in Table. 1. Next, the tracking accuracy and runtime for different region sizes of the correlation module is explored. Different building blocks are compared to illustrate the effectiveness and efficiency of the full correlation tracker. Spatial correlation. In order to evaluate the effectiveness Table 2. Ablation studies on MOT17 validation set. \"LT\" and \"Self\" denote using the proposed long-term memory and selfsupervised loss, respectively. of our spatial local correlation module (SLC), we compared our baseline model [58] and two relation methods, the nonlocal module [48] and CorrNet [46]. Directly use of nonlocal brings performance degradation on MOT since nonlocal module does not record the relative positional relationship between the targets and usually brings performance drops on small object [54]. In addition, non-local method has a huge overhead in memory and computation. In video recognition, a learnable correlation filter network [46] is proposed, and the grouped convolution is used to reduce the amount of calculation. Our method has achieved similar MOTA compared to CorrNet, but has a large improvement on IDF1, which comes from our multi-scale correlation design. Compared with the baseline, our IDF1 has increased by 2.9%, and identity switches have been reduced by 15%, demonstrating the discrimination of our spatial correlation. The Re-ID embeddings cannot easily distinguish similar distractor, our correlation feature models geometric information and is better suited for MOT. Temporal correlation. In the field of video object detection [38], temporal and global information are commonly used to improve performance [6]. The research of temporal detection for MOT is still preliminary. Recently, CenterTrack [64] concatenates the previous frame and the current frame, and Chained-Tracker [34] concatenates the high-level image features for two frames to fuse temporal information. We compare these two methods, dubbed concat-raw and concat-feat, with our temporal local correlation module (TLC). These two methods brought 0.2% and 1.3% MOTA improvements over the single frame baseline. Compared with these two methods, our temporal local correlation module achieves consistent improvements in both MOTA and IDF1. Our temporal correlation module helps for the temporal feature alignment around frames. At the same time, our method only adds a small overhead to feature-level concatenation, which proves the efficiency of our algorithm. Compared with the baseline FairMOT, employing both spatial and temporal local correlation (STLC) yields a IDF1 of 76.1% , which brings 3.2% improvement.\nLong-term dependences. We also analyze the performance improvement with long-term correlation. Compared with the method using two frames as the source cues, our long-term method achieves a large improvement of 0.6% MOTA, due to the increased capacity for object detections.\nThe improvement of MOTA also means an increase in the upper bound of our tracker. Self-supervised learning. For correlation learning, explicit supervision is usually not imposed [48,46]. We have proposed two supervision methods in section 3.5. It can be seen that the siamese tracking supervision imposed to training has achieved a relatively good improvement in IDF1.\nThere is no change in the run time of our algorithm, because the change in training loss does not change the inference processing. Self-supervised losses have also been appropriately improved on both MOTA and IDF1 due to more positive samples employed in correlation volume.\nChoice of local region. Figure 4 shows the MOTA and run time of our correlation module for different region size R ∈ {1, 2, ..., 8}. As expected, a larger local size R can cover a larger neighborhood while matching pixels, thus yields a higher accuracy. But the improvements become marginal beyond R = 5, possibly due to the low resolution of the feature maps. Note that non-local module usually doubles the run time of the backbone and the cost of explicitly computing optical flow [10] can be very high as well. This shows that our correlation module is more efficient by learning motion information from features directly. Region size R = 5 yields a good trade-off between speed and accuracy. The computation overhead is relatively small, compared to the complexity of the whole detection networks."},{"title":"Experiments on MOT Challenges","content":"To extensively evaluate the proposed method, we compare it with 8 state-of-the-art trackers, which cover most of current representative methods. There are 2 joint detection and embedding methods (JDE [50] and FairMOT [58]), 2 multi-frame prediction methods (Tube TK [32] and CTracker [34], 2 graph network based methods (MPN [5] and JDMOTGNN [49]), 2 offset prediction based methods (CenterTracker [64] and Tracktor++v2 [1]). The results are summarized in Table 3.\n2DMOT2015 [24]. The evaluation on 2DMOT2015 is performed by the official toolkit. As shown in Table 3, our correlation tracker outperforms the top private method of 2DMOT2015, (i.e., FairMOT [58]), by 1.7% in MOTA and 1.0% in IDF1. It is worth noting that the ID Switches are decreased by 13%, which shows the robustness of our cor- Table 3. Comparisons of our method with state-of-the-arts on MOT benchmarks [24,30,8]. We set new state-of-the-art results by a significant margin in terms of MOTA and IDF1. Our correlation tracker is more accurate while running with high speed. relation module. Moreover, our tracker is superior to the recent end-to-end graph trackers JDMOTGNN [49]. Our feature propagation approach can absorb both foreground and background information, which improves our tracker by 1.6% in terms of MOTA. MOT16 [30] and MOT17 [30]. Table 3 reports the evaluation results with the comparisons to recent prevailing trackers on MOT16. The recent proposed FairMOTv2 [58] achieves the second performance in MOTA and IDF1, while our method ranks first with 76.6% MOTA, outperforming other private approaches by a significant margin. Our CorrTracker achieves the best performance on MOT17, surpassing FairMOTv2 by 2.8% MOTA and 1.3% IDF1. Moreover, the FN of our CorrTracker surpasses FairMOTv2 by 15%, which means nearly 20,000 new bounding boxes are added to the association process. In this case, our algorithm still maintains comparable or even superior ID Switches, which actually proves that our method significantly improves the tracking association. As reported in Table 3, our CorrTracker, CenterTrack [64] and CTrack [34] all use multi-frame cues to predict detections, our FN is largely decreased by 30%.\nMOT20 [8]. To further evaluate the proposed models, we report the results on MOT20, which is more challenging than MOT17. The final results is presented in the bottom block of Table 3. Our CorrTracker achieves MOTA score of 65.2%, substantially outperforming FairMOTv2 [58] with MOTA of 61.8%. Although our approach is an order of magnitude faster than JDMOTGNN [49] in speed, our accuracy is slightly worse due to the anchor-free design. MOT20-06 MOT20-08 Figure 6. Qualitative results of our correlation tracker on MOT17 [30] and MOT20 [8]. The color of each bounding box indicates the target identity. The dotted line under each bounding box denotes the recent tracklet of each target. The proposed tracker predicts trajectories with substantially robust and temporally consistent."},{"title":"Visualization","content":"We visualize the tracking trajectories for prior methods, i.e., the center offset [64] and multi-frame bounding boxes [34,32], in Figure 5. We observe that our correlation map focuses on the entire context, while the regular appearance feature concentrates on the local region of the target. Our correlation module improves the reliability of recognition since it provides a global view of the target. Methods based on offset prediction, e.g., Cen-terTrack [64] and CTracker [34], can easily generate id switches when encountered with complex object interactions. Figure 6 shows qualitative results of our Correlation Tracker on MOT17 and MOT20, the advantage over existing method is most pronounced on the robust to occlusion and tiny objects."},{"title":"Conclusion","content":"In this work, we propose a novel correlation tracking framework based upon the observation that the relational structure helps to distinguish similar objects. Our correlation module densely matches all targets with their local context and learn a discriminative embeddings from the correlation volumes. Furthermore, we show how to extend the correlation module from spatial layout to the adjacent frames for strengthening the temporal modeling ability. We explore that self-supervised learning to impose a discriminative constraint on the correlation volume, which explicitly predicts a instance flow. Extensive experiments on four MOT challenges demonstrate that our CorrTracker achieves state-of-the-art performance and is efficient in inference."},{"title":"Figure 1 .","content":""},{"title":"Figure 2 .","content":""},{"title":"Figure 3 .","content":""},{"title":"Figure 4 .","content":""},{"title":"Figure 5 .","content":""},{"title":"Table 1 .","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Multi-Object Tracking (MOT) is an essential component for computer vision with many applications, such as video surveillance [31] and modern autonomous driving [19,41]. It aims to continuously locate trajectories of multiple targets in video frames. Decades of research efforts have led to impressive performance on challenging benchmarks [24,30,8].\\nMOT has traditionally adopted the tracking-by-detection paradigm [3,5,1,58], which capitalizes on the natural division of detection and data association tasks for the problem. These algorithms extract appearance features within each detection patches and record object location information for subsequent data association [51,5]. This tracking paradigm makes researchers mainly focus on optimizing detection [56,16], feature representation [27,15], or data association [3,5,17]. With the rapid progress of detection algorithms [13,14,36,35], the detection based tracking has achieved great performance improvement [56,16]. Although tremendous strides have been made in MOT, there still exists tough challenges in determining distractors and frequent occlusions, especially in complex interactive scenes [8]. Additionally, the above cascaded structure is inefficient and prevents the joint optimization between stages. One promising approach is to extend the end-to-end trainable detection framework [36,35,64] to jointly learn detection and appearance feature, which has largely advanced the state-of-the-art in MOT [44,50,29,58]. However as illustrated in Fig. 1, in the case of existing similar distractors, the appearance feature generates undistinctive and inaccurate matching confidences (Fig. 1a), severely affecting the performance of association. These methods are limited in local descriptors, and it is difficult to distinguish similar objects. While as shown in Fig. 1c, the context relation map can help to easily distinguish different targets.\\nBased on those observations, we propose a correlation network to learn the topological information of the object and context. Specifically, we use a spatial correlation layer to record the relationship between targets and relative spatial positions. While constructing a full correlation (e.g., non-local [48]) for all locations is computationally prohibitive for real-time MOT, this work constructs a local correlation volume by limiting the search range at each feature pyramid. Besides, our correlation learning is not limited for targets of interest category [53,49]. Background contexts, such as vehicles, are also modeled to help target recognition and relational reasoning (Fig. 1c). We establish dense correspondences of each spatial location and its context, and explicitly constrain the correlation volumes through self-supervised learning.\\nFurther, the detector in MOT usually uses independent frames as input and therefore does not make full use of temporal information. This detection method makes the algorithm suffer from missing detection in crowded scenes, and further increases the difficulty of subsequent data association. Recently, adjacent frames [64,34] or three frames [32] are adopted to enhance the temporal consistence. The performance of the algorithm in occlusion scenes has been improved to a certain extent, but these methods are still limited with fewer frames. CenterTrack [64] attempt to use an aggressive data augmentation to increase the ability of target alignment, but convolution networks itself are inherently limited in local receptive fields. To solve the above problem, we extend the spatial correlation module to the temporal dimension and incorporate the historical information to reduce ambiguities in object detections.\\nTo summarize, we make the following contributions:\\n• We propose CorrTracker, a unified correlation tracker to intensively model associations between objects and transmit information through associations. • We propose a local structure-aware network and enhance the discriminability of similar objects with selfsupervised learning. • We extend the local correlation network to model temporal information efficiently. • CorrTracker shows significant improvements over existing state-of-the-art results in four MOT benchmarks.\\nIn particular, we achieve 76.5% MOTA and IDF1 of 73.6% on MOT17.\"\n },\n {\n \"title\": \"Related Work\",\n \"content\": \"Real-time Tracking. As MOT has strong practical merit, the tracking speed attracts much attention. The researchers start from the simplest IOUTracker [4], which only uses the intersection-over-union of bounding boxes for tracking, to add the motion model of Kalman Filter [3] to predict the position of the rectangular boxes for matching. Although they have achieved amazing speed, stable tracking cannot be achieved under challenges such as target interleaving. Researchers [27,51] introduce Person Re-Identification (ReID) features as an appearance model to increase the discriminative power of the tracker. However, the individual calculation for patches makes the object classification and ReID feature extraction as a computational bottleneck. MOTDT [27] achieves real-time tracking by using RoI-pooling [14] on a shared feature map. In order to further decrease the computational cost of ReID feature extraction, JDE [50] adds a ReID branch in a single-stage detector YOLOv3 [35] to achieve efficient ReID feature calculation. FairMOT [58] explores the importance of detection and recognition tasks and uses anchor-free method [65] to reduce the ambiguity of anchors. We are mainly based on FairMOT, which achieves the state-of-the-art performance with a more balanced ReID and detection.\\nOther researchers [1,64,34,32] explore new tracking paradigms to remove ReID recognition. Tracktor [1] uses the the bounding boxes in the previous frame to directly regress the current locations. CenterTrack [64], Chained-Tracker [34], and TubeTK [32] use multiple frames to simultaneously predict the bounding boxes for adjacent frames to achieve short association, thereby merging to long-term tracks. However, these methods usually have many identity switches because they cannot model longterm dependencies. Tracking with Graph Model. MOT has traditionally been approached as a hand-crafted graph optimization problem [61,17], where the cropped targets are treated as nodes. Recently, graph neural network based methods [53,49,5] have been shown as a promising alternative to traditional optimization methods. State-of-the-art approach [53] utilizes graph convolutional network to propagate features in the spatial-temporal space. MPN [5] introduces the message passing network to dissect the information and associate detections through the edge classification. Different from these methods, feature propagation is carried out at the frame feature level, which can absorb the information of both the foreground and background and reduce the loss of contextual information. Tracking with Optical Flow. FlowTrack [60] introduces optical flow to predict the target location. But explicitly using optical flow is not only time-consuming, but also only encodes the pixel-level motion. CenterTrack [64] borrows the method of optical flow to directly predict the movement of the target center between two frames, which is called instance flow. However, directly predicting the offset on the concatenated feature map needs to provide training samples with all displacement, which requires excessive data augmentation. Our correlation method predicts a dense set of matching confidence for each target, which is intrinsically invariant to translation of the paired frames. Our correlation operation is similar to the correlation volume in optical flow [10,40,42] and correlation filter [7]. We both predict dense local correlation, and regard it as a part of the feature description. However, optical flow does not calculate the internal correlation of the image, nor does it have propagated the feature from multiple frames. D&T [12] also utilizes correlation layer to predict candidate motion between pair of consecutive frames. Compared with it, our anchor-free framework is more compact and efficient. Attention Mechanism. Our modeling of local correlation is similar to the self-attention mechanism and Transformer. Transformer has been a huge success in the NLP field [43] and has also been adapted to the computer vision [48,18,28] to capture long-range dependencies. In order to reduce the quadratic complexity of the non-local operation, the researchers propose to shrink the attention span with local region [33], or only along individual axes [47]. Different from these methods, we mainly encode the context identity through local correlation weighting, and use this cues to increase the model robustness. Figure 2 shows the overall pipeline of the proposed Cor-rTracker. Our method can be distilled down to three stages: (1) general feature extraction, (2) simultaneous learning correlation from spatial-temporal dependencies and predicting the detection, and (3) performing data association to assign detections into their most likely trajectories, where stage (1) and stage (2) are differentiable and composed into an end-to-end trainable architecture. We adopt a compact association technique that is similar to the one used by DeepSORT [51] to control the initialization and the termination of tracks. The main contribution is the highly efficient modeling for the correlation between dense location and their context on feature maps, which helps suppressing distractors in complex scenes.\"\n },\n {\n \"title\": \"Methodology\",\n \"content\": \"\"\n },\n {\n \"title\": \"Motivation\",\n \"content\": \"For each input video frame I t ∈ R H×W ×3 , an object detector is applied to find all candidate\\nappearing in this frame and we have existing trajectories\\nThen the affinity matrix A ∈ R N ×M is estimated by pair-wise comparisons of cropped patches and existing trajectories. The metric jointly considers both the appearance features f (·) ∈ R d and geometric representations.\\nThe discriminative featuref (T j t−1 ) of a trajectory is usually updated with a constant-weighting strategy to follow the appearance changes. Each confidence value for appearance feature is obtained in a distance metric, e.g., the inner product space. However, the sole reliance on person-to-person feature comparisons are often insufficient to disambiguate multiple similar regions in an image. As illustrated in Fig. 1, in the case of similar distractors, the feature extractor usually generates inaccurate and uninformative matching confidences ( Fig. 1a), severely affecting the performance of data association. This is the key limitation of appearance feature matching, since co-occurring similar objects are all pervasive in MOT.\\nPatch based feature extraction is applied as a prevalent scheme in MOT owing to its intuition. However the correlation information between the cropped image patches is lost directly, and the adjacency spatial relationship is only retained in coordinates d i t . Although the subsequent data association will be globally optimized, directly using ReID features without considering the context tends to introduce more identity switches, lagging the tracking robustness. To deal with this problem, we model the local structure of objects to distinguish it from distractors.\\nInspired by correlation volume from optical flow [10], we observe that a confidence value in the correlation vol-ume models the geometric structure of each target. We design a novel dense correlation module, aiming to explore the context information for MOT. The relative position is encoded in the correlation volumes, which can be used as an auxiliary discriminant information.\"\n },\n {\n \"title\": \"Spatial Local Correlation Layers\",\n \"content\": \"In this work, we use Spatial Local Correlation Layers to model the relational structure for associating a target with its neighbour. In our local correlation layer, the feature similarity is only evaluated in the neighbourhood of the target image coordinate. Formally, we let l denote the level in the feature pyramid and the correlation volume C l between the query feature F l q ∈ R H l ×W l ×d l and reference feature F l r ∈ R H l ×W l ×d l is defined as,\\nwhere x ∈ Z 2 is a coordinate in the query feature map and d ∈ Z 2 is the displacement from this location. The displacement is constrained to d ∞ ≤ R , i.e. the maximum motion in any direction is R. While most naturally thought of as a 4-dimensional tensor, the two displacement dimensions are usually vectorized into one to simplify further processing in the CNN. The resulting 3-d correlation volume C l thus is of size H l × W l × (2R + 1) 2 . We also introduce the dilation tricks [55], which can increase the receptive field without additional cost. We use element-wise addition to incorporate the correlation feature into a unified appearance representation. This context correlation features are encoded by a feed-forward Multilayer Perceptron (MLP) to match the number of channels d l in appearance features F l t .\\nThe non-local [48] module is to explicitly model all pairwise interactions between elements in a feature maps F l t ∈ R H l ×W l ×d l . The resulting four-dimensional correlation volume N L(F l t ) ∈ R H l ×W l ×H l ×W l captures dense matching confidences between every pair of image locations. They build a full connection volume at a single scale, which is both computationally expensive and memory intensive. By contrast, our work shows that constructing a local correlation volume leads to both effective and efficient models. In comparison with the global correlation method, our local correlation model adds less overhead to the latency (see Table 1).\"\n },\n {\n \"title\": \"Correlation at Multiple Pyramid Levels\",\n \"content\": \"In order to achieve long-range correlation, we propose to learn correlation at feature pyramids, as shown in Figure 3. On the one hand, we hope that our correlation module can obtain long-distance dependencies as much as possible, but as the local region size R increases, both calculation and storage increase significantly, which hinders the application. On the other hand, MOT naturally needs to deal with multi-scale targets. The two-stage detection [36] uses RoI pooling [14] to eliminate the difference in target scales, but this type of method usually suffers from high processing latency. In order to solve the above problems, we utilize the general pyramid structure in the convolutional network and learn correlation on the feature pyramids. Our multi-scale pyramid correlation can also be regarded as a comparison of multi-granularity features, covering the spatial context in the range of [0, R × D × 2 l ], where D refers to the dilation rate. And, we pass this correlation from the top layer to the bottom layer,\\nIn this way, we can obtain an approximate correlation between the target and the entire global context, while keep the compactness and efficiency. Our pyramid correlation leverages the natural spatial-temporal coherence in videos.\\nMulti-object tracking can be decomposed into multiple independent single-object tracking. Our method can be equivalent to a dense siamese network tracking [2] on the feature pyramid. On the other hand, from the perspective of set matching, global characteristics need to be considered. Our multi-scale correlation takes into account both aspects of information transmission.\"\n },\n {\n \"title\": \"Temporal Correlation Learning\",\n \"content\": \"The correlation between different frames are usually ignored by the MOT field, and trackers usually overcome occlusion through data association. Single frame detector is difficult to ensure a good temporal consistency [59]. This makes the algorithm's performance drop significantly in occlusion, motion blur and small target scenes, which becomes the bottleneck for MOT. We extend the spatial local correlation from Section 3.2 to the temporal dimension, and establish correlation for the targets in different frames. The correlation between two frames can be viewed as the establishment of motion information learning. We also use this correlation to enhance the feature representation, which can increase the detection accuracy.\\nSpecifically, we establish multi-scale correlation between different frames, and use reference images as memory to enhance image features. This method helps tracker overcome target occlusion and motion blur, and increases the consistency of detection and identity features.\\nSimilar to the multi-head attention [43], we adopt the embedded features and dot-product similarity. In our case, we set the normalization factor as (2R + 1) 2 and locally aggregate features. This shrinked region design also comes from the motion prior of the MOT scene. For the minimal memory consumption and fastest run-time, we can only save the previous features F t−1 in the memory. For the maximum accuracy, our long-term model saves the latest 5 frame image features by default.\"\n },\n {\n \"title\": \"Self-supervised Feature Learning\",\n \"content\": \"In Section 3.2 and Section 3.4, we present how we model the correlation in spatial and temporal dimension. We can simply use the proposed correlation module as a plugin module without explicitly adding constraints, similar to the non-local module, which has shown significant improvement. Here we investigate a multi-task learning approach that imposes a semantic supervision from visual object tracking [2] and self-supervised training from correspondence flow [45] on correlation volumes.\\nOur correlation module is interpretable, measuring the similarity between different objects. Actually, our method intensively performs M ×N siamese tracking operations [2] to increases the discrimination. In this view, we can explicitly impose a tracking supervision. Specifically, we set up the ground-truth label as\\nwhere y is the identity label of the corresponding position in feature maps. We ignore the position without objects y q (x) < 0 and use a class-balanced cross-entropy loss [2]. Inspired by the recent advances of self-supervised tracking [45], we use colorization as a proxy task for training our local correlation.\\nwe use the cross-entropy categorical loss after quantizing the color-space into discrete categories [45].\"\n },\n {\n \"title\": \"Tracking Framework\",\n \"content\": \"We modify the FairMOT [58] backbone by adding correlation module before the iterative deep aggregation module [57]. Our model retains the detection and ReID branches, and adding correlation loss in Sec. 3.5 for multitask learning. For the tracking inference, our tracker first calculates the similarity between the detections of the current frame and the previous trajectories according to Eq. (2), and use the Hungarian algorithm [23] for finding the optimal matching. The unmatched detections are used to initialize new trajectories. In order to reduce false positives, we mark these new trajectories as \\\"inactive\\\" until the next frame is matched again and confirmed as \\\"active\\\". The unmatched trajectories are set to the \\\"lost\\\" state. When the continuous lost time t loss of a trajectory exceeds the threshold τ loss , we put it in the remove set. If there is a success matching before removing, we restore the trajectory to the active state. We use Kalman Filter [20] to model the pedestrian motion and keep the same settings in FairMOT [58].\"\n },\n {\n \"title\": \"Experiments\",\n \"content\": \"To demonstrate the advantages of the proposed correlation tracker, we first compare the correlation module with other relational reasoning methods [48,46] and evaluate different settings to justify our design choices in Section 4.2. Then we show that our correlation tracker outperforms the state-of-the-art methods on four MOT benchmarks [24,30,8] in Section 4.3. Finally, we visualize the tracking trajectories in Section 4.4 and compare with other motion prediction based trackers [64,34,32].\"\n },\n {\n \"title\": \"Implementation Details\",\n \"content\": \"Network Setup.\\nThe implementation and hyperparameters mostly follow [58], we adopt CenterNet [65] detector with a variant of Deep Layer Aggregation (DLA-34) [57] as backbone and utilize the iterative deep aggregation module (IDA) to recover a high-resolution feature map with stride 4. We also add a 3 × 3 deformable convolution layer [66] before every upsampling stage. The backbone network is initialized with the parameters pre-trained on COCO [26] and then pre-trained on CrowdHuman [39] with self-supervised learning as FairMOT [58]. The proposed correlation module is augmented before IDA module to fuse multi-scale correlation. For the correlation module, we set local region size R = 5 and dilation rate D = 2.\\nTraining and Validation Datasets. For a fair comparison, we also use the default training datasets as Fair-MOT [58]. There are six training datasets including the ETH [11], CityPerson [62], CalTech [9], CUHK-SYSU [52], PRW [63] and MOT17 [30]. ETH and CityPerson only provide box annotations, so we ignore the ReID losses from these datasets. CalTech, CUHK-SYSU, PRW We train on an input resolution of 1088 × 608, which yields an output resolution of 272 × 152. We use random flip, random scaling (between 0.5 to 1.2), cropping, and color jittering as data augmentation, and use Adam [22] to optimize the overall objective. The learning rate is initialized as 1e −4 and then decayed to 1e −5 in the last 10 epochs. We train with a batch-size of 12 (on 2 GPUs) for 30 epochs. In the training phase, we sample 5 temporally ordered frames within a random interval of less than 3.\\nTest Datasets and Evaluation Metrics. We evaluate the performance of our correlation tracker on the 2DMOT2015 [24], MOT16 [30], MOT17 [30], and MOT20 [8]. In particular, 2DMOT2015 [24] contains 11 test videos. MOT16 [30] and MOT17 [30] contain same 7 test videos with part different annotations. The MOT20 [8] consists of 4 test videos on extremely crowded scenes, which makes them really challenging.\\nWe adopt the standard metrics of MOT Benchmarks for evaluation, including Multiple Object Tracking Accuracy (MOTA) [21], ID F1 Score [37], Mostly tracked targets (MT), Mostly lost targets (ML), the number of False Positives (FP), the number of False Negatives (FN), and the number of Identity Switch (ID Sw.) [25]. The run time is also provided and evaluated on a NVIDIA Tesla V100 GPU.\"\n },\n {\n \"title\": \"Ablation Studies\",\n \"content\": \"To elaborate on the effectiveness of the proposed approach, we conduct extensive ablation studies. First, we give detailed correlation analysis with different settings to justify our design choices, as presented in Table. 1. Next, the tracking accuracy and runtime for different region sizes of the correlation module is explored. Different building blocks are compared to illustrate the effectiveness and efficiency of the full correlation tracker. Spatial correlation. In order to evaluate the effectiveness Table 2. Ablation studies on MOT17 validation set. \\\"LT\\\" and \\\"Self\\\" denote using the proposed long-term memory and selfsupervised loss, respectively. of our spatial local correlation module (SLC), we compared our baseline model [58] and two relation methods, the nonlocal module [48] and CorrNet [46]. Directly use of nonlocal brings performance degradation on MOT since nonlocal module does not record the relative positional relationship between the targets and usually brings performance drops on small object [54]. In addition, non-local method has a huge overhead in memory and computation. In video recognition, a learnable correlation filter network [46] is proposed, and the grouped convolution is used to reduce the amount of calculation. Our method has achieved similar MOTA compared to CorrNet, but has a large improvement on IDF1, which comes from our multi-scale correlation design. Compared with the baseline, our IDF1 has increased by 2.9%, and identity switches have been reduced by 15%, demonstrating the discrimination of our spatial correlation. The Re-ID embeddings cannot easily distinguish similar distractor, our correlation feature models geometric information and is better suited for MOT. Temporal correlation. In the field of video object detection [38], temporal and global information are commonly used to improve performance [6]. The research of temporal detection for MOT is still preliminary. Recently, CenterTrack [64] concatenates the previous frame and the current frame, and Chained-Tracker [34] concatenates the high-level image features for two frames to fuse temporal information. We compare these two methods, dubbed concat-raw and concat-feat, with our temporal local correlation module (TLC). These two methods brought 0.2% and 1.3% MOTA improvements over the single frame baseline. Compared with these two methods, our temporal local correlation module achieves consistent improvements in both MOTA and IDF1. Our temporal correlation module helps for the temporal feature alignment around frames. At the same time, our method only adds a small overhead to feature-level concatenation, which proves the efficiency of our algorithm. Compared with the baseline FairMOT, employing both spatial and temporal local correlation (STLC) yields a IDF1 of 76.1% , which brings 3.2% improvement.\\nLong-term dependences. We also analyze the performance improvement with long-term correlation. Compared with the method using two frames as the source cues, our long-term method achieves a large improvement of 0.6% MOTA, due to the increased capacity for object detections.\\nThe improvement of MOTA also means an increase in the upper bound of our tracker. Self-supervised learning. For correlation learning, explicit supervision is usually not imposed [48,46]. We have proposed two supervision methods in section 3.5. It can be seen that the siamese tracking supervision imposed to training has achieved a relatively good improvement in IDF1.\\nThere is no change in the run time of our algorithm, because the change in training loss does not change the inference processing. Self-supervised losses have also been appropriately improved on both MOTA and IDF1 due to more positive samples employed in correlation volume.\\nChoice of local region. Figure 4 shows the MOTA and run time of our correlation module for different region size R ∈ {1, 2, ..., 8}. As expected, a larger local size R can cover a larger neighborhood while matching pixels, thus yields a higher accuracy. But the improvements become marginal beyond R = 5, possibly due to the low resolution of the feature maps. Note that non-local module usually doubles the run time of the backbone and the cost of explicitly computing optical flow [10] can be very high as well. This shows that our correlation module is more efficient by learning motion information from features directly. Region size R = 5 yields a good trade-off between speed and accuracy. The computation overhead is relatively small, compared to the complexity of the whole detection networks.\"\n },\n {\n \"title\": \"Experiments on MOT Challenges\",\n \"content\": \"To extensively evaluate the proposed method, we compare it with 8 state-of-the-art trackers, which cover most of current representative methods. There are 2 joint detection and embedding methods (JDE [50] and FairMOT [58]), 2 multi-frame prediction methods (Tube TK [32] and CTracker [34], 2 graph network based methods (MPN [5] and JDMOTGNN [49]), 2 offset prediction based methods (CenterTracker [64] and Tracktor++v2 [1]). The results are summarized in Table 3.\\n2DMOT2015 [24]. The evaluation on 2DMOT2015 is performed by the official toolkit. As shown in Table 3, our correlation tracker outperforms the top private method of 2DMOT2015, (i.e., FairMOT [58]), by 1.7% in MOTA and 1.0% in IDF1. It is worth noting that the ID Switches are decreased by 13%, which shows the robustness of our cor- Table 3. Comparisons of our method with state-of-the-arts on MOT benchmarks [24,30,8]. We set new state-of-the-art results by a significant margin in terms of MOTA and IDF1. Our correlation tracker is more accurate while running with high speed. relation module. Moreover, our tracker is superior to the recent end-to-end graph trackers JDMOTGNN [49]. Our feature propagation approach can absorb both foreground and background information, which improves our tracker by 1.6% in terms of MOTA. MOT16 [30] and MOT17 [30]. Table 3 reports the evaluation results with the comparisons to recent prevailing trackers on MOT16. The recent proposed FairMOTv2 [58] achieves the second performance in MOTA and IDF1, while our method ranks first with 76.6% MOTA, outperforming other private approaches by a significant margin. Our CorrTracker achieves the best performance on MOT17, surpassing FairMOTv2 by 2.8% MOTA and 1.3% IDF1. Moreover, the FN of our CorrTracker surpasses FairMOTv2 by 15%, which means nearly 20,000 new bounding boxes are added to the association process. In this case, our algorithm still maintains comparable or even superior ID Switches, which actually proves that our method significantly improves the tracking association. As reported in Table 3, our CorrTracker, CenterTrack [64] and CTrack [34] all use multi-frame cues to predict detections, our FN is largely decreased by 30%.\\nMOT20 [8]. To further evaluate the proposed models, we report the results on MOT20, which is more challenging than MOT17. The final results is presented in the bottom block of Table 3. Our CorrTracker achieves MOTA score of 65.2%, substantially outperforming FairMOTv2 [58] with MOTA of 61.8%. Although our approach is an order of magnitude faster than JDMOTGNN [49] in speed, our accuracy is slightly worse due to the anchor-free design. MOT20-06 MOT20-08 Figure 6. Qualitative results of our correlation tracker on MOT17 [30] and MOT20 [8]. The color of each bounding box indicates the target identity. The dotted line under each bounding box denotes the recent tracklet of each target. The proposed tracker predicts trajectories with substantially robust and temporally consistent.\"\n },\n {\n \"title\": \"Visualization\",\n \"content\": \"We visualize the tracking trajectories for prior methods, i.e., the center offset [64] and multi-frame bounding boxes [34,32], in Figure 5. We observe that our correlation map focuses on the entire context, while the regular appearance feature concentrates on the local region of the target. Our correlation module improves the reliability of recognition since it provides a global view of the target. Methods based on offset prediction, e.g., Cen-terTrack [64] and CTracker [34], can easily generate id switches when encountered with complex object interactions. Figure 6 shows qualitative results of our Correlation Tracker on MOT17 and MOT20, the advantage over existing method is most pronounced on the robust to occlusion and tiny objects.\"\n },\n {\n \"title\": \"Conclusion\",\n \"content\": \"In this work, we propose a novel correlation tracking framework based upon the observation that the relational structure helps to distinguish similar objects. Our correlation module densely matches all targets with their local context and learn a discriminative embeddings from the correlation volumes. Furthermore, we show how to extend the correlation module from spatial layout to the adjacent frames for strengthening the temporal modeling ability. We explore that self-supervised learning to impose a discriminative constraint on the correlation volume, which explicitly predicts a instance flow. Extensive experiments on four MOT challenges demonstrate that our CorrTracker achieves state-of-the-art performance and is efficient in inference.\"\n },\n {\n \"title\": \"Figure 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 1 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Visualization of the matching confidences (a)-(b) computed between the indicated target (white cross) in the reference image and all locations of the query image. The appearance feature based tracker[58] (a) generates undistinctive and inaccurate confidences due to the existing of similar distractors. In contrast, our Correlation Tracker (b) predicts a distinct high-confidence value at the correct location, with correlation learning (c).","content":"Figure 1 .\n1Visualization of the matching confidences (a)-(b) computed between the indicated target (white cross) in the reference image and all locations of the query image. The appearance feature based tracker[58] (a) generates undistinctive and inaccurate confidences due to the existing of similar distractors. In contrast, our Correlation Tracker (b) predicts a distinct high-confidence value at the correct location, with correlation learning (c)."},{"id":"","caption":"Overview. We enhance the appearance features with correlation layer, which densely encoding pairwise relation of object and their spatial-temporal neighbourhoods. The local correlation volumes are optimized in a self-supervised manner.","content":"Figure 2 .\n2Overview. We enhance the appearance features with correlation layer, which densely encoding pairwise relation of object and their spatial-temporal neighbourhoods. The local correlation volumes are optimized in a self-supervised manner."},{"id":"","caption":"Correlation at Multiple Pyramid Levels. For a feature tensor in F l , we take the inner product with local region (R = 1, D = 2) to generate a 3-d correlation volume W l × H l × (2R + 1) 2 , D is the dilation rate.","content":"Figure 3 .\n3Correlation at Multiple Pyramid Levels. For a feature tensor in F l , we take the inner product with local region (R = 1, D = 2) to generate a 3-d correlation volume W l × H l × (2R + 1) 2 , D is the dilation rate."},{"id":"","caption":"Effect of filter size R on speed and MOTA accuracy on MOT17-val.","content":"Figure 4 .\n4Effect of filter size R on speed and MOTA accuracy on MOT17-val."},{"id":"","caption":"Qualitative comparisons against several prior methods[58,64,34,32] in occlusion situations. Frames are sampled from MOT17-03. Our CorrTracker can identify objects via mining the context patterns around targets.","content":"Figure 5 .\n5Qualitative comparisons against several prior methods[58,64,34,32] in occlusion situations. Frames are sampled from MOT17-03. Our CorrTracker can identify objects via mining the context patterns around targets."},{"id":"","caption":"Evaluation of correlation architecture on the MOT17[30] validation set.","content":"Table 1 .\n1Evaluation of correlation architecture on the MOT17[30] validation set.Method \nTwo \nframes \nMOTA ↑ \nIDF1↑ \nID Sw. ↓ Speed↑ \n\nbaseline [58] \n\n69.1% \n72.9% \n299 \n25.6 \nnon-local [48] \n\n67.7% \n70.4% \n311 \n16.60 \nCorrNet [46] \n\n70.0% \n73.3% \n303 \n22.93 \nSLC (ours) \n\n70.3% \n75.8% \n258 \n20.19 \nconcat-raw [64] \n\n69.3% \n74.1% \n336 \n23.99 \nconcat-feat [34] \n\n70.4% \n74.0% \n308 \n19.77 \nTLC (ours) \n\n70.9% \n74.7% \n326 \n19.26 \nSTLC (ours) \n\n71.5% \n76.1% \n307 \n16.56 \n\nand MOT17 provide both box and identity annotations \nwhich allows us to train both branches. We de-duplicate \nthe overlaps between ETH datasets and MOT16 for fair \ncomparison. For all validation experiments, we use the \nsix datasets mentioned above and the first half frames of \nMOT17 as training, and the second half of MOT17 as the \nvalidation set."},{"id":"","caption":"Method MOTA ↑ IDF1 ↑ MT ↑ ML ↓ FP ↓ FN ↓ ID Sw. ↓ Hz ↑","content":"Method MOTA ↑ IDF1 ↑ MT ↑ ML ↓ FP ↓ FN ↓ ID Sw. ↓ Hz ↑2D MOT 2015 [24] \n\nCorrTracker (ours) \n62.3 \n65.7 49.0 12.9 6909 \n15728 \n513 \n17.9 \nJDMOTGNN [49] \n60.7 \n64.6 47.0 10.5 7334 \n16358 \n477 \n2.4 \nFairMOTv2 [58] \n60.6 \n64.7 47.6 11.0 7854 \n15785 \n591 \n30.5 \nTube TK POI [32] \n58.4 \n53.1 39.3 18.0 5756 \n18961 \n854 \n5.8 \nMPN [5] \n51.5 \n58.6 31.2 25.9 7620 \n21780 \n375 \n6.5 \nTracktor++v2 [1] \n46.6 \n47.6 18.2 27.9 4624 \n26896 \n1290 \n1.8 \n\nMOT16 [30] \n\nCorrTracker (ours) \n76.6 \n74.3 47.8 13.3 10860 30756 \n979 \n14.8 \nFairMOTv2 [58] \n74.9 \n72.8 44.7 15.9 10163 34484 \n1074 \n25.4 \nCTracker [34] \n67.6 \n57.2 32.9 19.0 8934 \n48305 \n1897 \n6.8 \nTube TK POI [32] \n66.9 \n62.2 39.0 16.1 11544 47502 \n1236 \n1.0 \nPOI [56] \n66.1 \n65.1 34.0 20.8 5061 \n55914 \n805 \n9.9 \nTracktor++v2 [1] \n56.2 \n54.9 20.7 35.8 2394 \n76844 \n617 \n1.8 \n\nMOT17 [30] \n\nCorrTracker (ours) \n76.5 \n73.6 47.6 12.7 29808 99510 \n3369 \n14.8 \nFairMOTv2 [58] \n73.7 \n72.3 43.2 17.3 27507 117477 \n3303 \n25.9 \nCenterTrack [64] \n67.8 \n64.7 34.6 24.6 18498 160,332 3039 \n3.8 \nCTracker [34] \n66.6 \n57.4 32.2 24.2 22284 160491 \n5529 \n6.8 \nTracktor++v2 [1] \n56.3 \n55.1 21.1 35.3 8866 235449 \n1987 \n1.8 \n\nMOT20 [8] \n\nCorrTracker (ours) \n65.2 \n69.1 66.4 8.9 79429 95855 \n5183 \n8.5 \nJDMOTGNN [49] \n67.1 \n67.5 53.1 13.2 31913 135409 \n3131 \n0.9 \nFairMOTv2 [58] \n61.8 \n67.3 68.8 7.6 103440 88901 \n5243 \n13.2"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Visualization of the matching confidences (a)-(b) computed between the indicated target (white cross) in the reference image and all locations of the query image. The appearance feature based tracker[58] (a) generates undistinctive and inaccurate confidences due to the existing of similar distractors. In contrast, our Correlation Tracker (b) predicts a distinct high-confidence value at the correct location, with correlation learning (c).\",\n \"content\": \"Figure 1 .\\n1Visualization of the matching confidences (a)-(b) computed between the indicated target (white cross) in the reference image and all locations of the query image. The appearance feature based tracker[58] (a) generates undistinctive and inaccurate confidences due to the existing of similar distractors. In contrast, our Correlation Tracker (b) predicts a distinct high-confidence value at the correct location, with correlation learning (c).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Overview. We enhance the appearance features with correlation layer, which densely encoding pairwise relation of object and their spatial-temporal neighbourhoods. The local correlation volumes are optimized in a self-supervised manner.\",\n \"content\": \"Figure 2 .\\n2Overview. We enhance the appearance features with correlation layer, which densely encoding pairwise relation of object and their spatial-temporal neighbourhoods. The local correlation volumes are optimized in a self-supervised manner.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Correlation at Multiple Pyramid Levels. For a feature tensor in F l , we take the inner product with local region (R = 1, D = 2) to generate a 3-d correlation volume W l × H l × (2R + 1) 2 , D is the dilation rate.\",\n \"content\": \"Figure 3 .\\n3Correlation at Multiple Pyramid Levels. For a feature tensor in F l , we take the inner product with local region (R = 1, D = 2) to generate a 3-d correlation volume W l × H l × (2R + 1) 2 , D is the dilation rate.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Effect of filter size R on speed and MOTA accuracy on MOT17-val.\",\n \"content\": \"Figure 4 .\\n4Effect of filter size R on speed and MOTA accuracy on MOT17-val.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Qualitative comparisons against several prior methods[58,64,34,32] in occlusion situations. Frames are sampled from MOT17-03. Our CorrTracker can identify objects via mining the context patterns around targets.\",\n \"content\": \"Figure 5 .\\n5Qualitative comparisons against several prior methods[58,64,34,32] in occlusion situations. Frames are sampled from MOT17-03. Our CorrTracker can identify objects via mining the context patterns around targets.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Evaluation of correlation architecture on the MOT17[30] validation set.\",\n \"content\": \"Table 1 .\\n1Evaluation of correlation architecture on the MOT17[30] validation set.Method \\nTwo \\nframes \\nMOTA ↑ \\nIDF1↑ \\nID Sw. ↓ Speed↑ \\n\\nbaseline [58] \\n\\n69.1% \\n72.9% \\n299 \\n25.6 \\nnon-local [48] \\n\\n67.7% \\n70.4% \\n311 \\n16.60 \\nCorrNet [46] \\n\\n70.0% \\n73.3% \\n303 \\n22.93 \\nSLC (ours) \\n\\n70.3% \\n75.8% \\n258 \\n20.19 \\nconcat-raw [64] \\n\\n69.3% \\n74.1% \\n336 \\n23.99 \\nconcat-feat [34] \\n\\n70.4% \\n74.0% \\n308 \\n19.77 \\nTLC (ours) \\n\\n70.9% \\n74.7% \\n326 \\n19.26 \\nSTLC (ours) \\n\\n71.5% \\n76.1% \\n307 \\n16.56 \\n\\nand MOT17 provide both box and identity annotations \\nwhich allows us to train both branches. We de-duplicate \\nthe overlaps between ETH datasets and MOT16 for fair \\ncomparison. For all validation experiments, we use the \\nsix datasets mentioned above and the first half frames of \\nMOT17 as training, and the second half of MOT17 as the \\nvalidation set.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Method MOTA ↑ IDF1 ↑ MT ↑ ML ↓ FP ↓ FN ↓ ID Sw. ↓ Hz ↑\",\n \"content\": \"Method MOTA ↑ IDF1 ↑ MT ↑ ML ↓ FP ↓ FN ↓ ID Sw. ↓ Hz ↑2D MOT 2015 [24] \\n\\nCorrTracker (ours) \\n62.3 \\n65.7 49.0 12.9 6909 \\n15728 \\n513 \\n17.9 \\nJDMOTGNN [49] \\n60.7 \\n64.6 47.0 10.5 7334 \\n16358 \\n477 \\n2.4 \\nFairMOTv2 [58] \\n60.6 \\n64.7 47.6 11.0 7854 \\n15785 \\n591 \\n30.5 \\nTube TK POI [32] \\n58.4 \\n53.1 39.3 18.0 5756 \\n18961 \\n854 \\n5.8 \\nMPN [5] \\n51.5 \\n58.6 31.2 25.9 7620 \\n21780 \\n375 \\n6.5 \\nTracktor++v2 [1] \\n46.6 \\n47.6 18.2 27.9 4624 \\n26896 \\n1290 \\n1.8 \\n\\nMOT16 [30] \\n\\nCorrTracker (ours) \\n76.6 \\n74.3 47.8 13.3 10860 30756 \\n979 \\n14.8 \\nFairMOTv2 [58] \\n74.9 \\n72.8 44.7 15.9 10163 34484 \\n1074 \\n25.4 \\nCTracker [34] \\n67.6 \\n57.2 32.9 19.0 8934 \\n48305 \\n1897 \\n6.8 \\nTube TK POI [32] \\n66.9 \\n62.2 39.0 16.1 11544 47502 \\n1236 \\n1.0 \\nPOI [56] \\n66.1 \\n65.1 34.0 20.8 5061 \\n55914 \\n805 \\n9.9 \\nTracktor++v2 [1] \\n56.2 \\n54.9 20.7 35.8 2394 \\n76844 \\n617 \\n1.8 \\n\\nMOT17 [30] \\n\\nCorrTracker (ours) \\n76.5 \\n73.6 47.6 12.7 29808 99510 \\n3369 \\n14.8 \\nFairMOTv2 [58] \\n73.7 \\n72.3 43.2 17.3 27507 117477 \\n3303 \\n25.9 \\nCenterTrack [64] \\n67.8 \\n64.7 34.6 24.6 18498 160,332 3039 \\n3.8 \\nCTracker [34] \\n66.6 \\n57.4 32.2 24.2 22284 160491 \\n5529 \\n6.8 \\nTracktor++v2 [1] \\n56.3 \\n55.1 21.1 35.3 8866 235449 \\n1987 \\n1.8 \\n\\nMOT20 [8] \\n\\nCorrTracker (ours) \\n65.2 \\n69.1 66.4 8.9 79429 95855 \\n5183 \\n8.5 \\nJDMOTGNN [49] \\n67.1 \\n67.5 53.1 13.2 31913 135409 \\n3131 \\n0.9 \\nFairMOTv2 [58] \\n61.8 \\n67.3 68.8 7.6 103440 88901 \\n5243 \\n13.2\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Tracking without bells and whistles. Philipp Bergmann, Tim Meinhardt, Laura Leal-Taixe, Proceedings of the IEEE international conference on computer vision. the IEEE international conference on computer visionPhilipp Bergmann, Tim Meinhardt, and Laura Leal-Taixe. Tracking without bells and whistles. In Proceedings of the IEEE international conference on computer vision, pages 941-951, 2019.","doi":""},{"text":"Fully-convolutional siamese networks for object tracking. Luca Bertinetto, Jack Valmadre, Joao F Henriques, Andrea Vedaldi, Philip Hs Torr, European conference on computer vision. SpringerLuca Bertinetto, Jack Valmadre, Joao F Henriques, Andrea Vedaldi, and Philip HS Torr. Fully-convolutional siamese networks for object tracking. In European conference on computer vision, pages 850-865. Springer, 2016.","doi":""},{"text":"Simple online and realtime tracking. Alex Bewley, Zongyuan Ge, Lionel Ott, Fabio Ramos, Ben Upcroft, 2016 IEEE International Conference on Image Processing (ICIP). IEEEAlex Bewley, Zongyuan Ge, Lionel Ott, Fabio Ramos, and Ben Upcroft. Simple online and realtime tracking. In 2016 IEEE International Conference on Image Processing (ICIP), pages 3464-3468. IEEE, 2016.","doi":""},{"text":"Highspeed tracking-by-detection without using image information. Erik Bochinski, Thomas Volker Eiselein, Sikora, 14th IEEE International Conference on Advanced Video and Signal Based Surveillance (AVSS). IEEEErik Bochinski, Volker Eiselein, and Thomas Sikora. High- speed tracking-by-detection without using image informa- tion. In 2017 14th IEEE International Conference on Ad- vanced Video and Signal Based Surveillance (AVSS), pages 1-6. IEEE.","doi":""},{"text":"Learning a neural solver for multiple object tracking. Guillem Brasó, Laura Leal-Taixé, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionGuillem Brasó and Laura Leal-Taixé. Learning a neu- ral solver for multiple object tracking. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 6247-6257, 2020.","doi":""},{"text":"Memory enhanced global-local aggregation for video object detection. Yihong Chen, Yue Cao, Han Hu, Liwei Wang, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionYihong Chen, Yue Cao, Han Hu, and Liwei Wang. Memory enhanced global-local aggregation for video object detec- tion. In Proceedings of the IEEE/CVF Conference on Com- puter Vision and Pattern Recognition, pages 10337-10346, 2020.","doi":""},{"text":"Famnet: Joint learning of feature, affinity and multi-dimensional assignment for online multiple object tracking. Peng Chu, Haibin Ling, Proceedings of the IEEE International Conference on Computer Vision. the IEEE International Conference on Computer VisionPeng Chu and Haibin Ling. Famnet: Joint learning of fea- ture, affinity and multi-dimensional assignment for online multiple object tracking. In Proceedings of the IEEE Inter- national Conference on Computer Vision, pages 6172-6181, 2019.","doi":""},{"text":"P Dendorfer, H Rezatofighi, A Milan, J Shi, D Cremers, I Reid, S Roth, K Schindler, L Leal-Taixé, arXiv:2003.09003arXiv: 2003.09003Mot20: A benchmark for multi object tracking in crowded scenes. P. Dendorfer, H. Rezatofighi, A. Milan, J. Shi, D. Cremers, I. Reid, S. Roth, K. Schindler, and L. Leal-Taixé. Mot20: A benchmark for multi object tracking in crowded scenes. arXiv:2003.09003[cs], Mar. 2020. arXiv: 2003.09003.","doi":"arXiv:2003.09003"},{"text":"Bernt Schiele, and Pietro Perona. Pedestrian detection: A benchmark. Piotr Dollár, Christian Wojek, IEEE Conference on Computer Vision and Pattern Recognition. IEEEPiotr Dollár, Christian Wojek, Bernt Schiele, and Pietro Per- ona. Pedestrian detection: A benchmark. In 2009 IEEE Con- ference on Computer Vision and Pattern Recognition, pages 304-311. IEEE, 2009.","doi":""},{"text":"Flownet: Learning optical flow with convolutional networks. Alexey Dosovitskiy, Philipp Fischer, Eddy Ilg, Philip Hausser, Caner Hazirbas, Vladimir Golkov, Patrick Van Der, Daniel Smagt, Thomas Cremers, Brox, Proceedings of the IEEE international conference on computer vision. the IEEE international conference on computer visionAlexey Dosovitskiy, Philipp Fischer, Eddy Ilg, Philip Hausser, Caner Hazirbas, Vladimir Golkov, Patrick Van Der Smagt, Daniel Cremers, and Thomas Brox. Flownet: Learning optical flow with convolutional networks. In Pro- ceedings of the IEEE international conference on computer vision, pages 2758-2766, 2015.","doi":""},{"text":"A mobile vision system for robust multi-person tracking. Andreas Ess, Bastian Leibe, Konrad Schindler, Luc Van Gool, 2008 IEEE Conference on Computer Vision and Pattern Recognition. IEEEAndreas Ess, Bastian Leibe, Konrad Schindler, and Luc Van Gool. A mobile vision system for robust multi-person tracking. In 2008 IEEE Conference on Computer Vision and Pattern Recognition, pages 1-8. IEEE, 2008.","doi":""},{"text":"Detect to track and track to detect. Christoph Feichtenhofer, Axel Pinz, Andrew Zisserman, Proceedings of the IEEE International Conference on Computer Vision. the IEEE International Conference on Computer VisionChristoph Feichtenhofer, Axel Pinz, and Andrew Zisserman. Detect to track and track to detect. In Proceedings of the IEEE International Conference on Computer Vision, pages 3038-3046, 2017.","doi":""},{"text":"Object detection with discriminatively trained part-based models. F Pedro, Ross B Felzenszwalb, David Girshick, Deva Mcallester, Ramanan, IEEE transactions on pattern analysis and machine intelligence. 32Pedro F Felzenszwalb, Ross B Girshick, David McAllester, and Deva Ramanan. Object detection with discriminatively trained part-based models. IEEE transactions on pattern analysis and machine intelligence, 32(9):1627-1645, 2009.","doi":""},{"text":"Fast r-cnn. Ross Girshick, Proceedings of the IEEE international conference on computer vision. the IEEE international conference on computer visionRoss Girshick. Fast r-cnn. In Proceedings of the IEEE inter- national conference on computer vision, pages 1440-1448, 2015.","doi":""},{"text":"Tracking with multi-level features. Roberto Henschel, Laura Leal-Taixé, Bodo Rosenhahn, Konrad Schindler, arXiv:1607.07304arXiv preprintRoberto Henschel, Laura Leal-Taixé, Bodo Rosenhahn, and Konrad Schindler. Tracking with multi-level features. arXiv preprint arXiv:1607.07304, 2016.","doi":""},{"text":"Multiple people tracking using body and joint detections. Roberto Henschel, Yunzhe Zou, Bodo Rosenhahn, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops. the IEEE Conference on Computer Vision and Pattern Recognition WorkshopsRoberto Henschel, Yunzhe Zou, and Bodo Rosenhahn. Mul- tiple people tracking using body and joint detections. In Pro- ceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, pages 0-0, 2019.","doi":""},{"text":"Lifted disjoint paths with application in multiple object tracking. Andrea Hornakova, Roberto Henschel, Bodo Rosenhahn, Paul Swoboda, The 37th International Conference on Machine Learning (ICML). Andrea Hornakova, Roberto Henschel, Bodo Rosenhahn, and Paul Swoboda. Lifted disjoint paths with application in multiple object tracking. In The 37th International Con- ference on Machine Learning (ICML), July 2020.","doi":""},{"text":"Relation networks for object detection. Han Hu, Jiayuan Gu, Zheng Zhang, Jifeng Dai, Yichen Wei, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionHan Hu, Jiayuan Gu, Zheng Zhang, Jifeng Dai, and Yichen Wei. Relation networks for object detection. In Proceed- ings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3588-3597, 2018.","doi":""},{"text":"Computer vision for autonomous vehicles: Problems, datasets and state of the art. Joel Janai, Fatma Güney, Aseem Behl, Andreas Geiger, Foundations and Trends® in Computer Graphics and Vision. 121-3Joel Janai, Fatma Güney, Aseem Behl, Andreas Geiger, et al. Computer vision for autonomous vehicles: Problems, datasets and state of the art. Foundations and Trends® in Computer Graphics and Vision, 12(1-3):1-308, 2020.","doi":""},{"text":"A new approach to linear filtering and prediction problems. R E Kalman, ASME Journal of Basic Engineering. R. E. Kalman. A new approach to linear filtering and predic- tion problems. ASME Journal of Basic Engineering, 1960.","doi":""},{"text":"Framework for performance evaluation of face, text, and vehicle detection and tracking in video: Data, metrics, and protocol. Rangachar Kasturi, Dmitry Goldgof, Padmanabhan Soundararajan, Vasant Manohar, John Garofolo, Rachel Bowers, Matthew Boonstra, Valentina Korzhova, Jing Zhang, IEEE Transactions on Pattern Analysis and Machine Intelligence. 312Rangachar Kasturi, Dmitry Goldgof, Padmanabhan Soundararajan, Vasant Manohar, John Garofolo, Rachel Bowers, Matthew Boonstra, Valentina Korzhova, and Jing Zhang. Framework for performance evaluation of face, text, and vehicle detection and tracking in video: Data, metrics, and protocol. IEEE Transactions on Pattern Analysis and Machine Intelligence, 31(2):319-336, 2008.","doi":""},{"text":"Adam: A method for stochastic optimization. P Diederik, Jimmy Kingma, Ba, arXiv:1412.6980arXiv preprintDiederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.","doi":""},{"text":"The hungarian method for the assignment problem. Harold W Kuhn, Naval research logistics quarterly. 2Harold W Kuhn. The hungarian method for the assignment problem. Naval research logistics quarterly, 2(1-2):83-97, 1955.","doi":""},{"text":"L Leal-Taixé, A Milan, I Reid, S Roth, K Schindler, Motchallenge, arXiv:1504.01942arXiv: 1504.01942Towards a benchmark for multitarget tracking. L. Leal-Taixé, A. Milan, I. Reid, S. Roth, and K. Schindler. MOTChallenge 2015: Towards a benchmark for multi- target tracking. arXiv:1504.01942 [cs], Apr. 2015. arXiv: 1504.01942.","doi":""},{"text":"Learning to associate: Hybridboosted multi-target tracker for crowded scene. Yuan Li, Chang Huang, Ram Nevatia, 2009 IEEE Conference on Computer Vision and Pattern Recognition. IEEEYuan Li, Chang Huang, and Ram Nevatia. Learning to asso- ciate: Hybridboosted multi-target tracker for crowded scene. In 2009 IEEE Conference on Computer Vision and Pattern Recognition, pages 2953-2960. IEEE, 2009.","doi":""},{"text":"Microsoft coco: Common objects in context. Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, C Lawrence Zitnick, European conference on computer vision. SpringerTsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C Lawrence Zitnick. Microsoft coco: Common objects in context. In European conference on computer vision, pages 740-755. Springer, 2014.","doi":""},{"text":"Real-time multiple people tracking with deeply learned candidate selection and person re-identification. Chen Long, Ai Haizhou, Zhuang Zijie, Shang Chong, ICME. Chen Long, Ai Haizhou, Zhuang Zijie, and Shang Chong. Real-time multiple people tracking with deeply learned can- didate selection and person re-identification. In ICME, 2018.","doi":""},{"text":"See more, know more: Unsupervised video object segmentation with co-attention siamese networks. Xiankai Lu, Wenguan Wang, Chao Ma, Jianbing Shen, Ling Shao, Fatih Porikli, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionXiankai Lu, Wenguan Wang, Chao Ma, Jianbing Shen, Ling Shao, and Fatih Porikli. See more, know more: Unsuper- vised video object segmentation with co-attention siamese networks. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3623- 3632, 2019.","doi":""},{"text":"Retinatrack: Online single stage joint detection and tracking. Zhichao Lu, Vivek Rathod, Ronny Votel, Jonathan Huang, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionZhichao Lu, Vivek Rathod, Ronny Votel, and Jonathan Huang. Retinatrack: Online single stage joint detection and tracking. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 14668- 14678, 2020.","doi":""},{"text":"A Milan, L Leal-Taixé, I Reid, S Roth, K Schindler, arXiv:1603.00831arXiv: 1603.00831MOT16: A benchmark for multi-object tracking. A. Milan, L. Leal-Taixé, I. Reid, S. Roth, and K. Schindler. MOT16: A benchmark for multi-object tracking. arXiv:1603.00831 [cs], Mar. 2016. arXiv: 1603.00831.","doi":""},{"text":"A largescale benchmark dataset for event recognition in surveillance video. Sangmin Oh, Anthony Hoogs, Amitha Perera, Naresh Cuntoor, Chia-Chih Chen, Jong Taek Lee, Saurajit Mukherjee, Hyungtae Aggarwal, Larry Lee, Davis, CVPR 2011. IEEESangmin Oh, Anthony Hoogs, Amitha Perera, Naresh Cun- toor, Chia-Chih Chen, Jong Taek Lee, Saurajit Mukherjee, JK Aggarwal, Hyungtae Lee, Larry Davis, et al. A large- scale benchmark dataset for event recognition in surveillance video. In CVPR 2011, pages 3153-3160. IEEE, 2011.","doi":""},{"text":"Tubetk: Adopting tubes to track multi-object in a one-step training model. Bo Pang, Yizhuo Li, Yifan Zhang, Muchen Li, Cewu Lu, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionBo Pang, Yizhuo Li, Yifan Zhang, Muchen Li, and Cewu Lu. Tubetk: Adopting tubes to track multi-object in a one-step training model. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 6308- 6318, 2020.","doi":""},{"text":"Image transformer. Niki Parmar, Ashish Vaswani, Jakob Uszkoreit, Lukasz Kaiser, Noam Shazeer, Alexander Ku, Dustin Tran, Proceedings of the 35th International Conference on Machine Learning. the 35th International Conference on Machine LearningPMLRNiki Parmar, Ashish Vaswani, Jakob Uszkoreit, Lukasz Kaiser, Noam Shazeer, Alexander Ku, and Dustin Tran. Im- age transformer. In Proceedings of the 35th International Conference on Machine Learning, pages 4055-4064. PMLR, 2018.","doi":""},{"text":"Chained-tracker: Chaining paired attentive regression results for end-to-end joint multiple-object detection and tracking. Jinlong Peng, Changan Wang, Fangbin Wan, Yang Wu, Yabiao Wang, Ying Tai, Chengjie Wang, Jilin Li, Feiyue Huang, Yanwei Fu, Proceedings of the European Conference on Computer Vision. the European Conference on Computer VisionJinlong Peng, Changan Wang, Fangbin Wan, Yang Wu, Yabiao Wang, Ying Tai, Chengjie Wang, Jilin Li, Feiyue Huang, and Yanwei Fu. Chained-tracker: Chaining paired at- tentive regression results for end-to-end joint multiple-object detection and tracking. In Proceedings of the European Con- ference on Computer Vision, 2020.","doi":""},{"text":"Joseph Redmon, Ali Farhadi, arXiv:1804.02767Yolov3: An incremental improvement. arXiv preprintJoseph Redmon and Ali Farhadi. Yolov3: An incremental improvement. arXiv preprint arXiv:1804.02767, 2018.","doi":""},{"text":"Faster r-cnn: Towards real-time object detection with region proposal networks. Kaiming Shaoqing Ren, Ross He, Jian Girshick, Sun, Advances in neural information processing systems. Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. Faster r-cnn: Towards real-time object detection with region proposal networks. In Advances in neural information pro- cessing systems, pages 91-99, 2015.","doi":""},{"text":"Performance measures and a data set for multi-target, multi-camera tracking. Ergys Ristani, Francesco Solera, Roger Zou, Rita Cucchiara, Carlo Tomasi, European Conference on Computer Vision. SpringerErgys Ristani, Francesco Solera, Roger Zou, Rita Cucchiara, and Carlo Tomasi. Performance measures and a data set for multi-target, multi-camera tracking. In European Conference on Computer Vision, pages 17-35. Springer, 2016.","doi":""},{"text":"Imagenet large scale visual recognition challenge. Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, International journal of computer vision. 1153Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, San- jeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, et al. Imagenet large scale visual recognition challenge. International journal of computer vision, 115(3):211-252, 2015.","doi":""},{"text":"Crowdhuman: A benchmark for detecting human in a crowd. Shuai Shao, Zijian Zhao, Boxun Li, Tete Xiao, Gang Yu, Xiangyu Zhang, Jian Sun, arXiv:1805.00123arXiv preprintShuai Shao, Zijian Zhao, Boxun Li, Tete Xiao, Gang Yu, Xiangyu Zhang, and Jian Sun. Crowdhuman: A bench- mark for detecting human in a crowd. arXiv preprint arXiv:1805.00123, 2018.","doi":""},{"text":"Pwc-net: Cnns for optical flow using pyramid, warping, and cost volume. Deqing Sun, Xiaodong Yang, Ming-Yu Liu, Jan Kautz, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionDeqing Sun, Xiaodong Yang, Ming-Yu Liu, and Jan Kautz. Pwc-net: Cnns for optical flow using pyramid, warping, and cost volume. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 8934-8943, 2018.","doi":""},{"text":"Jonathon Shlens, Zhifeng Chen, and Dragomir Anguelov. Scalability in perception for autonomous driving: Waymo open dataset. Pei Sun, Henrik Kretzschmar, Xerxes Dotiwalla, Aurelien Chouard, Vijaysai Patnaik, Paul Tsui, James Guo, Yin Zhou, Yuning Chai, Benjamin Caine, Vijay Vasudevan, Wei Han, Jiquan Ngiam, Hang Zhao, Aleksei Timofeev, Scott Ettinger, Maxim Krivokon, Amy Gao, Aditya Joshi, Yu Zhang, IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). Pei Sun, Henrik Kretzschmar, Xerxes Dotiwalla, Aurelien Chouard, Vijaysai Patnaik, Paul Tsui, James Guo, Yin Zhou, Yuning Chai, Benjamin Caine, Vijay Vasudevan, Wei Han, Jiquan Ngiam, Hang Zhao, Aleksei Timofeev, Scott Et- tinger, Maxim Krivokon, Amy Gao, Aditya Joshi, Yu Zhang, Jonathon Shlens, Zhifeng Chen, and Dragomir Anguelov. Scalability in perception for autonomous driving: Waymo open dataset. In IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2020.","doi":""},{"text":"Raft: Recurrent allpairs field transforms for optical flow. Zachary Teed, Jia Deng, arXiv:2003.12039arXiv preprintZachary Teed and Jia Deng. Raft: Recurrent all- pairs field transforms for optical flow. arXiv preprint arXiv:2003.12039, 2020.","doi":""},{"text":"Attention is all you need. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, Illia Polosukhin, Advances in neural information processing systems. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszko- reit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in neural information processing systems, pages 5998-6008, 2017.","doi":""},{"text":"Mots: Multi-object tracking and segmentation. Paul Voigtlaender, Michael Krause, Aljosa Osep, Jonathon Luiten, Berin Balachandar Gnana, Andreas Sekar, Bastian Geiger, Leibe, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionPaul Voigtlaender, Michael Krause, Aljosa Osep, Jonathon Luiten, Berin Balachandar Gnana Sekar, Andreas Geiger, and Bastian Leibe. Mots: Multi-object tracking and segmen- tation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7942-7951, 2019.","doi":""},{"text":"Tracking emerges by colorizing videos. Carl Vondrick, Abhinav Shrivastava, Alireza Fathi, Sergio Guadarrama, Kevin Murphy, Proceedings of the European conference on computer vision (ECCV). the European conference on computer vision (ECCV)Carl Vondrick, Abhinav Shrivastava, Alireza Fathi, Sergio Guadarrama, and Kevin Murphy. Tracking emerges by col- orizing videos. In Proceedings of the European conference on computer vision (ECCV), pages 391-408, 2018.","doi":""},{"text":"Video modeling with correlation networks. Heng Wang, Du Tran, Lorenzo Torresani, Matt Feiszli, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionHeng Wang, Du Tran, Lorenzo Torresani, and Matt Feiszli. Video modeling with correlation networks. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 352-361, 2020.","doi":""},{"text":"Axial-deeplab: Standalone axial-attention for panoptic segmentation. Huiyu Wang, Yukun Zhu, Bradley Green, Hartwig Adam, Alan Yuille, Liang-Chieh Chen, arXiv:2003.07853arXiv preprintHuiyu Wang, Yukun Zhu, Bradley Green, Hartwig Adam, Alan Yuille, and Liang-Chieh Chen. Axial-deeplab: Stand- alone axial-attention for panoptic segmentation. arXiv preprint arXiv:2003.07853, 2020.","doi":""},{"text":"Non-local neural networks. Xiaolong Wang, Ross Girshick, Abhinav Gupta, Kaiming He, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionXiaolong Wang, Ross Girshick, Abhinav Gupta, and Kaim- ing He. Non-local neural networks. In Proceedings of the IEEE conference on computer vision and pattern recogni- tion, pages 7794-7803, 2018.","doi":""},{"text":"Joint detection and multi-object tracking with graph neural networks. Yongxin Wang, Xinshuo Weng, Kris Kitani, arXiv:2006.13164arXiv preprintYongxin Wang, Xinshuo Weng, and Kris Kitani. Joint detec- tion and multi-object tracking with graph neural networks. arXiv preprint arXiv:2006.13164, 2020.","doi":""},{"text":"Towards real-time multi-object tracking. Zhongdao Wang, Liang Zheng, Yixuan Liu, Shengjin Wang, European Conference on Computer Vision. Zhongdao Wang, Liang Zheng, Yixuan Liu, and Shengjin Wang. Towards real-time multi-object tracking. In European Conference on Computer Vision, 2020.","doi":""},{"text":"Simple online and realtime tracking with a deep association metric. Nicolai Wojke, Alex Bewley, Dietrich Paulus, 2017 IEEE international conference on image processing (ICIP). IEEENicolai Wojke, Alex Bewley, and Dietrich Paulus. Simple online and realtime tracking with a deep association metric. In 2017 IEEE international conference on image processing (ICIP), pages 3645-3649. IEEE, 2017.","doi":""},{"text":"Joint detection and identification feature learning for person search. Tong Xiao, Shuang Li, Bochao Wang, Liang Lin, Xiaogang Wang, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionTong Xiao, Shuang Li, Bochao Wang, Liang Lin, and Xiao- gang Wang. Joint detection and identification feature learn- ing for person search. In Proceedings of the IEEE Con- ference on Computer Vision and Pattern Recognition, pages 3415-3424, 2017.","doi":""},{"text":"Spatialtemporal relation networks for multi-object tracking. Jiarui Xu, Yue Cao, Zheng Zhang, Han Hu, Proceedings of the IEEE International Conference on Computer Vision. the IEEE International Conference on Computer VisionJiarui Xu, Yue Cao, Zheng Zhang, and Han Hu. Spatial- temporal relation networks for multi-object tracking. In Pro- ceedings of the IEEE International Conference on Computer Vision, pages 3988-3998, 2019.","doi":""},{"text":"Disentangled non-local neural networks. Minghao Yin, Zhuliang Yao, Yue Cao, Xiu Li, Zheng Zhang, Stephen Lin, Han Hu, arXiv:2006.06668arXiv preprintMinghao Yin, Zhuliang Yao, Yue Cao, Xiu Li, Zheng Zhang, Stephen Lin, and Han Hu. Disentangled non-local neural networks. arXiv preprint arXiv:2006.06668, 2020.","doi":""},{"text":"Dilated residual networks. Fisher Yu, Vladlen Koltun, Thomas Funkhouser, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionFisher Yu, Vladlen Koltun, and Thomas Funkhouser. Dilated residual networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 472-480, 2017.","doi":""},{"text":"Poi: Multiple object tracking with high performance detection and appearance feature. Fengwei Yu, Wenbo Li, Quanquan Li, Yu Liu, Xiaohua Shi, Junjie Yan, European Conference on Computer Vision. SpringerFengwei Yu, Wenbo Li, Quanquan Li, Yu Liu, Xiaohua Shi, and Junjie Yan. Poi: Multiple object tracking with high per- formance detection and appearance feature. In European Conference on Computer Vision, pages 36-42. Springer, 2016.","doi":""},{"text":"Deep layer aggregation. Fisher Yu, Dequan Wang, Evan Shelhamer, Trevor Darrell, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionFisher Yu, Dequan Wang, Evan Shelhamer, and Trevor Darrell. Deep layer aggregation. In Proceedings of the IEEE conference on computer vision and pattern recogni- tion, pages 2403-2412, 2018.","doi":""},{"text":"A simple baseline for multi-object tracking. Yifu Zhan, Chunyu Wang, Xinggang Wang, Wenjun Zeng, Wenyu Liu, arXiv:2004.01888arXiv preprintYifu Zhan, Chunyu Wang, Xinggang Wang, Wenjun Zeng, and Wenyu Liu. A simple baseline for multi-object tracking. arXiv preprint arXiv:2004.01888, 2020.","doi":""},{"text":"On the stability of video detection and tracking. Hong Zhang, Naiyan Wang, arXiv:1611.06467arXiv preprintHong Zhang and Naiyan Wang. On the stability of video detection and tracking. arXiv preprint arXiv:1611.06467, 2016.","doi":""},{"text":"Jimuyang Zhang, Sanping Zhou, Xin Chang, Fangbin Wan, Jinjun Wang, Yang Wu, Dong Huang, arXiv:2001.11180Multiple object tracking by flowing and fusing. arXiv preprintJimuyang Zhang, Sanping Zhou, Xin Chang, Fangbin Wan, Jinjun Wang, Yang Wu, and Dong Huang. Multiple object tracking by flowing and fusing. arXiv preprint arXiv:2001.11180, 2020.","doi":""},{"text":"Global data association for multi-object tracking using network flows. Li Zhang, Yuan Li, Ramakant Nevatia, 2008 IEEE Conference on Computer Vision and Pattern Recognition. IEEELi Zhang, Yuan Li, and Ramakant Nevatia. Global data association for multi-object tracking using network flows. In 2008 IEEE Conference on Computer Vision and Pattern Recognition, pages 1-8. IEEE, 2008.","doi":""},{"text":"Citypersons: A diverse dataset for pedestrian detection. Shanshan Zhang, Rodrigo Benenson, Bernt Schiele, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionShanshan Zhang, Rodrigo Benenson, and Bernt Schiele. Citypersons: A diverse dataset for pedestrian detection. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3213-3221, 2017.","doi":""},{"text":"Person re-identification in the wild. Liang Zheng, Hengheng Zhang, Shaoyan Sun, Manmohan Chandraker, Yi Yang, Qi Tian, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionLiang Zheng, Hengheng Zhang, Shaoyan Sun, Manmohan Chandraker, Yi Yang, and Qi Tian. Person re-identification in the wild. In Proceedings of the IEEE Conference on Com- puter Vision and Pattern Recognition, pages 1367-1376, 2017.","doi":""},{"text":"Tracking objects as points. Xingyi Zhou, Vladlen Koltun, Philipp Krähenbühl, European Conference on Computer Vision. Xingyi Zhou, Vladlen Koltun, and Philipp Krähenbühl. Tracking objects as points. In European Conference on Com- puter Vision, 2020.","doi":""},{"text":"Objects as points. Xingyi Zhou, Dequan Wang, Philipp Krähenbühl, arXiv:1904.07850arXiv preprintXingyi Zhou, Dequan Wang, and Philipp Krähenbühl. Ob- jects as points. In arXiv preprint arXiv:1904.07850, 2019.","doi":""},{"text":"Deformable convnets v2: More deformable, better results. Xizhou Zhu, Han Hu, Stephen Lin, Jifeng Dai, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionXizhou Zhu, Han Hu, Stephen Lin, and Jifeng Dai. De- formable convnets v2: More deformable, better results. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 9308-9316, 2019.","doi":""}],"string":"[\n {\n \"text\": \"Tracking without bells and whistles. Philipp Bergmann, Tim Meinhardt, Laura Leal-Taixe, Proceedings of the IEEE international conference on computer vision. the IEEE international conference on computer visionPhilipp Bergmann, Tim Meinhardt, and Laura Leal-Taixe. Tracking without bells and whistles. In Proceedings of the IEEE international conference on computer vision, pages 941-951, 2019.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fully-convolutional siamese networks for object tracking. Luca Bertinetto, Jack Valmadre, Joao F Henriques, Andrea Vedaldi, Philip Hs Torr, European conference on computer vision. SpringerLuca Bertinetto, Jack Valmadre, Joao F Henriques, Andrea Vedaldi, and Philip HS Torr. Fully-convolutional siamese networks for object tracking. In European conference on computer vision, pages 850-865. Springer, 2016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Simple online and realtime tracking. Alex Bewley, Zongyuan Ge, Lionel Ott, Fabio Ramos, Ben Upcroft, 2016 IEEE International Conference on Image Processing (ICIP). IEEEAlex Bewley, Zongyuan Ge, Lionel Ott, Fabio Ramos, and Ben Upcroft. Simple online and realtime tracking. In 2016 IEEE International Conference on Image Processing (ICIP), pages 3464-3468. IEEE, 2016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Highspeed tracking-by-detection without using image information. Erik Bochinski, Thomas Volker Eiselein, Sikora, 14th IEEE International Conference on Advanced Video and Signal Based Surveillance (AVSS). IEEEErik Bochinski, Volker Eiselein, and Thomas Sikora. High- speed tracking-by-detection without using image informa- tion. In 2017 14th IEEE International Conference on Ad- vanced Video and Signal Based Surveillance (AVSS), pages 1-6. IEEE.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Learning a neural solver for multiple object tracking. Guillem Brasó, Laura Leal-Taixé, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionGuillem Brasó and Laura Leal-Taixé. Learning a neu- ral solver for multiple object tracking. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 6247-6257, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Memory enhanced global-local aggregation for video object detection. Yihong Chen, Yue Cao, Han Hu, Liwei Wang, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionYihong Chen, Yue Cao, Han Hu, and Liwei Wang. Memory enhanced global-local aggregation for video object detec- tion. In Proceedings of the IEEE/CVF Conference on Com- puter Vision and Pattern Recognition, pages 10337-10346, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Famnet: Joint learning of feature, affinity and multi-dimensional assignment for online multiple object tracking. Peng Chu, Haibin Ling, Proceedings of the IEEE International Conference on Computer Vision. the IEEE International Conference on Computer VisionPeng Chu and Haibin Ling. Famnet: Joint learning of fea- ture, affinity and multi-dimensional assignment for online multiple object tracking. In Proceedings of the IEEE Inter- national Conference on Computer Vision, pages 6172-6181, 2019.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"P Dendorfer, H Rezatofighi, A Milan, J Shi, D Cremers, I Reid, S Roth, K Schindler, L Leal-Taixé, arXiv:2003.09003arXiv: 2003.09003Mot20: A benchmark for multi object tracking in crowded scenes. P. Dendorfer, H. Rezatofighi, A. Milan, J. Shi, D. Cremers, I. Reid, S. Roth, K. Schindler, and L. Leal-Taixé. Mot20: A benchmark for multi object tracking in crowded scenes. arXiv:2003.09003[cs], Mar. 2020. arXiv: 2003.09003.\",\n \"doi\": \"arXiv:2003.09003\"\n },\n {\n \"text\": \"Bernt Schiele, and Pietro Perona. Pedestrian detection: A benchmark. Piotr Dollár, Christian Wojek, IEEE Conference on Computer Vision and Pattern Recognition. IEEEPiotr Dollár, Christian Wojek, Bernt Schiele, and Pietro Per- ona. Pedestrian detection: A benchmark. In 2009 IEEE Con- ference on Computer Vision and Pattern Recognition, pages 304-311. IEEE, 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Flownet: Learning optical flow with convolutional networks. Alexey Dosovitskiy, Philipp Fischer, Eddy Ilg, Philip Hausser, Caner Hazirbas, Vladimir Golkov, Patrick Van Der, Daniel Smagt, Thomas Cremers, Brox, Proceedings of the IEEE international conference on computer vision. the IEEE international conference on computer visionAlexey Dosovitskiy, Philipp Fischer, Eddy Ilg, Philip Hausser, Caner Hazirbas, Vladimir Golkov, Patrick Van Der Smagt, Daniel Cremers, and Thomas Brox. Flownet: Learning optical flow with convolutional networks. In Pro- ceedings of the IEEE international conference on computer vision, pages 2758-2766, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A mobile vision system for robust multi-person tracking. Andreas Ess, Bastian Leibe, Konrad Schindler, Luc Van Gool, 2008 IEEE Conference on Computer Vision and Pattern Recognition. IEEEAndreas Ess, Bastian Leibe, Konrad Schindler, and Luc Van Gool. A mobile vision system for robust multi-person tracking. In 2008 IEEE Conference on Computer Vision and Pattern Recognition, pages 1-8. IEEE, 2008.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Detect to track and track to detect. Christoph Feichtenhofer, Axel Pinz, Andrew Zisserman, Proceedings of the IEEE International Conference on Computer Vision. the IEEE International Conference on Computer VisionChristoph Feichtenhofer, Axel Pinz, and Andrew Zisserman. Detect to track and track to detect. In Proceedings of the IEEE International Conference on Computer Vision, pages 3038-3046, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Object detection with discriminatively trained part-based models. F Pedro, Ross B Felzenszwalb, David Girshick, Deva Mcallester, Ramanan, IEEE transactions on pattern analysis and machine intelligence. 32Pedro F Felzenszwalb, Ross B Girshick, David McAllester, and Deva Ramanan. Object detection with discriminatively trained part-based models. IEEE transactions on pattern analysis and machine intelligence, 32(9):1627-1645, 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fast r-cnn. Ross Girshick, Proceedings of the IEEE international conference on computer vision. the IEEE international conference on computer visionRoss Girshick. Fast r-cnn. In Proceedings of the IEEE inter- national conference on computer vision, pages 1440-1448, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Tracking with multi-level features. Roberto Henschel, Laura Leal-Taixé, Bodo Rosenhahn, Konrad Schindler, arXiv:1607.07304arXiv preprintRoberto Henschel, Laura Leal-Taixé, Bodo Rosenhahn, and Konrad Schindler. Tracking with multi-level features. arXiv preprint arXiv:1607.07304, 2016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Multiple people tracking using body and joint detections. Roberto Henschel, Yunzhe Zou, Bodo Rosenhahn, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops. the IEEE Conference on Computer Vision and Pattern Recognition WorkshopsRoberto Henschel, Yunzhe Zou, and Bodo Rosenhahn. Mul- tiple people tracking using body and joint detections. In Pro- ceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, pages 0-0, 2019.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Lifted disjoint paths with application in multiple object tracking. Andrea Hornakova, Roberto Henschel, Bodo Rosenhahn, Paul Swoboda, The 37th International Conference on Machine Learning (ICML). Andrea Hornakova, Roberto Henschel, Bodo Rosenhahn, and Paul Swoboda. Lifted disjoint paths with application in multiple object tracking. In The 37th International Con- ference on Machine Learning (ICML), July 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Relation networks for object detection. Han Hu, Jiayuan Gu, Zheng Zhang, Jifeng Dai, Yichen Wei, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionHan Hu, Jiayuan Gu, Zheng Zhang, Jifeng Dai, and Yichen Wei. Relation networks for object detection. In Proceed- ings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3588-3597, 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Computer vision for autonomous vehicles: Problems, datasets and state of the art. Joel Janai, Fatma Güney, Aseem Behl, Andreas Geiger, Foundations and Trends® in Computer Graphics and Vision. 121-3Joel Janai, Fatma Güney, Aseem Behl, Andreas Geiger, et al. Computer vision for autonomous vehicles: Problems, datasets and state of the art. Foundations and Trends® in Computer Graphics and Vision, 12(1-3):1-308, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A new approach to linear filtering and prediction problems. R E Kalman, ASME Journal of Basic Engineering. R. E. Kalman. A new approach to linear filtering and predic- tion problems. ASME Journal of Basic Engineering, 1960.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Framework for performance evaluation of face, text, and vehicle detection and tracking in video: Data, metrics, and protocol. Rangachar Kasturi, Dmitry Goldgof, Padmanabhan Soundararajan, Vasant Manohar, John Garofolo, Rachel Bowers, Matthew Boonstra, Valentina Korzhova, Jing Zhang, IEEE Transactions on Pattern Analysis and Machine Intelligence. 312Rangachar Kasturi, Dmitry Goldgof, Padmanabhan Soundararajan, Vasant Manohar, John Garofolo, Rachel Bowers, Matthew Boonstra, Valentina Korzhova, and Jing Zhang. Framework for performance evaluation of face, text, and vehicle detection and tracking in video: Data, metrics, and protocol. IEEE Transactions on Pattern Analysis and Machine Intelligence, 31(2):319-336, 2008.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Adam: A method for stochastic optimization. P Diederik, Jimmy Kingma, Ba, arXiv:1412.6980arXiv preprintDiederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The hungarian method for the assignment problem. Harold W Kuhn, Naval research logistics quarterly. 2Harold W Kuhn. The hungarian method for the assignment problem. Naval research logistics quarterly, 2(1-2):83-97, 1955.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"L Leal-Taixé, A Milan, I Reid, S Roth, K Schindler, Motchallenge, arXiv:1504.01942arXiv: 1504.01942Towards a benchmark for multitarget tracking. L. Leal-Taixé, A. Milan, I. Reid, S. Roth, and K. Schindler. MOTChallenge 2015: Towards a benchmark for multi- target tracking. arXiv:1504.01942 [cs], Apr. 2015. arXiv: 1504.01942.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Learning to associate: Hybridboosted multi-target tracker for crowded scene. Yuan Li, Chang Huang, Ram Nevatia, 2009 IEEE Conference on Computer Vision and Pattern Recognition. IEEEYuan Li, Chang Huang, and Ram Nevatia. Learning to asso- ciate: Hybridboosted multi-target tracker for crowded scene. In 2009 IEEE Conference on Computer Vision and Pattern Recognition, pages 2953-2960. IEEE, 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Microsoft coco: Common objects in context. Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, C Lawrence Zitnick, European conference on computer vision. SpringerTsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C Lawrence Zitnick. Microsoft coco: Common objects in context. In European conference on computer vision, pages 740-755. Springer, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Real-time multiple people tracking with deeply learned candidate selection and person re-identification. Chen Long, Ai Haizhou, Zhuang Zijie, Shang Chong, ICME. Chen Long, Ai Haizhou, Zhuang Zijie, and Shang Chong. Real-time multiple people tracking with deeply learned can- didate selection and person re-identification. In ICME, 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"See more, know more: Unsupervised video object segmentation with co-attention siamese networks. Xiankai Lu, Wenguan Wang, Chao Ma, Jianbing Shen, Ling Shao, Fatih Porikli, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionXiankai Lu, Wenguan Wang, Chao Ma, Jianbing Shen, Ling Shao, and Fatih Porikli. See more, know more: Unsuper- vised video object segmentation with co-attention siamese networks. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3623- 3632, 2019.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Retinatrack: Online single stage joint detection and tracking. Zhichao Lu, Vivek Rathod, Ronny Votel, Jonathan Huang, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionZhichao Lu, Vivek Rathod, Ronny Votel, and Jonathan Huang. Retinatrack: Online single stage joint detection and tracking. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 14668- 14678, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Milan, L Leal-Taixé, I Reid, S Roth, K Schindler, arXiv:1603.00831arXiv: 1603.00831MOT16: A benchmark for multi-object tracking. A. Milan, L. Leal-Taixé, I. Reid, S. Roth, and K. Schindler. MOT16: A benchmark for multi-object tracking. arXiv:1603.00831 [cs], Mar. 2016. arXiv: 1603.00831.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A largescale benchmark dataset for event recognition in surveillance video. Sangmin Oh, Anthony Hoogs, Amitha Perera, Naresh Cuntoor, Chia-Chih Chen, Jong Taek Lee, Saurajit Mukherjee, Hyungtae Aggarwal, Larry Lee, Davis, CVPR 2011. IEEESangmin Oh, Anthony Hoogs, Amitha Perera, Naresh Cun- toor, Chia-Chih Chen, Jong Taek Lee, Saurajit Mukherjee, JK Aggarwal, Hyungtae Lee, Larry Davis, et al. A large- scale benchmark dataset for event recognition in surveillance video. In CVPR 2011, pages 3153-3160. IEEE, 2011.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Tubetk: Adopting tubes to track multi-object in a one-step training model. Bo Pang, Yizhuo Li, Yifan Zhang, Muchen Li, Cewu Lu, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionBo Pang, Yizhuo Li, Yifan Zhang, Muchen Li, and Cewu Lu. Tubetk: Adopting tubes to track multi-object in a one-step training model. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 6308- 6318, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Image transformer. Niki Parmar, Ashish Vaswani, Jakob Uszkoreit, Lukasz Kaiser, Noam Shazeer, Alexander Ku, Dustin Tran, Proceedings of the 35th International Conference on Machine Learning. the 35th International Conference on Machine LearningPMLRNiki Parmar, Ashish Vaswani, Jakob Uszkoreit, Lukasz Kaiser, Noam Shazeer, Alexander Ku, and Dustin Tran. Im- age transformer. In Proceedings of the 35th International Conference on Machine Learning, pages 4055-4064. PMLR, 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Chained-tracker: Chaining paired attentive regression results for end-to-end joint multiple-object detection and tracking. Jinlong Peng, Changan Wang, Fangbin Wan, Yang Wu, Yabiao Wang, Ying Tai, Chengjie Wang, Jilin Li, Feiyue Huang, Yanwei Fu, Proceedings of the European Conference on Computer Vision. the European Conference on Computer VisionJinlong Peng, Changan Wang, Fangbin Wan, Yang Wu, Yabiao Wang, Ying Tai, Chengjie Wang, Jilin Li, Feiyue Huang, and Yanwei Fu. Chained-tracker: Chaining paired at- tentive regression results for end-to-end joint multiple-object detection and tracking. In Proceedings of the European Con- ference on Computer Vision, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Joseph Redmon, Ali Farhadi, arXiv:1804.02767Yolov3: An incremental improvement. arXiv preprintJoseph Redmon and Ali Farhadi. Yolov3: An incremental improvement. arXiv preprint arXiv:1804.02767, 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Faster r-cnn: Towards real-time object detection with region proposal networks. Kaiming Shaoqing Ren, Ross He, Jian Girshick, Sun, Advances in neural information processing systems. Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. Faster r-cnn: Towards real-time object detection with region proposal networks. In Advances in neural information pro- cessing systems, pages 91-99, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Performance measures and a data set for multi-target, multi-camera tracking. Ergys Ristani, Francesco Solera, Roger Zou, Rita Cucchiara, Carlo Tomasi, European Conference on Computer Vision. SpringerErgys Ristani, Francesco Solera, Roger Zou, Rita Cucchiara, and Carlo Tomasi. Performance measures and a data set for multi-target, multi-camera tracking. In European Conference on Computer Vision, pages 17-35. Springer, 2016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Imagenet large scale visual recognition challenge. Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, International journal of computer vision. 1153Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, San- jeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, et al. Imagenet large scale visual recognition challenge. International journal of computer vision, 115(3):211-252, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Crowdhuman: A benchmark for detecting human in a crowd. Shuai Shao, Zijian Zhao, Boxun Li, Tete Xiao, Gang Yu, Xiangyu Zhang, Jian Sun, arXiv:1805.00123arXiv preprintShuai Shao, Zijian Zhao, Boxun Li, Tete Xiao, Gang Yu, Xiangyu Zhang, and Jian Sun. Crowdhuman: A bench- mark for detecting human in a crowd. arXiv preprint arXiv:1805.00123, 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Pwc-net: Cnns for optical flow using pyramid, warping, and cost volume. Deqing Sun, Xiaodong Yang, Ming-Yu Liu, Jan Kautz, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionDeqing Sun, Xiaodong Yang, Ming-Yu Liu, and Jan Kautz. Pwc-net: Cnns for optical flow using pyramid, warping, and cost volume. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 8934-8943, 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Jonathon Shlens, Zhifeng Chen, and Dragomir Anguelov. Scalability in perception for autonomous driving: Waymo open dataset. Pei Sun, Henrik Kretzschmar, Xerxes Dotiwalla, Aurelien Chouard, Vijaysai Patnaik, Paul Tsui, James Guo, Yin Zhou, Yuning Chai, Benjamin Caine, Vijay Vasudevan, Wei Han, Jiquan Ngiam, Hang Zhao, Aleksei Timofeev, Scott Ettinger, Maxim Krivokon, Amy Gao, Aditya Joshi, Yu Zhang, IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). Pei Sun, Henrik Kretzschmar, Xerxes Dotiwalla, Aurelien Chouard, Vijaysai Patnaik, Paul Tsui, James Guo, Yin Zhou, Yuning Chai, Benjamin Caine, Vijay Vasudevan, Wei Han, Jiquan Ngiam, Hang Zhao, Aleksei Timofeev, Scott Et- tinger, Maxim Krivokon, Amy Gao, Aditya Joshi, Yu Zhang, Jonathon Shlens, Zhifeng Chen, and Dragomir Anguelov. Scalability in perception for autonomous driving: Waymo open dataset. In IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Raft: Recurrent allpairs field transforms for optical flow. Zachary Teed, Jia Deng, arXiv:2003.12039arXiv preprintZachary Teed and Jia Deng. Raft: Recurrent all- pairs field transforms for optical flow. arXiv preprint arXiv:2003.12039, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Attention is all you need. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, Illia Polosukhin, Advances in neural information processing systems. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszko- reit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in neural information processing systems, pages 5998-6008, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Mots: Multi-object tracking and segmentation. Paul Voigtlaender, Michael Krause, Aljosa Osep, Jonathon Luiten, Berin Balachandar Gnana, Andreas Sekar, Bastian Geiger, Leibe, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionPaul Voigtlaender, Michael Krause, Aljosa Osep, Jonathon Luiten, Berin Balachandar Gnana Sekar, Andreas Geiger, and Bastian Leibe. Mots: Multi-object tracking and segmen- tation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7942-7951, 2019.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Tracking emerges by colorizing videos. Carl Vondrick, Abhinav Shrivastava, Alireza Fathi, Sergio Guadarrama, Kevin Murphy, Proceedings of the European conference on computer vision (ECCV). the European conference on computer vision (ECCV)Carl Vondrick, Abhinav Shrivastava, Alireza Fathi, Sergio Guadarrama, and Kevin Murphy. Tracking emerges by col- orizing videos. In Proceedings of the European conference on computer vision (ECCV), pages 391-408, 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Video modeling with correlation networks. Heng Wang, Du Tran, Lorenzo Torresani, Matt Feiszli, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionHeng Wang, Du Tran, Lorenzo Torresani, and Matt Feiszli. Video modeling with correlation networks. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 352-361, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Axial-deeplab: Standalone axial-attention for panoptic segmentation. Huiyu Wang, Yukun Zhu, Bradley Green, Hartwig Adam, Alan Yuille, Liang-Chieh Chen, arXiv:2003.07853arXiv preprintHuiyu Wang, Yukun Zhu, Bradley Green, Hartwig Adam, Alan Yuille, and Liang-Chieh Chen. Axial-deeplab: Stand- alone axial-attention for panoptic segmentation. arXiv preprint arXiv:2003.07853, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Non-local neural networks. Xiaolong Wang, Ross Girshick, Abhinav Gupta, Kaiming He, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionXiaolong Wang, Ross Girshick, Abhinav Gupta, and Kaim- ing He. Non-local neural networks. In Proceedings of the IEEE conference on computer vision and pattern recogni- tion, pages 7794-7803, 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Joint detection and multi-object tracking with graph neural networks. Yongxin Wang, Xinshuo Weng, Kris Kitani, arXiv:2006.13164arXiv preprintYongxin Wang, Xinshuo Weng, and Kris Kitani. Joint detec- tion and multi-object tracking with graph neural networks. arXiv preprint arXiv:2006.13164, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Towards real-time multi-object tracking. Zhongdao Wang, Liang Zheng, Yixuan Liu, Shengjin Wang, European Conference on Computer Vision. Zhongdao Wang, Liang Zheng, Yixuan Liu, and Shengjin Wang. Towards real-time multi-object tracking. In European Conference on Computer Vision, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Simple online and realtime tracking with a deep association metric. Nicolai Wojke, Alex Bewley, Dietrich Paulus, 2017 IEEE international conference on image processing (ICIP). IEEENicolai Wojke, Alex Bewley, and Dietrich Paulus. Simple online and realtime tracking with a deep association metric. In 2017 IEEE international conference on image processing (ICIP), pages 3645-3649. IEEE, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Joint detection and identification feature learning for person search. Tong Xiao, Shuang Li, Bochao Wang, Liang Lin, Xiaogang Wang, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionTong Xiao, Shuang Li, Bochao Wang, Liang Lin, and Xiao- gang Wang. Joint detection and identification feature learn- ing for person search. In Proceedings of the IEEE Con- ference on Computer Vision and Pattern Recognition, pages 3415-3424, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Spatialtemporal relation networks for multi-object tracking. Jiarui Xu, Yue Cao, Zheng Zhang, Han Hu, Proceedings of the IEEE International Conference on Computer Vision. the IEEE International Conference on Computer VisionJiarui Xu, Yue Cao, Zheng Zhang, and Han Hu. Spatial- temporal relation networks for multi-object tracking. In Pro- ceedings of the IEEE International Conference on Computer Vision, pages 3988-3998, 2019.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Disentangled non-local neural networks. Minghao Yin, Zhuliang Yao, Yue Cao, Xiu Li, Zheng Zhang, Stephen Lin, Han Hu, arXiv:2006.06668arXiv preprintMinghao Yin, Zhuliang Yao, Yue Cao, Xiu Li, Zheng Zhang, Stephen Lin, and Han Hu. Disentangled non-local neural networks. arXiv preprint arXiv:2006.06668, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dilated residual networks. Fisher Yu, Vladlen Koltun, Thomas Funkhouser, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionFisher Yu, Vladlen Koltun, and Thomas Funkhouser. Dilated residual networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 472-480, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Poi: Multiple object tracking with high performance detection and appearance feature. Fengwei Yu, Wenbo Li, Quanquan Li, Yu Liu, Xiaohua Shi, Junjie Yan, European Conference on Computer Vision. SpringerFengwei Yu, Wenbo Li, Quanquan Li, Yu Liu, Xiaohua Shi, and Junjie Yan. Poi: Multiple object tracking with high per- formance detection and appearance feature. In European Conference on Computer Vision, pages 36-42. Springer, 2016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Deep layer aggregation. Fisher Yu, Dequan Wang, Evan Shelhamer, Trevor Darrell, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionFisher Yu, Dequan Wang, Evan Shelhamer, and Trevor Darrell. Deep layer aggregation. In Proceedings of the IEEE conference on computer vision and pattern recogni- tion, pages 2403-2412, 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A simple baseline for multi-object tracking. Yifu Zhan, Chunyu Wang, Xinggang Wang, Wenjun Zeng, Wenyu Liu, arXiv:2004.01888arXiv preprintYifu Zhan, Chunyu Wang, Xinggang Wang, Wenjun Zeng, and Wenyu Liu. A simple baseline for multi-object tracking. arXiv preprint arXiv:2004.01888, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the stability of video detection and tracking. Hong Zhang, Naiyan Wang, arXiv:1611.06467arXiv preprintHong Zhang and Naiyan Wang. On the stability of video detection and tracking. arXiv preprint arXiv:1611.06467, 2016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Jimuyang Zhang, Sanping Zhou, Xin Chang, Fangbin Wan, Jinjun Wang, Yang Wu, Dong Huang, arXiv:2001.11180Multiple object tracking by flowing and fusing. arXiv preprintJimuyang Zhang, Sanping Zhou, Xin Chang, Fangbin Wan, Jinjun Wang, Yang Wu, and Dong Huang. Multiple object tracking by flowing and fusing. arXiv preprint arXiv:2001.11180, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Global data association for multi-object tracking using network flows. Li Zhang, Yuan Li, Ramakant Nevatia, 2008 IEEE Conference on Computer Vision and Pattern Recognition. IEEELi Zhang, Yuan Li, and Ramakant Nevatia. Global data association for multi-object tracking using network flows. In 2008 IEEE Conference on Computer Vision and Pattern Recognition, pages 1-8. IEEE, 2008.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Citypersons: A diverse dataset for pedestrian detection. Shanshan Zhang, Rodrigo Benenson, Bernt Schiele, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionShanshan Zhang, Rodrigo Benenson, and Bernt Schiele. Citypersons: A diverse dataset for pedestrian detection. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3213-3221, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Person re-identification in the wild. Liang Zheng, Hengheng Zhang, Shaoyan Sun, Manmohan Chandraker, Yi Yang, Qi Tian, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionLiang Zheng, Hengheng Zhang, Shaoyan Sun, Manmohan Chandraker, Yi Yang, and Qi Tian. Person re-identification in the wild. In Proceedings of the IEEE Conference on Com- puter Vision and Pattern Recognition, pages 1367-1376, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Tracking objects as points. Xingyi Zhou, Vladlen Koltun, Philipp Krähenbühl, European Conference on Computer Vision. Xingyi Zhou, Vladlen Koltun, and Philipp Krähenbühl. Tracking objects as points. In European Conference on Com- puter Vision, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Objects as points. Xingyi Zhou, Dequan Wang, Philipp Krähenbühl, arXiv:1904.07850arXiv preprintXingyi Zhou, Dequan Wang, and Philipp Krähenbühl. Ob- jects as points. In arXiv preprint arXiv:1904.07850, 2019.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Deformable convnets v2: More deformable, better results. Xizhou Zhu, Han Hu, Stephen Lin, Jifeng Dai, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionXizhou Zhu, Han Hu, Stephen Lin, and Jifeng Dai. De- formable convnets v2: More deformable, better results. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 9308-9316, 2019.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":472,"cells":{"text":{"kind":"string","value":"\nComputers Can't Give Credit: How Automatic Attribution Falls Short in an Online Remixing Community\n\n\nAndrés Monroy-Hernández \nMassachusetts Institute of Technology\n02139CambridgeMA\n\nMicrosoft Research\n02142CambridgeMA\n\nBenjamin Mako Hill \nMassachusetts Institute of Technology\n02139CambridgeMA\n\nJazmin Gonzalez-Rivero \nMicrosoft Research\n02142CambridgeMA\n\nDanah Boyd \nMicrosoft Research\n02142CambridgeMA\n\nComputers Can't Give Credit: How Automatic Attribution Falls Short in an Online Remixing Community\nAuthor Keywords remixingattributioncredit-givinguser-generated contentonline communities ACM Classification Keywords H53 Group and Organization Interfaces General Terms DesignExperimentationHuman Factors\nIn this paper, we explore the role that attribution plays in shaping user reactions to content reuse, or remixing, in a large user-generated content community. We present two studies using data from the Scratch online communitya social media platform where hundreds of thousands of young people share and remix animations and video games. First, we present a quantitative analysis that examines the effects of a technological design intervention introducing automated attribution of remixes on users' reactions to being remixed. We compare this analysis to a parallel examination of \"manual\" credit-giving. Second, we present a qualitative analysis of twelve in-depth, semi-structured, interviews with Scratch participants on the subject of remixing and attribution. Results from both studies suggest that automatic attribution done by technological systems (i.e., the listing of names of contributors) plays a role that is distinct from, and less valuable than, credit which may superficially involve identical information but takes on new meaning when it is given by a human remixer. We discuss the implications of these findings for the designers of online communities and social media platforms.\n\nINTRODUCTION\n\nNetworked information technologies have changed the way people use and reuse creative -and frequently copyrightedmaterials. This change has generated excitement, and heated Best Paper Honorable Mention at CHI 2011. debate, among content-creators, technologists, legal academics, and media scholars. Media theorist Lev Manovich argues that remixing is an ancient cultural tradition (e.g., he has suggested that ancient Rome was a \"remix\" of ancient Greece) but that information technologies have accelerated these processes and made remixing more salient [14]. Sinnreich et al. argue that \"configurable culture\" has been significantly transformed by networked technologies which introduce perfect copying and allow people not only to be inspired by extant creations but to remix the original works themselves [25]. Legal scholars have stressed the importance of remixing in cultural creation broadly and warned that current copyright and intellectual property laws may hinder creativity and innovation [11,1].\n\nSeveral of the most influential scholarly explorations of remixing as a cultural phenomenon have focused on youth's remixing practices. For example, work on remixing by Jenkins [10] and Ito [9] has focused on young people's use and re-use of media. Palfrey and Gasser have suggested that the cultural practices of \"digital native\" youth have had a significant transformative effect on our culture [17]. Throughout his book \"Remix,\" Lessig uses youth's reuse practices to support an argument against what he considers excessive copyright legal protection [12].\n\nYet, despite a wide interest in remixing and authorship, researchers have only recently engaged in empirical research on the subject [4]. Several recent treatments have presented studies of video remixing communities [5,24], music remixing communities [4], collaborative video game communities [13] and social network sites [18]. There is also another quantitative study of our empirical setting [8] focused on characterizing the variety of responses to remixing. These studies have tended to be general and largely descriptive ex-arXiv:1507.01285v1 [cs.HC] 5 Jul 2015 aminations of remixing practice. This work has pointed to the existence of norms [5] and the territoriality of digital creators [26] and has considered issues of motivation [4]. However, empirical work has yet to unpack in detail the key social mechanisms that scholars have suggested drive behavior, norms, and motivation in remixing communities.\n\nPerhaps no mechanism has been more frequently cited as critical for remixing activity than attribution and the related phenomena of plagiarism, reputation, and status. For example, recent survey-based work has suggested that the \"authenticity and legitimacy\" of creative work \"are premised on the explicit acknowledgment of the source materials or 'original creator\"' and that such acknowledgment is a key component of how adults assess the fairness or ethical nature of content reuse [25]. Attribution, in this sense, can be seen as an important way that people distinguish remixing from \"theft.\"\n\nJudge and law professor Richard Posner stresses the importance of attribution and explains that this is important even when there is no monetary benefit to being attributed. For example, he explains that European copyright law is based on a doctrine of \"moral rights\" that \"entitles a writer or other artist to be credited for his original work and this 'attribution right', as it is called, would give him a legal claim against a plagiarist.\" Posner also explains that \"acknowledgment\" of another's contributions to a derivative negates any charge of plagiarism, although it may not establish originality [20]. Attribution plays such an important role in remix culture that Creative Commons made a requirement for attribution a component of all their licenses after more than 97% of licensors opted to require attribution when it was offered as a choice [2]. Young people's perceptions of attribution and complications around copying have also been examined. An article by Friedman describes that adolescents who allowed \"computer pirating\" -the unauthorized copying of computer programs -did so because technological affordances made it difficult for adolescents to identify \"harmful or unjust consequences of computer-mediated actions\" [7]. In a second study, psychologists Olson and Shaw have found that by five years old, \"children understand that others have ideas and dislike the copying of these ideas\" [16].\n\nYet, despite the fact that researchers in human computer interaction have begun to explore the complexity of attribution and cited its importance to remixing [13], many designers of online communities pay little attention to issues of attribution in their designs -a fact that is reflected in user behavior. For example, research on the use of photos from the photo sharing site Flickr [22], as well as a number of other usergenerated content communities [23], suggests that most reusers fail to attribute re-used content in ways that public-use licenses require. Although theory and survey based work points to a need to design for attribution in user-generated content communities, we still know very little about how attribution works or how designers might go about doing so. Indeed, our study suggests that the most obvious efforts to design for attribution are likely to be ineffective.\n\nIn this paper, we employ a mixed methods approach that combines qualitative and quantitative analyses to explore users' reactions to attribution and its absence in a large remixing community. First, we introduce our empirical setting; using qualitative data from users forums and comments, we present a rich description of remixing and evidence to support our core proposition that credit plays a central role in remixing in our environment. Second, we contextualize and describe a technological intervention in our setting, responding directly to several user suggestions, that automated the attribution of creators of antecedent projects when content was remixed. Third, we present a tentative quantitative analysis of the effect of this intervention along with a parallel analysis of the practice of manual creditgiving. We find that credit-giving, done manually, is associated with more positive reactions but that automatic attribution by the system is not associated with a similar effect. Fourth, we present analysis of a set of in-depth interviews with twelve users which helps confirm, and add nuance and depth to, our quantitative findings.\n\nOur results suggest that young users see an important, if currently under-appreciated and under-theorized, difference between credit and attribution. Credit represents more than a public reference to an \"upstream\" user's contributions. Coming from another human, credit can involve an explicit acknowledgment, an expression of gratitude, and an expression of deference, in a way that simple attribution can not. Our results suggest that identical attribution information means something very different to users when it comes from a computer, and when it comes from a human -and that users often feel that acknowledgment is worth much less when it comes from a system. We conclude that designers should create affordances that make it easier for users to credit each other, rather than to merely pursue automated means of acknowledgment.\n\nOur study offers two distinct contributions for social scientists and for technology designers. The first is an improved understanding of the way that attribution and credit work in user-generated content communities. The second is a broader contribution to the literature on design that suggests an important limitation to technologists' ability to support community norms and a suggestion for how designers might create affordances. Functionality that allows users to express information that a system might otherwise show automatically may play an important role in successful design for social media environments.\n\n\nSCRATCH: A COMMUNITY OF YOUNG REMIXERS\n\nThe Scratch online community is a free and publicly available website where young people share their own video games, animated stories, interactive art, and simulations [15]. Participants use the Scratch programming environment [21], a desktop application, to create these interactive projects by putting together images, music and sounds with programming command blocks (See Figure 1). The Scratch website was officially announced in 2007 and, as of September 2010, has more than 600,000 user accounts who have shared 1.3 million projects. At the time of writing, Scratch users share on average one new project per minute. Examples of projects range from an interactive virtual cake maker, to a simulation of an operating system, to a Pokemon-inspired video game, to an animation about climate change, to tutorials on how to draw cartoons. Like other user-generated content websites, such as YouTube or Flickr, Scratch projects are displayed on a webpage (See Figure 2) where people can interact with them, read metadata and give feedback. Visitors can use their mouse and/or keyboard to control a video game or other type of interactive projects or simply observe an animation play out in a web browser. Metadata displayed next to projects includes a text-based description of the project, the creator's name, the number of views, downloads, \"love its,\" remixes, and galleries (i.e., sets of projects) that the project belongs to. Users can interact with projects by giving feedback in the form of tags, comments, or clicks on the \"love it\" button, and can flag a project as \"inappropriate\" for review by site administrators.\n\nParticipants' self-reported ages range primarily from 8 to 17 years-old with 12 being the median. Thirty-six percent of users self-report as female. A large minority of users are from the United States (41%) while other countries prominently represented include the United Kingdom, Thailand, Australia, Canada, Brazil, South Korea, Taiwan, Colombia and Mexico. About 28% of all users -more than 170,000have uploaded at least one project.\n\n\nRemixing in Scratch\n\nScratch users can download any project shared on the website, open it up in the Scratch authoring environment, learn how it was made, and \"remix\" it. In Scratch, the term \"remixing\" refers to the creation of any new version of a Scratch program by adding, removing or changing the programming blocks, images or sounds. In this section we use qualitative data from the Scratch website to provide social context for remixing and to suggest that credit plays an important role in how users conceive of appropriate remixing practice.\n\nRemixing in Scratch is not only technically possible, it is something that the administrators of the website encourage and try to foster as a way for people to learn from others and collaborate. On every project page, the Scratch website displays a hyperlink with the text \"Some rights reserved\" that points to a child-friendly interpretation of the Creative Commons Attribution-Share Alike license under which all Scratch projects are licensed. 1 Even the name Scratch is a reference to hip hop DJs' practice of mixing records. A large portion of all projects shared on the Scratch website (28%) are remixes of other projects.\n\nThat said, remixing is not universally unproblematic in Scratch. Previous quantitative analysis of the the Scratch community showed that Scratch participants react both positively and negatively to the remixing of their projects and found that of those users who viewed a remix of their project, about one-fifth left positive comments while the same proportion of users accused the remixer of plagiarism [8]. This ambivalent reaction to remixing is echoed, and given additional texture, in the comments and complaints left by users on the Scratch website and sent to Scratch administrators.\n\nFor example, even before the Scratch website was publicly announced, a number of early adopters became upset when they found remixes of their projects on the website. Indeed, one of the very first complaints about Scratch occurred on the discussion forums where a 13 year-old boy asked:\n\nIs it allowed if someone uses your game, changes the theme, then calls it 'their creation'? Because I created a game called \"Paddling 1.5\" and a few days later, a user called \"julie\" redid the background, and called it 'her creation' and I am really annoyed with her for taking credit for MY project!! 2 A similar complaint was sent to the website administrators by a 14-year old boy: I think there should be a way to report plagiarized projects I've been seeing a lot of people's projects taken and renamed. This member, named kings651, has 44 projects, and most of them are made by other people. He even has one that I saw my friend make so I know he actually made it.\n\nIn other cases, the disagreements over remixing were more public and involved communication via projects and com-1 A copy of the current version of the kid-friendly license is available online at http://scratch.mit.edu/pages/license. The version available today encourages users to give credit manually in the project notes. A strong emphasis on credit-giving was added as a result of the findings reported here but was absent during the period of data collection for this study. 2 All usernames and quotes from the website were changed to disguise the identities of participants. ments. For example, user koolkid15 wrote the following message in a comment which was left is response to a remix that shows a cat frowning:\n\nHi i'm koolkid15 the original creator of luigi disco jay-man41 copied me!! and didn't even aknowladge me he didn't change anything !! I wrote or drew!! and jayman...if your reading this think about other people!!!! Despite the fact that Scratch was conceived, designed, and launched as a platform for remixing, these users expressed their displeasure at remixing. That said, none of these users complained directly about the reuse of their project in general, but in terms of unfair \"taking credit\", plagiarism, and a lack of acknowledgment. Remixing was seen as problematic for koolkid15, for example, because of the nontransformative nature of reuse, the lack of acknowledgment of antecedent contributors, and the confusion about credit that would result.\n\nOf course, other, more positive, scenarios around remixing also played out in Scratch. For example, jellogaliboo created a remix of Catham's project and wrote the following in the project notes: \"i kinda copied Catham's \"jetpackcat\" game. i used the kitty, the blocks (i added and changed some), and the fuel thingy.\" Catham later posted his approval of the remix saying, \"I like what you changed about my project!\" Like this example, many of these positive experiences involved explicit credit-giving by a remixer to the creator of the antecedent project.\n\n\nDESIGN INTERVENTION: AUTOMATING ATTRIBUTION\n\nSeveral user complaints about remixing and plagiarism also included suggestions for how Scratch's designers might address them. For example, in response to the forum thread mentioned in the previous section, a 16 year-old proposed two potential design-based solutions:\n\nMake it so you can only download a view of how your game/story/animation works. Or make it so downloadable Scratch files have read only protection. Maybe downloaded Scratch files, after being uploaded, are marked with the creators name at the bottom, and then any DIFFERENT people who edit it after are put on the list.\n\nInfluenced by these comments, Scratch administrators came to believe that negative responses towards remixing were often due to the fact that Scratch users did not acknowledge the sources of their remixes. As a result, these administrators implemented an architectural design change to the Scratch community along the lines suggested by the user in the second half of the quotation above.\n\nThe design change in question involved the introduction of a new technological facility that automatically identified and labeled remixes and inserted hyperlink pointers under each remix to the remix's antecedent and the antecedent's author (see Figure 3). Two days after the introduction of this feature, functionality was added to link to a comprehensive list of derivative works from the pages of antecedent projects (see Figure 4).\n\nThe new feature was announced in the discussion forums by an administrator of the website and user responses were positive. User terminator99 suggested that the change was, \"Awesome.\" Another user, marsUp, posted a comment saying, \"That's a very useful feature! I like that we can do pingpong like modding in Scratch.\" Users who did not visit the discussion forums also responded well to the the new feature. For example, user greekPlus posted a comment on a remix he created saying, \"i remixed it for you but i do not know how to ad credit to you for thinking of it in the first place.\" A few minutes later he realized that the remix automatically displayed the attribution and posted the a comment saying, \"never mind it did it for me. cool!\"\n\n\nSTUDY 1: HUMAN AND MACHINE ATTRIBUTION\n\nAlthough initial user feedback to the automatic attribution feature was positive, users continued to complain about remixing. In Study 1a, we present a quantitative analysis to more fully evaluate the effect of the technological design change described in the previous section. In Study 1b, we offer a parallel analysis of the relationship between manual crediting-giving by users and users' reactions to being remixed.\n\nBoth studies build on a dataset used in previous work by Hill, Monroy-Hernández, and Olson [8]. This dataset includes remix-pairs determined by an algorithm using detailed project metadata tracked by the Scratch online community. The dataset is limited in that it does not include projects whose concepts were copied by a user who had seen another's work but who did not actually copy code, graphics or sound. Similarly, the dataset contains no measure of the \"originality\" of projects or an indicator based on ideas that were taken from a source outside Scratch (e.g., a user may have created a Pacman clone which would not be considered a remix in our analysis).\n\nThe data presented here includes each coded reactions of the author of antecedent projects (i.e., originators) on remixes of their projects shared by other users in the site during a twelve week period after Scratch's launch from May 15 through October 28, 2007. Although 2,543 remixes were shared in this period, we limit our analysis to the 932 projects (37% of the total) that had been viewed at the time of data collection by the project originator -a necessary prerequisite to any response. Of these 932 remixes that were viewed by a project originator, 388 originators (42%) left comments on the remixes in question. The remaining were coded as \"silence.\" Comments left by originators were coded by two coders, blind to the hypotheses of the study and who were found to be reliable [8], as being positive, neutral, or negative. They were also coded as containing accusations of plagiarism (projects in which the the originator directly accused the remixer of copying, e.g., \"Hello mr plagiarist\", \"Copycat!\") or hinting plagiarism (projects in which the originator implied that the remixer had copied but did not state this explicitly, e.g., \"I mostly pretty much made this whole entire game\").\n\nUnless it also contained an explicitly negative reaction, an accusation of plagiarism was not coded as \"negative.\" However, because plagiarism tends to be viewed as negative within Scratch (as suggested by the quotations in the previous section) and more broadly in society [20], we re-coded accusations of plagiarism (both direct and hinting) as \"negative\" except, as was the case in several comments coded as \"hinting plagiarism,\" when these accusations were in comments that were also coded as positive. Previous published work using this dataset, and subsequent robustness checks, show that our results are substantively unchanged if we exclude these explicit charges of plagiarism from the \"negative\" category or exclude only the weaker \"hinting plagiarism\" accusations.\n\n\nStudy 1a: Automatic Attribution\n\nTo test the effectiveness of automatic attribution, we consider the effect of the design intervention described in the previous section. The design change took place six weeks after the public launch of the Scratch community and at the precise midpoint in our data collection window. The intervention affected all projects hosted on the Scratch online community including projects shared before the automatic attribution functionality was activated. As a result, we classify originators' reactions as occurring outside a technological regime of automatic attribution when a project was both uploaded and viewed by a project's originator before automatic attribution functionality was activated.\n\nA comparison of the distribution of coded comments between positive, neutral, negative, and silent in the periods before and after the intervention suggests that the introduction of automatic attribution had little effect on the distribution of reaction types (See Figure 5). Although the period after the intervention saw a larger proportion of users remaining silent and a smaller proportion of both positive and negative comments, χ 2 tests suggest that there is no statistically significant difference in originator reactions between remixes viewed before or after the introduction of automatic attribution (χ 2 = 3.94; df = 3; p = 0.27). As a result, we cannot conclude that the there is any relationship between the presence, or absence, of an automatic attribution system in Scratch and the distribution of different types of reactions.\n\nThese results suggest that automatic attribution systems may have limited effectiveness in communities like Scratch. Of course, our analysis is not without important limitations. For example, the existence of an automatic attribution regime may also affect the behavior of users preparing remixes. A remixer might avoid making perfect copies of projects if they know that their copies will be attributed and are more likely to be discovered.\n\n\nStudy 1b: Manual Crediting\n\nWhile the introduction of an automatic attribution feature to Scratch appears to have had a limited effect on originators responses to remixes of their projects, the presence or absence of credit was a recurring theme in discussions on Scratch online forums -as shown in the quotes in the previous section -and in many of the coded reactions from the periods both before and after the introduction of automatic attribution. Indeed, in project descriptions or notes from the periods both before and after the change, remixers frequently \"manually\" gave credit to the originators of their work. Even after remixes were automatically attributed to originators, remixers who did not also give credit manually -essentially producing information redundant to what was already being displayed by the system -were criticized.\n\nFor example, after the introduction of automatic attribution functionality, a user left the following comment on a remix of their project:\n\nBryan, you need to give me Pumaboy credit for this wonderful game that I mostly pretty much kinda totally made this whole entire game ... and that you need to give me some credit for it For this user, automatic attribution by the system did not represent a sufficient or valid form of credit-giving. In the following study, we test for this effect of \"manual\" creditgiving by remixers on coded response types using a method that parallels the analysis in Study 1a and that uses the same dataset.\n\nManual crediting can happen in multiple ways. Exploratory coding of 133 randomly selected projects showed that 35 (36%) of each remix pair gave credit. Of these 35 projects, 34 gave credit in the project description field while 1 project only gave credit in a \"credits\" screen inside the game. As a result, the authors of this study split the sample of projects used in the Study 1a and coded each of of the user-created descriptions for the presence or absence of explicit or manual credit-giving.\n\nTo first establish that we are examining distinct behaviors, we attempted to establish that automatic and manual attribution do not act as substitutes for each other. As suggested by our qualitative findings and our results in Study 1a, we found little difference in the rate of explicit credit giving between projects created in the presence or absence of automatic attribution. Overall, 276 (about 30%) of the 932 projects in our sample offered explicit credit in the description field of the project. Manual crediting-giving was a widespread practice both before automatic attribution, when 31% of projects in our sample offered explicit credit, and after, when 27% did so. The difference between these two periods was not statistically significant (χ 2 = 1.41; df = 1; p = 0.24). Previous work studying Jumpcut, a video remixing website, supports the idea that automatic and manual credit giving are not interchangable phenomena. One Jumpcut user with permission to creative derivative works commented that they, \"still feel a moral obligation to people as creators who have a moral right to be attributed (and notified) despite the physical design which accomplishes this automatically\" [5].\n\nWe measured effectiveness of manual credit giving using a parallel analysis to Study 1a. As in Study 1a, we compared the distribution of originator reactions in the presence, and absence, of manual credit-giving by remixers. We found that negative reactions are less common in the presence of manual credit but that this difference is very small (from 16% without manual credit to 14% with it). However, we see that the proportion of users who react positively almost doubles in the presence of credit-giving (from 16% with no crediting to 31% in its presence). A graph of these results are shown in Figure 6. Tests show that we can confidently reject the null hypothesis that these differences in the distribution of reactions are due to random variation (χ 2 = 27.60; df = 3; p < 0.001).\n\nAlso important to note is a difference in the number of users who are silent after viewing a project (62% in the absence of manual credit versus 49% in its presence). This larger proportion of commenting in general may have an important substantive effect on the discourse and behavior on the site because silent originators may, for obvious reasons, have a more limited effect on attitudes toward remixing and user experience than vocal users do. As a robustness check, we considered the reaction of only originators who left comments (n = 388) and found that even with a smaller sample, our result were stronger. In the restricted sample, 41% reacted negatively when they were not given credit. However, only 27% did so when they were credited. Similarly, 42% of users who left comments on projects that did not give credit manually left positive messages. Nearly two thirds of comments (61%) were positive when credit was given. These differences, in the reduced sample that includes only explicit reactions, were also statistically significantly different (χ 2 = 14.09; df = 2; p < 0.001). We include the large number of silent participants because we believe that nonresponse is an important type of reaction with real effects on the community. Understanding the reasons behind nonresponse and the effect of silence in response to different types of credit giving remains an opportunity for further research.\n\nAlthough not presented here due to limited space, we followed the general model of previous work using this dataset [8] and tested logistic regression models on dichotomous variables indicating the presence of negative and positive reactions and found that basic relationships described above were robust to the introduction of a control for the the intervention, to an interaction between these two variables, and to controls for the gender and age of originators and to the antecedent project's complexity. Both before or after the intervention, manual crediting resulted in more positive comments by the originators of remixed projects. Of course, the results presented here are uncontrolled, bivariate, relationships and we caution that these results, while provocative, should still be viewed as largely tentative. As we show in the subsequent qualitative analysis, attribution and credit-giving are complex social processes. We do not claim that the preceding analyses capture it fully.\n\n\nSTUDY 2: INTERVIEWS WITH PARTICIPANTS\n\nIn order to explore the reasoning behind young people's remixing behavior and attitudes toward attribution as we observed it in Study 1, we engaged in a second qualitative study and directly asked kids what role attribution and credit plays in their moral evaluations of remixing. \n\n\nMethodology\n\nWe conducted twelve one-hour semi-structured interviews with kids aged 8 to 17 years old. All of the interviewees had experience using computers and had access to the Internet at home. All the interviewees live in the United States except for one who lives in New Zealand. The participants were recruited via the Scratch website and during meet-ups with educators, teachers and young Scratch users. Eight of the interviews were conducted in person, in the Boston area, and the rest over the phone or voice over IP. The interviews were audio-recorded and transcribed before fully analyzing them. Nine of the interviewees were members of the Scratch community. The remaining three did not use Scratch but were included as a way to check if people who do not use Scratch have similar views about remixing, attribution, and credit. We found no substantive difference between the Scratch users and non-users in their answers to questions related to the hypothetical automatic and manual mechanism for attribution.\n\nBefore each interview, subjects completed a survey that elicited demographic information and posed questions about their familiarity with other technologies and which was primarily designed to get a sense of the interviewees' social and technical background. Interviews were structured around a protocol that included a set of nine fictional remixing cases intended to elicit conversations about remixing. 3 The cases were inspired by Sinnreich et al.'s theoretical work and from three years of experience moderating the Scratch community. They were designed to present cases where remixing could be controversial but where there is no clear \"correct\" answer. The goal of the cases was to offer a concrete, and common, set of dilemmas to stimulate broad conversations about attitudes toward remixing.\n\nThe cases were presented in the form of printed screenshots of different project pages from the Scratch website (anonymized to avoid referring to real cases that users might 3 Our interview protocol including example cases is available at http://www.media.mit.edu/∼andresmh/chi2011/interview.html.\n\nhave seen). The print outs were shown to the interviewees (or discussed over the phone) while explaining each case. All the cases included a remix and its corresponding antecedent project. The cases varied in the presence of automatic attribution, manual credit, and the degree of similarity between the remix and its antecedent. For example, the first three cases were:\n\n1. A remix and its antecedent are identical. The project notes only describe how to play the video game. The remix shows the automatic attribution but no manual credit on the notes.\n\n\n2.\n\nA remix and its antecedent are different (as seen visually and in project metadata) but one can clearly see the influence of its antecedent project. The project notes of the remix show manual credit but no automatic attribution. The interviewee was told to imagine the site had a glitch that prevented it from connecting it to its antecedent.\n\n3. The same set of remix and antecedent projects as in (2) but this time automatic attribution is displayed but manual credit is not.\n\nEach of the interview logs was coded using inductive codes and grounded theory [3]. The coded responses were analyzed based on categories related to how interviewees answered specific questions about the distinction between automatic attribution and manual credit.\n\n\nResults\n\nConfirming the results of Study 1, for users of Scratch, automatic attribution was generally seen as insincere and insufficient. Throughout the interviews, we found that for most of the kids, getting explicit credit from another person was preferred over attribution given automatically by the system. When asked why, kids often responded that knowing that another person had cared enough to give credit was valued more than what the computer system would do on its own. The fact that it takes some work, albeit minimal, to write an acknowledgment statement, sends a signal of empathy, authenticity and good intentions [6]. Amy articulated this when explaining why she preferred getting credit from another person:\n\nI would like it even more if the person did it [gave credit] on their own accord, because it would mean that [...] they weren't trying to copy it, pirate it.\n\nSimilarly, Jon explained, \"No [the \"Based on\" is not enough], because he [the remixer] didn't put that, it always says that.\" For Jon, automatic attribution is not authentic because it is always there and, as a result, it is clear that is not coming from the person doing the remix.\n\nMost of the interviewees seemed to have a clear notion of what they think a moral remix should be. For some, it is all about making something different. Jake for example, defines a \"good\" remix as, \"if it has a bunch of differences then it's a good remix. If it has like two, then it's bad.\" In addition to the differences between the remix and its antecedent project, for some, manual credit is part of what makes it moral. Charles said, \"[remixing] is taking somebody else's project and then changing a lot of it and sharing it and giving credit.\" Continuing, Charles explained:\n\nIf Green had actually said in the project notes, \"This is a remix of Red's project, full credit goes to him,\" then I would consider it a remix. But this [pointing at a remix without manual credit] is definitely a copy.\n\nLikewise, Ryan mentions that a fictional remix was, \"perfectly fine because they gave credit in the project notes.\"\n\nInterviewees suggested that manual credit also allows users to be more expressive. For example, Susie explained that expressiveness is the reason that she prefers manual credit through the project notes saying, \"I think the manual one is better because you can say 'thank you' and things like that. The automatic one just says 'it's based on.\"' Susie also notes that for her, the project notes are a space where a creator can express her wishes in regards to her intellectual property, independent, and even in contradiction to, the license of the projects:\n\nIf I do a project that has music that I really like, I often download the project, take the music. Unless it says in the project notes, \"Do not take the music.\"\n\nFor Susie and other users of Scratch, the project notes are a space for more than just instructions on how to interact with one's project; they are an expressive space where one can communicate with an audience without having to encumber the creative piece of work with it.\n\nOthers point at the fact that people do not pay as much attention to automatic attribution statement as much they do to the manual credit left in project descriptions. Jake, for example, explains that, while he agrees there is some usefulness to having both, project notes still are more important, \"because, you know, sometimes people just like skim through a project and you don't see it 'til the end.\" Jake continued to say that creators that do not have both should get a \"warning.\"\n\nEven though interviewees value manual credit, they still see the usefulness of the automatic mechanism as some sort of community-building prosthetic device -an explanation for the positive reactions to the feature's initial introduction. For example, Nicole argues that while manual credit on the notes has more value for her, the automatic attribution is useful as a backup and because it provides a link:\n\nWell, I think that they should probably write in the notes that -then it should also say \"Based on blank's project,\" just in case they forget, and also because it gives a link to the original project and it gives a link to the user so you don't have to search for it.\n\nA similar explanation was articulated on a comment exchange on one the website's galleries. A teenage girl that actively participates in Scratch explained the pragmatic value of automatic attribution saying, \"the 'based on' thingy, it gives a link, and we all luv links, less typing,\" before reiterating that manual credit is more valuable:\n\nat the beginning i thought that you don't have to give credit when the \"based on\" thingy is in there, but i realized a lot of people don't look at that, and i noticed people confused the remix with the original.\n\nCreating a Scratch project is a complicated task. A project's sources can be diverse and the creator can easily forget to acknowledge some, as Paul explains, when asked to choose between a system of manual credit or automatic attribution:\n\nThe thing is, it would be a lot better if they had both. Because, sometimes people probably just forget to do that. And then people would not know.\n\nThere are also situations where interviewees recognize what Posner calls the \"awkwardness of acknowledgment,\" that is, situations where credit is not really needed and it can be an unnecessary burden or go against the aesthetics of the work [20]. For example, Paul mentioned that sometimes, there are some projects in Scratch that are remixed so much -like the sample projects that come with Scratch or some \"remix chains\" 4 -where credit is not necessary:\n\nThere's this one called \"perfect platformer base\" which a lot of people remix. So I don't think that needs any credit. It's not actually a real game. It's all the levels and stuff are just demonstrations.\n\nSince manual crediting has a higher emotional value, some kids mentioned that conflicts over remixing could be addressed by the administrators of the site by editing the project of the remix in question, as a way to enforce credit without transforming it into attribution. Doing so would make it appear that a remixer had credited an antecedent when they had not. Susie offers a suggestion along these lines when asked about how the administrators of the website should deal with a case of a complaint over a remix that is a parody of someone else's project. Susie suggested that, \"I might remove the project but I might not, you know, maybe I would edit the notes to to give credit.\" Similarly, Charles described his approach for solving conflicts if he was the administrator of the website suggesting that, \"I probably just would stay out of the argument. Although not designed to be a random sample, these interviews support the proposition that both Scratch participants and other young people share a set of norms about characteristics that determine what a \"good\" or moral remix is. Among these norms, acknowledging one's sources seems to play a central role. However, participants also seem to share the opinion that this norm is not satisfied through an automated process. They clearly understand the pragmatic value of automating acknowledgment-giving, but they do not see it as a substitute for adherence to the social norm of creditgiving. They also see it as void of emotion and expressiveness. For Scratch users, normative constraints are separate from architectural constraints and one cannot replace the other. These findings support and enrich the results from our first study and help us understand better how Scratch participants, and perhaps kids in general, experience authorship norms and automation in online spaces.\n\n\nCONCLUSIONS\n\nOur results from Study 1a called into the question the effectiveness of automatic attribution functionality in encouraging more positive user reactions in Scratch. We build on these results in Study 1b to suggest that manual crediting may do the work that Scratch's designers had hoped automatic attribution would. Results from the analysis of user interviews presented in Study 2 help to answer the question of \"why?\" and suggest that users find manual credit to be more authentic and more meaningful to users because it takes more time and effort. Usually, UI improvements are designed to help reduce the time and effort involved in using a system. But in trying to help users by attributing automatically, Scratch's designers misunderstood the way that attribution as a social mechanism worked for Scratch's users. Our fundamental insight is that while both attribution and credit may be important, they are distinct concepts and that credit is, socially, worth more. A system can attribute the work of a user but credit, which is seen as much more important by users and which has a greater effect on user behavior, cannot be done automatically. Computers can attribute. Crediting, however, takes a human.\n\nAs we suggested in our introduction, this fundamental result leads to two distinct contributions. First, and more specifically, our analysis offers an improved understanding of the way that attribution and credit works in user-generated content communities over what has been available in previous work. Our two studies suggest that scholars are correct to argue that credit plays an important role in social media communities and offer empirical confirmation for the important role that authenticity plays in how users conceptualize credit. In our in-depth interviews, we explain some of the reasons why this may be the case. Second, through our evaluation of an unsuccessful technological design, our work offers a broader, if more preliminary, contribution in suggesting an important limit of designers' ability to support community norms in social media systems. As the literature on design and social media grows, the importance of good support for communities with healthy norms promoting positive interactions is likely to increase. In attempting to design for these norms, we suspect that researchers will increasingly encounter similar challenges.\n\nWe argue that designers should approach interventions iteratively. This design approach can be understood through the theoretical lens of the social construction of technology [19]: designers can't control technological outcomes which must be built through a close relationship between designers and users. Designers must move away from seeing their profession as providing solutions. They must channel users, work closely with them, and iterate together, to negotiate and achieve a set of shared goals.\n\nThe prevalence of user-generated content sites stresses the importance of how online social spaces should deal with issues of attribution and our results are likely to be immediately relevant to designers. For example, the Semantic Clipboard is a tool built as a system of automatic attribution for content reuse [22]. Developed by researchers who found a high degree of Creative Commons license violations around the re-use of Flickr images, the tool is a Firefox plugin that provides, \"license awareness of Web media,\" and enables people to automatically, \"copy [media] along with the appropriate license metadata.\" Our results suggest one way that this approach may fall short.\n\nHowever, automatic attribution is not the only way that technologists can design to acknowledge others' contributions. Indeed, our results suggest that there may be gains from design changes which encourage credit-giving without simply automating attribution. For example, Scratch's designers might present users with a metadata field that prompts users to credit others and suggests antecedent authors whose work the system has determined may have played a role. This affordance might remind users to credit others, and might increase the amount of crediting, while maintaining a human role in the process and the extra effort that, our research has suggested, imbues manual credit giving with its value. We suggest that in other social media communities, similar affordances that help prompt or remind users to do things that a system might do automatically represent a class of increasingly important design patterns and a template for successful design interventions in support of community norms.\n\nFigure 1 .\n1An example Scratch program written using programming \"blocks\".\n\nFigure 2 .\n2Screenshot of a Scratch remix project highlighting automatic (the area inside the bottom left circle) and manual (the area in the top right circle) attribution.\n\nFigure 3 .\n3Example of the automatic attribution-giving statement displayed under remixes in the period after the design intervention.\n\nFigure 4 .\n4Example of the link pointing to remixes of a project displayed under antecedent projects in the period after the design intervention.\n\nFigure 5 .\n5Mosaic plot showing the distribution of reactions of original creators who had viewed remixes of their project during the six week periods before and after the introduction of automatic attribution. The proportion of response types is shown along the y-axis. The proportion of projects viewed with, and without, automatic attribution is shown along the x-axis. (n = 932)\n\nFigure 6 .\n6Mosaic plot showing the distribution of reactions of original creators who had viewed remixes of their project and indicating whether they left manual credit. The proportion of response types is shown along the y-axis. The proportion of projects including and omitting manual credit is shown along the x-axis. (n = 932)\n\n\nI probably wouldn't remove it [the remix], I'd just add something in the project notes [like] 'based on Gray's project.\"'This phenomena of giving less value to technologically simplified social signals is experienced in other social platforms. For example, Amy expressed how on the social network site Facebook, she loves to get comments on her photographs but dislikes those who do not leave comments or opt instead to press the \"I like it\" button:I love when people comment on my pictures. Everybody sees them, because they tell me they have. I'm like, \"Oh really? That's great. Why didn't you comment?\" I don't like it when people just \"like it\", because you know they have something to say about it; they just don't. It's like, if they like it, then [they should] take the time to say something.\nRemix chains typically start with someone sharing a project inviting others to remix (i.e. \"add your animated avatar to the park.\")\nACKNOWLEDGMENTSThis research was supported by Microsoft Research. Scratch is a project of the Lifelong Kindergarten Group at the MIT Media Lab with financial support from the National Science Foundation award ITR-0325828, Microsoft Corp., Intel Foundation, Google, the MacArthur Foundation and the MIT Media Lab research consortia.\nThe Wealth of Networks: How Social Production Transforms Markets and Freedom. Y Benkler, Yale University PressBenkler, Y. The Wealth of Networks: How Social Production Transforms Markets and Freedom. Yale University Press, Oct. 2007.\n\nAnnouncing (and explaining) our new 2.0 licenses. G O Brown, Brown, G. O. Announcing (and explaining) our new 2.0 licenses. http://creativecommons.org/weblog /entry/4216, May 2004.\n\nConstructing Grounded Theory: A Practical Guide Through Qualitative Analysis. K Charmaz, Sage PublicationsLondonCharmaz, K. Constructing Grounded Theory: A Practical Guide Through Qualitative Analysis. Sage Publications, London, 2006.\n\nAn analysis of the social structure of remix culture. G Cheliotis, J Yew, Proc. C&T 2009. C&T 2009University Park, PA, USAACMCheliotis, G., and Yew, J. An analysis of the social structure of remix culture. In Proc. C&T 2009 (University Park, PA, USA, 2009), ACM, 165-174.\n\nThe evolution of authorship in a remix society. N Diakopoulos, K Luther, Y E Medynskiy, Essa , I , Proc. HT. HTManchester, UKACMDiakopoulos, N., Luther, K., Medynskiy, Y. E., and Essa, I. The evolution of authorship in a remix society. In Proc. HT 2007 (Manchester, UK, 2007), ACM, 133-136.\n\nSignals in social supernets. J Donath, Journal of Computer-Mediated Communication. 13Donath, J. Signals in social supernets. Journal of Computer-Mediated Communication 13, 1 (2008), 231-251.\n\nSocial judgments and technological innovation: Adolescents' understanding of property, privacy, and electronic information. B Friedman, Computers in Human Behavior. 13Friedman, B. Social judgments and technological innovation: Adolescents' understanding of property, privacy, and electronic information. Computers in Human Behavior 13, 3 (Aug. 1997), 327-351.\n\nResponses to remixing on a social media sharing website. B M H Hill, A Monroy-Hernández, K Olson, Proc. ICWSM 2010. ICWSM 2010Washington, D.C.AAAIHill, B. M. H., Monroy-Hernández, A., and Olson, K. Responses to remixing on a social media sharing website. In Proc. ICWSM 2010 (Washington, D.C., 2010), AAAI, 74-81.\n\nPedestrian: Mobile Phones in Japanese Life. M Ito, D Okabe, M Matsuda, Personal, Portable, The MIT PressIto, M., Okabe, D., and Matsuda, M. Personal, Portable, Pedestrian: Mobile Phones in Japanese Life. The MIT Press, 2006.\n\nConvergence Culture: Where Old and New Media Collide. H Jenkins, New York University PressNew YorkJenkins, H. Convergence Culture: Where Old and New Media Collide. New York University Press, New York, 2006.\n\nCode: And Other Laws of Cyberspace, Version 2.0. Basic Books. L Lessig, Lessig, L. Code: And Other Laws of Cyberspace, Version 2.0. Basic Books, Dec. 2006.\n\nMaking Art and Commerce Thrive in the Hybrid Economy. L Lessig, Remix, Penguin Press HC, TheLessig, L. Remix: Making Art and Commerce Thrive in the Hybrid Economy. Penguin Press HC, The, Oct. 2008.\n\nEdits & credits: exploring integration and attribution in online creative collaboration. K Luther, N Diakopoulos, A Bruckman, Proc. CHI 2010. CHI 2010Atlanta, Georgia, USAACMLuther, K., Diakopoulos, N., and Bruckman, A. Edits & credits: exploring integration and attribution in online creative collaboration. In Proc. CHI 2010 (Atlanta, Georgia, USA, 2010), ACM, 2823-2832.\n\n. L Manovich, Remix, Remixability, Manovich, L. Remix and remixability. http://www.manovich.net/DOCS/Remix modular.doc, 2005.\n\nEmpowering kids to create and share programmable media. interactions. A Monroy-Hernández, M Resnick, 15Monroy-Hernández, A., and Resnick, M. Empowering kids to create and share programmable media. interactions 15, 2 (2008), 50-53.\n\nNo fair, copycat!': what children's response to plagiarism tells us about their understanding of ideas. K R Olson, A Shaw, Developmental Science. Olson, K. R., and Shaw, A. 'No fair, copycat!': what children's response to plagiarism tells us about their understanding of ideas. Developmental Science (2010).\n\nUnderstanding the First Generation of Digital Natives. J Palfrey, U Gasser, Born, Digital, Basic BooksPalfrey, J., and Gasser, U. Born Digital: Understanding the First Generation of Digital Natives. Basic Books, Aug. 2008.\n\nCopy and paste literacy: Literacy practices in the production of a MySpace profile. D Perkel, Informal Learning and Digital Media. Perkel, D. Copy and paste literacy: Literacy practices in the production of a MySpace profile. Informal Learning and Digital Media (2006).\n\nThe social construction of facts and artefacts: Or how the sociology of science and the sociology of technology might benefit each other. T J Pinch, W E Bijker, Social Studies of Science. 14Pinch, T. J., and Bijker, W. E. The social construction of facts and artefacts: Or how the sociology of science and the sociology of technology might benefit each other. Social Studies of Science 14, 3 (Aug. 1984), 399-441.\n\nThe little book of plagiarism. R A Posner, Pantheon BooksPosner, R. A. The little book of plagiarism. Pantheon Books, Jan. 2007.\n\nScratch: programming for all. M Resnick, J Maloney, A Monroy-Hernández, N Rusk, E Eastmond, K Brennan, A Millner, E Rosenbaum, J Silver, B Silverman, Y Kafai, Commun. ACM. 52Resnick, M., Maloney, J., Monroy-Hernández, A., Rusk, N., Eastmond, E., Brennan, K., Millner, A., Rosenbaum, E., Silver, J., Silverman, B., and Kafai, Y. Scratch: programming for all. Commun. ACM 52, 11 (2009), 60-67.\n\nPolicy-Aware content reuse on the web. O Seneviratne, L Kagal, T Berners-Lee, The Semantic Web -ISWC. Seneviratne, O., Kagal, L., and Berners-Lee, T. Policy-Aware content reuse on the web. In The Semantic Web -ISWC 2009. 2009, 553-568.\n\nRemix culture on the web: A survey of content reuse on different User-Generated content websites. O Seneviratne, A Monroy-Hernández, Web Science. Raleigh, NCSeneviratne, O., and Monroy-Hernández, A. Remix culture on the web: A survey of content reuse on different User-Generated content websites. In Web Science (Raleigh, NC, Apr. 2010).\n\nCommunity annotation and remix: a research platform and pilot deployment. R Shaw, P Schmitz, Proc. MM. MMSanta Barbara, California, USAACMShaw, R., and Schmitz, P. Community annotation and remix: a research platform and pilot deployment. In Proc. MM 2006 (Santa Barbara, California, USA, 2006), ACM, 89-98.\n\n. A Sinnreich, M Latonero, M Gluck, Ethics reconfigured. Information, Communication &. Sinnreich, A., Latonero, M., and Gluck, M. Ethics reconfigured. Information, Communication &\n\n. Society. 121242Society 12, 8 (2009), 1242.\n\nWhat's mine is mine: territoriality in collaborative authoring. J Thom-Santelli, D R Cosley, G Gay, Proc. CHI. CHIBoston, MA, USAACMThom-Santelli, J., Cosley, D. R., and Gay, G. What's mine is mine: territoriality in collaborative authoring. In Proc. CHI 2009 (Boston, MA, USA, 2009), ACM, 1481-1484.\n\n. ¡/Pre¿¡/Body¿¡/ Html¿, ¡/PRE¿¡/BODY¿¡/HTML¿\n"},"corpus_id":{"kind":"number","value":7494330,"string":"7,494,330"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1507.01285\",\n \"mag\": 3123727583,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"journals/corr/Monroy-Hernandez15a\",\n \"doi\": \"10.1145/1978942.1979452\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1507.01285v1.pdf\"\n ],\n \"pdf_sha\": \"d825d5aeb21acfc950efee6211bb130f7c701083\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"https://arxiv.org/pdf/1507.01285\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Computers Can't Give Credit: How Automatic Attribution Falls Short in an Online Remixing Community"},"authors":{"kind":"list like","value":["Andrés Monroy-Hernández \nMassachusetts Institute of Technology\n02139CambridgeMA\n\nMicrosoft Research\n02142CambridgeMA","Benjamin Mako Hill \nMassachusetts Institute of Technology\n02139CambridgeMA","Jazmin Gonzalez-Rivero \nMicrosoft Research\n02142CambridgeMA","Danah Boyd \nMicrosoft Research\n02142CambridgeMA"],"string":"[\n \"Andrés Monroy-Hernández \\nMassachusetts Institute of Technology\\n02139CambridgeMA\\n\\nMicrosoft Research\\n02142CambridgeMA\",\n \"Benjamin Mako Hill \\nMassachusetts Institute of Technology\\n02139CambridgeMA\",\n \"Jazmin Gonzalez-Rivero \\nMicrosoft Research\\n02142CambridgeMA\",\n \"Danah Boyd \\nMicrosoft Research\\n02142CambridgeMA\"\n]","isConversational":false},"abstract":{"kind":"string","value":"In this paper, we explore the role that attribution plays in shaping user reactions to content reuse, or remixing, in a large user-generated content community. We present two studies using data from the Scratch online communitya social media platform where hundreds of thousands of young people share and remix animations and video games. First, we present a quantitative analysis that examines the effects of a technological design intervention introducing automated attribution of remixes on users' reactions to being remixed. We compare this analysis to a parallel examination of \"manual\" credit-giving. Second, we present a qualitative analysis of twelve in-depth, semi-structured, interviews with Scratch participants on the subject of remixing and attribution. Results from both studies suggest that automatic attribution done by technological systems (i.e., the listing of names of contributors) plays a role that is distinct from, and less valuable than, credit which may superficially involve identical information but takes on new meaning when it is given by a human remixer. We discuss the implications of these findings for the designers of online communities and social media platforms."},"sections":{"kind":"list like","value":[{"title":"INTRODUCTION","content":"Networked information technologies have changed the way people use and reuse creative -and frequently copyrightedmaterials. This change has generated excitement, and heated Best Paper Honorable Mention at CHI 2011. debate, among content-creators, technologists, legal academics, and media scholars. Media theorist Lev Manovich argues that remixing is an ancient cultural tradition (e.g., he has suggested that ancient Rome was a \"remix\" of ancient Greece) but that information technologies have accelerated these processes and made remixing more salient [14]. Sinnreich et al. argue that \"configurable culture\" has been significantly transformed by networked technologies which introduce perfect copying and allow people not only to be inspired by extant creations but to remix the original works themselves [25]. Legal scholars have stressed the importance of remixing in cultural creation broadly and warned that current copyright and intellectual property laws may hinder creativity and innovation [11,1].\nSeveral of the most influential scholarly explorations of remixing as a cultural phenomenon have focused on youth's remixing practices. For example, work on remixing by Jenkins [10] and Ito [9] has focused on young people's use and re-use of media. Palfrey and Gasser have suggested that the cultural practices of \"digital native\" youth have had a significant transformative effect on our culture [17]. Throughout his book \"Remix,\" Lessig uses youth's reuse practices to support an argument against what he considers excessive copyright legal protection [12].\nYet, despite a wide interest in remixing and authorship, researchers have only recently engaged in empirical research on the subject [4]. Several recent treatments have presented studies of video remixing communities [5,24], music remixing communities [4], collaborative video game communities [13] and social network sites [18]. There is also another quantitative study of our empirical setting [8] focused on characterizing the variety of responses to remixing. These studies have tended to be general and largely descriptive ex-arXiv:1507.01285v1 [cs.HC] 5 Jul 2015 aminations of remixing practice. This work has pointed to the existence of norms [5] and the territoriality of digital creators [26] and has considered issues of motivation [4]. However, empirical work has yet to unpack in detail the key social mechanisms that scholars have suggested drive behavior, norms, and motivation in remixing communities.\nPerhaps no mechanism has been more frequently cited as critical for remixing activity than attribution and the related phenomena of plagiarism, reputation, and status. For example, recent survey-based work has suggested that the \"authenticity and legitimacy\" of creative work \"are premised on the explicit acknowledgment of the source materials or 'original creator\"' and that such acknowledgment is a key component of how adults assess the fairness or ethical nature of content reuse [25]. Attribution, in this sense, can be seen as an important way that people distinguish remixing from \"theft.\"\nJudge and law professor Richard Posner stresses the importance of attribution and explains that this is important even when there is no monetary benefit to being attributed. For example, he explains that European copyright law is based on a doctrine of \"moral rights\" that \"entitles a writer or other artist to be credited for his original work and this 'attribution right', as it is called, would give him a legal claim against a plagiarist.\" Posner also explains that \"acknowledgment\" of another's contributions to a derivative negates any charge of plagiarism, although it may not establish originality [20]. Attribution plays such an important role in remix culture that Creative Commons made a requirement for attribution a component of all their licenses after more than 97% of licensors opted to require attribution when it was offered as a choice [2]. Young people's perceptions of attribution and complications around copying have also been examined. An article by Friedman describes that adolescents who allowed \"computer pirating\" -the unauthorized copying of computer programs -did so because technological affordances made it difficult for adolescents to identify \"harmful or unjust consequences of computer-mediated actions\" [7]. In a second study, psychologists Olson and Shaw have found that by five years old, \"children understand that others have ideas and dislike the copying of these ideas\" [16].\nYet, despite the fact that researchers in human computer interaction have begun to explore the complexity of attribution and cited its importance to remixing [13], many designers of online communities pay little attention to issues of attribution in their designs -a fact that is reflected in user behavior. For example, research on the use of photos from the photo sharing site Flickr [22], as well as a number of other usergenerated content communities [23], suggests that most reusers fail to attribute re-used content in ways that public-use licenses require. Although theory and survey based work points to a need to design for attribution in user-generated content communities, we still know very little about how attribution works or how designers might go about doing so. Indeed, our study suggests that the most obvious efforts to design for attribution are likely to be ineffective.\nIn this paper, we employ a mixed methods approach that combines qualitative and quantitative analyses to explore users' reactions to attribution and its absence in a large remixing community. First, we introduce our empirical setting; using qualitative data from users forums and comments, we present a rich description of remixing and evidence to support our core proposition that credit plays a central role in remixing in our environment. Second, we contextualize and describe a technological intervention in our setting, responding directly to several user suggestions, that automated the attribution of creators of antecedent projects when content was remixed. Third, we present a tentative quantitative analysis of the effect of this intervention along with a parallel analysis of the practice of manual creditgiving. We find that credit-giving, done manually, is associated with more positive reactions but that automatic attribution by the system is not associated with a similar effect. Fourth, we present analysis of a set of in-depth interviews with twelve users which helps confirm, and add nuance and depth to, our quantitative findings.\nOur results suggest that young users see an important, if currently under-appreciated and under-theorized, difference between credit and attribution. Credit represents more than a public reference to an \"upstream\" user's contributions. Coming from another human, credit can involve an explicit acknowledgment, an expression of gratitude, and an expression of deference, in a way that simple attribution can not. Our results suggest that identical attribution information means something very different to users when it comes from a computer, and when it comes from a human -and that users often feel that acknowledgment is worth much less when it comes from a system. We conclude that designers should create affordances that make it easier for users to credit each other, rather than to merely pursue automated means of acknowledgment.\nOur study offers two distinct contributions for social scientists and for technology designers. The first is an improved understanding of the way that attribution and credit work in user-generated content communities. The second is a broader contribution to the literature on design that suggests an important limitation to technologists' ability to support community norms and a suggestion for how designers might create affordances. Functionality that allows users to express information that a system might otherwise show automatically may play an important role in successful design for social media environments."},{"title":"SCRATCH: A COMMUNITY OF YOUNG REMIXERS","content":"The Scratch online community is a free and publicly available website where young people share their own video games, animated stories, interactive art, and simulations [15]. Participants use the Scratch programming environment [21], a desktop application, to create these interactive projects by putting together images, music and sounds with programming command blocks (See Figure 1). The Scratch website was officially announced in 2007 and, as of September 2010, has more than 600,000 user accounts who have shared 1.3 million projects. At the time of writing, Scratch users share on average one new project per minute. Examples of projects range from an interactive virtual cake maker, to a simulation of an operating system, to a Pokemon-inspired video game, to an animation about climate change, to tutorials on how to draw cartoons. Like other user-generated content websites, such as YouTube or Flickr, Scratch projects are displayed on a webpage (See Figure 2) where people can interact with them, read metadata and give feedback. Visitors can use their mouse and/or keyboard to control a video game or other type of interactive projects or simply observe an animation play out in a web browser. Metadata displayed next to projects includes a text-based description of the project, the creator's name, the number of views, downloads, \"love its,\" remixes, and galleries (i.e., sets of projects) that the project belongs to. Users can interact with projects by giving feedback in the form of tags, comments, or clicks on the \"love it\" button, and can flag a project as \"inappropriate\" for review by site administrators.\nParticipants' self-reported ages range primarily from 8 to 17 years-old with 12 being the median. Thirty-six percent of users self-report as female. A large minority of users are from the United States (41%) while other countries prominently represented include the United Kingdom, Thailand, Australia, Canada, Brazil, South Korea, Taiwan, Colombia and Mexico. About 28% of all users -more than 170,000have uploaded at least one project."},{"title":"Remixing in Scratch","content":"Scratch users can download any project shared on the website, open it up in the Scratch authoring environment, learn how it was made, and \"remix\" it. In Scratch, the term \"remixing\" refers to the creation of any new version of a Scratch program by adding, removing or changing the programming blocks, images or sounds. In this section we use qualitative data from the Scratch website to provide social context for remixing and to suggest that credit plays an important role in how users conceive of appropriate remixing practice.\nRemixing in Scratch is not only technically possible, it is something that the administrators of the website encourage and try to foster as a way for people to learn from others and collaborate. On every project page, the Scratch website displays a hyperlink with the text \"Some rights reserved\" that points to a child-friendly interpretation of the Creative Commons Attribution-Share Alike license under which all Scratch projects are licensed. 1 Even the name Scratch is a reference to hip hop DJs' practice of mixing records. A large portion of all projects shared on the Scratch website (28%) are remixes of other projects.\nThat said, remixing is not universally unproblematic in Scratch. Previous quantitative analysis of the the Scratch community showed that Scratch participants react both positively and negatively to the remixing of their projects and found that of those users who viewed a remix of their project, about one-fifth left positive comments while the same proportion of users accused the remixer of plagiarism [8]. This ambivalent reaction to remixing is echoed, and given additional texture, in the comments and complaints left by users on the Scratch website and sent to Scratch administrators.\nFor example, even before the Scratch website was publicly announced, a number of early adopters became upset when they found remixes of their projects on the website. Indeed, one of the very first complaints about Scratch occurred on the discussion forums where a 13 year-old boy asked:\nIs it allowed if someone uses your game, changes the theme, then calls it 'their creation'? Because I created a game called \"Paddling 1.5\" and a few days later, a user called \"julie\" redid the background, and called it 'her creation' and I am really annoyed with her for taking credit for MY project!! 2 A similar complaint was sent to the website administrators by a 14-year old boy: I think there should be a way to report plagiarized projects I've been seeing a lot of people's projects taken and renamed. This member, named kings651, has 44 projects, and most of them are made by other people. He even has one that I saw my friend make so I know he actually made it.\nIn other cases, the disagreements over remixing were more public and involved communication via projects and com-1 A copy of the current version of the kid-friendly license is available online at http://scratch.mit.edu/pages/license. The version available today encourages users to give credit manually in the project notes. A strong emphasis on credit-giving was added as a result of the findings reported here but was absent during the period of data collection for this study. 2 All usernames and quotes from the website were changed to disguise the identities of participants. ments. For example, user koolkid15 wrote the following message in a comment which was left is response to a remix that shows a cat frowning:\nHi i'm koolkid15 the original creator of luigi disco jay-man41 copied me!! and didn't even aknowladge me he didn't change anything !! I wrote or drew!! and jayman...if your reading this think about other people!!!! Despite the fact that Scratch was conceived, designed, and launched as a platform for remixing, these users expressed their displeasure at remixing. That said, none of these users complained directly about the reuse of their project in general, but in terms of unfair \"taking credit\", plagiarism, and a lack of acknowledgment. Remixing was seen as problematic for koolkid15, for example, because of the nontransformative nature of reuse, the lack of acknowledgment of antecedent contributors, and the confusion about credit that would result.\nOf course, other, more positive, scenarios around remixing also played out in Scratch. For example, jellogaliboo created a remix of Catham's project and wrote the following in the project notes: \"i kinda copied Catham's \"jetpackcat\" game. i used the kitty, the blocks (i added and changed some), and the fuel thingy.\" Catham later posted his approval of the remix saying, \"I like what you changed about my project!\" Like this example, many of these positive experiences involved explicit credit-giving by a remixer to the creator of the antecedent project."},{"title":"DESIGN INTERVENTION: AUTOMATING ATTRIBUTION","content":"Several user complaints about remixing and plagiarism also included suggestions for how Scratch's designers might address them. For example, in response to the forum thread mentioned in the previous section, a 16 year-old proposed two potential design-based solutions:\nMake it so you can only download a view of how your game/story/animation works. Or make it so downloadable Scratch files have read only protection. Maybe downloaded Scratch files, after being uploaded, are marked with the creators name at the bottom, and then any DIFFERENT people who edit it after are put on the list.\nInfluenced by these comments, Scratch administrators came to believe that negative responses towards remixing were often due to the fact that Scratch users did not acknowledge the sources of their remixes. As a result, these administrators implemented an architectural design change to the Scratch community along the lines suggested by the user in the second half of the quotation above.\nThe design change in question involved the introduction of a new technological facility that automatically identified and labeled remixes and inserted hyperlink pointers under each remix to the remix's antecedent and the antecedent's author (see Figure 3). Two days after the introduction of this feature, functionality was added to link to a comprehensive list of derivative works from the pages of antecedent projects (see Figure 4).\nThe new feature was announced in the discussion forums by an administrator of the website and user responses were positive. User terminator99 suggested that the change was, \"Awesome.\" Another user, marsUp, posted a comment saying, \"That's a very useful feature! I like that we can do pingpong like modding in Scratch.\" Users who did not visit the discussion forums also responded well to the the new feature. For example, user greekPlus posted a comment on a remix he created saying, \"i remixed it for you but i do not know how to ad credit to you for thinking of it in the first place.\" A few minutes later he realized that the remix automatically displayed the attribution and posted the a comment saying, \"never mind it did it for me. cool!\""},{"title":"STUDY 1: HUMAN AND MACHINE ATTRIBUTION","content":"Although initial user feedback to the automatic attribution feature was positive, users continued to complain about remixing. In Study 1a, we present a quantitative analysis to more fully evaluate the effect of the technological design change described in the previous section. In Study 1b, we offer a parallel analysis of the relationship between manual crediting-giving by users and users' reactions to being remixed.\nBoth studies build on a dataset used in previous work by Hill, Monroy-Hernández, and Olson [8]. This dataset includes remix-pairs determined by an algorithm using detailed project metadata tracked by the Scratch online community. The dataset is limited in that it does not include projects whose concepts were copied by a user who had seen another's work but who did not actually copy code, graphics or sound. Similarly, the dataset contains no measure of the \"originality\" of projects or an indicator based on ideas that were taken from a source outside Scratch (e.g., a user may have created a Pacman clone which would not be considered a remix in our analysis).\nThe data presented here includes each coded reactions of the author of antecedent projects (i.e., originators) on remixes of their projects shared by other users in the site during a twelve week period after Scratch's launch from May 15 through October 28, 2007. Although 2,543 remixes were shared in this period, we limit our analysis to the 932 projects (37% of the total) that had been viewed at the time of data collection by the project originator -a necessary prerequisite to any response. Of these 932 remixes that were viewed by a project originator, 388 originators (42%) left comments on the remixes in question. The remaining were coded as \"silence.\" Comments left by originators were coded by two coders, blind to the hypotheses of the study and who were found to be reliable [8], as being positive, neutral, or negative. They were also coded as containing accusations of plagiarism (projects in which the the originator directly accused the remixer of copying, e.g., \"Hello mr plagiarist\", \"Copycat!\") or hinting plagiarism (projects in which the originator implied that the remixer had copied but did not state this explicitly, e.g., \"I mostly pretty much made this whole entire game\").\nUnless it also contained an explicitly negative reaction, an accusation of plagiarism was not coded as \"negative.\" However, because plagiarism tends to be viewed as negative within Scratch (as suggested by the quotations in the previous section) and more broadly in society [20], we re-coded accusations of plagiarism (both direct and hinting) as \"negative\" except, as was the case in several comments coded as \"hinting plagiarism,\" when these accusations were in comments that were also coded as positive. Previous published work using this dataset, and subsequent robustness checks, show that our results are substantively unchanged if we exclude these explicit charges of plagiarism from the \"negative\" category or exclude only the weaker \"hinting plagiarism\" accusations."},{"title":"Study 1a: Automatic Attribution","content":"To test the effectiveness of automatic attribution, we consider the effect of the design intervention described in the previous section. The design change took place six weeks after the public launch of the Scratch community and at the precise midpoint in our data collection window. The intervention affected all projects hosted on the Scratch online community including projects shared before the automatic attribution functionality was activated. As a result, we classify originators' reactions as occurring outside a technological regime of automatic attribution when a project was both uploaded and viewed by a project's originator before automatic attribution functionality was activated.\nA comparison of the distribution of coded comments between positive, neutral, negative, and silent in the periods before and after the intervention suggests that the introduction of automatic attribution had little effect on the distribution of reaction types (See Figure 5). Although the period after the intervention saw a larger proportion of users remaining silent and a smaller proportion of both positive and negative comments, χ 2 tests suggest that there is no statistically significant difference in originator reactions between remixes viewed before or after the introduction of automatic attribution (χ 2 = 3.94; df = 3; p = 0.27). As a result, we cannot conclude that the there is any relationship between the presence, or absence, of an automatic attribution system in Scratch and the distribution of different types of reactions.\nThese results suggest that automatic attribution systems may have limited effectiveness in communities like Scratch. Of course, our analysis is not without important limitations. For example, the existence of an automatic attribution regime may also affect the behavior of users preparing remixes. A remixer might avoid making perfect copies of projects if they know that their copies will be attributed and are more likely to be discovered."},{"title":"Study 1b: Manual Crediting","content":"While the introduction of an automatic attribution feature to Scratch appears to have had a limited effect on originators responses to remixes of their projects, the presence or absence of credit was a recurring theme in discussions on Scratch online forums -as shown in the quotes in the previous section -and in many of the coded reactions from the periods both before and after the introduction of automatic attribution. Indeed, in project descriptions or notes from the periods both before and after the change, remixers frequently \"manually\" gave credit to the originators of their work. Even after remixes were automatically attributed to originators, remixers who did not also give credit manually -essentially producing information redundant to what was already being displayed by the system -were criticized.\nFor example, after the introduction of automatic attribution functionality, a user left the following comment on a remix of their project:\nBryan, you need to give me Pumaboy credit for this wonderful game that I mostly pretty much kinda totally made this whole entire game ... and that you need to give me some credit for it For this user, automatic attribution by the system did not represent a sufficient or valid form of credit-giving. In the following study, we test for this effect of \"manual\" creditgiving by remixers on coded response types using a method that parallels the analysis in Study 1a and that uses the same dataset.\nManual crediting can happen in multiple ways. Exploratory coding of 133 randomly selected projects showed that 35 (36%) of each remix pair gave credit. Of these 35 projects, 34 gave credit in the project description field while 1 project only gave credit in a \"credits\" screen inside the game. As a result, the authors of this study split the sample of projects used in the Study 1a and coded each of of the user-created descriptions for the presence or absence of explicit or manual credit-giving.\nTo first establish that we are examining distinct behaviors, we attempted to establish that automatic and manual attribution do not act as substitutes for each other. As suggested by our qualitative findings and our results in Study 1a, we found little difference in the rate of explicit credit giving between projects created in the presence or absence of automatic attribution. Overall, 276 (about 30%) of the 932 projects in our sample offered explicit credit in the description field of the project. Manual crediting-giving was a widespread practice both before automatic attribution, when 31% of projects in our sample offered explicit credit, and after, when 27% did so. The difference between these two periods was not statistically significant (χ 2 = 1.41; df = 1; p = 0.24). Previous work studying Jumpcut, a video remixing website, supports the idea that automatic and manual credit giving are not interchangable phenomena. One Jumpcut user with permission to creative derivative works commented that they, \"still feel a moral obligation to people as creators who have a moral right to be attributed (and notified) despite the physical design which accomplishes this automatically\" [5].\nWe measured effectiveness of manual credit giving using a parallel analysis to Study 1a. As in Study 1a, we compared the distribution of originator reactions in the presence, and absence, of manual credit-giving by remixers. We found that negative reactions are less common in the presence of manual credit but that this difference is very small (from 16% without manual credit to 14% with it). However, we see that the proportion of users who react positively almost doubles in the presence of credit-giving (from 16% with no crediting to 31% in its presence). A graph of these results are shown in Figure 6. Tests show that we can confidently reject the null hypothesis that these differences in the distribution of reactions are due to random variation (χ 2 = 27.60; df = 3; p < 0.001).\nAlso important to note is a difference in the number of users who are silent after viewing a project (62% in the absence of manual credit versus 49% in its presence). This larger proportion of commenting in general may have an important substantive effect on the discourse and behavior on the site because silent originators may, for obvious reasons, have a more limited effect on attitudes toward remixing and user experience than vocal users do. As a robustness check, we considered the reaction of only originators who left comments (n = 388) and found that even with a smaller sample, our result were stronger. In the restricted sample, 41% reacted negatively when they were not given credit. However, only 27% did so when they were credited. Similarly, 42% of users who left comments on projects that did not give credit manually left positive messages. Nearly two thirds of comments (61%) were positive when credit was given. These differences, in the reduced sample that includes only explicit reactions, were also statistically significantly different (χ 2 = 14.09; df = 2; p < 0.001). We include the large number of silent participants because we believe that nonresponse is an important type of reaction with real effects on the community. Understanding the reasons behind nonresponse and the effect of silence in response to different types of credit giving remains an opportunity for further research.\nAlthough not presented here due to limited space, we followed the general model of previous work using this dataset [8] and tested logistic regression models on dichotomous variables indicating the presence of negative and positive reactions and found that basic relationships described above were robust to the introduction of a control for the the intervention, to an interaction between these two variables, and to controls for the gender and age of originators and to the antecedent project's complexity. Both before or after the intervention, manual crediting resulted in more positive comments by the originators of remixed projects. Of course, the results presented here are uncontrolled, bivariate, relationships and we caution that these results, while provocative, should still be viewed as largely tentative. As we show in the subsequent qualitative analysis, attribution and credit-giving are complex social processes. We do not claim that the preceding analyses capture it fully."},{"title":"STUDY 2: INTERVIEWS WITH PARTICIPANTS","content":"In order to explore the reasoning behind young people's remixing behavior and attitudes toward attribution as we observed it in Study 1, we engaged in a second qualitative study and directly asked kids what role attribution and credit plays in their moral evaluations of remixing."},{"title":"Methodology","content":"We conducted twelve one-hour semi-structured interviews with kids aged 8 to 17 years old. All of the interviewees had experience using computers and had access to the Internet at home. All the interviewees live in the United States except for one who lives in New Zealand. The participants were recruited via the Scratch website and during meet-ups with educators, teachers and young Scratch users. Eight of the interviews were conducted in person, in the Boston area, and the rest over the phone or voice over IP. The interviews were audio-recorded and transcribed before fully analyzing them. Nine of the interviewees were members of the Scratch community. The remaining three did not use Scratch but were included as a way to check if people who do not use Scratch have similar views about remixing, attribution, and credit. We found no substantive difference between the Scratch users and non-users in their answers to questions related to the hypothetical automatic and manual mechanism for attribution.\nBefore each interview, subjects completed a survey that elicited demographic information and posed questions about their familiarity with other technologies and which was primarily designed to get a sense of the interviewees' social and technical background. Interviews were structured around a protocol that included a set of nine fictional remixing cases intended to elicit conversations about remixing. 3 The cases were inspired by Sinnreich et al.'s theoretical work and from three years of experience moderating the Scratch community. They were designed to present cases where remixing could be controversial but where there is no clear \"correct\" answer. The goal of the cases was to offer a concrete, and common, set of dilemmas to stimulate broad conversations about attitudes toward remixing.\nThe cases were presented in the form of printed screenshots of different project pages from the Scratch website (anonymized to avoid referring to real cases that users might 3 Our interview protocol including example cases is available at http://www.media.mit.edu/∼andresmh/chi2011/interview.html.\nhave seen). The print outs were shown to the interviewees (or discussed over the phone) while explaining each case. All the cases included a remix and its corresponding antecedent project. The cases varied in the presence of automatic attribution, manual credit, and the degree of similarity between the remix and its antecedent. For example, the first three cases were:\n1. A remix and its antecedent are identical. The project notes only describe how to play the video game. The remix shows the automatic attribution but no manual credit on the notes."},{"title":"2.","content":"A remix and its antecedent are different (as seen visually and in project metadata) but one can clearly see the influence of its antecedent project. The project notes of the remix show manual credit but no automatic attribution. The interviewee was told to imagine the site had a glitch that prevented it from connecting it to its antecedent.\n3. The same set of remix and antecedent projects as in (2) but this time automatic attribution is displayed but manual credit is not.\nEach of the interview logs was coded using inductive codes and grounded theory [3]. The coded responses were analyzed based on categories related to how interviewees answered specific questions about the distinction between automatic attribution and manual credit."},{"title":"Results","content":"Confirming the results of Study 1, for users of Scratch, automatic attribution was generally seen as insincere and insufficient. Throughout the interviews, we found that for most of the kids, getting explicit credit from another person was preferred over attribution given automatically by the system. When asked why, kids often responded that knowing that another person had cared enough to give credit was valued more than what the computer system would do on its own. The fact that it takes some work, albeit minimal, to write an acknowledgment statement, sends a signal of empathy, authenticity and good intentions [6]. Amy articulated this when explaining why she preferred getting credit from another person:\nI would like it even more if the person did it [gave credit] on their own accord, because it would mean that [...] they weren't trying to copy it, pirate it.\nSimilarly, Jon explained, \"No [the \"Based on\" is not enough], because he [the remixer] didn't put that, it always says that.\" For Jon, automatic attribution is not authentic because it is always there and, as a result, it is clear that is not coming from the person doing the remix.\nMost of the interviewees seemed to have a clear notion of what they think a moral remix should be. For some, it is all about making something different. Jake for example, defines a \"good\" remix as, \"if it has a bunch of differences then it's a good remix. If it has like two, then it's bad.\" In addition to the differences between the remix and its antecedent project, for some, manual credit is part of what makes it moral. Charles said, \"[remixing] is taking somebody else's project and then changing a lot of it and sharing it and giving credit.\" Continuing, Charles explained:\nIf Green had actually said in the project notes, \"This is a remix of Red's project, full credit goes to him,\" then I would consider it a remix. But this [pointing at a remix without manual credit] is definitely a copy.\nLikewise, Ryan mentions that a fictional remix was, \"perfectly fine because they gave credit in the project notes.\"\nInterviewees suggested that manual credit also allows users to be more expressive. For example, Susie explained that expressiveness is the reason that she prefers manual credit through the project notes saying, \"I think the manual one is better because you can say 'thank you' and things like that. The automatic one just says 'it's based on.\"' Susie also notes that for her, the project notes are a space where a creator can express her wishes in regards to her intellectual property, independent, and even in contradiction to, the license of the projects:\nIf I do a project that has music that I really like, I often download the project, take the music. Unless it says in the project notes, \"Do not take the music.\"\nFor Susie and other users of Scratch, the project notes are a space for more than just instructions on how to interact with one's project; they are an expressive space where one can communicate with an audience without having to encumber the creative piece of work with it.\nOthers point at the fact that people do not pay as much attention to automatic attribution statement as much they do to the manual credit left in project descriptions. Jake, for example, explains that, while he agrees there is some usefulness to having both, project notes still are more important, \"because, you know, sometimes people just like skim through a project and you don't see it 'til the end.\" Jake continued to say that creators that do not have both should get a \"warning.\"\nEven though interviewees value manual credit, they still see the usefulness of the automatic mechanism as some sort of community-building prosthetic device -an explanation for the positive reactions to the feature's initial introduction. For example, Nicole argues that while manual credit on the notes has more value for her, the automatic attribution is useful as a backup and because it provides a link:\nWell, I think that they should probably write in the notes that -then it should also say \"Based on blank's project,\" just in case they forget, and also because it gives a link to the original project and it gives a link to the user so you don't have to search for it.\nA similar explanation was articulated on a comment exchange on one the website's galleries. A teenage girl that actively participates in Scratch explained the pragmatic value of automatic attribution saying, \"the 'based on' thingy, it gives a link, and we all luv links, less typing,\" before reiterating that manual credit is more valuable:\nat the beginning i thought that you don't have to give credit when the \"based on\" thingy is in there, but i realized a lot of people don't look at that, and i noticed people confused the remix with the original.\nCreating a Scratch project is a complicated task. A project's sources can be diverse and the creator can easily forget to acknowledge some, as Paul explains, when asked to choose between a system of manual credit or automatic attribution:\nThe thing is, it would be a lot better if they had both. Because, sometimes people probably just forget to do that. And then people would not know.\nThere are also situations where interviewees recognize what Posner calls the \"awkwardness of acknowledgment,\" that is, situations where credit is not really needed and it can be an unnecessary burden or go against the aesthetics of the work [20]. For example, Paul mentioned that sometimes, there are some projects in Scratch that are remixed so much -like the sample projects that come with Scratch or some \"remix chains\" 4 -where credit is not necessary:\nThere's this one called \"perfect platformer base\" which a lot of people remix. So I don't think that needs any credit. It's not actually a real game. It's all the levels and stuff are just demonstrations.\nSince manual crediting has a higher emotional value, some kids mentioned that conflicts over remixing could be addressed by the administrators of the site by editing the project of the remix in question, as a way to enforce credit without transforming it into attribution. Doing so would make it appear that a remixer had credited an antecedent when they had not. Susie offers a suggestion along these lines when asked about how the administrators of the website should deal with a case of a complaint over a remix that is a parody of someone else's project. Susie suggested that, \"I might remove the project but I might not, you know, maybe I would edit the notes to to give credit.\" Similarly, Charles described his approach for solving conflicts if he was the administrator of the website suggesting that, \"I probably just would stay out of the argument. Although not designed to be a random sample, these interviews support the proposition that both Scratch participants and other young people share a set of norms about characteristics that determine what a \"good\" or moral remix is. Among these norms, acknowledging one's sources seems to play a central role. However, participants also seem to share the opinion that this norm is not satisfied through an automated process. They clearly understand the pragmatic value of automating acknowledgment-giving, but they do not see it as a substitute for adherence to the social norm of creditgiving. They also see it as void of emotion and expressiveness. For Scratch users, normative constraints are separate from architectural constraints and one cannot replace the other. These findings support and enrich the results from our first study and help us understand better how Scratch participants, and perhaps kids in general, experience authorship norms and automation in online spaces."},{"title":"CONCLUSIONS","content":"Our results from Study 1a called into the question the effectiveness of automatic attribution functionality in encouraging more positive user reactions in Scratch. We build on these results in Study 1b to suggest that manual crediting may do the work that Scratch's designers had hoped automatic attribution would. Results from the analysis of user interviews presented in Study 2 help to answer the question of \"why?\" and suggest that users find manual credit to be more authentic and more meaningful to users because it takes more time and effort. Usually, UI improvements are designed to help reduce the time and effort involved in using a system. But in trying to help users by attributing automatically, Scratch's designers misunderstood the way that attribution as a social mechanism worked for Scratch's users. Our fundamental insight is that while both attribution and credit may be important, they are distinct concepts and that credit is, socially, worth more. A system can attribute the work of a user but credit, which is seen as much more important by users and which has a greater effect on user behavior, cannot be done automatically. Computers can attribute. Crediting, however, takes a human.\nAs we suggested in our introduction, this fundamental result leads to two distinct contributions. First, and more specifically, our analysis offers an improved understanding of the way that attribution and credit works in user-generated content communities over what has been available in previous work. Our two studies suggest that scholars are correct to argue that credit plays an important role in social media communities and offer empirical confirmation for the important role that authenticity plays in how users conceptualize credit. In our in-depth interviews, we explain some of the reasons why this may be the case. Second, through our evaluation of an unsuccessful technological design, our work offers a broader, if more preliminary, contribution in suggesting an important limit of designers' ability to support community norms in social media systems. As the literature on design and social media grows, the importance of good support for communities with healthy norms promoting positive interactions is likely to increase. In attempting to design for these norms, we suspect that researchers will increasingly encounter similar challenges.\nWe argue that designers should approach interventions iteratively. This design approach can be understood through the theoretical lens of the social construction of technology [19]: designers can't control technological outcomes which must be built through a close relationship between designers and users. Designers must move away from seeing their profession as providing solutions. They must channel users, work closely with them, and iterate together, to negotiate and achieve a set of shared goals.\nThe prevalence of user-generated content sites stresses the importance of how online social spaces should deal with issues of attribution and our results are likely to be immediately relevant to designers. For example, the Semantic Clipboard is a tool built as a system of automatic attribution for content reuse [22]. Developed by researchers who found a high degree of Creative Commons license violations around the re-use of Flickr images, the tool is a Firefox plugin that provides, \"license awareness of Web media,\" and enables people to automatically, \"copy [media] along with the appropriate license metadata.\" Our results suggest one way that this approach may fall short.\nHowever, automatic attribution is not the only way that technologists can design to acknowledge others' contributions. Indeed, our results suggest that there may be gains from design changes which encourage credit-giving without simply automating attribution. For example, Scratch's designers might present users with a metadata field that prompts users to credit others and suggests antecedent authors whose work the system has determined may have played a role. This affordance might remind users to credit others, and might increase the amount of crediting, while maintaining a human role in the process and the extra effort that, our research has suggested, imbues manual credit giving with its value. We suggest that in other social media communities, similar affordances that help prompt or remind users to do things that a system might do automatically represent a class of increasingly important design patterns and a template for successful design interventions in support of community norms."},{"title":"Figure 1 .","content":""},{"title":"Figure 2 .","content":""},{"title":"Figure 3 .","content":""},{"title":"Figure 4 .","content":""},{"title":"Figure 5 .","content":""},{"title":"Figure 6 .","content":""}],"string":"[\n {\n \"title\": \"INTRODUCTION\",\n \"content\": \"Networked information technologies have changed the way people use and reuse creative -and frequently copyrightedmaterials. This change has generated excitement, and heated Best Paper Honorable Mention at CHI 2011. debate, among content-creators, technologists, legal academics, and media scholars. Media theorist Lev Manovich argues that remixing is an ancient cultural tradition (e.g., he has suggested that ancient Rome was a \\\"remix\\\" of ancient Greece) but that information technologies have accelerated these processes and made remixing more salient [14]. Sinnreich et al. argue that \\\"configurable culture\\\" has been significantly transformed by networked technologies which introduce perfect copying and allow people not only to be inspired by extant creations but to remix the original works themselves [25]. Legal scholars have stressed the importance of remixing in cultural creation broadly and warned that current copyright and intellectual property laws may hinder creativity and innovation [11,1].\\nSeveral of the most influential scholarly explorations of remixing as a cultural phenomenon have focused on youth's remixing practices. For example, work on remixing by Jenkins [10] and Ito [9] has focused on young people's use and re-use of media. Palfrey and Gasser have suggested that the cultural practices of \\\"digital native\\\" youth have had a significant transformative effect on our culture [17]. Throughout his book \\\"Remix,\\\" Lessig uses youth's reuse practices to support an argument against what he considers excessive copyright legal protection [12].\\nYet, despite a wide interest in remixing and authorship, researchers have only recently engaged in empirical research on the subject [4]. Several recent treatments have presented studies of video remixing communities [5,24], music remixing communities [4], collaborative video game communities [13] and social network sites [18]. There is also another quantitative study of our empirical setting [8] focused on characterizing the variety of responses to remixing. These studies have tended to be general and largely descriptive ex-arXiv:1507.01285v1 [cs.HC] 5 Jul 2015 aminations of remixing practice. This work has pointed to the existence of norms [5] and the territoriality of digital creators [26] and has considered issues of motivation [4]. However, empirical work has yet to unpack in detail the key social mechanisms that scholars have suggested drive behavior, norms, and motivation in remixing communities.\\nPerhaps no mechanism has been more frequently cited as critical for remixing activity than attribution and the related phenomena of plagiarism, reputation, and status. For example, recent survey-based work has suggested that the \\\"authenticity and legitimacy\\\" of creative work \\\"are premised on the explicit acknowledgment of the source materials or 'original creator\\\"' and that such acknowledgment is a key component of how adults assess the fairness or ethical nature of content reuse [25]. Attribution, in this sense, can be seen as an important way that people distinguish remixing from \\\"theft.\\\"\\nJudge and law professor Richard Posner stresses the importance of attribution and explains that this is important even when there is no monetary benefit to being attributed. For example, he explains that European copyright law is based on a doctrine of \\\"moral rights\\\" that \\\"entitles a writer or other artist to be credited for his original work and this 'attribution right', as it is called, would give him a legal claim against a plagiarist.\\\" Posner also explains that \\\"acknowledgment\\\" of another's contributions to a derivative negates any charge of plagiarism, although it may not establish originality [20]. Attribution plays such an important role in remix culture that Creative Commons made a requirement for attribution a component of all their licenses after more than 97% of licensors opted to require attribution when it was offered as a choice [2]. Young people's perceptions of attribution and complications around copying have also been examined. An article by Friedman describes that adolescents who allowed \\\"computer pirating\\\" -the unauthorized copying of computer programs -did so because technological affordances made it difficult for adolescents to identify \\\"harmful or unjust consequences of computer-mediated actions\\\" [7]. In a second study, psychologists Olson and Shaw have found that by five years old, \\\"children understand that others have ideas and dislike the copying of these ideas\\\" [16].\\nYet, despite the fact that researchers in human computer interaction have begun to explore the complexity of attribution and cited its importance to remixing [13], many designers of online communities pay little attention to issues of attribution in their designs -a fact that is reflected in user behavior. For example, research on the use of photos from the photo sharing site Flickr [22], as well as a number of other usergenerated content communities [23], suggests that most reusers fail to attribute re-used content in ways that public-use licenses require. Although theory and survey based work points to a need to design for attribution in user-generated content communities, we still know very little about how attribution works or how designers might go about doing so. Indeed, our study suggests that the most obvious efforts to design for attribution are likely to be ineffective.\\nIn this paper, we employ a mixed methods approach that combines qualitative and quantitative analyses to explore users' reactions to attribution and its absence in a large remixing community. First, we introduce our empirical setting; using qualitative data from users forums and comments, we present a rich description of remixing and evidence to support our core proposition that credit plays a central role in remixing in our environment. Second, we contextualize and describe a technological intervention in our setting, responding directly to several user suggestions, that automated the attribution of creators of antecedent projects when content was remixed. Third, we present a tentative quantitative analysis of the effect of this intervention along with a parallel analysis of the practice of manual creditgiving. We find that credit-giving, done manually, is associated with more positive reactions but that automatic attribution by the system is not associated with a similar effect. Fourth, we present analysis of a set of in-depth interviews with twelve users which helps confirm, and add nuance and depth to, our quantitative findings.\\nOur results suggest that young users see an important, if currently under-appreciated and under-theorized, difference between credit and attribution. Credit represents more than a public reference to an \\\"upstream\\\" user's contributions. Coming from another human, credit can involve an explicit acknowledgment, an expression of gratitude, and an expression of deference, in a way that simple attribution can not. Our results suggest that identical attribution information means something very different to users when it comes from a computer, and when it comes from a human -and that users often feel that acknowledgment is worth much less when it comes from a system. We conclude that designers should create affordances that make it easier for users to credit each other, rather than to merely pursue automated means of acknowledgment.\\nOur study offers two distinct contributions for social scientists and for technology designers. The first is an improved understanding of the way that attribution and credit work in user-generated content communities. The second is a broader contribution to the literature on design that suggests an important limitation to technologists' ability to support community norms and a suggestion for how designers might create affordances. Functionality that allows users to express information that a system might otherwise show automatically may play an important role in successful design for social media environments.\"\n },\n {\n \"title\": \"SCRATCH: A COMMUNITY OF YOUNG REMIXERS\",\n \"content\": \"The Scratch online community is a free and publicly available website where young people share their own video games, animated stories, interactive art, and simulations [15]. Participants use the Scratch programming environment [21], a desktop application, to create these interactive projects by putting together images, music and sounds with programming command blocks (See Figure 1). The Scratch website was officially announced in 2007 and, as of September 2010, has more than 600,000 user accounts who have shared 1.3 million projects. At the time of writing, Scratch users share on average one new project per minute. Examples of projects range from an interactive virtual cake maker, to a simulation of an operating system, to a Pokemon-inspired video game, to an animation about climate change, to tutorials on how to draw cartoons. Like other user-generated content websites, such as YouTube or Flickr, Scratch projects are displayed on a webpage (See Figure 2) where people can interact with them, read metadata and give feedback. Visitors can use their mouse and/or keyboard to control a video game or other type of interactive projects or simply observe an animation play out in a web browser. Metadata displayed next to projects includes a text-based description of the project, the creator's name, the number of views, downloads, \\\"love its,\\\" remixes, and galleries (i.e., sets of projects) that the project belongs to. Users can interact with projects by giving feedback in the form of tags, comments, or clicks on the \\\"love it\\\" button, and can flag a project as \\\"inappropriate\\\" for review by site administrators.\\nParticipants' self-reported ages range primarily from 8 to 17 years-old with 12 being the median. Thirty-six percent of users self-report as female. A large minority of users are from the United States (41%) while other countries prominently represented include the United Kingdom, Thailand, Australia, Canada, Brazil, South Korea, Taiwan, Colombia and Mexico. About 28% of all users -more than 170,000have uploaded at least one project.\"\n },\n {\n \"title\": \"Remixing in Scratch\",\n \"content\": \"Scratch users can download any project shared on the website, open it up in the Scratch authoring environment, learn how it was made, and \\\"remix\\\" it. In Scratch, the term \\\"remixing\\\" refers to the creation of any new version of a Scratch program by adding, removing or changing the programming blocks, images or sounds. In this section we use qualitative data from the Scratch website to provide social context for remixing and to suggest that credit plays an important role in how users conceive of appropriate remixing practice.\\nRemixing in Scratch is not only technically possible, it is something that the administrators of the website encourage and try to foster as a way for people to learn from others and collaborate. On every project page, the Scratch website displays a hyperlink with the text \\\"Some rights reserved\\\" that points to a child-friendly interpretation of the Creative Commons Attribution-Share Alike license under which all Scratch projects are licensed. 1 Even the name Scratch is a reference to hip hop DJs' practice of mixing records. A large portion of all projects shared on the Scratch website (28%) are remixes of other projects.\\nThat said, remixing is not universally unproblematic in Scratch. Previous quantitative analysis of the the Scratch community showed that Scratch participants react both positively and negatively to the remixing of their projects and found that of those users who viewed a remix of their project, about one-fifth left positive comments while the same proportion of users accused the remixer of plagiarism [8]. This ambivalent reaction to remixing is echoed, and given additional texture, in the comments and complaints left by users on the Scratch website and sent to Scratch administrators.\\nFor example, even before the Scratch website was publicly announced, a number of early adopters became upset when they found remixes of their projects on the website. Indeed, one of the very first complaints about Scratch occurred on the discussion forums where a 13 year-old boy asked:\\nIs it allowed if someone uses your game, changes the theme, then calls it 'their creation'? Because I created a game called \\\"Paddling 1.5\\\" and a few days later, a user called \\\"julie\\\" redid the background, and called it 'her creation' and I am really annoyed with her for taking credit for MY project!! 2 A similar complaint was sent to the website administrators by a 14-year old boy: I think there should be a way to report plagiarized projects I've been seeing a lot of people's projects taken and renamed. This member, named kings651, has 44 projects, and most of them are made by other people. He even has one that I saw my friend make so I know he actually made it.\\nIn other cases, the disagreements over remixing were more public and involved communication via projects and com-1 A copy of the current version of the kid-friendly license is available online at http://scratch.mit.edu/pages/license. The version available today encourages users to give credit manually in the project notes. A strong emphasis on credit-giving was added as a result of the findings reported here but was absent during the period of data collection for this study. 2 All usernames and quotes from the website were changed to disguise the identities of participants. ments. For example, user koolkid15 wrote the following message in a comment which was left is response to a remix that shows a cat frowning:\\nHi i'm koolkid15 the original creator of luigi disco jay-man41 copied me!! and didn't even aknowladge me he didn't change anything !! I wrote or drew!! and jayman...if your reading this think about other people!!!! Despite the fact that Scratch was conceived, designed, and launched as a platform for remixing, these users expressed their displeasure at remixing. That said, none of these users complained directly about the reuse of their project in general, but in terms of unfair \\\"taking credit\\\", plagiarism, and a lack of acknowledgment. Remixing was seen as problematic for koolkid15, for example, because of the nontransformative nature of reuse, the lack of acknowledgment of antecedent contributors, and the confusion about credit that would result.\\nOf course, other, more positive, scenarios around remixing also played out in Scratch. For example, jellogaliboo created a remix of Catham's project and wrote the following in the project notes: \\\"i kinda copied Catham's \\\"jetpackcat\\\" game. i used the kitty, the blocks (i added and changed some), and the fuel thingy.\\\" Catham later posted his approval of the remix saying, \\\"I like what you changed about my project!\\\" Like this example, many of these positive experiences involved explicit credit-giving by a remixer to the creator of the antecedent project.\"\n },\n {\n \"title\": \"DESIGN INTERVENTION: AUTOMATING ATTRIBUTION\",\n \"content\": \"Several user complaints about remixing and plagiarism also included suggestions for how Scratch's designers might address them. For example, in response to the forum thread mentioned in the previous section, a 16 year-old proposed two potential design-based solutions:\\nMake it so you can only download a view of how your game/story/animation works. Or make it so downloadable Scratch files have read only protection. Maybe downloaded Scratch files, after being uploaded, are marked with the creators name at the bottom, and then any DIFFERENT people who edit it after are put on the list.\\nInfluenced by these comments, Scratch administrators came to believe that negative responses towards remixing were often due to the fact that Scratch users did not acknowledge the sources of their remixes. As a result, these administrators implemented an architectural design change to the Scratch community along the lines suggested by the user in the second half of the quotation above.\\nThe design change in question involved the introduction of a new technological facility that automatically identified and labeled remixes and inserted hyperlink pointers under each remix to the remix's antecedent and the antecedent's author (see Figure 3). Two days after the introduction of this feature, functionality was added to link to a comprehensive list of derivative works from the pages of antecedent projects (see Figure 4).\\nThe new feature was announced in the discussion forums by an administrator of the website and user responses were positive. User terminator99 suggested that the change was, \\\"Awesome.\\\" Another user, marsUp, posted a comment saying, \\\"That's a very useful feature! I like that we can do pingpong like modding in Scratch.\\\" Users who did not visit the discussion forums also responded well to the the new feature. For example, user greekPlus posted a comment on a remix he created saying, \\\"i remixed it for you but i do not know how to ad credit to you for thinking of it in the first place.\\\" A few minutes later he realized that the remix automatically displayed the attribution and posted the a comment saying, \\\"never mind it did it for me. cool!\\\"\"\n },\n {\n \"title\": \"STUDY 1: HUMAN AND MACHINE ATTRIBUTION\",\n \"content\": \"Although initial user feedback to the automatic attribution feature was positive, users continued to complain about remixing. In Study 1a, we present a quantitative analysis to more fully evaluate the effect of the technological design change described in the previous section. In Study 1b, we offer a parallel analysis of the relationship between manual crediting-giving by users and users' reactions to being remixed.\\nBoth studies build on a dataset used in previous work by Hill, Monroy-Hernández, and Olson [8]. This dataset includes remix-pairs determined by an algorithm using detailed project metadata tracked by the Scratch online community. The dataset is limited in that it does not include projects whose concepts were copied by a user who had seen another's work but who did not actually copy code, graphics or sound. Similarly, the dataset contains no measure of the \\\"originality\\\" of projects or an indicator based on ideas that were taken from a source outside Scratch (e.g., a user may have created a Pacman clone which would not be considered a remix in our analysis).\\nThe data presented here includes each coded reactions of the author of antecedent projects (i.e., originators) on remixes of their projects shared by other users in the site during a twelve week period after Scratch's launch from May 15 through October 28, 2007. Although 2,543 remixes were shared in this period, we limit our analysis to the 932 projects (37% of the total) that had been viewed at the time of data collection by the project originator -a necessary prerequisite to any response. Of these 932 remixes that were viewed by a project originator, 388 originators (42%) left comments on the remixes in question. The remaining were coded as \\\"silence.\\\" Comments left by originators were coded by two coders, blind to the hypotheses of the study and who were found to be reliable [8], as being positive, neutral, or negative. They were also coded as containing accusations of plagiarism (projects in which the the originator directly accused the remixer of copying, e.g., \\\"Hello mr plagiarist\\\", \\\"Copycat!\\\") or hinting plagiarism (projects in which the originator implied that the remixer had copied but did not state this explicitly, e.g., \\\"I mostly pretty much made this whole entire game\\\").\\nUnless it also contained an explicitly negative reaction, an accusation of plagiarism was not coded as \\\"negative.\\\" However, because plagiarism tends to be viewed as negative within Scratch (as suggested by the quotations in the previous section) and more broadly in society [20], we re-coded accusations of plagiarism (both direct and hinting) as \\\"negative\\\" except, as was the case in several comments coded as \\\"hinting plagiarism,\\\" when these accusations were in comments that were also coded as positive. Previous published work using this dataset, and subsequent robustness checks, show that our results are substantively unchanged if we exclude these explicit charges of plagiarism from the \\\"negative\\\" category or exclude only the weaker \\\"hinting plagiarism\\\" accusations.\"\n },\n {\n \"title\": \"Study 1a: Automatic Attribution\",\n \"content\": \"To test the effectiveness of automatic attribution, we consider the effect of the design intervention described in the previous section. The design change took place six weeks after the public launch of the Scratch community and at the precise midpoint in our data collection window. The intervention affected all projects hosted on the Scratch online community including projects shared before the automatic attribution functionality was activated. As a result, we classify originators' reactions as occurring outside a technological regime of automatic attribution when a project was both uploaded and viewed by a project's originator before automatic attribution functionality was activated.\\nA comparison of the distribution of coded comments between positive, neutral, negative, and silent in the periods before and after the intervention suggests that the introduction of automatic attribution had little effect on the distribution of reaction types (See Figure 5). Although the period after the intervention saw a larger proportion of users remaining silent and a smaller proportion of both positive and negative comments, χ 2 tests suggest that there is no statistically significant difference in originator reactions between remixes viewed before or after the introduction of automatic attribution (χ 2 = 3.94; df = 3; p = 0.27). As a result, we cannot conclude that the there is any relationship between the presence, or absence, of an automatic attribution system in Scratch and the distribution of different types of reactions.\\nThese results suggest that automatic attribution systems may have limited effectiveness in communities like Scratch. Of course, our analysis is not without important limitations. For example, the existence of an automatic attribution regime may also affect the behavior of users preparing remixes. A remixer might avoid making perfect copies of projects if they know that their copies will be attributed and are more likely to be discovered.\"\n },\n {\n \"title\": \"Study 1b: Manual Crediting\",\n \"content\": \"While the introduction of an automatic attribution feature to Scratch appears to have had a limited effect on originators responses to remixes of their projects, the presence or absence of credit was a recurring theme in discussions on Scratch online forums -as shown in the quotes in the previous section -and in many of the coded reactions from the periods both before and after the introduction of automatic attribution. Indeed, in project descriptions or notes from the periods both before and after the change, remixers frequently \\\"manually\\\" gave credit to the originators of their work. Even after remixes were automatically attributed to originators, remixers who did not also give credit manually -essentially producing information redundant to what was already being displayed by the system -were criticized.\\nFor example, after the introduction of automatic attribution functionality, a user left the following comment on a remix of their project:\\nBryan, you need to give me Pumaboy credit for this wonderful game that I mostly pretty much kinda totally made this whole entire game ... and that you need to give me some credit for it For this user, automatic attribution by the system did not represent a sufficient or valid form of credit-giving. In the following study, we test for this effect of \\\"manual\\\" creditgiving by remixers on coded response types using a method that parallels the analysis in Study 1a and that uses the same dataset.\\nManual crediting can happen in multiple ways. Exploratory coding of 133 randomly selected projects showed that 35 (36%) of each remix pair gave credit. Of these 35 projects, 34 gave credit in the project description field while 1 project only gave credit in a \\\"credits\\\" screen inside the game. As a result, the authors of this study split the sample of projects used in the Study 1a and coded each of of the user-created descriptions for the presence or absence of explicit or manual credit-giving.\\nTo first establish that we are examining distinct behaviors, we attempted to establish that automatic and manual attribution do not act as substitutes for each other. As suggested by our qualitative findings and our results in Study 1a, we found little difference in the rate of explicit credit giving between projects created in the presence or absence of automatic attribution. Overall, 276 (about 30%) of the 932 projects in our sample offered explicit credit in the description field of the project. Manual crediting-giving was a widespread practice both before automatic attribution, when 31% of projects in our sample offered explicit credit, and after, when 27% did so. The difference between these two periods was not statistically significant (χ 2 = 1.41; df = 1; p = 0.24). Previous work studying Jumpcut, a video remixing website, supports the idea that automatic and manual credit giving are not interchangable phenomena. One Jumpcut user with permission to creative derivative works commented that they, \\\"still feel a moral obligation to people as creators who have a moral right to be attributed (and notified) despite the physical design which accomplishes this automatically\\\" [5].\\nWe measured effectiveness of manual credit giving using a parallel analysis to Study 1a. As in Study 1a, we compared the distribution of originator reactions in the presence, and absence, of manual credit-giving by remixers. We found that negative reactions are less common in the presence of manual credit but that this difference is very small (from 16% without manual credit to 14% with it). However, we see that the proportion of users who react positively almost doubles in the presence of credit-giving (from 16% with no crediting to 31% in its presence). A graph of these results are shown in Figure 6. Tests show that we can confidently reject the null hypothesis that these differences in the distribution of reactions are due to random variation (χ 2 = 27.60; df = 3; p < 0.001).\\nAlso important to note is a difference in the number of users who are silent after viewing a project (62% in the absence of manual credit versus 49% in its presence). This larger proportion of commenting in general may have an important substantive effect on the discourse and behavior on the site because silent originators may, for obvious reasons, have a more limited effect on attitudes toward remixing and user experience than vocal users do. As a robustness check, we considered the reaction of only originators who left comments (n = 388) and found that even with a smaller sample, our result were stronger. In the restricted sample, 41% reacted negatively when they were not given credit. However, only 27% did so when they were credited. Similarly, 42% of users who left comments on projects that did not give credit manually left positive messages. Nearly two thirds of comments (61%) were positive when credit was given. These differences, in the reduced sample that includes only explicit reactions, were also statistically significantly different (χ 2 = 14.09; df = 2; p < 0.001). We include the large number of silent participants because we believe that nonresponse is an important type of reaction with real effects on the community. Understanding the reasons behind nonresponse and the effect of silence in response to different types of credit giving remains an opportunity for further research.\\nAlthough not presented here due to limited space, we followed the general model of previous work using this dataset [8] and tested logistic regression models on dichotomous variables indicating the presence of negative and positive reactions and found that basic relationships described above were robust to the introduction of a control for the the intervention, to an interaction between these two variables, and to controls for the gender and age of originators and to the antecedent project's complexity. Both before or after the intervention, manual crediting resulted in more positive comments by the originators of remixed projects. Of course, the results presented here are uncontrolled, bivariate, relationships and we caution that these results, while provocative, should still be viewed as largely tentative. As we show in the subsequent qualitative analysis, attribution and credit-giving are complex social processes. We do not claim that the preceding analyses capture it fully.\"\n },\n {\n \"title\": \"STUDY 2: INTERVIEWS WITH PARTICIPANTS\",\n \"content\": \"In order to explore the reasoning behind young people's remixing behavior and attitudes toward attribution as we observed it in Study 1, we engaged in a second qualitative study and directly asked kids what role attribution and credit plays in their moral evaluations of remixing.\"\n },\n {\n \"title\": \"Methodology\",\n \"content\": \"We conducted twelve one-hour semi-structured interviews with kids aged 8 to 17 years old. All of the interviewees had experience using computers and had access to the Internet at home. All the interviewees live in the United States except for one who lives in New Zealand. The participants were recruited via the Scratch website and during meet-ups with educators, teachers and young Scratch users. Eight of the interviews were conducted in person, in the Boston area, and the rest over the phone or voice over IP. The interviews were audio-recorded and transcribed before fully analyzing them. Nine of the interviewees were members of the Scratch community. The remaining three did not use Scratch but were included as a way to check if people who do not use Scratch have similar views about remixing, attribution, and credit. We found no substantive difference between the Scratch users and non-users in their answers to questions related to the hypothetical automatic and manual mechanism for attribution.\\nBefore each interview, subjects completed a survey that elicited demographic information and posed questions about their familiarity with other technologies and which was primarily designed to get a sense of the interviewees' social and technical background. Interviews were structured around a protocol that included a set of nine fictional remixing cases intended to elicit conversations about remixing. 3 The cases were inspired by Sinnreich et al.'s theoretical work and from three years of experience moderating the Scratch community. They were designed to present cases where remixing could be controversial but where there is no clear \\\"correct\\\" answer. The goal of the cases was to offer a concrete, and common, set of dilemmas to stimulate broad conversations about attitudes toward remixing.\\nThe cases were presented in the form of printed screenshots of different project pages from the Scratch website (anonymized to avoid referring to real cases that users might 3 Our interview protocol including example cases is available at http://www.media.mit.edu/∼andresmh/chi2011/interview.html.\\nhave seen). The print outs were shown to the interviewees (or discussed over the phone) while explaining each case. All the cases included a remix and its corresponding antecedent project. The cases varied in the presence of automatic attribution, manual credit, and the degree of similarity between the remix and its antecedent. For example, the first three cases were:\\n1. A remix and its antecedent are identical. The project notes only describe how to play the video game. The remix shows the automatic attribution but no manual credit on the notes.\"\n },\n {\n \"title\": \"2.\",\n \"content\": \"A remix and its antecedent are different (as seen visually and in project metadata) but one can clearly see the influence of its antecedent project. The project notes of the remix show manual credit but no automatic attribution. The interviewee was told to imagine the site had a glitch that prevented it from connecting it to its antecedent.\\n3. The same set of remix and antecedent projects as in (2) but this time automatic attribution is displayed but manual credit is not.\\nEach of the interview logs was coded using inductive codes and grounded theory [3]. The coded responses were analyzed based on categories related to how interviewees answered specific questions about the distinction between automatic attribution and manual credit.\"\n },\n {\n \"title\": \"Results\",\n \"content\": \"Confirming the results of Study 1, for users of Scratch, automatic attribution was generally seen as insincere and insufficient. Throughout the interviews, we found that for most of the kids, getting explicit credit from another person was preferred over attribution given automatically by the system. When asked why, kids often responded that knowing that another person had cared enough to give credit was valued more than what the computer system would do on its own. The fact that it takes some work, albeit minimal, to write an acknowledgment statement, sends a signal of empathy, authenticity and good intentions [6]. Amy articulated this when explaining why she preferred getting credit from another person:\\nI would like it even more if the person did it [gave credit] on their own accord, because it would mean that [...] they weren't trying to copy it, pirate it.\\nSimilarly, Jon explained, \\\"No [the \\\"Based on\\\" is not enough], because he [the remixer] didn't put that, it always says that.\\\" For Jon, automatic attribution is not authentic because it is always there and, as a result, it is clear that is not coming from the person doing the remix.\\nMost of the interviewees seemed to have a clear notion of what they think a moral remix should be. For some, it is all about making something different. Jake for example, defines a \\\"good\\\" remix as, \\\"if it has a bunch of differences then it's a good remix. If it has like two, then it's bad.\\\" In addition to the differences between the remix and its antecedent project, for some, manual credit is part of what makes it moral. Charles said, \\\"[remixing] is taking somebody else's project and then changing a lot of it and sharing it and giving credit.\\\" Continuing, Charles explained:\\nIf Green had actually said in the project notes, \\\"This is a remix of Red's project, full credit goes to him,\\\" then I would consider it a remix. But this [pointing at a remix without manual credit] is definitely a copy.\\nLikewise, Ryan mentions that a fictional remix was, \\\"perfectly fine because they gave credit in the project notes.\\\"\\nInterviewees suggested that manual credit also allows users to be more expressive. For example, Susie explained that expressiveness is the reason that she prefers manual credit through the project notes saying, \\\"I think the manual one is better because you can say 'thank you' and things like that. The automatic one just says 'it's based on.\\\"' Susie also notes that for her, the project notes are a space where a creator can express her wishes in regards to her intellectual property, independent, and even in contradiction to, the license of the projects:\\nIf I do a project that has music that I really like, I often download the project, take the music. Unless it says in the project notes, \\\"Do not take the music.\\\"\\nFor Susie and other users of Scratch, the project notes are a space for more than just instructions on how to interact with one's project; they are an expressive space where one can communicate with an audience without having to encumber the creative piece of work with it.\\nOthers point at the fact that people do not pay as much attention to automatic attribution statement as much they do to the manual credit left in project descriptions. Jake, for example, explains that, while he agrees there is some usefulness to having both, project notes still are more important, \\\"because, you know, sometimes people just like skim through a project and you don't see it 'til the end.\\\" Jake continued to say that creators that do not have both should get a \\\"warning.\\\"\\nEven though interviewees value manual credit, they still see the usefulness of the automatic mechanism as some sort of community-building prosthetic device -an explanation for the positive reactions to the feature's initial introduction. For example, Nicole argues that while manual credit on the notes has more value for her, the automatic attribution is useful as a backup and because it provides a link:\\nWell, I think that they should probably write in the notes that -then it should also say \\\"Based on blank's project,\\\" just in case they forget, and also because it gives a link to the original project and it gives a link to the user so you don't have to search for it.\\nA similar explanation was articulated on a comment exchange on one the website's galleries. A teenage girl that actively participates in Scratch explained the pragmatic value of automatic attribution saying, \\\"the 'based on' thingy, it gives a link, and we all luv links, less typing,\\\" before reiterating that manual credit is more valuable:\\nat the beginning i thought that you don't have to give credit when the \\\"based on\\\" thingy is in there, but i realized a lot of people don't look at that, and i noticed people confused the remix with the original.\\nCreating a Scratch project is a complicated task. A project's sources can be diverse and the creator can easily forget to acknowledge some, as Paul explains, when asked to choose between a system of manual credit or automatic attribution:\\nThe thing is, it would be a lot better if they had both. Because, sometimes people probably just forget to do that. And then people would not know.\\nThere are also situations where interviewees recognize what Posner calls the \\\"awkwardness of acknowledgment,\\\" that is, situations where credit is not really needed and it can be an unnecessary burden or go against the aesthetics of the work [20]. For example, Paul mentioned that sometimes, there are some projects in Scratch that are remixed so much -like the sample projects that come with Scratch or some \\\"remix chains\\\" 4 -where credit is not necessary:\\nThere's this one called \\\"perfect platformer base\\\" which a lot of people remix. So I don't think that needs any credit. It's not actually a real game. It's all the levels and stuff are just demonstrations.\\nSince manual crediting has a higher emotional value, some kids mentioned that conflicts over remixing could be addressed by the administrators of the site by editing the project of the remix in question, as a way to enforce credit without transforming it into attribution. Doing so would make it appear that a remixer had credited an antecedent when they had not. Susie offers a suggestion along these lines when asked about how the administrators of the website should deal with a case of a complaint over a remix that is a parody of someone else's project. Susie suggested that, \\\"I might remove the project but I might not, you know, maybe I would edit the notes to to give credit.\\\" Similarly, Charles described his approach for solving conflicts if he was the administrator of the website suggesting that, \\\"I probably just would stay out of the argument. Although not designed to be a random sample, these interviews support the proposition that both Scratch participants and other young people share a set of norms about characteristics that determine what a \\\"good\\\" or moral remix is. Among these norms, acknowledging one's sources seems to play a central role. However, participants also seem to share the opinion that this norm is not satisfied through an automated process. They clearly understand the pragmatic value of automating acknowledgment-giving, but they do not see it as a substitute for adherence to the social norm of creditgiving. They also see it as void of emotion and expressiveness. For Scratch users, normative constraints are separate from architectural constraints and one cannot replace the other. These findings support and enrich the results from our first study and help us understand better how Scratch participants, and perhaps kids in general, experience authorship norms and automation in online spaces.\"\n },\n {\n \"title\": \"CONCLUSIONS\",\n \"content\": \"Our results from Study 1a called into the question the effectiveness of automatic attribution functionality in encouraging more positive user reactions in Scratch. We build on these results in Study 1b to suggest that manual crediting may do the work that Scratch's designers had hoped automatic attribution would. Results from the analysis of user interviews presented in Study 2 help to answer the question of \\\"why?\\\" and suggest that users find manual credit to be more authentic and more meaningful to users because it takes more time and effort. Usually, UI improvements are designed to help reduce the time and effort involved in using a system. But in trying to help users by attributing automatically, Scratch's designers misunderstood the way that attribution as a social mechanism worked for Scratch's users. Our fundamental insight is that while both attribution and credit may be important, they are distinct concepts and that credit is, socially, worth more. A system can attribute the work of a user but credit, which is seen as much more important by users and which has a greater effect on user behavior, cannot be done automatically. Computers can attribute. Crediting, however, takes a human.\\nAs we suggested in our introduction, this fundamental result leads to two distinct contributions. First, and more specifically, our analysis offers an improved understanding of the way that attribution and credit works in user-generated content communities over what has been available in previous work. Our two studies suggest that scholars are correct to argue that credit plays an important role in social media communities and offer empirical confirmation for the important role that authenticity plays in how users conceptualize credit. In our in-depth interviews, we explain some of the reasons why this may be the case. Second, through our evaluation of an unsuccessful technological design, our work offers a broader, if more preliminary, contribution in suggesting an important limit of designers' ability to support community norms in social media systems. As the literature on design and social media grows, the importance of good support for communities with healthy norms promoting positive interactions is likely to increase. In attempting to design for these norms, we suspect that researchers will increasingly encounter similar challenges.\\nWe argue that designers should approach interventions iteratively. This design approach can be understood through the theoretical lens of the social construction of technology [19]: designers can't control technological outcomes which must be built through a close relationship between designers and users. Designers must move away from seeing their profession as providing solutions. They must channel users, work closely with them, and iterate together, to negotiate and achieve a set of shared goals.\\nThe prevalence of user-generated content sites stresses the importance of how online social spaces should deal with issues of attribution and our results are likely to be immediately relevant to designers. For example, the Semantic Clipboard is a tool built as a system of automatic attribution for content reuse [22]. Developed by researchers who found a high degree of Creative Commons license violations around the re-use of Flickr images, the tool is a Firefox plugin that provides, \\\"license awareness of Web media,\\\" and enables people to automatically, \\\"copy [media] along with the appropriate license metadata.\\\" Our results suggest one way that this approach may fall short.\\nHowever, automatic attribution is not the only way that technologists can design to acknowledge others' contributions. Indeed, our results suggest that there may be gains from design changes which encourage credit-giving without simply automating attribution. For example, Scratch's designers might present users with a metadata field that prompts users to credit others and suggests antecedent authors whose work the system has determined may have played a role. This affordance might remind users to credit others, and might increase the amount of crediting, while maintaining a human role in the process and the extra effort that, our research has suggested, imbues manual credit giving with its value. We suggest that in other social media communities, similar affordances that help prompt or remind users to do things that a system might do automatically represent a class of increasingly important design patterns and a template for successful design interventions in support of community norms.\"\n },\n {\n \"title\": \"Figure 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 6 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"An example Scratch program written using programming \"blocks\".","content":"Figure 1 .\n1An example Scratch program written using programming \"blocks\"."},{"id":"","caption":"Screenshot of a Scratch remix project highlighting automatic (the area inside the bottom left circle) and manual (the area in the top right circle) attribution.","content":"Figure 2 .\n2Screenshot of a Scratch remix project highlighting automatic (the area inside the bottom left circle) and manual (the area in the top right circle) attribution."},{"id":"","caption":"Example of the automatic attribution-giving statement displayed under remixes in the period after the design intervention.","content":"Figure 3 .\n3Example of the automatic attribution-giving statement displayed under remixes in the period after the design intervention."},{"id":"","caption":"Example of the link pointing to remixes of a project displayed under antecedent projects in the period after the design intervention.","content":"Figure 4 .\n4Example of the link pointing to remixes of a project displayed under antecedent projects in the period after the design intervention."},{"id":"","caption":"Mosaic plot showing the distribution of reactions of original creators who had viewed remixes of their project during the six week periods before and after the introduction of automatic attribution. The proportion of response types is shown along the y-axis. The proportion of projects viewed with, and without, automatic attribution is shown along the x-axis. (n = 932)","content":"Figure 5 .\n5Mosaic plot showing the distribution of reactions of original creators who had viewed remixes of their project during the six week periods before and after the introduction of automatic attribution. The proportion of response types is shown along the y-axis. The proportion of projects viewed with, and without, automatic attribution is shown along the x-axis. (n = 932)"},{"id":"","caption":"Mosaic plot showing the distribution of reactions of original creators who had viewed remixes of their project and indicating whether they left manual credit. The proportion of response types is shown along the y-axis. The proportion of projects including and omitting manual credit is shown along the x-axis. (n = 932)","content":"Figure 6 .\n6Mosaic plot showing the distribution of reactions of original creators who had viewed remixes of their project and indicating whether they left manual credit. The proportion of response types is shown along the y-axis. The proportion of projects including and omitting manual credit is shown along the x-axis. (n = 932)"},{"id":"","caption":"I probably wouldn't remove it [the remix], I'd just add something in the project notes [like] 'based on Gray's project.\"'This phenomena of giving less value to technologically simplified social signals is experienced in other social platforms. For example, Amy expressed how on the social network site Facebook, she loves to get comments on her photographs but dislikes those who do not leave comments or opt instead to press the \"I like it\" button:I love when people comment on my pictures. Everybody sees them, because they tell me they have. I'm like, \"Oh really? That's great. Why didn't you comment?\" I don't like it when people just \"like it\", because you know they have something to say about it; they just don't. It's like, if they like it, then [they should] take the time to say something.","content":"I probably wouldn't remove it [the remix], I'd just add something in the project notes [like] 'based on Gray's project.\"'This phenomena of giving less value to technologically simplified social signals is experienced in other social platforms. For example, Amy expressed how on the social network site Facebook, she loves to get comments on her photographs but dislikes those who do not leave comments or opt instead to press the \"I like it\" button:I love when people comment on my pictures. Everybody sees them, because they tell me they have. I'm like, \"Oh really? That's great. Why didn't you comment?\" I don't like it when people just \"like it\", because you know they have something to say about it; they just don't. It's like, if they like it, then [they should] take the time to say something."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"An example Scratch program written using programming \\\"blocks\\\".\",\n \"content\": \"Figure 1 .\\n1An example Scratch program written using programming \\\"blocks\\\".\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Screenshot of a Scratch remix project highlighting automatic (the area inside the bottom left circle) and manual (the area in the top right circle) attribution.\",\n \"content\": \"Figure 2 .\\n2Screenshot of a Scratch remix project highlighting automatic (the area inside the bottom left circle) and manual (the area in the top right circle) attribution.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Example of the automatic attribution-giving statement displayed under remixes in the period after the design intervention.\",\n \"content\": \"Figure 3 .\\n3Example of the automatic attribution-giving statement displayed under remixes in the period after the design intervention.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Example of the link pointing to remixes of a project displayed under antecedent projects in the period after the design intervention.\",\n \"content\": \"Figure 4 .\\n4Example of the link pointing to remixes of a project displayed under antecedent projects in the period after the design intervention.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Mosaic plot showing the distribution of reactions of original creators who had viewed remixes of their project during the six week periods before and after the introduction of automatic attribution. The proportion of response types is shown along the y-axis. The proportion of projects viewed with, and without, automatic attribution is shown along the x-axis. (n = 932)\",\n \"content\": \"Figure 5 .\\n5Mosaic plot showing the distribution of reactions of original creators who had viewed remixes of their project during the six week periods before and after the introduction of automatic attribution. The proportion of response types is shown along the y-axis. The proportion of projects viewed with, and without, automatic attribution is shown along the x-axis. (n = 932)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Mosaic plot showing the distribution of reactions of original creators who had viewed remixes of their project and indicating whether they left manual credit. The proportion of response types is shown along the y-axis. The proportion of projects including and omitting manual credit is shown along the x-axis. (n = 932)\",\n \"content\": \"Figure 6 .\\n6Mosaic plot showing the distribution of reactions of original creators who had viewed remixes of their project and indicating whether they left manual credit. The proportion of response types is shown along the y-axis. The proportion of projects including and omitting manual credit is shown along the x-axis. (n = 932)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"I probably wouldn't remove it [the remix], I'd just add something in the project notes [like] 'based on Gray's project.\\\"'This phenomena of giving less value to technologically simplified social signals is experienced in other social platforms. For example, Amy expressed how on the social network site Facebook, she loves to get comments on her photographs but dislikes those who do not leave comments or opt instead to press the \\\"I like it\\\" button:I love when people comment on my pictures. Everybody sees them, because they tell me they have. I'm like, \\\"Oh really? That's great. Why didn't you comment?\\\" I don't like it when people just \\\"like it\\\", because you know they have something to say about it; they just don't. It's like, if they like it, then [they should] take the time to say something.\",\n \"content\": \"I probably wouldn't remove it [the remix], I'd just add something in the project notes [like] 'based on Gray's project.\\\"'This phenomena of giving less value to technologically simplified social signals is experienced in other social platforms. For example, Amy expressed how on the social network site Facebook, she loves to get comments on her photographs but dislikes those who do not leave comments or opt instead to press the \\\"I like it\\\" button:I love when people comment on my pictures. Everybody sees them, because they tell me they have. I'm like, \\\"Oh really? That's great. Why didn't you comment?\\\" I don't like it when people just \\\"like it\\\", because you know they have something to say about it; they just don't. It's like, if they like it, then [they should] take the time to say something.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"The Wealth of Networks: How Social Production Transforms Markets and Freedom. Y Benkler, Yale University PressBenkler, Y. The Wealth of Networks: How Social Production Transforms Markets and Freedom. Yale University Press, Oct. 2007.","doi":""},{"text":"Announcing (and explaining) our new 2.0 licenses. G O Brown, Brown, G. O. Announcing (and explaining) our new 2.0 licenses. http://creativecommons.org/weblog /entry/4216, May 2004.","doi":""},{"text":"Constructing Grounded Theory: A Practical Guide Through Qualitative Analysis. K Charmaz, Sage PublicationsLondonCharmaz, K. Constructing Grounded Theory: A Practical Guide Through Qualitative Analysis. Sage Publications, London, 2006.","doi":""},{"text":"An analysis of the social structure of remix culture. G Cheliotis, J Yew, Proc. C&T 2009. C&T 2009University Park, PA, USAACMCheliotis, G., and Yew, J. An analysis of the social structure of remix culture. In Proc. C&T 2009 (University Park, PA, USA, 2009), ACM, 165-174.","doi":""},{"text":"The evolution of authorship in a remix society. N Diakopoulos, K Luther, Y E Medynskiy, Essa , I , Proc. HT. HTManchester, UKACMDiakopoulos, N., Luther, K., Medynskiy, Y. E., and Essa, I. The evolution of authorship in a remix society. In Proc. HT 2007 (Manchester, UK, 2007), ACM, 133-136.","doi":""},{"text":"Signals in social supernets. J Donath, Journal of Computer-Mediated Communication. 13Donath, J. Signals in social supernets. Journal of Computer-Mediated Communication 13, 1 (2008), 231-251.","doi":""},{"text":"Social judgments and technological innovation: Adolescents' understanding of property, privacy, and electronic information. B Friedman, Computers in Human Behavior. 13Friedman, B. Social judgments and technological innovation: Adolescents' understanding of property, privacy, and electronic information. Computers in Human Behavior 13, 3 (Aug. 1997), 327-351.","doi":""},{"text":"Responses to remixing on a social media sharing website. B M H Hill, A Monroy-Hernández, K Olson, Proc. ICWSM 2010. ICWSM 2010Washington, D.C.AAAIHill, B. M. H., Monroy-Hernández, A., and Olson, K. Responses to remixing on a social media sharing website. In Proc. ICWSM 2010 (Washington, D.C., 2010), AAAI, 74-81.","doi":""},{"text":"Pedestrian: Mobile Phones in Japanese Life. M Ito, D Okabe, M Matsuda, Personal, Portable, The MIT PressIto, M., Okabe, D., and Matsuda, M. Personal, Portable, Pedestrian: Mobile Phones in Japanese Life. The MIT Press, 2006.","doi":""},{"text":"Convergence Culture: Where Old and New Media Collide. H Jenkins, New York University PressNew YorkJenkins, H. Convergence Culture: Where Old and New Media Collide. New York University Press, New York, 2006.","doi":""},{"text":"Code: And Other Laws of Cyberspace, Version 2.0. Basic Books. L Lessig, Lessig, L. Code: And Other Laws of Cyberspace, Version 2.0. Basic Books, Dec. 2006.","doi":""},{"text":"Making Art and Commerce Thrive in the Hybrid Economy. L Lessig, Remix, Penguin Press HC, TheLessig, L. Remix: Making Art and Commerce Thrive in the Hybrid Economy. Penguin Press HC, The, Oct. 2008.","doi":""},{"text":"Edits & credits: exploring integration and attribution in online creative collaboration. K Luther, N Diakopoulos, A Bruckman, Proc. CHI 2010. CHI 2010Atlanta, Georgia, USAACMLuther, K., Diakopoulos, N., and Bruckman, A. Edits & credits: exploring integration and attribution in online creative collaboration. In Proc. CHI 2010 (Atlanta, Georgia, USA, 2010), ACM, 2823-2832.","doi":""},{"text":". L Manovich, Remix, Remixability, Manovich, L. Remix and remixability. http://www.manovich.net/DOCS/Remix modular.doc, 2005.","doi":""},{"text":"Empowering kids to create and share programmable media. interactions. A Monroy-Hernández, M Resnick, 15Monroy-Hernández, A., and Resnick, M. Empowering kids to create and share programmable media. interactions 15, 2 (2008), 50-53.","doi":""},{"text":"No fair, copycat!': what children's response to plagiarism tells us about their understanding of ideas. K R Olson, A Shaw, Developmental Science. Olson, K. R., and Shaw, A. 'No fair, copycat!': what children's response to plagiarism tells us about their understanding of ideas. Developmental Science (2010).","doi":""},{"text":"Understanding the First Generation of Digital Natives. J Palfrey, U Gasser, Born, Digital, Basic BooksPalfrey, J., and Gasser, U. Born Digital: Understanding the First Generation of Digital Natives. Basic Books, Aug. 2008.","doi":""},{"text":"Copy and paste literacy: Literacy practices in the production of a MySpace profile. D Perkel, Informal Learning and Digital Media. Perkel, D. Copy and paste literacy: Literacy practices in the production of a MySpace profile. Informal Learning and Digital Media (2006).","doi":""},{"text":"The social construction of facts and artefacts: Or how the sociology of science and the sociology of technology might benefit each other. T J Pinch, W E Bijker, Social Studies of Science. 14Pinch, T. J., and Bijker, W. E. The social construction of facts and artefacts: Or how the sociology of science and the sociology of technology might benefit each other. Social Studies of Science 14, 3 (Aug. 1984), 399-441.","doi":""},{"text":"The little book of plagiarism. R A Posner, Pantheon BooksPosner, R. A. The little book of plagiarism. Pantheon Books, Jan. 2007.","doi":""},{"text":"Scratch: programming for all. M Resnick, J Maloney, A Monroy-Hernández, N Rusk, E Eastmond, K Brennan, A Millner, E Rosenbaum, J Silver, B Silverman, Y Kafai, Commun. ACM. 52Resnick, M., Maloney, J., Monroy-Hernández, A., Rusk, N., Eastmond, E., Brennan, K., Millner, A., Rosenbaum, E., Silver, J., Silverman, B., and Kafai, Y. Scratch: programming for all. Commun. ACM 52, 11 (2009), 60-67.","doi":""},{"text":"Policy-Aware content reuse on the web. O Seneviratne, L Kagal, T Berners-Lee, The Semantic Web -ISWC. Seneviratne, O., Kagal, L., and Berners-Lee, T. Policy-Aware content reuse on the web. In The Semantic Web -ISWC 2009. 2009, 553-568.","doi":""},{"text":"Remix culture on the web: A survey of content reuse on different User-Generated content websites. O Seneviratne, A Monroy-Hernández, Web Science. Raleigh, NCSeneviratne, O., and Monroy-Hernández, A. Remix culture on the web: A survey of content reuse on different User-Generated content websites. In Web Science (Raleigh, NC, Apr. 2010).","doi":""},{"text":"Community annotation and remix: a research platform and pilot deployment. R Shaw, P Schmitz, Proc. MM. MMSanta Barbara, California, USAACMShaw, R., and Schmitz, P. Community annotation and remix: a research platform and pilot deployment. In Proc. MM 2006 (Santa Barbara, California, USA, 2006), ACM, 89-98.","doi":""},{"text":". A Sinnreich, M Latonero, M Gluck, Ethics reconfigured. Information, Communication &. Sinnreich, A., Latonero, M., and Gluck, M. Ethics reconfigured. Information, Communication &","doi":""},{"text":". Society. 121242Society 12, 8 (2009), 1242.","doi":""},{"text":"What's mine is mine: territoriality in collaborative authoring. J Thom-Santelli, D R Cosley, G Gay, Proc. CHI. CHIBoston, MA, USAACMThom-Santelli, J., Cosley, D. R., and Gay, G. What's mine is mine: territoriality in collaborative authoring. In Proc. CHI 2009 (Boston, MA, USA, 2009), ACM, 1481-1484.","doi":""},{"text":". ¡/Pre¿¡/Body¿¡/ Html¿, ¡/PRE¿¡/BODY¿¡/HTML¿","doi":""}],"string":"[\n {\n \"text\": \"The Wealth of Networks: How Social Production Transforms Markets and Freedom. Y Benkler, Yale University PressBenkler, Y. The Wealth of Networks: How Social Production Transforms Markets and Freedom. Yale University Press, Oct. 2007.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Announcing (and explaining) our new 2.0 licenses. G O Brown, Brown, G. O. Announcing (and explaining) our new 2.0 licenses. http://creativecommons.org/weblog /entry/4216, May 2004.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Constructing Grounded Theory: A Practical Guide Through Qualitative Analysis. K Charmaz, Sage PublicationsLondonCharmaz, K. Constructing Grounded Theory: A Practical Guide Through Qualitative Analysis. Sage Publications, London, 2006.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An analysis of the social structure of remix culture. G Cheliotis, J Yew, Proc. C&T 2009. C&T 2009University Park, PA, USAACMCheliotis, G., and Yew, J. An analysis of the social structure of remix culture. In Proc. C&T 2009 (University Park, PA, USA, 2009), ACM, 165-174.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The evolution of authorship in a remix society. N Diakopoulos, K Luther, Y E Medynskiy, Essa , I , Proc. HT. HTManchester, UKACMDiakopoulos, N., Luther, K., Medynskiy, Y. E., and Essa, I. The evolution of authorship in a remix society. In Proc. HT 2007 (Manchester, UK, 2007), ACM, 133-136.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Signals in social supernets. J Donath, Journal of Computer-Mediated Communication. 13Donath, J. Signals in social supernets. Journal of Computer-Mediated Communication 13, 1 (2008), 231-251.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Social judgments and technological innovation: Adolescents' understanding of property, privacy, and electronic information. B Friedman, Computers in Human Behavior. 13Friedman, B. Social judgments and technological innovation: Adolescents' understanding of property, privacy, and electronic information. Computers in Human Behavior 13, 3 (Aug. 1997), 327-351.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Responses to remixing on a social media sharing website. B M H Hill, A Monroy-Hernández, K Olson, Proc. ICWSM 2010. ICWSM 2010Washington, D.C.AAAIHill, B. M. H., Monroy-Hernández, A., and Olson, K. Responses to remixing on a social media sharing website. In Proc. ICWSM 2010 (Washington, D.C., 2010), AAAI, 74-81.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Pedestrian: Mobile Phones in Japanese Life. M Ito, D Okabe, M Matsuda, Personal, Portable, The MIT PressIto, M., Okabe, D., and Matsuda, M. Personal, Portable, Pedestrian: Mobile Phones in Japanese Life. The MIT Press, 2006.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Convergence Culture: Where Old and New Media Collide. H Jenkins, New York University PressNew YorkJenkins, H. Convergence Culture: Where Old and New Media Collide. New York University Press, New York, 2006.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Code: And Other Laws of Cyberspace, Version 2.0. Basic Books. L Lessig, Lessig, L. Code: And Other Laws of Cyberspace, Version 2.0. Basic Books, Dec. 2006.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Making Art and Commerce Thrive in the Hybrid Economy. L Lessig, Remix, Penguin Press HC, TheLessig, L. Remix: Making Art and Commerce Thrive in the Hybrid Economy. Penguin Press HC, The, Oct. 2008.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Edits & credits: exploring integration and attribution in online creative collaboration. K Luther, N Diakopoulos, A Bruckman, Proc. CHI 2010. CHI 2010Atlanta, Georgia, USAACMLuther, K., Diakopoulos, N., and Bruckman, A. Edits & credits: exploring integration and attribution in online creative collaboration. In Proc. CHI 2010 (Atlanta, Georgia, USA, 2010), ACM, 2823-2832.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Manovich, Remix, Remixability, Manovich, L. Remix and remixability. http://www.manovich.net/DOCS/Remix modular.doc, 2005.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Empowering kids to create and share programmable media. interactions. A Monroy-Hernández, M Resnick, 15Monroy-Hernández, A., and Resnick, M. Empowering kids to create and share programmable media. interactions 15, 2 (2008), 50-53.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"No fair, copycat!': what children's response to plagiarism tells us about their understanding of ideas. K R Olson, A Shaw, Developmental Science. Olson, K. R., and Shaw, A. 'No fair, copycat!': what children's response to plagiarism tells us about their understanding of ideas. Developmental Science (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Understanding the First Generation of Digital Natives. J Palfrey, U Gasser, Born, Digital, Basic BooksPalfrey, J., and Gasser, U. Born Digital: Understanding the First Generation of Digital Natives. Basic Books, Aug. 2008.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Copy and paste literacy: Literacy practices in the production of a MySpace profile. D Perkel, Informal Learning and Digital Media. Perkel, D. Copy and paste literacy: Literacy practices in the production of a MySpace profile. Informal Learning and Digital Media (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The social construction of facts and artefacts: Or how the sociology of science and the sociology of technology might benefit each other. T J Pinch, W E Bijker, Social Studies of Science. 14Pinch, T. J., and Bijker, W. E. The social construction of facts and artefacts: Or how the sociology of science and the sociology of technology might benefit each other. Social Studies of Science 14, 3 (Aug. 1984), 399-441.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The little book of plagiarism. R A Posner, Pantheon BooksPosner, R. A. The little book of plagiarism. Pantheon Books, Jan. 2007.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Scratch: programming for all. M Resnick, J Maloney, A Monroy-Hernández, N Rusk, E Eastmond, K Brennan, A Millner, E Rosenbaum, J Silver, B Silverman, Y Kafai, Commun. ACM. 52Resnick, M., Maloney, J., Monroy-Hernández, A., Rusk, N., Eastmond, E., Brennan, K., Millner, A., Rosenbaum, E., Silver, J., Silverman, B., and Kafai, Y. Scratch: programming for all. Commun. ACM 52, 11 (2009), 60-67.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Policy-Aware content reuse on the web. O Seneviratne, L Kagal, T Berners-Lee, The Semantic Web -ISWC. Seneviratne, O., Kagal, L., and Berners-Lee, T. Policy-Aware content reuse on the web. In The Semantic Web -ISWC 2009. 2009, 553-568.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Remix culture on the web: A survey of content reuse on different User-Generated content websites. O Seneviratne, A Monroy-Hernández, Web Science. Raleigh, NCSeneviratne, O., and Monroy-Hernández, A. Remix culture on the web: A survey of content reuse on different User-Generated content websites. In Web Science (Raleigh, NC, Apr. 2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Community annotation and remix: a research platform and pilot deployment. R Shaw, P Schmitz, Proc. MM. MMSanta Barbara, California, USAACMShaw, R., and Schmitz, P. Community annotation and remix: a research platform and pilot deployment. In Proc. MM 2006 (Santa Barbara, California, USA, 2006), ACM, 89-98.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Sinnreich, M Latonero, M Gluck, Ethics reconfigured. Information, Communication &. Sinnreich, A., Latonero, M., and Gluck, M. Ethics reconfigured. Information, Communication &\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Society. 121242Society 12, 8 (2009), 1242.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"What's mine is mine: territoriality in collaborative authoring. J Thom-Santelli, D R Cosley, G Gay, Proc. CHI. CHIBoston, MA, USAACMThom-Santelli, J., Cosley, D. R., and Gay, G. What's mine is mine: territoriality in collaborative authoring. In Proc. CHI 2009 (Boston, MA, USA, 2009), ACM, 1481-1484.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". ¡/Pre¿¡/Body¿¡/ Html¿, ¡/PRE¿¡/BODY¿¡/HTML¿\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":473,"cells":{"text":{"kind":"string","value":"\nA quantum topological phase transition at the microscopic level\n1 Mar 2008\n\nClaudio Castelnovo \nRudolf Peierls Centre for Theoretical Physics\nUniversity of Oxford\nOX1 3NPOxfordUK\n\nClaudio Chamon \nPhysics Department\nBoston University\n02215BostonMAUSA\n\nA quantum topological phase transition at the microscopic level\n1 Mar 2008(Dated: March 3, 2008)arXiv:0707.2084v3 [cond-mat.str-el]\nWe study a quantum phase transition between a phase which is topologically ordered and one which is not. We focus on a spin model, an extension of the toric code, for which we obtain the exact ground state for all values of the coupling constant that takes the system across the phase transition. We compute the entanglement and the topological entropy of the system as a function of this coupling constant, and show that the topological entropy remains constant all the way up to the critical point, and jumps to zero beyond it. Despite the jump in the topological entropy, the transition is second order as detected via local observables.\n\nI. INTRODUCTION\n\nSome strongly correlated quantum many body systems display a type of order which cannot be characterized by any local order parameter. Instead, such order is topological in nature 1 , with the fractional quantum Hall systems being the primary example so far: they are liquid states that exhibit exotic properties such as a ground state (GS) degeneracy that cannot be lifted by any local perturbations 2,3 and fractionalized degrees of freedom 4 . Topologically ordered states are also interesting in that their robustness against local perturbations might be of use for decoherence-free quantum computation 5 .\n\nAn example of an exactly solvable lattice spin model that is topologically ordered was presented by Kitaev in Ref. 5, and the system was argued to be robust against small perturbations that tend to order the systemà la Landau-Ginzburg and take it away from its topological phase. The departure from the topologically ordered phase should occur through a quantum phase transition. Such quantum phase transition, however, cannot be entirely captured by ordinary methods based on local Landau-Ginzburg order parameters, and new methods need to be devised in order to investigate the fate of topological order across the phase transition. These novel methods must be based on the fundamental properties of topologically ordered phases, such as the GS degeneracy in presence of a gap, and the presence of a non-vanishing topological entropy.\n\nRecent efforts to understand quantum phase transitions in topologically ordered states include a mean-field approach for these exotic states 6 , and analytical and numerical studies 7,8,9 of the Kitaev model in the presence of a field. The numerical analysis presented in Refs. 8,9 leads to the conclusion that topological order survives unchanged up to the second order phase transition at β c = 0.32847(6) (in the notation of Eq. (18)), while the system is no longer topologically ordered for β > β c . (Here β stands for the coupling constant that drives the T = 0 quantum phase transition -the notation will become apparent shortly, and is chosen because of a close relation to a classical model.)\n\nIn this paper, we investigate analytically a quantum phase transition out of a topological phase. We show that the recently defined topological entropy 10,11 works well as an \"order parameter\" across the transition. We study the transition using a model -see Eq. (5) -that is shown to behave much like the Kitaev model in a magnetic field for small values of the field.\n\nThe advantage of this model is that the ground state can be obtained exactly, from which we can then compute the topological entropy explicitly, and show that it remains constant in the topologically ordered phase (β < β c ≃ 0.4406868), dropping abruptly to zero in the non-topologically-ordered phase (β > β c ), despite the continuous (second order) character of the transition.\n\nWe find that in this model, even though one cannot identify a local order parameter that vanishes in one phase and not in the other, one can show that the (continuous) local magnetization has a singularity at the critical point. In the model, we show that the magnetization equals the energy E Ising (β) of a 2D classical Ising model with N spins evaluated at an inverse (classical) temperature equal to the value of the coupling constant β that drives the system through the T = 0 phase transition:\nm(β) = 1 N i σ z i = 1 N E Ising (β) .(1)\nFrom this relation, it becomes evident that the magnetization m(β), although continuous and non-vanishing across the transition at β c (much as the energy of the classical Ising model across the classical transition), has a singularity in its first derivative, since ∂m ∂β\n= 1 N ∂E Ising ∂β = −β 2 1 N C Ising (β) ,(2)\nand the Ising model heat capacity C Ising diverges logarithmically at β c . Hence, although there is no local order parameter that can detect either the topological or the non-topological phase in this system, one can expose the topological quantum phase transition through the singularity in the derivative of a local quantity. This is contrasted, for example, with the case discussed in Sec. III B, where a similar topological transition is accompanied by a simultaneous Z 2 symmetry breaking phase transition. In that case, the same transition is captured both by the non-local topological entropy, and by a local (Landau-Ginzburg) order parameter.\n\n\nII. THE MODEL\n\nThe model that we consider is a deformation of the Kitaev model 5 , and it is defined on a square lattice with spin-1/2 de-grees of freedom living on the bonds, as shown in Fig. 1. The pure Kitaev model is written in terms of star and plaquette operators (see Fig. 1). Star operators are defined as\nA s = i∈star(s)σ x i ≡ i∈sσ x i ,(3)\nwhere i labels the four spins on the bonds departing from some vertex s of the square lattice. Plaquette operators are defined as\nB p = i∈plaquette(p)σ z i ≡ i∈pσ z i ,(4)\nwhere i labels the four spins on the bonds around some plaquette p of the square lattice. The Hamiltonian we consider in this paper is\nH = −λ 0 p B p − λ 1 s A s + λ 1 s e −β P i∈sσ z i = H Kitaev + λ 1 s e −β P i∈sσ z i ,(5)\nwhere λ 0,1 > 0 and β is a parameter that we use to tune the system across a topological quantum phase transition. Notice that for β = 0 the Hamiltonian Eq. (5) Take G to be the (Abelian) group of all spin flip operations obtained as products of star type operators. Notice that g 2 = 1 1 for any element g of the group G. By acting with elements of G on a given reference configuration i |σ z i one generates a manifold of states, which however does not encompass the whole basis. For example, the action of a star operator A s cannot change the sign of the product of σ z 's around any square plaquette in the lattice 5,12 (see Fig. 1). Therefore, there is a non-trivial (and non-unique) minimal set {|Ψ α } of reference configurations that generates the full σ z -basis under the action of the group G. (In particular, one of the elements in this set is the reference configuration |0 that is fully magnetized in the z-direction, say σ z i = 1, ∀i.) Consider then the family of Hamiltonians\nH 1 (β) = s e −β P i∈sσ z i − i∈sσ x i ,(6)\nfor some real-valued parameter β. The ground state of any such Hamiltonian can be obtained exactly and it can be written in the form\n|GS 1 = α ψ α g∈G e β P i σ z i (g,α)/2 Z α g |Ψ α ,(7)Z α = g∈G e β P i σ z i (g,α)(8)\nwhere α labels the different block-diagonal sectors corresponding to the states in the minimal set {|Ψ α }; σ z i (g, α) is the z-component of the spin at site i in state g |Ψ α ; and the coefficients ψ α can be chosen at will, subject to the normalization condition α |ψ α | 2 = 1. Although the choice of minimal set is non-unique, one can show that Eq. (8) is independent of such choice, modulo an irrelevant permutation of the α indices. Within each block-diagonal sector, the GS of Eq. (6) is unique. Instead of proving directly that (8) is the GS of (6), it is more convenient to notice that the family of Hamiltonians in Eq. (6) is a particular choice of Stochastic Matrix Form decompositions of quantum Hamiltonians that exhibit precisely Eq. (8) as their GS. 13 One can verify this by showing that each of the operators\nQ s = e −β P i∈sσ z i − i∈sσ x i(9)\nbetween square brackets in Eq. (6) annihilates the inner sum in |GS 1 , independently of the index α. That the GS energy is zero follows because\n\nFor larger values of β, the many-body terms in Eq. (19) are no longer negligible and the equivalence is lost, although the form of the GS (15) suggests that the system gets deeper and deeper into the magnetized phase -as one would expect upon increasing the strength of the magnetic field in the Kitaev model. As we discuss in Section III A, our model undergoes a second-order phase transition at β c = (1/2) ln( √ 2 + 1) ≃ 0.4406868, where it displays a dimensionality reduction that places the transition in a different universality class than the one studied in Refs. 7,8,9. One can use the decomposition\ne −β P i∈sσ z i = i∈s cosh(β) −σ z i sinh(β) = cosh 4 (β) − cosh 3 (β) sinh(β) i∈sσ z i + cosh 2 (β) sinh 2 (β) i =j∈sσ z iσ z j − cosh(β) sinh 3 (β) i =j =k∈sσ z iσ z jσ z k + sinh 4 (β) i∈sσ z i ,(19)\nto estimate the limit of validity of Eq. (18) to be given by the condition\ncosh 3 (β) sinh(β) cosh 2 (β) sinh 2 (β) = cosh(β) sinh(β) 2.(20)\nThis corresponds to a ratio between the coupling to the magnetic field and the coupling to the cooperative transverse field ( s i∈sσ\nx i ) 2λ 1 cosh 3 (β) sinh(β) λ 1 16 9 ≃ 1.78.(21)\nThe detailed numerical analysis presented in Refs. 8,9 lead the authors to conclude that topological order survives up to the second order phase transition at finite β c (in the notation of Eq. (18)), while the system is no longer topologically ordered for β > β c . In the following, we investigate this phase and the relative phase transition using the exact ground state of our model (5) to compute the topological entropy 10,11 across the transition. Using an exact derivation from the microscopic degrees of freedom, we show that the topological entropy is able to detect a transition from a topologically ordered phase (β < β c ) to a non-topologically-ordered phase (β > β c ). Indeed, it remains constant at its known β → 0 value up to the transition and drops abruptly to zero afterwards, despite the continuous character of the transition.\n\n\nIII. THE TOPOLOGICAL ENTROPY OF FACTORIZABLE (LOCAL) WAVEFUNCTIONS\n\nUsing the definition in Refs. 10,11, the topological entropy can be obtained as a linear combination of Von Neumann entanglement entropies S VN of different bipartitions of the system into subsystems A and B:\nS A VN ≡ −Tr [ρ A log 2 ρ A ] = S B VN ,(22)\nwhere ρ A = Tr B (ρ) is the reduced density matrix obtained from the full density matrix ρ by tracing out the degrees of freedom of subsystem B, and the last equality holds whenever the full density matrix ρ is a pure-state density matrix.\n\nThe different bipartitions are aimed at removing all the extensive (boundary) contributions to uncover the sole topological contribution. A particular choice of the four bipartitions 10 is illustrated in Fig. 2 as:\nS topo = lim r,R→∞ −S (1A) VN + S (2A) VN + S (3A) VN − S (4A) VN . (23)\nIn order to compute the topological entropy as a function of the parameter β, let us first notice that the Hamiltonian (14) constructed above, with GS given by (15), belongs to a class of Hamiltonians whose GS wavefunctions\n|Ψ = |Z| −1/2 g∈G e −βE g /2 g|0 have non-negative, fac- torizable amplitudes, i.e., E g = E A g A +E B g B , with g = g A ⊗g B for all bipartitions (A, B)\n. For this type of Hamiltonians, one can compute the entanglement entropy as follows.\n\nConsider a given bipartition (A, B) of the system. The reduced density matrix ρ A = Tr B (ρ), obtained by tracing over all degrees of freedom in B, is given by 12\nρ A = 1 Z g,g∈G e −β(E g +E gg )/2 × 0 B |g BgB g B |0 B g A |0 A 0 A |g AgA = 1 Z g∈G, g ′ ∈G A e −βE B g B e −β(E A g A +E A g A g ′ A )/2 × g A |0 A 0 A |g A g ′ A .(24)where g = g A ⊗ g B , |0 = |0 A ⊗ |0 B and G A ⊂ G (G B ⊂ G)\nis the subgroup of transformations acting solely on A (B) and leaving B (A) invariant:\nG A = {g ∈ G | g B = 1 1 B } G B = {g ∈ G | g A = 1 1 A }\nNotice that we used the group property to rewrite a generic element of G as gg, ∃!g ∈ G, as well as the additive property of E g . We can then compute the trace of the n-th power of the reduced density matrix Tr (ρ A ) n and use the identity\n− lim n→1 ∂ ∂n Tr [(ρ A ) n ] = −Tr [ρ A ln ρ A ](25)\nto obtain the Von Neumann entropy S\n(A) VN = −Tr [ρ A log 2 ρ A ]: Tr [ρ n A ] = 1 Z n g 1 ,...,g n ∈G g ′ 1 ,...,g ′ n ∈G A e −β P n i=1 E B g i,B × e −β P n i=1 (E A g i,A +E A g i,A g ′ i,A )/2 × 0 A |g 1,A g ′ 1,A g 2,A |0 A 0 A |g 2,A g ′ 2,A g 3,A |0 A . . . × . . . 0 A |g n,A g ′ n,A g 1,A |0 A = 1 Z n g 1 ,...,g n ∈G g ′ 1 ,...,g ′ n ∈G A e −β P n i=1 E g i × 0 A |g 1,A g ′ 1,A g 2,A |0 A 0 A |g 2,A g ′ 2,A g 3,A |0 A . . . × . . . 0 A |g n,A g ′ n,A g 1,A |0 A ,(26)\nwhere we used the fact that the inner products in Eq. (26) impose\ng i+1,A = g i,A g ′ i,A ,(27)\nfor i = 1, . . . , n, with the identification n + 1 ≡ 1, and therefore\nE B g i,B + 1 2 (E A g i,A + E A g i−1,A g ′ i−1,A ) = E g i .(28)\nThe condition in Eq. (27) can be satisfied if and only if\ng ′ i = g i+1,A g i,A ⊗ 1 1 B ∈ G A .(29)\nThus, the summation over all g ′ i of the inner products in Eq. (26) yields a constraint over the allowed values of g 1 , . . . , g n ∈ G:\ng i+1,A g i,A ⊗ 1 1 B ∈ G A , ∀ i = 1, . . . , n (n + 1 ≡ 1) g i,A g j,A ⊗ 1 1 B ∈ G A , ∀ i, j = 1, . . . , n g i g j ∈ G A G B , ∀ i, j = 1, . . . , n.\n(30) In particular, the last line can be recast as\ng i = h i g 1 k i ∃! h i ∈ G A , k i ∈ G B , ∀ i = 2, . . . , n.\n(31) The physical meaning of these conditions will become clear in the next section, for the specific case of the system considered in this paper, although the form of Eq. (31) already suggests that they require all the elements g i to agree at the boundary of the bipartition (A, B).\n\nWe can finally use Eq. (31) to simplify Eq. (26)\nTr [ρ n A ] = 1 Z n g∈G e −βE g h 2 ,...,h n ∈G A k 2 ,...,k n ∈G B e −β P n i=2 E h i gk i = 1 Z g∈G e −βE g h∈G A , k∈G B e −βE hgk Z n−1(32)\nand obtain, via Eq. (25),\nS (A) VN = − 1 Z g∈G e −βE g log 2 h∈G A , k∈G B e −βE hgk Z = log 2 h∈G A , k∈G B e −βE hgk Z = − log 2Zg + log 2 Z = β F g − F ,(33)\nwhereF g is the partial free energy given by all the configurations that can be obtained from g via products of spin flip operators that act solely on subsystem A or subsystem B (i.e., having the same 'boundary' as g), and . . . denotes the ensemble average over g ∈ G with weight e −βE g . Notice that our result in Eq. (33) is the lattice equivalent of the Von Neumann entropy obtained by Fradkin and Moore in Ref. 14 for continuous systems. Alternatively, Eq. (33) can be interpreted as the entropy of mixing (or configurational entropy) of the allowed bipartition boundaries in G. This can be made more transparent by introducing the quotient group Q = G/(G A G B ), and by rewriting Eq. (33) as\nS (A) VN = − q∈Q h∈G A k∈G B e −βE hqk Z × log 2 h ∈G A ,k∈G B e −βEh (hqk)k Z = − q∈Q P q log 2 P q ,(34)\nwhere we used the fact that the term in square brackets is independent of h and k, and where we introduced the notation\nP q = h∈G A , k∈G B e −βE hqk Z(35)\nfor the probability of boundary q to appear in G, for a given inverse temperature β and energy E g . In order to proceed further, let us focus for simplicity on the specific GS of our system (15). The generic case of a wavefunction with factorizable amplitudes can be inferred with minor modifications.\n\nA. The case of 1-body potentials All of the above results apply straightforwardly to the GS in Eq. (15). Notice that (i) the topological entropy in each block-diagonal sector of the pure Kitaev model is the same, 12 and (ii) it is reasonable to make the working assumption that the relevant sector for the transition to the fully magnetized state |0 is the one that contains this state, and that is therefore obtained upon applying the group G to |0 . For the purpose of computing the topological entropy, one can thus replace Eq. (15) by\n|GS = 1 √ Z g∈G e β P i σ z i (g)/2 g |0 .(36)\nand obtain\nS (A) VN = − 1 Z g∈G e β P i σ z i (g) × log 2 h∈G A , k∈G B e β P i σ z i (hgk) Z (37) where Z = g∈G e β P i σ z i (g)\n. In order to simplify Eq. (37) with the purpose of computing the topological entropy of the system (23), it is convenient to do the following change of variables. Recall that a generic configuration g|0 is uniquely specified by the set of star operators acting on the reference configuration |0 , which we chose to be the ferromagnetic state with all the σ spins pointing up, modulo the action of the product of all the star operators (which is equal to the identity). Thus, there is a 1to-2 mapping between G = {g} and the configuration space Θ = {θ} of an Ising model with degrees of freedom θ s living on the sites s of the square lattice, where for example θ s = −1 (+1) means that the corresponding star operator is (not) acting in the associated g. Since each σ spin can be flipped only by its two neighboring θ spins, then σ i ≡ θ s θ s ′ , where i labels the bond between the two neighboring sites s, s ′ , and g∈G e β\nP i σ z i (g) ≡ 1 2 θ∈Θ e β P s,s ′ θ s θ s ′ .(38)\nNotice that, using the above mapping, the GS wavefunction of our model, Eq. (15), can be rewritten as\n|GS = θ∈Θ e β P s,s ′ θ s θ s ′ /2 √ Z g(θ) |0 ,(39)\nwhere Z = θ∈Θ e β P s,s ′ θ s θ s ′ . Thus, all equal-time correlation functions that can be expressed in terms of the θ s variables are the same as those of a 2D classical Ising model with reduced nearest-neighbor coupling J/T = β, implying that the critical point of the latter β c = (1/2) ln( √ 2 + 1) ≃ 0.4406868 corresponds precisely to the critical point of our quantum system. Notice also that the magnetization in the original σ spin language is indeed the nearest-neighbor spinspin correlation (i.e., the energy) in the θ spin language,\nm(β) = 1 N i GS|σ z i |GS = 1 Z θ∈Θ e β P s,s ′ θ s θ s ′ 1 N i σ z i (g(θ)) = 1 Z θ∈Θ e β P s,s ′ θ s θ s ′ 1 N s,s ′ θ s θ s ′ = 1 N E Ising (β) .(40)\nTherefore, one concludes that the magnetization m(β) is continuous across the transition at β c but there is a singularity in its first derivative\n∂m ∂β = 1 N ∂E Ising ∂β = −β 2 1 N C Ising (β) ,(41)\nas the Ising model heat capacity C Ising diverges logarithmically at β c . In the following, we will show how such continuous phase transition is accompanied by a sudden, discontinuous vanishing of the topological entropy of the system.\n\nThe case of a configuration of the form hgk, with h ∈ G A and k ∈ G B requires a few additional steps. First of all, notice that the composition of any two elements g,g ∈ G is represented in the θ spin language by the site-by-site product of the two configurations corresponding to g andg, respectively:\nθ s (gg) = θ s (g) θ s (g). In particular, θ s (hgk) = θ s (hk) θ s (g).\nMoreover, using similar arguments as in Ref. 15, the star operators of a bipartite system (A, B) can be divided into bulk star operators, i.e., those acting solely on subsystem A or subsystem B, and boundary star operators acting simultaneously on A and B spins. The boundary star operators can be further subdivided into different sets according to the different boundaries around each connected component of A and B (for a total of m A + m B − 1 boundaries, m A and m B being the number of connected components of A and B, respectively).\n\nLet us define a collective operation as the product of all the star operators in one of these sets. That is, the product of all the stars around a connected boundary of the bipartition (A, B). Clearly, the number of such collective operations is given by the number of sets, m A + m B − 1.\n\nOne can show that the subgroup G A G B ⊂ G, to which the product hk belongs, can be generated by all the bulk star operators together with all but one of the collective operators (all but one, independently of which one is chosen to be left out, is required because the product of all boundary star operators is equivalent to the product of all bulk star operators). For example, G A G B is generated by the bulk star operators alone in bipartitions 2 and 3 in Fig. 2, while the product of all boundary star operators along one of the two boundaries must be included to generate G A G B for bipartitions 1 and 4.\n\nLet\nus define Θ b = {θ b }, 'b'\nfor 'bulk', to be the set of Ising spin configurations on the sites of the square lattice where all θ b s corresponding to boundary sites s are fixed to equal +1. Let us also define Θ δ = {θ δ }, 'δ' for 'boundary', to be the set of Ising configurations where θ δ s = +1 for all bulk star operators, θ δ s = +1 for all boundary star operators belonging to one chosen boundary, and θ δ s = ±1 for the remaining boundary star operators, so long as all θ δ s spins belonging to the same boundary have the same sign. Notice that Θ δ = {1 1} for bipartitions 2 and 3 in Fig. 2, where 1 1 is the configuration with all the spins θ δ s = +1. One can finally show that there is a one-to-one correspondence between the elements of G A G B and the Ising configurations in\n{θ b θ δ , ∀ θ b ∈ Θ b , θ δ ∈ Θ δ },\nwhere θ b θ δ represents the site-by-site product of the two configurations (i.e., (θ b θ δ ) s = θ b s θ δ s ). Therefore,\nh∈G A , k∈G B e β P i σ z i (hgk) ≡ ≡ θ b ∈Θ b , θ δ ∈Θ δ e β P s,s ′ θ b s θ δ s θ s (g)θ s ′ (g)θ δ s ′ θ b s ′ ,(42)\nand in particular,\ns,s ′ θ b s θ δ s θ s (g)θ s ′ (g)θ δ s ′ θ b s ′ = = s,s ′ bulk s,s ′ θ b s θ s (g)θ s ′ (g)θ b s ′ (43) + s ′ boundary s bulk s,s ′ θ b s θ s (g)θ s ′ (g)θ δ s ′ (44) + s,s ′ different boundaries s,s ′ θ δ s θ s (g)θ s ′ (g)θ δ s ′ (45) + s,s ′ same boundary s,s ′ θ s (g)θ s ′ (g),(46)\nwhere we used the fact that if s is in the bulk then θ δ s = +1, if s belongs to a boundary then θ b s = +1, and if both s and s ′ belong to the same boundary then θ δ s θ δ s ′ = +1. Let us focus on the bipartitions of interest to compute the topological entropy (23). First of all, in the limit r, R → ∞ there are no nearest-neighboring stars s and s ′ belonging to two different boundaries. Therefore, the term (45) vanishes identically. For bipartitions 2 and 3, Θ δ = {1 1} and\nh∈G A k∈G B e β P i σ z i (hgk) ≡ e β Ps,s ′ boundary s,s ′ θ s (g)θ s ′ (g) θ b ∈Θ b e β Ps,s ′ bulk s,s ′ θ b s θ s (g)θ s ′ (g)θ b s ′ e β Ps ′ boundary,s bulk s,s ′ θ b s θ s (g)θ s ′ (g) .(47)\nThe r.h.s. of the above equation can be interpreted as the partition function of an Ising model with nearest-neighbor interactions, where only the bulk degrees of freedom are allowed to flip starting from a given configuration θ(g). Clearly such partition function is invariant upon changing the initial configuration as long as the new one is in the same ergodic sector.\n\nFor example, one can equivalently choosẽ θ(g) = +1 if s belongs to the bulk θ s (g) if s belongs to the boundary, (48) and the expression above simplifies to\nh∈G A k∈G B e β P i σ z i (hgk) ≡ e β Ps,s ′ boundary s,s ′ θ s (g)θ s ′ (g) θ b ∈Θ b e β Ps,s ′ bulk s,s ′ θ b s θ b s ′ e β Ps ′ boundary,s bulk s,s ′ θ b s θ s ′ (g) = Z ∂ 2,3 (g).(49)\nHere Z ∂ 2,3 (g) represents the partition function of an Ising model with nearest-neighbor interaction of reduced strength J/T = β, and with fixed spins along the boundary of bipartitions 2 and 3, respectively. The values of the spins at the boundary are determined by g.\n\nFor bipartitions 1 and 4, Θ δ = {1 1, f }, where the configuration f has all the spins equal to +1 except for those belonging to the chosen boundary, say boundary 2 in Fig. 2, which are equal to −1. In this case\nh∈G A k∈G B e β P i σ z i (hgk) ≡ e β Ps,s ′ same boundary s,s ′ θ s (g)θ s ′ (g) θ b ∈Θ b e β Ps,s ′ bulk s,s ′ θ b s θ b s ′ e β Ps ′ boundary 1,s bulk s,s ′ θ b s θ s ′ (g) × e β Ps ′ boundary 2,s bulk s,s ′ θ b s θ s ′ (g) + e −β Ps ′ boundary 2,s bulk s,s ′ θ b s θ s ′ (g) = Z ∂ 1,4 (g) + Z ∂, twisted 1,4 (g).(50)\nHere Z ∂ 1,4 (g) are the analog of Z ∂ 2,3 (g) for bipartitions 1 and 4, respectively, while Z ∂, twisted 1,4 (g) differ from the former by the fact that all the (fixed) spins belonging to boundary 2 in bipartitions 1 and 4 respectively have been flipped. In other words, Z ∂ 1,4 (g) represents the partition function of an Ising model with nearest-neighbor interaction of reduced strength J/T = β, and with fixed spins along the bound-ary of bipartitions 1 and 4, respectively. The partition functions Z ∂, twisted 1,4 (g) differ in that the spins along one of the two boundaries have been flipped with respect to their values in Z ∂ 1,4 (g). Again, the values of the spins at the boundary are determined by g.\n\nIn this notation, the topological entropy of the system can be written as\nS topo = lim r,R→∞ 1 Z g∈G e β P i σ z i (g) log 2 Z ∂ 1 (g) + Z ∂, twisted 1 (g) Z ∂ 4 (g) + Z ∂, twisted 4 (g) Z ∂ 2 (g)Z ∂ 3 (g) ,(51)\nwhere the sum over g acts as a weighed average of the logarithmic term over all possible values of the spins at the boundary. Notice that in Eq. (51) the partitions with two boundaries, and hence with non-trivial topology, are those that appear with two contributions (bipartitions 1 and 4), corresponding to some relative boundary conditions (BCs) and their twisted counterparts. These contributions, as we show below in detail, are responsible for the non-vanishing topological entropy.\n\nIn the topological phase, the two partition functions for the twisted and untwisted BCs contribute equally, and in the nontopological phase, one partition function is exponentially suppressed when compared to the other, in the thermodynamic limit. Therefore, there is an extra entropy contribution in one of the phases depending on whether the boundaries of topologically non-trivial bipartitions are twisted or not relative to one another. From Eq. (51), the behavior of the topological entropy can be qualitatively argued as follows. Deep in the disordered phase, where the correlations are short ranged, the choice of boundary conditions is likely to affect the partition function of the system only with exponentially small corrections. Thus, we can expect to have\nZ ∂ 1 (g)Z ∂ 4 (g) ≃ Z ∂, twisted 1 (g)Z ∂ 4 (g) ≃ . . . ≃ Z ∂ 2 (g)Z ∂ 3 (g) and S topo = 2.\nOn the other hand, deep in the (ferromagnetically) ordered phase the partition function of a system with twisted boundary conditions is exponentially suppressed with respect to the one without the twist. Thus,\nZ ∂ 1 (g) ≫ Z ∂, twisted 1 (g), Z ∂ 4 (g) ≫ Z ∂, twisted 4 (g), while Z ∂ 1 (g)Z ∂ 4 (g) ≃ Z ∂ 2 (g)Z ∂ 3 (g) still holds\n. This leads to S topo = 0.\n\nIn the following two sections we will show with rigorous arguments that the behavior of the topological entropy across the transition is strongly first order, with a sudden jump from S topo = 2 to S topo = 0.\n\n\nThe disordered phase (β < β c )\n\nIn the limit of small β, namely above the ordering transition, one can compute S topo via the high-temperature expansion of the Ising model with fixed spins at the boundary.\n\nLet us rewrite,\nZ ∂ 2 (g) = θ b ∈Θ b bonds i cosh β + sinh β θ s i θ s ′ i = 2 N/2−δ c N G t ℓ(G) s∈EG θ s (g),(52)\nwhere s i , s ′ i are the sites at the ends of bond i, N is the total number of bonds on the lattice, δ is the length of the boundary in number of θ spins, c = cosh β, t = tanh β. The sum over G runs over all possible graphs on the bonds of the square lattice, composed entirely of closed loops and open strings connecting two boundary spins. The product s∈EG θ s (g) encompasses all the boundary spins that appear as end points (the set E G ) of open strings in G. Finally, ℓ(G) is the total length of the closed loops and open strings in G.\n\nAnalogously for Z ∂ 3 (g). The case of Z ∂ 1 (g) and Z ∂ 4 (g) differs from Z ∂ 2 (g) and Z ∂ 3 (g) in that there are now two types of open strings: those going from one boundary to itself and those connecting the two boundaries (see Fig. 2).\n\nNext, let us compare the product Z ∂ 2 (g)Z ∂ 3 (g) with the product Z ∂ 1 (g)Z ∂ 4 (g). Notice that bipartitions 2 and 3 have precisely the same total combined boundary as bipartitions 1 and 4. In order for a graph to appear in one of the two products and not in the other, it needs to comprise loops or strings that are able to tell the difference between to two possible origins (2 + 3 vs. 1 + 4) of the total combined boundary.\n\nExamples of such open strings are shown in Fig. 3. One can show that these telltale strings, and the analogous closed loops, cannot be arbitrarily short, and their length is bounded from below by R − 2r. As a consequence, the corresponding graphs are exponentially suppressed at least as t R−2r , and in the limit r, R → ∞ with R − 2r → ∞, implicit in the definition of the topological entropy, one obtains\nZ ∂ 1 (g)Z ∂ 4 (g) Z ∂ 2 (g)Z ∂ 3 (g) → 1.(53)\nSimilar considerations apply when comparing the product Z ∂ 1 (g)Z ∂ 4 (g) with products of the kind Z ∂, twisted 1 (g)Z ∂ 4 (g). In this case, the boundaries involved are exaclty the same, and the relevant telltale elements of the graph are open strings connecting one of the two components of the boundary with the other. Such strings are in fact the only elements that are sensitive to the twisted boundary conditions. Clearly the length of these strings is bounded from below by R − 2r, and\nZ ∂, twisted 1 (g)Z ∂ 4 (g) Z ∂ 1 (g)Z ∂ 4 (g) → 1(54)\nexponentially fast, at least as t R−2r , with (R − 2r) → ∞.\n\nOf course, our reasoning is correct up to the point where the high temperature expansion breaks down, and entropic contributions balance the exponential suppression. Said differently, this is the case when the correlation length in the Ising model goes to infinity, and the large r, R limit does not guarantee that the ratios of products of partition functions above tend to one.\n\nGiven Eqs. (53,54), we can finally use Eq. (51) to obtain the topological entropy of the system throughout the disordered phase β < β c ≃ 0.4406868,\nS topo = 1 Z g∈G e β P i σ z i (g) log 2 4\n= log 2 4 = 2.\n\n\nThe Landau-Ginzburg ordered phase (β > β c )\n\nWhat happens below this transition? Rather than attempting a low-temperature expansion, it is convenient to use the duality relations derived by A. Bugrij and V. Shadura in Ref. 16 for the inhomogeneous, finite-size Ising model. In particular, they obtained the duality relations for a square lattice Ising model wrapped around a cylinder of finite length, with fixed, free and mixed boundary conditions. Following the usual convention, let us labelβ the coupling constant of the dual Ising model (defined on the plaquettes of the original lattice), which is related to β by the duality relation sinh β sinhβ = 1. Let us also indicate with Z(δ, δ ′ ) andZ(δ, δ ′ ) the partition functions of the system on the finite cylinder and its dual, with δ, δ ′ specifying the boundary conditions, namely δ, δ ′ = , × for free and fixed boundary spins, respectively. With this notation in mind, the results by Bugrij and Shadura -to the purpose of the present paper -can be summarized by 16\nZ( , ) = K Z(×, ×) + Z twisted (×, ×) (55) Z(×, ) = K Z( , ×) (56) Z( , ×) = K Z(×, ),(57)\nwhere Z twisted (×, ×) differs from Z(×, ×) by the fact that the fixed boundary spins at one end of the cylinder have been flipped. Notice that the proportionality coefficient K is the same in all the equations, and that Z( , ×) = Z(×, ) and Z(×, ) =Z( , ×).\n\nLet us then consider Z ∂ 2 in Eq. (49). Thanks to the nearestneighbor character of the interaction between θ spins, subsystem A interacts only with itself and with the boundary δ 2 , and so does subsystem B. Thus, one can factorize the two subsystems and obtain (cfr. Eq.(49))\nh∈G 2A k∈G 2B e β P i σ z i (hgk) ≡ e β Ps,s ′ ∈δ 2 s,s ′ θ s (g)θ s ′ (g) × θ b 2A ∈Θ b 2A e β Ps,s ′ bulk s,s ′ θ b s θ b s ′ e β Ps ′ boundary,s bulk s,s ′ θ b s θ s ′ (g) × θ b 2B ∈Θ b 2B e β Ps,s ′ bulk s,s ′ θ b s θ b s ′ e β Ps ′ boundary,s bulk s,s ′ θ b s θ s ′ (g) = e β Ps,s ′ ∈δ 2 s,s ′ θ s (g)θ s ′ (g) Z ∂ 2A (g) Z ∂ 2B (g).(58)\nSimilar arguments apply to bipartition 3,\nh∈G 3A k∈G 3B e β P i σ z i (hgk) = = e β Ps,s ′ ∈δ 3 s,s ′ θ s (g)θ s ′ (g) Z ∂ 3A (g) Z ∂ 3B (g),(59)\nand with a few more steps, to bipartitions 1 and 4 as well (cfr. Eq.(50)), \nh∈G 1A k∈G 1B e β P i σ z i (hgk) = = e β Ps,s ′ ∈δ 1 , boundary 1 s,s ′ θ s (g)θ s ′ (g) × e β Ps,s ′ ∈δ 1 , boundary 2 s,s ′ θ s (g)θ s ′ (g) × Z ∂ 1A (g) + Z ∂, twisted 1A (g) Z ∂ 1B 1 (g) Z ∂ 1B 2 (g),(60)×Z ∂ 4A 1 (g) Z ∂ 4A 2 (g) Z ∂ 4B (g) + Z ∂, twisted 4B (g) ,(61)\nwhere 1B 1 and 1B 2 refer to the two connected components of subsystem B in bipartition 1, i.e., the component inside boundary 1 and the component outside boundary 2, and analogously for 4A 1 and 4A 2 . In order to apply Eqs. (55-57) to the present case, some further considerations on the bipartitions in Fig. 2 are needed. Recall that, although S topo is indeed a quantity of order one, we expressed it in Eq. (51) in terms of a ratio of extensive partition functions Z ∂ i (g). Thus, any sub-extensive correction to these partition functions (i.e., O(2 N α ), with α < 1, N being the number of degrees of freedom in the system) will only amount to an exponentially small correction to S topo , that vanishes in the thermodynamic limit. In this context, the partition function Z ∂ 1A (g) (see Fig. 4) is 'equivalent', in the thermodynamic limit, to the partition function of an Ising model on an infinite cylinder with fixed boundaries at the edges (bound- ary 1 and 2, respectively). Similarly, the partition function Z ∂ 1B 1 (g) can be regarded as that of an Ising model on an infinite cylinder with fixed boundary conditions on one edge (boundary 1) and open boundary conditions on a suitably introduced boundary γ 1 . Finally, the same approach can be used for Z ∂ 1B 2 (g), with fixed boundary conditions on one edge (boundary 2) and open boundary conditions on another suitably introduced boundary γ 4 . Qualitatively, this is illustrated in Fig. 4a, where the spins on boundary 1 and boundary 2 are fixed and those belonging to γ 1 and γ 4 are free. Notice that the mapping onto infinite cylinders requires the distance between any of the γ i boundaries introduced in Fig. 4 and any of the original boundaries in Fig. 2 to diverge with system size. Similar arguments apply to bipartitions 2, 3 and 4. This leads to a correspondence between our factorized partition functions and those used in Ref. 16, namely\nZ ∂ 1B 1 (g), Z ∂ 1B 2 (g), Z ∂ 2A (g), Z ∂ 2B (g), Z ∂ 3A (g), Z ∂ 3B (g), Z ∂ 4A 1 (g), Z ∂ 4A 2 (g) ∼ Z( , ×) Z ∂ 1A (g), Z ∂ 4B (g) ∼ Z(×, ×) Z ∂, twisted 1A (g), Z ∂, twisted 4B (g) ∼ Z twisted (×, ×).\nThe results in Ref. 16 can then be applied to our systems and lead to the following equations:\nZ ∂ 1A ( , ) ∝ Z ∂ 1A (g) + Z ∂, twisted 1A (g) (62a) Z ∂ 1B 1 (×, ) ∝ Z ∂ 1B 1 (g) (62b) Z ∂ 1B 2 (×, ) ∝ Z ∂ 1B 2 (g) (62c) Z ∂ 2A (×, ) ∝ Z ∂ 2A (g) (62d) Z ∂ 2B (×, ) ∝ Z ∂ 2B (g) (62e) Z ∂ 3A (×, ) ∝ Z ∂ 3A (g) (62f) Z ∂ 3B (×, ) ∝ Z ∂ 3B (g) (62g) Z ∂ 4B ( , ) ∝ Z ∂ 4B (g) + Z ∂, twisted 4B (g) (62h) Z ∂ 4A 1 (×, ) ∝ Z ∂ 4A 1 (g) (62i) Z ∂ 4A 2 (×, ) ∝ Z ∂ 4A 2 (g).(62j)\nFor convenience of notation, let us define the dual partition functions for the whole system in the different bipartitions\nZ ∂ 1 ≡Z ∂ 1A ( , )Z ∂ 1B 1 (×, )Z ∂ 1B 2 (×, )(63)Z ∂ 2 ≡Z ∂ 2A (×, )Z ∂ 2B (×, ) (64) Z ∂ 3 ≡Z ∂ 3A (×, )Z ∂ 3B (×, )(65)Z ∂ 4 ≡Z ∂ 4A 1 (×, )Z ∂ 4A 2 (×, )Z ∂ 4B ( , )(66)\nFinally, we have all the ingredients to evaluate the topological entropy for β > β c . Let us first rewrite Eq. (51) using Eqs. (58-61) instead of Eqs. (49-50):\nS topo = lim r,R→∞ 1 Z g∈G e β P i σ z i (g) × log 2 Z ∂ 1A (g) + Z ∂, twisted 1A (g) Z ∂ 1B 1 (g)Z ∂ 1B 2 (g)Z ∂ 4A 1 (g)Z ∂ 4A 2 (g) Z ∂ 4B (g) + Z ∂, twisted 4B (g) Z ∂ 2A (g)Z ∂ 2B (g)Z ∂ 3A (g)Z ∂ 3B (g) + s,s ′ ∈δ 1 +δ 4 s,s ′ θ s (g)θ s ′ (g) − s,s ′ ∈δ 2 +δ 3 s,s ′ θ s (g)θ s ′ (g) ,(67)\nwhere the last two terms inside the square brackets come from the exponential factors in Eqs. (58-61). Using the duality relations (62), we can identify\nZ ∂ 1A (g) + Z ∂, twisted 1A (g) Z ∂ 1B 1 (g)Z ∂ 1B 2 (g)Z ∂ 4A 1 (g)Z ∂ 4A 2 (g) Z ∂ 4B (g) + Z ∂, twisted 4B (g) Z ∂ 2A (g)Z ∂ 2B (g)Z ∂ 3A (g)Z ∂ 3B (g) = =Z ∂ 1A ( , )Z ∂ 1B 1 (×, )Z ∂ 1B 2 (×, )Z ∂ 4A 1 (×, )Z ∂ 4A 2 (×, )Z ∂ 4B ( , ) Z ∂ 2A (×, )Z ∂ 2B (×, )Z ∂ 3A (×, )Z ∂ 3B (×, ) ≡Z ∂ 1Z ∂ 4 Z ∂ 2Z ∂ 3 .\nFor β > β c , the dual Ising models are in the disordered phase and one can perform a high-temperature expansion to calculate the ratio (Z ∂ \ns,s ′ θ s (g)θ s ′ (g) − s,s ′ ∈δ 2 +δ 3 s,s ′ θ s (g)θ s ′ (g) = = i∈δ 1 +δ 4 σ i (g) − i∈δ 2 +δ 3 σ i (g),\nwhere i labels the bonds of the square lattice, and i ∈ δ means that the bond i connects two sites s and s ′ belonging to δ. This contribution can be shown to vanish identically since the set of boundary σ spins in bipartitions 1 and 4 is identical to the set of boundary spins in bipartitions 2 and 3 (see Fig. (2)).\n\nIn the end we find that\nS topo = lim r,R→∞ 1 Z g∈G e β P i σ z i (g) log 2Z ∂ 1Z ∂ 4 Z ∂ 2Z ∂ 3 = 0\nidentically in the ordered phase β > β c .\n\n\nB. Beyond 1-body potentials\n\nAs we already mentioned, the calculations carried out in Sec. III A for the specific model presented in this paper can be straightforwardly extended to the case of any factorizable wavefunction. All one needs to do is identify a proper set of local generators (i.e., acting on the σ spins contained within a disc of finite radius) for the group G, and the equivalent of the collective boundary flip operators. The rest of the derivation follows essentially unchanged, in the limit r, R → ∞.\n\nWhat happens if we attempt to generalize our approach further and we consider non-factorizable wavefunctions? For simplicity, take once again the Kitaev-like GS wavefunction in Eq. (15), but replace the argument of the exponential β i σ z i (g, α)/2 with some generic function −βE g /2. As we can see immediately from Eq. (24), our approach to compute the topological entropy can no longer be used from the very first stage. On the other hand, it is tempting to conjecture that, so long as E g is short ranged (i.e., it can be written as the sum of terms involving σ spins within a disc of finite radius on the lattice), the error that one makes by neglecting the terms involving spins across the boundary of a bipartition (E ∂ g ) does not give topological contributions to S topo . Under this assumption, one can then set E ∂ g = 0 and use the approximate equality E g ≃ E A g A + E B g B to re-establish the factorability needed to carry on with the calculations. The result obtained for S topo in Eq. (51), which employed this approximation, nonetheless shows no explicit dependence on it in the final expression, and one could then reinstate the full E g at that stage. If the conjecture above is correct, the formula in Eq. (51) gives the exact topological entropy for a generic GS wavefunction that satis-fies (i) the positive amplitude condition, (ii) the group condition for G, 17 and (iii) the locality (i.e., short ranged) condition on E g .\n\nA simple example where this conjecture can be applied rather straightforwardly is the case where Eqs. (14) and (15) are replaced by\nH = H Kitaev + λ 1 s e −β P i∈s P ij ,j / ∈sσ z iσ z j (68) |GS = 1 √ Z g∈G e β P ij σ z i (g)σ z j (g)/2 g |0 .(69)\n(For a discussion of the general construction scheme of such type of Hamiltonians, see Ref. 13.) Here |GS is the GS wavefunction of H in the topological sector where i∈pσ z i = +1. The notation ij , j / ∈ s stands for j nearest-neighbor of i but not adjacent to the same vertex s. Without loss of generality, we consider the range β ∈ (0, ∞), where the new term in the Hamiltonian favors ferromagnetic order in the σ spins.\n\nLet us then introduce the same description in terms of the θ spins, as in the previous section. Given that the product of two nearest-neighboring σ spins translates into the product of two next-nearest-neighboring θ spins, we obtain\ng∈G e β P ij σ z i (g)σ z j (g) ≡ 1 2 θ∈Θ e 2β P s,s ′ θ s θ s ′\n(notice the additional factor of 2 in the exponent due to the fact that the same product θ s θ s ′ corresponds to two distinct products σ z i σ z j ), and\n|GS = θ∈Θ e β P s,s ′ θ s θ s ′ √ Z g(θ) |0 ,(70)\nwhere Z = θ∈Θ e 2β P s,s ′ θ s θ s ′ . The latter is the partition function of a square-lattice Ising model with sole nextnearest-neighbor interactions, which factorizes into the product of two decoupled Ising models with nearest-neighbor interactions (namely corresponding to the θ spins on each of the two sublattices). In this case, all equal-time correlators in the GS of the quantum system can be written in terms of classical correlators of two decoupled Ising models. As before, we expect the system to undergo a phase transition when the two Ising models become critical at β c = (1/4) ln( √ 2 + 1) ≃ 0.2203434. However, contrarily to the previous case, the new model undergoes a spontaneous symmetry-breaking (Landau-Ginzburg) phase transition! This is best seen by mapping the system onto a quantum eight-vertex model, as discussed below. The local order parameter that captures the transition is the magnetization of the σ spins, whose expectation value can be written as an ensemble average of the product of two neighboring θ spins, i.e., belonging to two decoupled Ising models. Clearly such average vanishes identically in the hightemperature phase β < β c , while it becomes finite in the ordered phase β > β c . Notice that this local order parameter that acquires an expectation value does so in the nontopologically ordered phase, as expected from the fact that no local order parameter exists that resolves the topological phase.\n\nWhat is the fate of the topological entropy across this Landau-Ginzburg phase transition? According to the conjecture above, we can directly substitute i σ z i → ij σ z i σ z j into Eq. (51) and compute S topo . This amounts to replacing the boundary Ising partition functions in the argument of the logarithm with the partition functions of two decoupled boundary Ising models. As a result, all calculations carried out in the previous section remain essentially unchanged and one arrives to the identical result that S topo = 2 throughout the high-temperature phase, and vanishes otherwise. This scenario is in agreement with previous results on a quantum version of the eight-vertex model by Ardonne et al., 18 whose GS is a generalization of the one in our model. Consider indeed the wavefunction in Eq. (69). Given the nature of the group G, the four spins belonging to any plaquette of the square lattice can assume only eight distinct configurations ( i∈pσ z i = +1), illustrated in Fig. 5. Such configurations map naturally onto the vertices of an eight-vertex model upon replacing each positive spin with an arrow along the corresponding bond of the dual lattice, pointing, say, from sublattice A to sublattice B, and vice versa for the negative spins (as shown in Fig. 5). Given that ij · · · ≡ s ij ∈s · · · , the amplitudes in the GS wavefunction (69) factorize into products of vertex fugacities exp(β ij ∈s σ z i σ z j /2). In the notation of Fig. 5, the vertex fugacities assume the values a = b = 1, c = e −2β and d = e 2β . The GS spatial properties of our model are therefore captured by a classical eight-vertex model with the appropriate fugacities 18 (but see Ref. 13 for a general discussion of such quantum-to-classical correspondence), and one can then use Baxter's exact solution 19 to obtain the phase diagram as well as the scaling exponents at the critical point. All this is discussed in detail in Ref. 18: the model undergoes a second-order, Z 2 -symmetry-breaking phase transition when d 2 = c 2 + 2 (i.e., β c = (1/4) ln( √ 2 + 1)), separating a topologically ordered liquid phase from a Landau-Ginzburg ordered phase. The local order parameter across the transition is indeed the magnetization in the original σ spins.\n\n\nIV. CONCLUSIONS\n\nIn this paper we studied a topological quantum phase transition in a microscopic model that can be examined analytically. For this system, an extension of the toric code, the ground state wavefunction can be written exactly as a function of the parameter β that drives the system across the quantum phase transition. We computed the topological entropy for this system as a function of β, and showed that it remains at a constant non-zero value throughout the topologically ordered phase (β < β c ≃ 0.4406868). Immediately after the quantum phase transition at β c , the topological entropy drops to zero and remains so in the non-topologically ordered phase (β > β c ).\n\nThe GS wavefunction of our quantum system has positive amplitudes in the basis of choice. This property allows us to relate many quantities that are relevant in characterizing the (2+1 D) quantum system to those of a simple (2D, not 3D) classical Ising model at an inverse (classical) temperature equal to the value of the coupling constant β that drives the quantum system through the T = 0 phase transition. For example, the magnetization of the quantum system equals the energy E Ising (β) of the classical Ising model. While the magnetization is continuous and non-vanishing across the quantum phase transition (much as the energy E Ising (β) is across the classical Ising transition), its derivative with respect to β diverges logarithmically at β c (much as the Ising model heat capacity C Ising diverges logarithmically at β c ).\n\nDespite the relation to the 2D classical Ising model, the quantum phase transition does not have a local order parameter that vanishes on one side and not on the other. Of course one expects that no local order parameter can characterize the topological phase, but in this particular example, there is no order parameter that characterizes the non-topologically ordered phase either. One can indeed identify from the mapping to the Ising model a parameter that orders in the nontopological phase; however, this variable is non-local in the physical spin variables used to define the local Hamiltonian. Specifically, the order parameter is, in the language used in this paper, the expectation value of the θ s variables defined on the sites of the square lattice, such that σ ss ′ = θ s θ ′ s , for nearest neighboring sites s, s ′ . While σ ss ′ is obviously local in terms of the θ s , θ ′ s , the inversion needed to write the θ's in terms of the σ's is non-local. Hence, θ s may detect the transition into the non-topological phase, but as it is non-local it is not an order parameter in the usual sense. That there is no order parameter for the non-topologically ordered phase is not generic (see the example in Sec. III B), as perhaps the most obvious exit from a topological phase is by escaping into a locally ordered phase due to spontaneous symmetry breaking. Hence, the main example studied in this paper is particularly interesting in that one has no local order parameter in either phases.\n\nRecently, P. Zanardi et al. proposed a new approach to study quantum phase transitions through the behavior of a metric tensor in parameter space, derived from a fidelity-based notion of distance between states. 20 In this approach, one does not require any a priori knowledge of an order parameter to detect a phase transition (one may argue that the fidelity encompasses correlations of both local and non-local operators). Thus, these ideas may be particularly useful to detect topological quantum phase transitions (see Ref. 9).\n\nWe end with a speculative note: topological quantum phase transitions should share the feature that, even in the case when there is no local order parameter in either the topological or the non-topological phase, there should be, generically, detectable singularities in high enough derivatives of local observables with respect to the coupling that takes the system across the transition (as in the case study presented here). 21 After all, what would be a phase transition without singularities in any physical observable?\n\nUpon completion of this work, we became aware of similar work being pursued from a gauge theoretical perspective by S. Papanikolaou, K. Raman and E. Fradkin on the quantum eight-vertex model, 22 to which we refer the reader for a complementary approach.\n\nFIG. 1 :\n1(Color online) -Examples of star and plaquette operators, centered at a lattice site s (blue open circle) and at a dual lattice site p (red open square), respectively. The solid black dots represent the spin-1/2 degrees of freedom living on the bonds of the lattice, and the dashed lines connect the spins involved in the definition of each of the above operators.\n\nFIG. 3 :\n3(Color online)-Examples of open strings that appear in the expansion of Z ∂ 2 (g)Z ∂ 3 (g) but are not present in the expansion of Z ∂ 1 (g)Z ∂ 4 (g) (Top), and vice versa (Bottom). Top panel: The thick yellow and blue lines correspond to the boundaries in bipartitions 2 and 3 respectively. Boundaries belonging to both are shown in a thick dashed yellow-blue pattern. The strings in question are symbolically represented by thin black lines. Bottom panel: Same color coding, with yellow corresponding to bipartition 4 and blue corresponding to biparition 1. Notice that strings appearing in one expansion and not in the other must connect boundaries of the same solid color, and therefore cannot be shorter than R − 2r.\n\nFIG. 4 :\n4(Color online) -Illustration of the four bipartitions used to compute the topological entropy in Ref.10, with a possible choice for the additional boundaries needed to map each partition onto an infinite cylinder in the thermodynamic limit.\n\n.\nUsing the same loop description as for the original system, with t replaced byt = tanh(β), one can show that (terms in Eq. (67) can be dealt with more promptly by reverting back to the original σ spin degrees of freedom, s,s ′ ∈δ 1 +δ 4\n\nFIG. 5 :\n5Illustration of the eight allowed spin configurations around a plaquette of the square lattice, in any basis state g|0 , g ∈ G. These can in turn be mapped onto the configurations of an eight vertex model by replacing each positive spin with an arrow along the corresponding bond of the dual lattice (dashed lines), pointing, say, from sublattice A to sublattice B (vice versa for the negative spins). Clearly, the corresponding eight vertex configurations differ depending on the location of the plaquette in the dual lattice: sublattice A (middle), or sublattice B (bottom). The letters a, b, c, d correspond to the usual labeling of the vertex fugacities in the eight vertex model.\n\n\nis simply the Kitaev Hamiltonian H Kitaev in Ref.5, up to a trivial overall constant shift of the energy.The exact ground state wavefunction of this Hamiltonian can be obtained by deconstructing H into two pieces, H = λ 0 H 0 + λ 1 H 1 , as follows.\n\n\n, and the topological entropy is then defined FIG. 2: Illustration of the four bipartitions used to compute the topological entropy in Ref. 10.00000000000000 \n00000000000000 00000000000000 \n00000000000000 00000000000000 \n00000000000000 \n00000000000000 00000000000000 \n00000000000000 00000000000000 \n00000000000000 \n00000000000000 00000000000000 \n00000000000000 \n\n11111111111111 \n11111111111111 11111111111111 \n11111111111111 11111111111111 \n11111111111111 \n11111111111111 11111111111111 \n11111111111111 11111111111111 \n11111111111111 \n11111111111111 11111111111111 \n11111111111111 \n\n00000000000000 \n00000000000000 00000000000000 \n00000000000000 00000000000000 \n00000000000000 \n00000000000000 00000000000000 \n00000000000000 00000000000000 \n00000000000000 \n00000000000000 00000000000000 \n00000000000000 \n\n11111111111111 \n11111111111111 11111111111111 \n11111111111111 11111111111111 \n11111111111111 \n11111111111111 11111111111111 \n11111111111111 11111111111111 \n11111111111111 \n11111111111111 11111111111111 \n11111111111111 \n\n00000 \n00000 00000 \n00000 00000 \n00000 \n00000 00000 \n00000 00000 \n00000 \n00000 00000 \n00000 \n\n11111 \n11111 11111 \n11111 11111 \n11111 \n11111 11111 \n11111 11111 \n11111 \n11111 11111 \n11111 \n\n000000000000000 \n000000000000000 000000000000000 \n000000000000000 000000000000000 \n000000000000000 \n000000000000000 000000000000000 \n000000000000000 000000000000000 \n000000000000000 \n000000000000000 000000000000000 \n000000000000000 \n\n111111111111111 \n111111111111111 111111111111111 \n111111111111111 111111111111111 \n111111111111111 \n111111111111111 111111111111111 \n111111111111111 111111111111111 \n111111111111111 \n111111111111111 111111111111111 \n111111111111111 \n\n0000 \n0000 0000 \n0000 0000 \n0000 \n0000 0000 \n0000 0000 \n0000 \n0000 0000 \n0000 \n\n1111 \n1111 1111 \n1111 1111 \n1111 \n1111 1111 \n1111 1111 \n1111 \n1111 1111 \n1111 \n\n(4) \n(2) \n(3) \n(1) \n\nr \n\nR \n\nboundary 1 \nboundary 1 \n\nboundary 2 \nboundary 2 \n\n\nAcknowledgmentsWe would like to thank Xiao-Gang Wen, Eduardo Fradkin and Stefanos Papanikolaou for enlightening discussions. We are particularly indebted to Paul Fendley, whose comments brought us to consider the model discussed in Sec. III B. This work is supported in part by the NSF Grants DMR-0305482\nDMR-0403997 (CC and CC), and by EPSRC Grant No. DMR-0403997 (CC and CC), and by EPSRC Grant No.\n\nCastelnovo). C. Castelnovo would like to acknowledge the I2CAM NSF Grant DMR No. 0645461 for travel support, during which part of this work was carried out. GR/R83712/01GR/R83712/01 (C. Castelnovo). C. Castelnovo would like to acknowledge the I2CAM NSF Grant DMR No. 0645461 for travel support, during which part of this work was carried out.\n\n. X.-G Wen, Int. J. Mod. Phys. B. 4239X.-G. Wen, Int. J. Mod. Phys. B 4, 239 (1990);\n\n. Adv. in Phys. 44405Adv. in Phys. 44, 405 (1995);\n\n. Phys. Rev. B. 65165113Phys. Rev. B 65, 165113 (2002).\n\n. F D M Haldane, E H Rezayi, Phys. Rev. B. 312529F. D. M. Haldane, and E. H. Rezayi, Phys. Rev. B 31, 2529 (1985).\n\n. X.-G Wen, Q Niu, Phys. Rev. B. 419377X.-G. Wen, and Q. Niu, Phys. Rev. B 41, 9377 (1990).\n\n. D Arovas, J R Schrieffer, F Wilczek, Phys. Rev. Lett. 53722D. Arovas, J. R. Schrieffer, and F. Wilczek, Phys. Rev. Lett. 53, 722 (1984).\n\n. A Y Kitaev, Ann. Phys. (N.Y.). 303A. Y. Kitaev, Ann. Phys. (N.Y.) 303, 2 (2003).\n\n. M Levin, X.-G Wen, Phys. Rev. B. 7575116M. Levin, and X.-G. Wen, Phys. Rev. B 75, 075116 (2007).\n\n. A Hamma, D A Lidar, arXiv:quant-ph/0607145v4Phys. Rev. Lett.). A. Hamma and D. A. Lidar, (accepted for publication on Phys. Rev. Lett.), arXiv:quant-ph/0607145v4 (2006).\n\n. S Trebst, P Werner, M Troyer, K Shtengel, C Nayak, Phys. Rev. Lett. 9870602S. Trebst, P. Werner, M. Troyer, K. Shtengel, and C. Nayak, Phys. Rev. Lett. 98, 070602 (2007).\n\n. A Hamma, W Zhang, S Haas, D A Lidar, arXiv:0705.0026v1A. Hamma, W. Zhang, S. Haas, and D. A. Lidar, arXiv:0705.0026v1 (2007).\n\n. M Levin, X.-G Wen, Phys. Rev. Lett. 96110405M. Levin, and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006).\n\n. A Y Kitaev, J Preskill, Phys. Rev. Lett. 96110404A. Y. Kitaev, and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006).\n\n. A Hamma, R Ionicioiu, P Zanardi, Phys. Rev. A. 7122315A. Hamma, R. Ionicioiu, and P. Zanardi, Phys. Rev. A 71, 022315 (2005).\n\n. C Castelnovo, C Chamon, C Mudry, P Pujol, Ann. Phys. (N.Y.). 318316C. Castelnovo, C. Chamon, C. Mudry, and P. Pujol, Ann. Phys. (N.Y.) 318, 316 (2005).\n\n. E Fradkin, J E Moore, Phys. Rev. Lett. 9750404E. Fradkin, and J. E. Moore, Phys. Rev. Lett. 97, 050404 (2006).\n\n. C Castelnovo, C Chamon, Phys. Rev. B. 76174416C. Castelnovo and C. Chamon, Phys. Rev. B 76, 174416 (2007).\n\n. A Bugrij, V N Shadura, arXiv:hep-th/9601106JETP Lett. 63384A. Bugrij, and V. N. Shadura, JETP Lett. 63, 384 (1996); arXiv:hep-th/9601106.\n\nThis condition is actually immaterial, and can be avoided using an approach similar to the one by Furukawa and Misguich. Phys. Rev. B. 75214407This condition is actually immaterial, and can be avoided us- ing an approach similar to the one by Furukawa and Misguich [Phys. Rev. B 75, 214407 (2007)].\n\n. E Ardonne, P Fendley, E Fradkin, Ann. Phys. (N.Y.). 310493E. Ardonne, P. Fendley, and E. Fradkin, Ann. Phys. (N.Y.) 310, 493 (2004).\n\nR Baxter, Exactly Solved Models in Statistical Mechanics. LondonAcademic PressR. Baxter, Exactly Solved Models in Statistical Mechanics (Aca- demic Press, London, 1982).\n\nand references therein. The behavior of fidelity across a topological phase transition in the Kitaev model in presence of a magnetic field was studied numerically in Ref. P Zanardi, L Venuti, P Giorda, Phys. Rev. 7662318P. Zanardi, L. Campos Venuti, P. Giorda, Phys. Rev. A76, 062318 (2007), and references therein. The behavior of fidelity across a topological phase transition in the Kitaev model in presence of a magnetic field was studied numerically in Ref. 9.\n\nthe lack of a local order parameter but display of local non-analyticities -was recently argued by Z. Nussinov and G. Ortiz to characterize as well a temperature-driven topological phase. arXiv:cond-mat/0702377A similar behavior -i.e.. transition in the Kitaev modelA similar behavior -i.e., the lack of a local order param- eter but display of local non-analyticities -was recently ar- gued by Z. Nussinov and G. Ortiz to characterize as well a temperature-driven topological phase transition in the Kitaev model [arXiv:cond-mat/0702377].\n\n. S Papanikolaou, K Raman, E Fradkin, Phys. Rev. B. 76224421S. Papanikolaou, K. Raman, and E. Fradkin, Phys. Rev. B 76, 224421 (2007).\n"},"corpus_id":{"kind":"number","value":16676596,"string":"16,676,596"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"0707.2084\",\n \"mag\": 2108036131,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1103/physrevb.77.054433\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/0707.2084v3.pdf\"\n ],\n \"pdf_sha\": \"5b78ee6cbb5d6deb5ac5c4b40cd7e8e336e5e181\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/0707.2084\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"A quantum topological phase transition at the microscopic level"},"authors":{"kind":"list like","value":["Claudio Castelnovo \nRudolf Peierls Centre for Theoretical Physics\nUniversity of Oxford\nOX1 3NPOxfordUK","Claudio Chamon \nPhysics Department\nBoston University\n02215BostonMAUSA"],"string":"[\n \"Claudio Castelnovo \\nRudolf Peierls Centre for Theoretical Physics\\nUniversity of Oxford\\nOX1 3NPOxfordUK\",\n \"Claudio Chamon \\nPhysics Department\\nBoston University\\n02215BostonMAUSA\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We study a quantum phase transition between a phase which is topologically ordered and one which is not. We focus on a spin model, an extension of the toric code, for which we obtain the exact ground state for all values of the coupling constant that takes the system across the phase transition. We compute the entanglement and the topological entropy of the system as a function of this coupling constant, and show that the topological entropy remains constant all the way up to the critical point, and jumps to zero beyond it. Despite the jump in the topological entropy, the transition is second order as detected via local observables."},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":"Some strongly correlated quantum many body systems display a type of order which cannot be characterized by any local order parameter. Instead, such order is topological in nature 1 , with the fractional quantum Hall systems being the primary example so far: they are liquid states that exhibit exotic properties such as a ground state (GS) degeneracy that cannot be lifted by any local perturbations 2,3 and fractionalized degrees of freedom 4 . Topologically ordered states are also interesting in that their robustness against local perturbations might be of use for decoherence-free quantum computation 5 .\nAn example of an exactly solvable lattice spin model that is topologically ordered was presented by Kitaev in Ref. 5, and the system was argued to be robust against small perturbations that tend to order the systemà la Landau-Ginzburg and take it away from its topological phase. The departure from the topologically ordered phase should occur through a quantum phase transition. Such quantum phase transition, however, cannot be entirely captured by ordinary methods based on local Landau-Ginzburg order parameters, and new methods need to be devised in order to investigate the fate of topological order across the phase transition. These novel methods must be based on the fundamental properties of topologically ordered phases, such as the GS degeneracy in presence of a gap, and the presence of a non-vanishing topological entropy.\nRecent efforts to understand quantum phase transitions in topologically ordered states include a mean-field approach for these exotic states 6 , and analytical and numerical studies 7,8,9 of the Kitaev model in the presence of a field. The numerical analysis presented in Refs. 8,9 leads to the conclusion that topological order survives unchanged up to the second order phase transition at β c = 0.32847(6) (in the notation of Eq. (18)), while the system is no longer topologically ordered for β > β c . (Here β stands for the coupling constant that drives the T = 0 quantum phase transition -the notation will become apparent shortly, and is chosen because of a close relation to a classical model.)\nIn this paper, we investigate analytically a quantum phase transition out of a topological phase. We show that the recently defined topological entropy 10,11 works well as an \"order parameter\" across the transition. We study the transition using a model -see Eq. (5) -that is shown to behave much like the Kitaev model in a magnetic field for small values of the field.\nThe advantage of this model is that the ground state can be obtained exactly, from which we can then compute the topological entropy explicitly, and show that it remains constant in the topologically ordered phase (β < β c ≃ 0.4406868), dropping abruptly to zero in the non-topologically-ordered phase (β > β c ), despite the continuous (second order) character of the transition.\nWe find that in this model, even though one cannot identify a local order parameter that vanishes in one phase and not in the other, one can show that the (continuous) local magnetization has a singularity at the critical point. In the model, we show that the magnetization equals the energy E Ising (β) of a 2D classical Ising model with N spins evaluated at an inverse (classical) temperature equal to the value of the coupling constant β that drives the system through the T = 0 phase transition:\nFrom this relation, it becomes evident that the magnetization m(β), although continuous and non-vanishing across the transition at β c (much as the energy of the classical Ising model across the classical transition), has a singularity in its first derivative, since ∂m ∂β\nand the Ising model heat capacity C Ising diverges logarithmically at β c . Hence, although there is no local order parameter that can detect either the topological or the non-topological phase in this system, one can expose the topological quantum phase transition through the singularity in the derivative of a local quantity. This is contrasted, for example, with the case discussed in Sec. III B, where a similar topological transition is accompanied by a simultaneous Z 2 symmetry breaking phase transition. In that case, the same transition is captured both by the non-local topological entropy, and by a local (Landau-Ginzburg) order parameter."},{"title":"II. THE MODEL","content":"The model that we consider is a deformation of the Kitaev model 5 , and it is defined on a square lattice with spin-1/2 de-grees of freedom living on the bonds, as shown in Fig. 1. The pure Kitaev model is written in terms of star and plaquette operators (see Fig. 1). Star operators are defined as\nwhere i labels the four spins on the bonds departing from some vertex s of the square lattice. Plaquette operators are defined as\nwhere i labels the four spins on the bonds around some plaquette p of the square lattice. The Hamiltonian we consider in this paper is\nwhere λ 0,1 > 0 and β is a parameter that we use to tune the system across a topological quantum phase transition. Notice that for β = 0 the Hamiltonian Eq. (5) Take G to be the (Abelian) group of all spin flip operations obtained as products of star type operators. Notice that g 2 = 1 1 for any element g of the group G. By acting with elements of G on a given reference configuration i |σ z i one generates a manifold of states, which however does not encompass the whole basis. For example, the action of a star operator A s cannot change the sign of the product of σ z 's around any square plaquette in the lattice 5,12 (see Fig. 1). Therefore, there is a non-trivial (and non-unique) minimal set {|Ψ α } of reference configurations that generates the full σ z -basis under the action of the group G. (In particular, one of the elements in this set is the reference configuration |0 that is fully magnetized in the z-direction, say σ z i = 1, ∀i.) Consider then the family of Hamiltonians\nfor some real-valued parameter β. The ground state of any such Hamiltonian can be obtained exactly and it can be written in the form\nwhere α labels the different block-diagonal sectors corresponding to the states in the minimal set {|Ψ α }; σ z i (g, α) is the z-component of the spin at site i in state g |Ψ α ; and the coefficients ψ α can be chosen at will, subject to the normalization condition α |ψ α | 2 = 1. Although the choice of minimal set is non-unique, one can show that Eq. (8) is independent of such choice, modulo an irrelevant permutation of the α indices. Within each block-diagonal sector, the GS of Eq. (6) is unique. Instead of proving directly that (8) is the GS of (6), it is more convenient to notice that the family of Hamiltonians in Eq. (6) is a particular choice of Stochastic Matrix Form decompositions of quantum Hamiltonians that exhibit precisely Eq. (8) as their GS. 13 One can verify this by showing that each of the operators\nbetween square brackets in Eq. (6) annihilates the inner sum in |GS 1 , independently of the index α. That the GS energy is zero follows because\nFor larger values of β, the many-body terms in Eq. (19) are no longer negligible and the equivalence is lost, although the form of the GS (15) suggests that the system gets deeper and deeper into the magnetized phase -as one would expect upon increasing the strength of the magnetic field in the Kitaev model. As we discuss in Section III A, our model undergoes a second-order phase transition at β c = (1/2) ln( √ 2 + 1) ≃ 0.4406868, where it displays a dimensionality reduction that places the transition in a different universality class than the one studied in Refs. 7,8,9. One can use the decomposition\nto estimate the limit of validity of Eq. (18) to be given by the condition\nThis corresponds to a ratio between the coupling to the magnetic field and the coupling to the cooperative transverse field ( s i∈sσ\nThe detailed numerical analysis presented in Refs. 8,9 lead the authors to conclude that topological order survives up to the second order phase transition at finite β c (in the notation of Eq. (18)), while the system is no longer topologically ordered for β > β c . In the following, we investigate this phase and the relative phase transition using the exact ground state of our model (5) to compute the topological entropy 10,11 across the transition. Using an exact derivation from the microscopic degrees of freedom, we show that the topological entropy is able to detect a transition from a topologically ordered phase (β < β c ) to a non-topologically-ordered phase (β > β c ). Indeed, it remains constant at its known β → 0 value up to the transition and drops abruptly to zero afterwards, despite the continuous character of the transition."},{"title":"III. THE TOPOLOGICAL ENTROPY OF FACTORIZABLE (LOCAL) WAVEFUNCTIONS","content":"Using the definition in Refs. 10,11, the topological entropy can be obtained as a linear combination of Von Neumann entanglement entropies S VN of different bipartitions of the system into subsystems A and B:\nwhere ρ A = Tr B (ρ) is the reduced density matrix obtained from the full density matrix ρ by tracing out the degrees of freedom of subsystem B, and the last equality holds whenever the full density matrix ρ is a pure-state density matrix.\nThe different bipartitions are aimed at removing all the extensive (boundary) contributions to uncover the sole topological contribution. A particular choice of the four bipartitions 10 is illustrated in Fig. 2 as:\nIn order to compute the topological entropy as a function of the parameter β, let us first notice that the Hamiltonian (14) constructed above, with GS given by (15), belongs to a class of Hamiltonians whose GS wavefunctions\n. For this type of Hamiltonians, one can compute the entanglement entropy as follows.\nConsider a given bipartition (A, B) of the system. The reduced density matrix ρ A = Tr B (ρ), obtained by tracing over all degrees of freedom in B, is given by 12\nis the subgroup of transformations acting solely on A (B) and leaving B (A) invariant:\nNotice that we used the group property to rewrite a generic element of G as gg, ∃!g ∈ G, as well as the additive property of E g . We can then compute the trace of the n-th power of the reduced density matrix Tr (ρ A ) n and use the identity\nto obtain the Von Neumann entropy S\nwhere we used the fact that the inner products in Eq. (26) impose\nfor i = 1, . . . , n, with the identification n + 1 ≡ 1, and therefore\nThe condition in Eq. (27) can be satisfied if and only if\nThus, the summation over all g ′ i of the inner products in Eq. (26) yields a constraint over the allowed values of g 1 , . . . , g n ∈ G:\n(30) In particular, the last line can be recast as\n(31) The physical meaning of these conditions will become clear in the next section, for the specific case of the system considered in this paper, although the form of Eq. (31) already suggests that they require all the elements g i to agree at the boundary of the bipartition (A, B).\nWe can finally use Eq. (31) to simplify Eq. (26)\nand obtain, via Eq. (25),\nwhereF g is the partial free energy given by all the configurations that can be obtained from g via products of spin flip operators that act solely on subsystem A or subsystem B (i.e., having the same 'boundary' as g), and . . . denotes the ensemble average over g ∈ G with weight e −βE g . Notice that our result in Eq. (33) is the lattice equivalent of the Von Neumann entropy obtained by Fradkin and Moore in Ref. 14 for continuous systems. Alternatively, Eq. (33) can be interpreted as the entropy of mixing (or configurational entropy) of the allowed bipartition boundaries in G. This can be made more transparent by introducing the quotient group Q = G/(G A G B ), and by rewriting Eq. (33) as\nwhere we used the fact that the term in square brackets is independent of h and k, and where we introduced the notation\nfor the probability of boundary q to appear in G, for a given inverse temperature β and energy E g . In order to proceed further, let us focus for simplicity on the specific GS of our system (15). The generic case of a wavefunction with factorizable amplitudes can be inferred with minor modifications.\nA. The case of 1-body potentials All of the above results apply straightforwardly to the GS in Eq. (15). Notice that (i) the topological entropy in each block-diagonal sector of the pure Kitaev model is the same, 12 and (ii) it is reasonable to make the working assumption that the relevant sector for the transition to the fully magnetized state |0 is the one that contains this state, and that is therefore obtained upon applying the group G to |0 . For the purpose of computing the topological entropy, one can thus replace Eq. (15) by\nand obtain\n. In order to simplify Eq. (37) with the purpose of computing the topological entropy of the system (23), it is convenient to do the following change of variables. Recall that a generic configuration g|0 is uniquely specified by the set of star operators acting on the reference configuration |0 , which we chose to be the ferromagnetic state with all the σ spins pointing up, modulo the action of the product of all the star operators (which is equal to the identity). Thus, there is a 1to-2 mapping between G = {g} and the configuration space Θ = {θ} of an Ising model with degrees of freedom θ s living on the sites s of the square lattice, where for example θ s = −1 (+1) means that the corresponding star operator is (not) acting in the associated g. Since each σ spin can be flipped only by its two neighboring θ spins, then σ i ≡ θ s θ s ′ , where i labels the bond between the two neighboring sites s, s ′ , and g∈G e β\nNotice that, using the above mapping, the GS wavefunction of our model, Eq. (15), can be rewritten as\nwhere Z = θ∈Θ e β P s,s ′ θ s θ s ′ . Thus, all equal-time correlation functions that can be expressed in terms of the θ s variables are the same as those of a 2D classical Ising model with reduced nearest-neighbor coupling J/T = β, implying that the critical point of the latter β c = (1/2) ln( √ 2 + 1) ≃ 0.4406868 corresponds precisely to the critical point of our quantum system. Notice also that the magnetization in the original σ spin language is indeed the nearest-neighbor spinspin correlation (i.e., the energy) in the θ spin language,\nTherefore, one concludes that the magnetization m(β) is continuous across the transition at β c but there is a singularity in its first derivative\nas the Ising model heat capacity C Ising diverges logarithmically at β c . In the following, we will show how such continuous phase transition is accompanied by a sudden, discontinuous vanishing of the topological entropy of the system.\nThe case of a configuration of the form hgk, with h ∈ G A and k ∈ G B requires a few additional steps. First of all, notice that the composition of any two elements g,g ∈ G is represented in the θ spin language by the site-by-site product of the two configurations corresponding to g andg, respectively:\nMoreover, using similar arguments as in Ref. 15, the star operators of a bipartite system (A, B) can be divided into bulk star operators, i.e., those acting solely on subsystem A or subsystem B, and boundary star operators acting simultaneously on A and B spins. The boundary star operators can be further subdivided into different sets according to the different boundaries around each connected component of A and B (for a total of m A + m B − 1 boundaries, m A and m B being the number of connected components of A and B, respectively).\nLet us define a collective operation as the product of all the star operators in one of these sets. That is, the product of all the stars around a connected boundary of the bipartition (A, B). Clearly, the number of such collective operations is given by the number of sets, m A + m B − 1.\nOne can show that the subgroup G A G B ⊂ G, to which the product hk belongs, can be generated by all the bulk star operators together with all but one of the collective operators (all but one, independently of which one is chosen to be left out, is required because the product of all boundary star operators is equivalent to the product of all bulk star operators). For example, G A G B is generated by the bulk star operators alone in bipartitions 2 and 3 in Fig. 2, while the product of all boundary star operators along one of the two boundaries must be included to generate G A G B for bipartitions 1 and 4.\nLet\nfor 'bulk', to be the set of Ising spin configurations on the sites of the square lattice where all θ b s corresponding to boundary sites s are fixed to equal +1. Let us also define Θ δ = {θ δ }, 'δ' for 'boundary', to be the set of Ising configurations where θ δ s = +1 for all bulk star operators, θ δ s = +1 for all boundary star operators belonging to one chosen boundary, and θ δ s = ±1 for the remaining boundary star operators, so long as all θ δ s spins belonging to the same boundary have the same sign. Notice that Θ δ = {1 1} for bipartitions 2 and 3 in Fig. 2, where 1 1 is the configuration with all the spins θ δ s = +1. One can finally show that there is a one-to-one correspondence between the elements of G A G B and the Ising configurations in\nwhere θ b θ δ represents the site-by-site product of the two configurations (i.e., (θ b θ δ ) s = θ b s θ δ s ). Therefore,\nand in particular,\nwhere we used the fact that if s is in the bulk then θ δ s = +1, if s belongs to a boundary then θ b s = +1, and if both s and s ′ belong to the same boundary then θ δ s θ δ s ′ = +1. Let us focus on the bipartitions of interest to compute the topological entropy (23). First of all, in the limit r, R → ∞ there are no nearest-neighboring stars s and s ′ belonging to two different boundaries. Therefore, the term (45) vanishes identically. For bipartitions 2 and 3, Θ δ = {1 1} and\nThe r.h.s. of the above equation can be interpreted as the partition function of an Ising model with nearest-neighbor interactions, where only the bulk degrees of freedom are allowed to flip starting from a given configuration θ(g). Clearly such partition function is invariant upon changing the initial configuration as long as the new one is in the same ergodic sector.\nFor example, one can equivalently choosẽ θ(g) = +1 if s belongs to the bulk θ s (g) if s belongs to the boundary, (48) and the expression above simplifies to\nHere Z ∂ 2,3 (g) represents the partition function of an Ising model with nearest-neighbor interaction of reduced strength J/T = β, and with fixed spins along the boundary of bipartitions 2 and 3, respectively. The values of the spins at the boundary are determined by g.\nFor bipartitions 1 and 4, Θ δ = {1 1, f }, where the configuration f has all the spins equal to +1 except for those belonging to the chosen boundary, say boundary 2 in Fig. 2, which are equal to −1. In this case\nHere Z ∂ 1,4 (g) are the analog of Z ∂ 2,3 (g) for bipartitions 1 and 4, respectively, while Z ∂, twisted 1,4 (g) differ from the former by the fact that all the (fixed) spins belonging to boundary 2 in bipartitions 1 and 4 respectively have been flipped. In other words, Z ∂ 1,4 (g) represents the partition function of an Ising model with nearest-neighbor interaction of reduced strength J/T = β, and with fixed spins along the bound-ary of bipartitions 1 and 4, respectively. The partition functions Z ∂, twisted 1,4 (g) differ in that the spins along one of the two boundaries have been flipped with respect to their values in Z ∂ 1,4 (g). Again, the values of the spins at the boundary are determined by g.\nIn this notation, the topological entropy of the system can be written as\nwhere the sum over g acts as a weighed average of the logarithmic term over all possible values of the spins at the boundary. Notice that in Eq. (51) the partitions with two boundaries, and hence with non-trivial topology, are those that appear with two contributions (bipartitions 1 and 4), corresponding to some relative boundary conditions (BCs) and their twisted counterparts. These contributions, as we show below in detail, are responsible for the non-vanishing topological entropy.\nIn the topological phase, the two partition functions for the twisted and untwisted BCs contribute equally, and in the nontopological phase, one partition function is exponentially suppressed when compared to the other, in the thermodynamic limit. Therefore, there is an extra entropy contribution in one of the phases depending on whether the boundaries of topologically non-trivial bipartitions are twisted or not relative to one another. From Eq. (51), the behavior of the topological entropy can be qualitatively argued as follows. Deep in the disordered phase, where the correlations are short ranged, the choice of boundary conditions is likely to affect the partition function of the system only with exponentially small corrections. Thus, we can expect to have\nOn the other hand, deep in the (ferromagnetically) ordered phase the partition function of a system with twisted boundary conditions is exponentially suppressed with respect to the one without the twist. Thus,\n. This leads to S topo = 0.\nIn the following two sections we will show with rigorous arguments that the behavior of the topological entropy across the transition is strongly first order, with a sudden jump from S topo = 2 to S topo = 0."},{"title":"The disordered phase (β < β c )","content":"In the limit of small β, namely above the ordering transition, one can compute S topo via the high-temperature expansion of the Ising model with fixed spins at the boundary.\nLet us rewrite,\nwhere s i , s ′ i are the sites at the ends of bond i, N is the total number of bonds on the lattice, δ is the length of the boundary in number of θ spins, c = cosh β, t = tanh β. The sum over G runs over all possible graphs on the bonds of the square lattice, composed entirely of closed loops and open strings connecting two boundary spins. The product s∈EG θ s (g) encompasses all the boundary spins that appear as end points (the set E G ) of open strings in G. Finally, ℓ(G) is the total length of the closed loops and open strings in G.\nAnalogously for Z ∂ 3 (g). The case of Z ∂ 1 (g) and Z ∂ 4 (g) differs from Z ∂ 2 (g) and Z ∂ 3 (g) in that there are now two types of open strings: those going from one boundary to itself and those connecting the two boundaries (see Fig. 2).\nNext, let us compare the product Z ∂ 2 (g)Z ∂ 3 (g) with the product Z ∂ 1 (g)Z ∂ 4 (g). Notice that bipartitions 2 and 3 have precisely the same total combined boundary as bipartitions 1 and 4. In order for a graph to appear in one of the two products and not in the other, it needs to comprise loops or strings that are able to tell the difference between to two possible origins (2 + 3 vs. 1 + 4) of the total combined boundary.\nExamples of such open strings are shown in Fig. 3. One can show that these telltale strings, and the analogous closed loops, cannot be arbitrarily short, and their length is bounded from below by R − 2r. As a consequence, the corresponding graphs are exponentially suppressed at least as t R−2r , and in the limit r, R → ∞ with R − 2r → ∞, implicit in the definition of the topological entropy, one obtains\nSimilar considerations apply when comparing the product Z ∂ 1 (g)Z ∂ 4 (g) with products of the kind Z ∂, twisted 1 (g)Z ∂ 4 (g). In this case, the boundaries involved are exaclty the same, and the relevant telltale elements of the graph are open strings connecting one of the two components of the boundary with the other. Such strings are in fact the only elements that are sensitive to the twisted boundary conditions. Clearly the length of these strings is bounded from below by R − 2r, and\nexponentially fast, at least as t R−2r , with (R − 2r) → ∞.\nOf course, our reasoning is correct up to the point where the high temperature expansion breaks down, and entropic contributions balance the exponential suppression. Said differently, this is the case when the correlation length in the Ising model goes to infinity, and the large r, R limit does not guarantee that the ratios of products of partition functions above tend to one.\nGiven Eqs. (53,54), we can finally use Eq. (51) to obtain the topological entropy of the system throughout the disordered phase β < β c ≃ 0.4406868,\n= log 2 4 = 2."},{"title":"The Landau-Ginzburg ordered phase (β > β c )","content":"What happens below this transition? Rather than attempting a low-temperature expansion, it is convenient to use the duality relations derived by A. Bugrij and V. Shadura in Ref. 16 for the inhomogeneous, finite-size Ising model. In particular, they obtained the duality relations for a square lattice Ising model wrapped around a cylinder of finite length, with fixed, free and mixed boundary conditions. Following the usual convention, let us labelβ the coupling constant of the dual Ising model (defined on the plaquettes of the original lattice), which is related to β by the duality relation sinh β sinhβ = 1. Let us also indicate with Z(δ, δ ′ ) andZ(δ, δ ′ ) the partition functions of the system on the finite cylinder and its dual, with δ, δ ′ specifying the boundary conditions, namely δ, δ ′ = , × for free and fixed boundary spins, respectively. With this notation in mind, the results by Bugrij and Shadura -to the purpose of the present paper -can be summarized by 16\nwhere Z twisted (×, ×) differs from Z(×, ×) by the fact that the fixed boundary spins at one end of the cylinder have been flipped. Notice that the proportionality coefficient K is the same in all the equations, and that Z( , ×) = Z(×, ) and Z(×, ) =Z( , ×).\nLet us then consider Z ∂ 2 in Eq. (49). Thanks to the nearestneighbor character of the interaction between θ spins, subsystem A interacts only with itself and with the boundary δ 2 , and so does subsystem B. Thus, one can factorize the two subsystems and obtain (cfr. Eq.(49))\nSimilar arguments apply to bipartition 3,\nand with a few more steps, to bipartitions 1 and 4 as well (cfr. Eq.(50)),\nwhere 1B 1 and 1B 2 refer to the two connected components of subsystem B in bipartition 1, i.e., the component inside boundary 1 and the component outside boundary 2, and analogously for 4A 1 and 4A 2 . In order to apply Eqs. (55-57) to the present case, some further considerations on the bipartitions in Fig. 2 are needed. Recall that, although S topo is indeed a quantity of order one, we expressed it in Eq. (51) in terms of a ratio of extensive partition functions Z ∂ i (g). Thus, any sub-extensive correction to these partition functions (i.e., O(2 N α ), with α < 1, N being the number of degrees of freedom in the system) will only amount to an exponentially small correction to S topo , that vanishes in the thermodynamic limit. In this context, the partition function Z ∂ 1A (g) (see Fig. 4) is 'equivalent', in the thermodynamic limit, to the partition function of an Ising model on an infinite cylinder with fixed boundaries at the edges (bound- ary 1 and 2, respectively). Similarly, the partition function Z ∂ 1B 1 (g) can be regarded as that of an Ising model on an infinite cylinder with fixed boundary conditions on one edge (boundary 1) and open boundary conditions on a suitably introduced boundary γ 1 . Finally, the same approach can be used for Z ∂ 1B 2 (g), with fixed boundary conditions on one edge (boundary 2) and open boundary conditions on another suitably introduced boundary γ 4 . Qualitatively, this is illustrated in Fig. 4a, where the spins on boundary 1 and boundary 2 are fixed and those belonging to γ 1 and γ 4 are free. Notice that the mapping onto infinite cylinders requires the distance between any of the γ i boundaries introduced in Fig. 4 and any of the original boundaries in Fig. 2 to diverge with system size. Similar arguments apply to bipartitions 2, 3 and 4. This leads to a correspondence between our factorized partition functions and those used in Ref. 16, namely\nThe results in Ref. 16 can then be applied to our systems and lead to the following equations:\nFor convenience of notation, let us define the dual partition functions for the whole system in the different bipartitions\nFinally, we have all the ingredients to evaluate the topological entropy for β > β c . Let us first rewrite Eq. (51) using Eqs. (58-61) instead of Eqs. (49-50):\nwhere the last two terms inside the square brackets come from the exponential factors in Eqs. (58-61). Using the duality relations (62), we can identify\nFor β > β c , the dual Ising models are in the disordered phase and one can perform a high-temperature expansion to calculate the ratio (Z ∂\nwhere i labels the bonds of the square lattice, and i ∈ δ means that the bond i connects two sites s and s ′ belonging to δ. This contribution can be shown to vanish identically since the set of boundary σ spins in bipartitions 1 and 4 is identical to the set of boundary spins in bipartitions 2 and 3 (see Fig. (2)).\nIn the end we find that\nidentically in the ordered phase β > β c ."},{"title":"B. Beyond 1-body potentials","content":"As we already mentioned, the calculations carried out in Sec. III A for the specific model presented in this paper can be straightforwardly extended to the case of any factorizable wavefunction. All one needs to do is identify a proper set of local generators (i.e., acting on the σ spins contained within a disc of finite radius) for the group G, and the equivalent of the collective boundary flip operators. The rest of the derivation follows essentially unchanged, in the limit r, R → ∞.\nWhat happens if we attempt to generalize our approach further and we consider non-factorizable wavefunctions? For simplicity, take once again the Kitaev-like GS wavefunction in Eq. (15), but replace the argument of the exponential β i σ z i (g, α)/2 with some generic function −βE g /2. As we can see immediately from Eq. (24), our approach to compute the topological entropy can no longer be used from the very first stage. On the other hand, it is tempting to conjecture that, so long as E g is short ranged (i.e., it can be written as the sum of terms involving σ spins within a disc of finite radius on the lattice), the error that one makes by neglecting the terms involving spins across the boundary of a bipartition (E ∂ g ) does not give topological contributions to S topo . Under this assumption, one can then set E ∂ g = 0 and use the approximate equality E g ≃ E A g A + E B g B to re-establish the factorability needed to carry on with the calculations. The result obtained for S topo in Eq. (51), which employed this approximation, nonetheless shows no explicit dependence on it in the final expression, and one could then reinstate the full E g at that stage. If the conjecture above is correct, the formula in Eq. (51) gives the exact topological entropy for a generic GS wavefunction that satis-fies (i) the positive amplitude condition, (ii) the group condition for G, 17 and (iii) the locality (i.e., short ranged) condition on E g .\nA simple example where this conjecture can be applied rather straightforwardly is the case where Eqs. (14) and (15) are replaced by\n(For a discussion of the general construction scheme of such type of Hamiltonians, see Ref. 13.) Here |GS is the GS wavefunction of H in the topological sector where i∈pσ z i = +1. The notation ij , j / ∈ s stands for j nearest-neighbor of i but not adjacent to the same vertex s. Without loss of generality, we consider the range β ∈ (0, ∞), where the new term in the Hamiltonian favors ferromagnetic order in the σ spins.\nLet us then introduce the same description in terms of the θ spins, as in the previous section. Given that the product of two nearest-neighboring σ spins translates into the product of two next-nearest-neighboring θ spins, we obtain\n(notice the additional factor of 2 in the exponent due to the fact that the same product θ s θ s ′ corresponds to two distinct products σ z i σ z j ), and\nwhere Z = θ∈Θ e 2β P s,s ′ θ s θ s ′ . The latter is the partition function of a square-lattice Ising model with sole nextnearest-neighbor interactions, which factorizes into the product of two decoupled Ising models with nearest-neighbor interactions (namely corresponding to the θ spins on each of the two sublattices). In this case, all equal-time correlators in the GS of the quantum system can be written in terms of classical correlators of two decoupled Ising models. As before, we expect the system to undergo a phase transition when the two Ising models become critical at β c = (1/4) ln( √ 2 + 1) ≃ 0.2203434. However, contrarily to the previous case, the new model undergoes a spontaneous symmetry-breaking (Landau-Ginzburg) phase transition! This is best seen by mapping the system onto a quantum eight-vertex model, as discussed below. The local order parameter that captures the transition is the magnetization of the σ spins, whose expectation value can be written as an ensemble average of the product of two neighboring θ spins, i.e., belonging to two decoupled Ising models. Clearly such average vanishes identically in the hightemperature phase β < β c , while it becomes finite in the ordered phase β > β c . Notice that this local order parameter that acquires an expectation value does so in the nontopologically ordered phase, as expected from the fact that no local order parameter exists that resolves the topological phase.\nWhat is the fate of the topological entropy across this Landau-Ginzburg phase transition? According to the conjecture above, we can directly substitute i σ z i → ij σ z i σ z j into Eq. (51) and compute S topo . This amounts to replacing the boundary Ising partition functions in the argument of the logarithm with the partition functions of two decoupled boundary Ising models. As a result, all calculations carried out in the previous section remain essentially unchanged and one arrives to the identical result that S topo = 2 throughout the high-temperature phase, and vanishes otherwise. This scenario is in agreement with previous results on a quantum version of the eight-vertex model by Ardonne et al., 18 whose GS is a generalization of the one in our model. Consider indeed the wavefunction in Eq. (69). Given the nature of the group G, the four spins belonging to any plaquette of the square lattice can assume only eight distinct configurations ( i∈pσ z i = +1), illustrated in Fig. 5. Such configurations map naturally onto the vertices of an eight-vertex model upon replacing each positive spin with an arrow along the corresponding bond of the dual lattice, pointing, say, from sublattice A to sublattice B, and vice versa for the negative spins (as shown in Fig. 5). Given that ij · · · ≡ s ij ∈s · · · , the amplitudes in the GS wavefunction (69) factorize into products of vertex fugacities exp(β ij ∈s σ z i σ z j /2). In the notation of Fig. 5, the vertex fugacities assume the values a = b = 1, c = e −2β and d = e 2β . The GS spatial properties of our model are therefore captured by a classical eight-vertex model with the appropriate fugacities 18 (but see Ref. 13 for a general discussion of such quantum-to-classical correspondence), and one can then use Baxter's exact solution 19 to obtain the phase diagram as well as the scaling exponents at the critical point. All this is discussed in detail in Ref. 18: the model undergoes a second-order, Z 2 -symmetry-breaking phase transition when d 2 = c 2 + 2 (i.e., β c = (1/4) ln( √ 2 + 1)), separating a topologically ordered liquid phase from a Landau-Ginzburg ordered phase. The local order parameter across the transition is indeed the magnetization in the original σ spins."},{"title":"IV. CONCLUSIONS","content":"In this paper we studied a topological quantum phase transition in a microscopic model that can be examined analytically. For this system, an extension of the toric code, the ground state wavefunction can be written exactly as a function of the parameter β that drives the system across the quantum phase transition. We computed the topological entropy for this system as a function of β, and showed that it remains at a constant non-zero value throughout the topologically ordered phase (β < β c ≃ 0.4406868). Immediately after the quantum phase transition at β c , the topological entropy drops to zero and remains so in the non-topologically ordered phase (β > β c ).\nThe GS wavefunction of our quantum system has positive amplitudes in the basis of choice. This property allows us to relate many quantities that are relevant in characterizing the (2+1 D) quantum system to those of a simple (2D, not 3D) classical Ising model at an inverse (classical) temperature equal to the value of the coupling constant β that drives the quantum system through the T = 0 phase transition. For example, the magnetization of the quantum system equals the energy E Ising (β) of the classical Ising model. While the magnetization is continuous and non-vanishing across the quantum phase transition (much as the energy E Ising (β) is across the classical Ising transition), its derivative with respect to β diverges logarithmically at β c (much as the Ising model heat capacity C Ising diverges logarithmically at β c ).\nDespite the relation to the 2D classical Ising model, the quantum phase transition does not have a local order parameter that vanishes on one side and not on the other. Of course one expects that no local order parameter can characterize the topological phase, but in this particular example, there is no order parameter that characterizes the non-topologically ordered phase either. One can indeed identify from the mapping to the Ising model a parameter that orders in the nontopological phase; however, this variable is non-local in the physical spin variables used to define the local Hamiltonian. Specifically, the order parameter is, in the language used in this paper, the expectation value of the θ s variables defined on the sites of the square lattice, such that σ ss ′ = θ s θ ′ s , for nearest neighboring sites s, s ′ . While σ ss ′ is obviously local in terms of the θ s , θ ′ s , the inversion needed to write the θ's in terms of the σ's is non-local. Hence, θ s may detect the transition into the non-topological phase, but as it is non-local it is not an order parameter in the usual sense. That there is no order parameter for the non-topologically ordered phase is not generic (see the example in Sec. III B), as perhaps the most obvious exit from a topological phase is by escaping into a locally ordered phase due to spontaneous symmetry breaking. Hence, the main example studied in this paper is particularly interesting in that one has no local order parameter in either phases.\nRecently, P. Zanardi et al. proposed a new approach to study quantum phase transitions through the behavior of a metric tensor in parameter space, derived from a fidelity-based notion of distance between states. 20 In this approach, one does not require any a priori knowledge of an order parameter to detect a phase transition (one may argue that the fidelity encompasses correlations of both local and non-local operators). Thus, these ideas may be particularly useful to detect topological quantum phase transitions (see Ref. 9).\nWe end with a speculative note: topological quantum phase transitions should share the feature that, even in the case when there is no local order parameter in either the topological or the non-topological phase, there should be, generically, detectable singularities in high enough derivatives of local observables with respect to the coupling that takes the system across the transition (as in the case study presented here). 21 After all, what would be a phase transition without singularities in any physical observable?\nUpon completion of this work, we became aware of similar work being pursued from a gauge theoretical perspective by S. Papanikolaou, K. Raman and E. Fradkin on the quantum eight-vertex model, 22 to which we refer the reader for a complementary approach."},{"title":"FIG. 1 :","content":""},{"title":"FIG. 3 :","content":""},{"title":"FIG. 4 :","content":""},{"title":".","content":""},{"title":"FIG. 5 :","content":""}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"Some strongly correlated quantum many body systems display a type of order which cannot be characterized by any local order parameter. Instead, such order is topological in nature 1 , with the fractional quantum Hall systems being the primary example so far: they are liquid states that exhibit exotic properties such as a ground state (GS) degeneracy that cannot be lifted by any local perturbations 2,3 and fractionalized degrees of freedom 4 . Topologically ordered states are also interesting in that their robustness against local perturbations might be of use for decoherence-free quantum computation 5 .\\nAn example of an exactly solvable lattice spin model that is topologically ordered was presented by Kitaev in Ref. 5, and the system was argued to be robust against small perturbations that tend to order the systemà la Landau-Ginzburg and take it away from its topological phase. The departure from the topologically ordered phase should occur through a quantum phase transition. Such quantum phase transition, however, cannot be entirely captured by ordinary methods based on local Landau-Ginzburg order parameters, and new methods need to be devised in order to investigate the fate of topological order across the phase transition. These novel methods must be based on the fundamental properties of topologically ordered phases, such as the GS degeneracy in presence of a gap, and the presence of a non-vanishing topological entropy.\\nRecent efforts to understand quantum phase transitions in topologically ordered states include a mean-field approach for these exotic states 6 , and analytical and numerical studies 7,8,9 of the Kitaev model in the presence of a field. The numerical analysis presented in Refs. 8,9 leads to the conclusion that topological order survives unchanged up to the second order phase transition at β c = 0.32847(6) (in the notation of Eq. (18)), while the system is no longer topologically ordered for β > β c . (Here β stands for the coupling constant that drives the T = 0 quantum phase transition -the notation will become apparent shortly, and is chosen because of a close relation to a classical model.)\\nIn this paper, we investigate analytically a quantum phase transition out of a topological phase. We show that the recently defined topological entropy 10,11 works well as an \\\"order parameter\\\" across the transition. We study the transition using a model -see Eq. (5) -that is shown to behave much like the Kitaev model in a magnetic field for small values of the field.\\nThe advantage of this model is that the ground state can be obtained exactly, from which we can then compute the topological entropy explicitly, and show that it remains constant in the topologically ordered phase (β < β c ≃ 0.4406868), dropping abruptly to zero in the non-topologically-ordered phase (β > β c ), despite the continuous (second order) character of the transition.\\nWe find that in this model, even though one cannot identify a local order parameter that vanishes in one phase and not in the other, one can show that the (continuous) local magnetization has a singularity at the critical point. In the model, we show that the magnetization equals the energy E Ising (β) of a 2D classical Ising model with N spins evaluated at an inverse (classical) temperature equal to the value of the coupling constant β that drives the system through the T = 0 phase transition:\\nFrom this relation, it becomes evident that the magnetization m(β), although continuous and non-vanishing across the transition at β c (much as the energy of the classical Ising model across the classical transition), has a singularity in its first derivative, since ∂m ∂β\\nand the Ising model heat capacity C Ising diverges logarithmically at β c . Hence, although there is no local order parameter that can detect either the topological or the non-topological phase in this system, one can expose the topological quantum phase transition through the singularity in the derivative of a local quantity. This is contrasted, for example, with the case discussed in Sec. III B, where a similar topological transition is accompanied by a simultaneous Z 2 symmetry breaking phase transition. In that case, the same transition is captured both by the non-local topological entropy, and by a local (Landau-Ginzburg) order parameter.\"\n },\n {\n \"title\": \"II. THE MODEL\",\n \"content\": \"The model that we consider is a deformation of the Kitaev model 5 , and it is defined on a square lattice with spin-1/2 de-grees of freedom living on the bonds, as shown in Fig. 1. The pure Kitaev model is written in terms of star and plaquette operators (see Fig. 1). Star operators are defined as\\nwhere i labels the four spins on the bonds departing from some vertex s of the square lattice. Plaquette operators are defined as\\nwhere i labels the four spins on the bonds around some plaquette p of the square lattice. The Hamiltonian we consider in this paper is\\nwhere λ 0,1 > 0 and β is a parameter that we use to tune the system across a topological quantum phase transition. Notice that for β = 0 the Hamiltonian Eq. (5) Take G to be the (Abelian) group of all spin flip operations obtained as products of star type operators. Notice that g 2 = 1 1 for any element g of the group G. By acting with elements of G on a given reference configuration i |σ z i one generates a manifold of states, which however does not encompass the whole basis. For example, the action of a star operator A s cannot change the sign of the product of σ z 's around any square plaquette in the lattice 5,12 (see Fig. 1). Therefore, there is a non-trivial (and non-unique) minimal set {|Ψ α } of reference configurations that generates the full σ z -basis under the action of the group G. (In particular, one of the elements in this set is the reference configuration |0 that is fully magnetized in the z-direction, say σ z i = 1, ∀i.) Consider then the family of Hamiltonians\\nfor some real-valued parameter β. The ground state of any such Hamiltonian can be obtained exactly and it can be written in the form\\nwhere α labels the different block-diagonal sectors corresponding to the states in the minimal set {|Ψ α }; σ z i (g, α) is the z-component of the spin at site i in state g |Ψ α ; and the coefficients ψ α can be chosen at will, subject to the normalization condition α |ψ α | 2 = 1. Although the choice of minimal set is non-unique, one can show that Eq. (8) is independent of such choice, modulo an irrelevant permutation of the α indices. Within each block-diagonal sector, the GS of Eq. (6) is unique. Instead of proving directly that (8) is the GS of (6), it is more convenient to notice that the family of Hamiltonians in Eq. (6) is a particular choice of Stochastic Matrix Form decompositions of quantum Hamiltonians that exhibit precisely Eq. (8) as their GS. 13 One can verify this by showing that each of the operators\\nbetween square brackets in Eq. (6) annihilates the inner sum in |GS 1 , independently of the index α. That the GS energy is zero follows because\\nFor larger values of β, the many-body terms in Eq. (19) are no longer negligible and the equivalence is lost, although the form of the GS (15) suggests that the system gets deeper and deeper into the magnetized phase -as one would expect upon increasing the strength of the magnetic field in the Kitaev model. As we discuss in Section III A, our model undergoes a second-order phase transition at β c = (1/2) ln( √ 2 + 1) ≃ 0.4406868, where it displays a dimensionality reduction that places the transition in a different universality class than the one studied in Refs. 7,8,9. One can use the decomposition\\nto estimate the limit of validity of Eq. (18) to be given by the condition\\nThis corresponds to a ratio between the coupling to the magnetic field and the coupling to the cooperative transverse field ( s i∈sσ\\nThe detailed numerical analysis presented in Refs. 8,9 lead the authors to conclude that topological order survives up to the second order phase transition at finite β c (in the notation of Eq. (18)), while the system is no longer topologically ordered for β > β c . In the following, we investigate this phase and the relative phase transition using the exact ground state of our model (5) to compute the topological entropy 10,11 across the transition. Using an exact derivation from the microscopic degrees of freedom, we show that the topological entropy is able to detect a transition from a topologically ordered phase (β < β c ) to a non-topologically-ordered phase (β > β c ). Indeed, it remains constant at its known β → 0 value up to the transition and drops abruptly to zero afterwards, despite the continuous character of the transition.\"\n },\n {\n \"title\": \"III. THE TOPOLOGICAL ENTROPY OF FACTORIZABLE (LOCAL) WAVEFUNCTIONS\",\n \"content\": \"Using the definition in Refs. 10,11, the topological entropy can be obtained as a linear combination of Von Neumann entanglement entropies S VN of different bipartitions of the system into subsystems A and B:\\nwhere ρ A = Tr B (ρ) is the reduced density matrix obtained from the full density matrix ρ by tracing out the degrees of freedom of subsystem B, and the last equality holds whenever the full density matrix ρ is a pure-state density matrix.\\nThe different bipartitions are aimed at removing all the extensive (boundary) contributions to uncover the sole topological contribution. A particular choice of the four bipartitions 10 is illustrated in Fig. 2 as:\\nIn order to compute the topological entropy as a function of the parameter β, let us first notice that the Hamiltonian (14) constructed above, with GS given by (15), belongs to a class of Hamiltonians whose GS wavefunctions\\n. For this type of Hamiltonians, one can compute the entanglement entropy as follows.\\nConsider a given bipartition (A, B) of the system. The reduced density matrix ρ A = Tr B (ρ), obtained by tracing over all degrees of freedom in B, is given by 12\\nis the subgroup of transformations acting solely on A (B) and leaving B (A) invariant:\\nNotice that we used the group property to rewrite a generic element of G as gg, ∃!g ∈ G, as well as the additive property of E g . We can then compute the trace of the n-th power of the reduced density matrix Tr (ρ A ) n and use the identity\\nto obtain the Von Neumann entropy S\\nwhere we used the fact that the inner products in Eq. (26) impose\\nfor i = 1, . . . , n, with the identification n + 1 ≡ 1, and therefore\\nThe condition in Eq. (27) can be satisfied if and only if\\nThus, the summation over all g ′ i of the inner products in Eq. (26) yields a constraint over the allowed values of g 1 , . . . , g n ∈ G:\\n(30) In particular, the last line can be recast as\\n(31) The physical meaning of these conditions will become clear in the next section, for the specific case of the system considered in this paper, although the form of Eq. (31) already suggests that they require all the elements g i to agree at the boundary of the bipartition (A, B).\\nWe can finally use Eq. (31) to simplify Eq. (26)\\nand obtain, via Eq. (25),\\nwhereF g is the partial free energy given by all the configurations that can be obtained from g via products of spin flip operators that act solely on subsystem A or subsystem B (i.e., having the same 'boundary' as g), and . . . denotes the ensemble average over g ∈ G with weight e −βE g . Notice that our result in Eq. (33) is the lattice equivalent of the Von Neumann entropy obtained by Fradkin and Moore in Ref. 14 for continuous systems. Alternatively, Eq. (33) can be interpreted as the entropy of mixing (or configurational entropy) of the allowed bipartition boundaries in G. This can be made more transparent by introducing the quotient group Q = G/(G A G B ), and by rewriting Eq. (33) as\\nwhere we used the fact that the term in square brackets is independent of h and k, and where we introduced the notation\\nfor the probability of boundary q to appear in G, for a given inverse temperature β and energy E g . In order to proceed further, let us focus for simplicity on the specific GS of our system (15). The generic case of a wavefunction with factorizable amplitudes can be inferred with minor modifications.\\nA. The case of 1-body potentials All of the above results apply straightforwardly to the GS in Eq. (15). Notice that (i) the topological entropy in each block-diagonal sector of the pure Kitaev model is the same, 12 and (ii) it is reasonable to make the working assumption that the relevant sector for the transition to the fully magnetized state |0 is the one that contains this state, and that is therefore obtained upon applying the group G to |0 . For the purpose of computing the topological entropy, one can thus replace Eq. (15) by\\nand obtain\\n. In order to simplify Eq. (37) with the purpose of computing the topological entropy of the system (23), it is convenient to do the following change of variables. Recall that a generic configuration g|0 is uniquely specified by the set of star operators acting on the reference configuration |0 , which we chose to be the ferromagnetic state with all the σ spins pointing up, modulo the action of the product of all the star operators (which is equal to the identity). Thus, there is a 1to-2 mapping between G = {g} and the configuration space Θ = {θ} of an Ising model with degrees of freedom θ s living on the sites s of the square lattice, where for example θ s = −1 (+1) means that the corresponding star operator is (not) acting in the associated g. Since each σ spin can be flipped only by its two neighboring θ spins, then σ i ≡ θ s θ s ′ , where i labels the bond between the two neighboring sites s, s ′ , and g∈G e β\\nNotice that, using the above mapping, the GS wavefunction of our model, Eq. (15), can be rewritten as\\nwhere Z = θ∈Θ e β P s,s ′ θ s θ s ′ . Thus, all equal-time correlation functions that can be expressed in terms of the θ s variables are the same as those of a 2D classical Ising model with reduced nearest-neighbor coupling J/T = β, implying that the critical point of the latter β c = (1/2) ln( √ 2 + 1) ≃ 0.4406868 corresponds precisely to the critical point of our quantum system. Notice also that the magnetization in the original σ spin language is indeed the nearest-neighbor spinspin correlation (i.e., the energy) in the θ spin language,\\nTherefore, one concludes that the magnetization m(β) is continuous across the transition at β c but there is a singularity in its first derivative\\nas the Ising model heat capacity C Ising diverges logarithmically at β c . In the following, we will show how such continuous phase transition is accompanied by a sudden, discontinuous vanishing of the topological entropy of the system.\\nThe case of a configuration of the form hgk, with h ∈ G A and k ∈ G B requires a few additional steps. First of all, notice that the composition of any two elements g,g ∈ G is represented in the θ spin language by the site-by-site product of the two configurations corresponding to g andg, respectively:\\nMoreover, using similar arguments as in Ref. 15, the star operators of a bipartite system (A, B) can be divided into bulk star operators, i.e., those acting solely on subsystem A or subsystem B, and boundary star operators acting simultaneously on A and B spins. The boundary star operators can be further subdivided into different sets according to the different boundaries around each connected component of A and B (for a total of m A + m B − 1 boundaries, m A and m B being the number of connected components of A and B, respectively).\\nLet us define a collective operation as the product of all the star operators in one of these sets. That is, the product of all the stars around a connected boundary of the bipartition (A, B). Clearly, the number of such collective operations is given by the number of sets, m A + m B − 1.\\nOne can show that the subgroup G A G B ⊂ G, to which the product hk belongs, can be generated by all the bulk star operators together with all but one of the collective operators (all but one, independently of which one is chosen to be left out, is required because the product of all boundary star operators is equivalent to the product of all bulk star operators). For example, G A G B is generated by the bulk star operators alone in bipartitions 2 and 3 in Fig. 2, while the product of all boundary star operators along one of the two boundaries must be included to generate G A G B for bipartitions 1 and 4.\\nLet\\nfor 'bulk', to be the set of Ising spin configurations on the sites of the square lattice where all θ b s corresponding to boundary sites s are fixed to equal +1. Let us also define Θ δ = {θ δ }, 'δ' for 'boundary', to be the set of Ising configurations where θ δ s = +1 for all bulk star operators, θ δ s = +1 for all boundary star operators belonging to one chosen boundary, and θ δ s = ±1 for the remaining boundary star operators, so long as all θ δ s spins belonging to the same boundary have the same sign. Notice that Θ δ = {1 1} for bipartitions 2 and 3 in Fig. 2, where 1 1 is the configuration with all the spins θ δ s = +1. One can finally show that there is a one-to-one correspondence between the elements of G A G B and the Ising configurations in\\nwhere θ b θ δ represents the site-by-site product of the two configurations (i.e., (θ b θ δ ) s = θ b s θ δ s ). Therefore,\\nand in particular,\\nwhere we used the fact that if s is in the bulk then θ δ s = +1, if s belongs to a boundary then θ b s = +1, and if both s and s ′ belong to the same boundary then θ δ s θ δ s ′ = +1. Let us focus on the bipartitions of interest to compute the topological entropy (23). First of all, in the limit r, R → ∞ there are no nearest-neighboring stars s and s ′ belonging to two different boundaries. Therefore, the term (45) vanishes identically. For bipartitions 2 and 3, Θ δ = {1 1} and\\nThe r.h.s. of the above equation can be interpreted as the partition function of an Ising model with nearest-neighbor interactions, where only the bulk degrees of freedom are allowed to flip starting from a given configuration θ(g). Clearly such partition function is invariant upon changing the initial configuration as long as the new one is in the same ergodic sector.\\nFor example, one can equivalently choosẽ θ(g) = +1 if s belongs to the bulk θ s (g) if s belongs to the boundary, (48) and the expression above simplifies to\\nHere Z ∂ 2,3 (g) represents the partition function of an Ising model with nearest-neighbor interaction of reduced strength J/T = β, and with fixed spins along the boundary of bipartitions 2 and 3, respectively. The values of the spins at the boundary are determined by g.\\nFor bipartitions 1 and 4, Θ δ = {1 1, f }, where the configuration f has all the spins equal to +1 except for those belonging to the chosen boundary, say boundary 2 in Fig. 2, which are equal to −1. In this case\\nHere Z ∂ 1,4 (g) are the analog of Z ∂ 2,3 (g) for bipartitions 1 and 4, respectively, while Z ∂, twisted 1,4 (g) differ from the former by the fact that all the (fixed) spins belonging to boundary 2 in bipartitions 1 and 4 respectively have been flipped. In other words, Z ∂ 1,4 (g) represents the partition function of an Ising model with nearest-neighbor interaction of reduced strength J/T = β, and with fixed spins along the bound-ary of bipartitions 1 and 4, respectively. The partition functions Z ∂, twisted 1,4 (g) differ in that the spins along one of the two boundaries have been flipped with respect to their values in Z ∂ 1,4 (g). Again, the values of the spins at the boundary are determined by g.\\nIn this notation, the topological entropy of the system can be written as\\nwhere the sum over g acts as a weighed average of the logarithmic term over all possible values of the spins at the boundary. Notice that in Eq. (51) the partitions with two boundaries, and hence with non-trivial topology, are those that appear with two contributions (bipartitions 1 and 4), corresponding to some relative boundary conditions (BCs) and their twisted counterparts. These contributions, as we show below in detail, are responsible for the non-vanishing topological entropy.\\nIn the topological phase, the two partition functions for the twisted and untwisted BCs contribute equally, and in the nontopological phase, one partition function is exponentially suppressed when compared to the other, in the thermodynamic limit. Therefore, there is an extra entropy contribution in one of the phases depending on whether the boundaries of topologically non-trivial bipartitions are twisted or not relative to one another. From Eq. (51), the behavior of the topological entropy can be qualitatively argued as follows. Deep in the disordered phase, where the correlations are short ranged, the choice of boundary conditions is likely to affect the partition function of the system only with exponentially small corrections. Thus, we can expect to have\\nOn the other hand, deep in the (ferromagnetically) ordered phase the partition function of a system with twisted boundary conditions is exponentially suppressed with respect to the one without the twist. Thus,\\n. This leads to S topo = 0.\\nIn the following two sections we will show with rigorous arguments that the behavior of the topological entropy across the transition is strongly first order, with a sudden jump from S topo = 2 to S topo = 0.\"\n },\n {\n \"title\": \"The disordered phase (β < β c )\",\n \"content\": \"In the limit of small β, namely above the ordering transition, one can compute S topo via the high-temperature expansion of the Ising model with fixed spins at the boundary.\\nLet us rewrite,\\nwhere s i , s ′ i are the sites at the ends of bond i, N is the total number of bonds on the lattice, δ is the length of the boundary in number of θ spins, c = cosh β, t = tanh β. The sum over G runs over all possible graphs on the bonds of the square lattice, composed entirely of closed loops and open strings connecting two boundary spins. The product s∈EG θ s (g) encompasses all the boundary spins that appear as end points (the set E G ) of open strings in G. Finally, ℓ(G) is the total length of the closed loops and open strings in G.\\nAnalogously for Z ∂ 3 (g). The case of Z ∂ 1 (g) and Z ∂ 4 (g) differs from Z ∂ 2 (g) and Z ∂ 3 (g) in that there are now two types of open strings: those going from one boundary to itself and those connecting the two boundaries (see Fig. 2).\\nNext, let us compare the product Z ∂ 2 (g)Z ∂ 3 (g) with the product Z ∂ 1 (g)Z ∂ 4 (g). Notice that bipartitions 2 and 3 have precisely the same total combined boundary as bipartitions 1 and 4. In order for a graph to appear in one of the two products and not in the other, it needs to comprise loops or strings that are able to tell the difference between to two possible origins (2 + 3 vs. 1 + 4) of the total combined boundary.\\nExamples of such open strings are shown in Fig. 3. One can show that these telltale strings, and the analogous closed loops, cannot be arbitrarily short, and their length is bounded from below by R − 2r. As a consequence, the corresponding graphs are exponentially suppressed at least as t R−2r , and in the limit r, R → ∞ with R − 2r → ∞, implicit in the definition of the topological entropy, one obtains\\nSimilar considerations apply when comparing the product Z ∂ 1 (g)Z ∂ 4 (g) with products of the kind Z ∂, twisted 1 (g)Z ∂ 4 (g). In this case, the boundaries involved are exaclty the same, and the relevant telltale elements of the graph are open strings connecting one of the two components of the boundary with the other. Such strings are in fact the only elements that are sensitive to the twisted boundary conditions. Clearly the length of these strings is bounded from below by R − 2r, and\\nexponentially fast, at least as t R−2r , with (R − 2r) → ∞.\\nOf course, our reasoning is correct up to the point where the high temperature expansion breaks down, and entropic contributions balance the exponential suppression. Said differently, this is the case when the correlation length in the Ising model goes to infinity, and the large r, R limit does not guarantee that the ratios of products of partition functions above tend to one.\\nGiven Eqs. (53,54), we can finally use Eq. (51) to obtain the topological entropy of the system throughout the disordered phase β < β c ≃ 0.4406868,\\n= log 2 4 = 2.\"\n },\n {\n \"title\": \"The Landau-Ginzburg ordered phase (β > β c )\",\n \"content\": \"What happens below this transition? Rather than attempting a low-temperature expansion, it is convenient to use the duality relations derived by A. Bugrij and V. Shadura in Ref. 16 for the inhomogeneous, finite-size Ising model. In particular, they obtained the duality relations for a square lattice Ising model wrapped around a cylinder of finite length, with fixed, free and mixed boundary conditions. Following the usual convention, let us labelβ the coupling constant of the dual Ising model (defined on the plaquettes of the original lattice), which is related to β by the duality relation sinh β sinhβ = 1. Let us also indicate with Z(δ, δ ′ ) andZ(δ, δ ′ ) the partition functions of the system on the finite cylinder and its dual, with δ, δ ′ specifying the boundary conditions, namely δ, δ ′ = , × for free and fixed boundary spins, respectively. With this notation in mind, the results by Bugrij and Shadura -to the purpose of the present paper -can be summarized by 16\\nwhere Z twisted (×, ×) differs from Z(×, ×) by the fact that the fixed boundary spins at one end of the cylinder have been flipped. Notice that the proportionality coefficient K is the same in all the equations, and that Z( , ×) = Z(×, ) and Z(×, ) =Z( , ×).\\nLet us then consider Z ∂ 2 in Eq. (49). Thanks to the nearestneighbor character of the interaction between θ spins, subsystem A interacts only with itself and with the boundary δ 2 , and so does subsystem B. Thus, one can factorize the two subsystems and obtain (cfr. Eq.(49))\\nSimilar arguments apply to bipartition 3,\\nand with a few more steps, to bipartitions 1 and 4 as well (cfr. Eq.(50)),\\nwhere 1B 1 and 1B 2 refer to the two connected components of subsystem B in bipartition 1, i.e., the component inside boundary 1 and the component outside boundary 2, and analogously for 4A 1 and 4A 2 . In order to apply Eqs. (55-57) to the present case, some further considerations on the bipartitions in Fig. 2 are needed. Recall that, although S topo is indeed a quantity of order one, we expressed it in Eq. (51) in terms of a ratio of extensive partition functions Z ∂ i (g). Thus, any sub-extensive correction to these partition functions (i.e., O(2 N α ), with α < 1, N being the number of degrees of freedom in the system) will only amount to an exponentially small correction to S topo , that vanishes in the thermodynamic limit. In this context, the partition function Z ∂ 1A (g) (see Fig. 4) is 'equivalent', in the thermodynamic limit, to the partition function of an Ising model on an infinite cylinder with fixed boundaries at the edges (bound- ary 1 and 2, respectively). Similarly, the partition function Z ∂ 1B 1 (g) can be regarded as that of an Ising model on an infinite cylinder with fixed boundary conditions on one edge (boundary 1) and open boundary conditions on a suitably introduced boundary γ 1 . Finally, the same approach can be used for Z ∂ 1B 2 (g), with fixed boundary conditions on one edge (boundary 2) and open boundary conditions on another suitably introduced boundary γ 4 . Qualitatively, this is illustrated in Fig. 4a, where the spins on boundary 1 and boundary 2 are fixed and those belonging to γ 1 and γ 4 are free. Notice that the mapping onto infinite cylinders requires the distance between any of the γ i boundaries introduced in Fig. 4 and any of the original boundaries in Fig. 2 to diverge with system size. Similar arguments apply to bipartitions 2, 3 and 4. This leads to a correspondence between our factorized partition functions and those used in Ref. 16, namely\\nThe results in Ref. 16 can then be applied to our systems and lead to the following equations:\\nFor convenience of notation, let us define the dual partition functions for the whole system in the different bipartitions\\nFinally, we have all the ingredients to evaluate the topological entropy for β > β c . Let us first rewrite Eq. (51) using Eqs. (58-61) instead of Eqs. (49-50):\\nwhere the last two terms inside the square brackets come from the exponential factors in Eqs. (58-61). Using the duality relations (62), we can identify\\nFor β > β c , the dual Ising models are in the disordered phase and one can perform a high-temperature expansion to calculate the ratio (Z ∂\\nwhere i labels the bonds of the square lattice, and i ∈ δ means that the bond i connects two sites s and s ′ belonging to δ. This contribution can be shown to vanish identically since the set of boundary σ spins in bipartitions 1 and 4 is identical to the set of boundary spins in bipartitions 2 and 3 (see Fig. (2)).\\nIn the end we find that\\nidentically in the ordered phase β > β c .\"\n },\n {\n \"title\": \"B. Beyond 1-body potentials\",\n \"content\": \"As we already mentioned, the calculations carried out in Sec. III A for the specific model presented in this paper can be straightforwardly extended to the case of any factorizable wavefunction. All one needs to do is identify a proper set of local generators (i.e., acting on the σ spins contained within a disc of finite radius) for the group G, and the equivalent of the collective boundary flip operators. The rest of the derivation follows essentially unchanged, in the limit r, R → ∞.\\nWhat happens if we attempt to generalize our approach further and we consider non-factorizable wavefunctions? For simplicity, take once again the Kitaev-like GS wavefunction in Eq. (15), but replace the argument of the exponential β i σ z i (g, α)/2 with some generic function −βE g /2. As we can see immediately from Eq. (24), our approach to compute the topological entropy can no longer be used from the very first stage. On the other hand, it is tempting to conjecture that, so long as E g is short ranged (i.e., it can be written as the sum of terms involving σ spins within a disc of finite radius on the lattice), the error that one makes by neglecting the terms involving spins across the boundary of a bipartition (E ∂ g ) does not give topological contributions to S topo . Under this assumption, one can then set E ∂ g = 0 and use the approximate equality E g ≃ E A g A + E B g B to re-establish the factorability needed to carry on with the calculations. The result obtained for S topo in Eq. (51), which employed this approximation, nonetheless shows no explicit dependence on it in the final expression, and one could then reinstate the full E g at that stage. If the conjecture above is correct, the formula in Eq. (51) gives the exact topological entropy for a generic GS wavefunction that satis-fies (i) the positive amplitude condition, (ii) the group condition for G, 17 and (iii) the locality (i.e., short ranged) condition on E g .\\nA simple example where this conjecture can be applied rather straightforwardly is the case where Eqs. (14) and (15) are replaced by\\n(For a discussion of the general construction scheme of such type of Hamiltonians, see Ref. 13.) Here |GS is the GS wavefunction of H in the topological sector where i∈pσ z i = +1. The notation ij , j / ∈ s stands for j nearest-neighbor of i but not adjacent to the same vertex s. Without loss of generality, we consider the range β ∈ (0, ∞), where the new term in the Hamiltonian favors ferromagnetic order in the σ spins.\\nLet us then introduce the same description in terms of the θ spins, as in the previous section. Given that the product of two nearest-neighboring σ spins translates into the product of two next-nearest-neighboring θ spins, we obtain\\n(notice the additional factor of 2 in the exponent due to the fact that the same product θ s θ s ′ corresponds to two distinct products σ z i σ z j ), and\\nwhere Z = θ∈Θ e 2β P s,s ′ θ s θ s ′ . The latter is the partition function of a square-lattice Ising model with sole nextnearest-neighbor interactions, which factorizes into the product of two decoupled Ising models with nearest-neighbor interactions (namely corresponding to the θ spins on each of the two sublattices). In this case, all equal-time correlators in the GS of the quantum system can be written in terms of classical correlators of two decoupled Ising models. As before, we expect the system to undergo a phase transition when the two Ising models become critical at β c = (1/4) ln( √ 2 + 1) ≃ 0.2203434. However, contrarily to the previous case, the new model undergoes a spontaneous symmetry-breaking (Landau-Ginzburg) phase transition! This is best seen by mapping the system onto a quantum eight-vertex model, as discussed below. The local order parameter that captures the transition is the magnetization of the σ spins, whose expectation value can be written as an ensemble average of the product of two neighboring θ spins, i.e., belonging to two decoupled Ising models. Clearly such average vanishes identically in the hightemperature phase β < β c , while it becomes finite in the ordered phase β > β c . Notice that this local order parameter that acquires an expectation value does so in the nontopologically ordered phase, as expected from the fact that no local order parameter exists that resolves the topological phase.\\nWhat is the fate of the topological entropy across this Landau-Ginzburg phase transition? According to the conjecture above, we can directly substitute i σ z i → ij σ z i σ z j into Eq. (51) and compute S topo . This amounts to replacing the boundary Ising partition functions in the argument of the logarithm with the partition functions of two decoupled boundary Ising models. As a result, all calculations carried out in the previous section remain essentially unchanged and one arrives to the identical result that S topo = 2 throughout the high-temperature phase, and vanishes otherwise. This scenario is in agreement with previous results on a quantum version of the eight-vertex model by Ardonne et al., 18 whose GS is a generalization of the one in our model. Consider indeed the wavefunction in Eq. (69). Given the nature of the group G, the four spins belonging to any plaquette of the square lattice can assume only eight distinct configurations ( i∈pσ z i = +1), illustrated in Fig. 5. Such configurations map naturally onto the vertices of an eight-vertex model upon replacing each positive spin with an arrow along the corresponding bond of the dual lattice, pointing, say, from sublattice A to sublattice B, and vice versa for the negative spins (as shown in Fig. 5). Given that ij · · · ≡ s ij ∈s · · · , the amplitudes in the GS wavefunction (69) factorize into products of vertex fugacities exp(β ij ∈s σ z i σ z j /2). In the notation of Fig. 5, the vertex fugacities assume the values a = b = 1, c = e −2β and d = e 2β . The GS spatial properties of our model are therefore captured by a classical eight-vertex model with the appropriate fugacities 18 (but see Ref. 13 for a general discussion of such quantum-to-classical correspondence), and one can then use Baxter's exact solution 19 to obtain the phase diagram as well as the scaling exponents at the critical point. All this is discussed in detail in Ref. 18: the model undergoes a second-order, Z 2 -symmetry-breaking phase transition when d 2 = c 2 + 2 (i.e., β c = (1/4) ln( √ 2 + 1)), separating a topologically ordered liquid phase from a Landau-Ginzburg ordered phase. The local order parameter across the transition is indeed the magnetization in the original σ spins.\"\n },\n {\n \"title\": \"IV. CONCLUSIONS\",\n \"content\": \"In this paper we studied a topological quantum phase transition in a microscopic model that can be examined analytically. For this system, an extension of the toric code, the ground state wavefunction can be written exactly as a function of the parameter β that drives the system across the quantum phase transition. We computed the topological entropy for this system as a function of β, and showed that it remains at a constant non-zero value throughout the topologically ordered phase (β < β c ≃ 0.4406868). Immediately after the quantum phase transition at β c , the topological entropy drops to zero and remains so in the non-topologically ordered phase (β > β c ).\\nThe GS wavefunction of our quantum system has positive amplitudes in the basis of choice. This property allows us to relate many quantities that are relevant in characterizing the (2+1 D) quantum system to those of a simple (2D, not 3D) classical Ising model at an inverse (classical) temperature equal to the value of the coupling constant β that drives the quantum system through the T = 0 phase transition. For example, the magnetization of the quantum system equals the energy E Ising (β) of the classical Ising model. While the magnetization is continuous and non-vanishing across the quantum phase transition (much as the energy E Ising (β) is across the classical Ising transition), its derivative with respect to β diverges logarithmically at β c (much as the Ising model heat capacity C Ising diverges logarithmically at β c ).\\nDespite the relation to the 2D classical Ising model, the quantum phase transition does not have a local order parameter that vanishes on one side and not on the other. Of course one expects that no local order parameter can characterize the topological phase, but in this particular example, there is no order parameter that characterizes the non-topologically ordered phase either. One can indeed identify from the mapping to the Ising model a parameter that orders in the nontopological phase; however, this variable is non-local in the physical spin variables used to define the local Hamiltonian. Specifically, the order parameter is, in the language used in this paper, the expectation value of the θ s variables defined on the sites of the square lattice, such that σ ss ′ = θ s θ ′ s , for nearest neighboring sites s, s ′ . While σ ss ′ is obviously local in terms of the θ s , θ ′ s , the inversion needed to write the θ's in terms of the σ's is non-local. Hence, θ s may detect the transition into the non-topological phase, but as it is non-local it is not an order parameter in the usual sense. That there is no order parameter for the non-topologically ordered phase is not generic (see the example in Sec. III B), as perhaps the most obvious exit from a topological phase is by escaping into a locally ordered phase due to spontaneous symmetry breaking. Hence, the main example studied in this paper is particularly interesting in that one has no local order parameter in either phases.\\nRecently, P. Zanardi et al. proposed a new approach to study quantum phase transitions through the behavior of a metric tensor in parameter space, derived from a fidelity-based notion of distance between states. 20 In this approach, one does not require any a priori knowledge of an order parameter to detect a phase transition (one may argue that the fidelity encompasses correlations of both local and non-local operators). Thus, these ideas may be particularly useful to detect topological quantum phase transitions (see Ref. 9).\\nWe end with a speculative note: topological quantum phase transitions should share the feature that, even in the case when there is no local order parameter in either the topological or the non-topological phase, there should be, generically, detectable singularities in high enough derivatives of local observables with respect to the coupling that takes the system across the transition (as in the case study presented here). 21 After all, what would be a phase transition without singularities in any physical observable?\\nUpon completion of this work, we became aware of similar work being pursued from a gauge theoretical perspective by S. Papanikolaou, K. Raman and E. Fradkin on the quantum eight-vertex model, 22 to which we refer the reader for a complementary approach.\"\n },\n {\n \"title\": \"FIG. 1 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 3 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 4 :\",\n \"content\": \"\"\n },\n {\n \"title\": \".\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 5 :\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"(Color online) -Examples of star and plaquette operators, centered at a lattice site s (blue open circle) and at a dual lattice site p (red open square), respectively. The solid black dots represent the spin-1/2 degrees of freedom living on the bonds of the lattice, and the dashed lines connect the spins involved in the definition of each of the above operators.","content":"FIG. 1 :\n1(Color online) -Examples of star and plaquette operators, centered at a lattice site s (blue open circle) and at a dual lattice site p (red open square), respectively. The solid black dots represent the spin-1/2 degrees of freedom living on the bonds of the lattice, and the dashed lines connect the spins involved in the definition of each of the above operators."},{"id":"","caption":"(Color online)-Examples of open strings that appear in the expansion of Z ∂ 2 (g)Z ∂ 3 (g) but are not present in the expansion of Z ∂ 1 (g)Z ∂ 4 (g) (Top), and vice versa (Bottom). Top panel: The thick yellow and blue lines correspond to the boundaries in bipartitions 2 and 3 respectively. Boundaries belonging to both are shown in a thick dashed yellow-blue pattern. The strings in question are symbolically represented by thin black lines. Bottom panel: Same color coding, with yellow corresponding to bipartition 4 and blue corresponding to biparition 1. Notice that strings appearing in one expansion and not in the other must connect boundaries of the same solid color, and therefore cannot be shorter than R − 2r.","content":"FIG. 3 :\n3(Color online)-Examples of open strings that appear in the expansion of Z ∂ 2 (g)Z ∂ 3 (g) but are not present in the expansion of Z ∂ 1 (g)Z ∂ 4 (g) (Top), and vice versa (Bottom). Top panel: The thick yellow and blue lines correspond to the boundaries in bipartitions 2 and 3 respectively. Boundaries belonging to both are shown in a thick dashed yellow-blue pattern. The strings in question are symbolically represented by thin black lines. Bottom panel: Same color coding, with yellow corresponding to bipartition 4 and blue corresponding to biparition 1. Notice that strings appearing in one expansion and not in the other must connect boundaries of the same solid color, and therefore cannot be shorter than R − 2r."},{"id":"","caption":"(Color online) -Illustration of the four bipartitions used to compute the topological entropy in Ref.10, with a possible choice for the additional boundaries needed to map each partition onto an infinite cylinder in the thermodynamic limit.","content":"FIG. 4 :\n4(Color online) -Illustration of the four bipartitions used to compute the topological entropy in Ref.10, with a possible choice for the additional boundaries needed to map each partition onto an infinite cylinder in the thermodynamic limit."},{"id":"","caption":"Using the same loop description as for the original system, with t replaced byt = tanh(β), one can show that (terms in Eq. (67) can be dealt with more promptly by reverting back to the original σ spin degrees of freedom, s,s ′ ∈δ 1 +δ 4","content":".\nUsing the same loop description as for the original system, with t replaced byt = tanh(β), one can show that (terms in Eq. (67) can be dealt with more promptly by reverting back to the original σ spin degrees of freedom, s,s ′ ∈δ 1 +δ 4"},{"id":"","caption":"Illustration of the eight allowed spin configurations around a plaquette of the square lattice, in any basis state g|0 , g ∈ G. These can in turn be mapped onto the configurations of an eight vertex model by replacing each positive spin with an arrow along the corresponding bond of the dual lattice (dashed lines), pointing, say, from sublattice A to sublattice B (vice versa for the negative spins). Clearly, the corresponding eight vertex configurations differ depending on the location of the plaquette in the dual lattice: sublattice A (middle), or sublattice B (bottom). The letters a, b, c, d correspond to the usual labeling of the vertex fugacities in the eight vertex model.","content":"FIG. 5 :\n5Illustration of the eight allowed spin configurations around a plaquette of the square lattice, in any basis state g|0 , g ∈ G. These can in turn be mapped onto the configurations of an eight vertex model by replacing each positive spin with an arrow along the corresponding bond of the dual lattice (dashed lines), pointing, say, from sublattice A to sublattice B (vice versa for the negative spins). Clearly, the corresponding eight vertex configurations differ depending on the location of the plaquette in the dual lattice: sublattice A (middle), or sublattice B (bottom). The letters a, b, c, d correspond to the usual labeling of the vertex fugacities in the eight vertex model."},{"id":"","caption":"is simply the Kitaev Hamiltonian H Kitaev in Ref.5, up to a trivial overall constant shift of the energy.The exact ground state wavefunction of this Hamiltonian can be obtained by deconstructing H into two pieces, H = λ 0 H 0 + λ 1 H 1 , as follows.","content":"is simply the Kitaev Hamiltonian H Kitaev in Ref.5, up to a trivial overall constant shift of the energy.The exact ground state wavefunction of this Hamiltonian can be obtained by deconstructing H into two pieces, H = λ 0 H 0 + λ 1 H 1 , as follows."},{"id":"","caption":", and the topological entropy is then defined FIG. 2: Illustration of the four bipartitions used to compute the topological entropy in Ref. 10.","content":", and the topological entropy is then defined FIG. 2: Illustration of the four bipartitions used to compute the topological entropy in Ref. 10.00000000000000 \n00000000000000 00000000000000 \n00000000000000 00000000000000 \n00000000000000 \n00000000000000 00000000000000 \n00000000000000 00000000000000 \n00000000000000 \n00000000000000 00000000000000 \n00000000000000 \n\n11111111111111 \n11111111111111 11111111111111 \n11111111111111 11111111111111 \n11111111111111 \n11111111111111 11111111111111 \n11111111111111 11111111111111 \n11111111111111 \n11111111111111 11111111111111 \n11111111111111 \n\n00000000000000 \n00000000000000 00000000000000 \n00000000000000 00000000000000 \n00000000000000 \n00000000000000 00000000000000 \n00000000000000 00000000000000 \n00000000000000 \n00000000000000 00000000000000 \n00000000000000 \n\n11111111111111 \n11111111111111 11111111111111 \n11111111111111 11111111111111 \n11111111111111 \n11111111111111 11111111111111 \n11111111111111 11111111111111 \n11111111111111 \n11111111111111 11111111111111 \n11111111111111 \n\n00000 \n00000 00000 \n00000 00000 \n00000 \n00000 00000 \n00000 00000 \n00000 \n00000 00000 \n00000 \n\n11111 \n11111 11111 \n11111 11111 \n11111 \n11111 11111 \n11111 11111 \n11111 \n11111 11111 \n11111 \n\n000000000000000 \n000000000000000 000000000000000 \n000000000000000 000000000000000 \n000000000000000 \n000000000000000 000000000000000 \n000000000000000 000000000000000 \n000000000000000 \n000000000000000 000000000000000 \n000000000000000 \n\n111111111111111 \n111111111111111 111111111111111 \n111111111111111 111111111111111 \n111111111111111 \n111111111111111 111111111111111 \n111111111111111 111111111111111 \n111111111111111 \n111111111111111 111111111111111 \n111111111111111 \n\n0000 \n0000 0000 \n0000 0000 \n0000 \n0000 0000 \n0000 0000 \n0000 \n0000 0000 \n0000 \n\n1111 \n1111 1111 \n1111 1111 \n1111 \n1111 1111 \n1111 1111 \n1111 \n1111 1111 \n1111 \n\n(4) \n(2) \n(3) \n(1) \n\nr \n\nR \n\nboundary 1 \nboundary 1 \n\nboundary 2 \nboundary 2"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"(Color online) -Examples of star and plaquette operators, centered at a lattice site s (blue open circle) and at a dual lattice site p (red open square), respectively. The solid black dots represent the spin-1/2 degrees of freedom living on the bonds of the lattice, and the dashed lines connect the spins involved in the definition of each of the above operators.\",\n \"content\": \"FIG. 1 :\\n1(Color online) -Examples of star and plaquette operators, centered at a lattice site s (blue open circle) and at a dual lattice site p (red open square), respectively. The solid black dots represent the spin-1/2 degrees of freedom living on the bonds of the lattice, and the dashed lines connect the spins involved in the definition of each of the above operators.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(Color online)-Examples of open strings that appear in the expansion of Z ∂ 2 (g)Z ∂ 3 (g) but are not present in the expansion of Z ∂ 1 (g)Z ∂ 4 (g) (Top), and vice versa (Bottom). Top panel: The thick yellow and blue lines correspond to the boundaries in bipartitions 2 and 3 respectively. Boundaries belonging to both are shown in a thick dashed yellow-blue pattern. The strings in question are symbolically represented by thin black lines. Bottom panel: Same color coding, with yellow corresponding to bipartition 4 and blue corresponding to biparition 1. Notice that strings appearing in one expansion and not in the other must connect boundaries of the same solid color, and therefore cannot be shorter than R − 2r.\",\n \"content\": \"FIG. 3 :\\n3(Color online)-Examples of open strings that appear in the expansion of Z ∂ 2 (g)Z ∂ 3 (g) but are not present in the expansion of Z ∂ 1 (g)Z ∂ 4 (g) (Top), and vice versa (Bottom). Top panel: The thick yellow and blue lines correspond to the boundaries in bipartitions 2 and 3 respectively. Boundaries belonging to both are shown in a thick dashed yellow-blue pattern. The strings in question are symbolically represented by thin black lines. Bottom panel: Same color coding, with yellow corresponding to bipartition 4 and blue corresponding to biparition 1. Notice that strings appearing in one expansion and not in the other must connect boundaries of the same solid color, and therefore cannot be shorter than R − 2r.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(Color online) -Illustration of the four bipartitions used to compute the topological entropy in Ref.10, with a possible choice for the additional boundaries needed to map each partition onto an infinite cylinder in the thermodynamic limit.\",\n \"content\": \"FIG. 4 :\\n4(Color online) -Illustration of the four bipartitions used to compute the topological entropy in Ref.10, with a possible choice for the additional boundaries needed to map each partition onto an infinite cylinder in the thermodynamic limit.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Using the same loop description as for the original system, with t replaced byt = tanh(β), one can show that (terms in Eq. (67) can be dealt with more promptly by reverting back to the original σ spin degrees of freedom, s,s ′ ∈δ 1 +δ 4\",\n \"content\": \".\\nUsing the same loop description as for the original system, with t replaced byt = tanh(β), one can show that (terms in Eq. (67) can be dealt with more promptly by reverting back to the original σ spin degrees of freedom, s,s ′ ∈δ 1 +δ 4\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Illustration of the eight allowed spin configurations around a plaquette of the square lattice, in any basis state g|0 , g ∈ G. These can in turn be mapped onto the configurations of an eight vertex model by replacing each positive spin with an arrow along the corresponding bond of the dual lattice (dashed lines), pointing, say, from sublattice A to sublattice B (vice versa for the negative spins). Clearly, the corresponding eight vertex configurations differ depending on the location of the plaquette in the dual lattice: sublattice A (middle), or sublattice B (bottom). The letters a, b, c, d correspond to the usual labeling of the vertex fugacities in the eight vertex model.\",\n \"content\": \"FIG. 5 :\\n5Illustration of the eight allowed spin configurations around a plaquette of the square lattice, in any basis state g|0 , g ∈ G. These can in turn be mapped onto the configurations of an eight vertex model by replacing each positive spin with an arrow along the corresponding bond of the dual lattice (dashed lines), pointing, say, from sublattice A to sublattice B (vice versa for the negative spins). Clearly, the corresponding eight vertex configurations differ depending on the location of the plaquette in the dual lattice: sublattice A (middle), or sublattice B (bottom). The letters a, b, c, d correspond to the usual labeling of the vertex fugacities in the eight vertex model.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"is simply the Kitaev Hamiltonian H Kitaev in Ref.5, up to a trivial overall constant shift of the energy.The exact ground state wavefunction of this Hamiltonian can be obtained by deconstructing H into two pieces, H = λ 0 H 0 + λ 1 H 1 , as follows.\",\n \"content\": \"is simply the Kitaev Hamiltonian H Kitaev in Ref.5, up to a trivial overall constant shift of the energy.The exact ground state wavefunction of this Hamiltonian can be obtained by deconstructing H into two pieces, H = λ 0 H 0 + λ 1 H 1 , as follows.\"\n },\n {\n \"id\": \"\",\n \"caption\": \", and the topological entropy is then defined FIG. 2: Illustration of the four bipartitions used to compute the topological entropy in Ref. 10.\",\n \"content\": \", and the topological entropy is then defined FIG. 2: Illustration of the four bipartitions used to compute the topological entropy in Ref. 10.00000000000000 \\n00000000000000 00000000000000 \\n00000000000000 00000000000000 \\n00000000000000 \\n00000000000000 00000000000000 \\n00000000000000 00000000000000 \\n00000000000000 \\n00000000000000 00000000000000 \\n00000000000000 \\n\\n11111111111111 \\n11111111111111 11111111111111 \\n11111111111111 11111111111111 \\n11111111111111 \\n11111111111111 11111111111111 \\n11111111111111 11111111111111 \\n11111111111111 \\n11111111111111 11111111111111 \\n11111111111111 \\n\\n00000000000000 \\n00000000000000 00000000000000 \\n00000000000000 00000000000000 \\n00000000000000 \\n00000000000000 00000000000000 \\n00000000000000 00000000000000 \\n00000000000000 \\n00000000000000 00000000000000 \\n00000000000000 \\n\\n11111111111111 \\n11111111111111 11111111111111 \\n11111111111111 11111111111111 \\n11111111111111 \\n11111111111111 11111111111111 \\n11111111111111 11111111111111 \\n11111111111111 \\n11111111111111 11111111111111 \\n11111111111111 \\n\\n00000 \\n00000 00000 \\n00000 00000 \\n00000 \\n00000 00000 \\n00000 00000 \\n00000 \\n00000 00000 \\n00000 \\n\\n11111 \\n11111 11111 \\n11111 11111 \\n11111 \\n11111 11111 \\n11111 11111 \\n11111 \\n11111 11111 \\n11111 \\n\\n000000000000000 \\n000000000000000 000000000000000 \\n000000000000000 000000000000000 \\n000000000000000 \\n000000000000000 000000000000000 \\n000000000000000 000000000000000 \\n000000000000000 \\n000000000000000 000000000000000 \\n000000000000000 \\n\\n111111111111111 \\n111111111111111 111111111111111 \\n111111111111111 111111111111111 \\n111111111111111 \\n111111111111111 111111111111111 \\n111111111111111 111111111111111 \\n111111111111111 \\n111111111111111 111111111111111 \\n111111111111111 \\n\\n0000 \\n0000 0000 \\n0000 0000 \\n0000 \\n0000 0000 \\n0000 0000 \\n0000 \\n0000 0000 \\n0000 \\n\\n1111 \\n1111 1111 \\n1111 1111 \\n1111 \\n1111 1111 \\n1111 1111 \\n1111 \\n1111 1111 \\n1111 \\n\\n(4) \\n(2) \\n(3) \\n(1) \\n\\nr \\n\\nR \\n\\nboundary 1 \\nboundary 1 \\n\\nboundary 2 \\nboundary 2\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"DMR-0403997 (CC and CC), and by EPSRC Grant No. DMR-0403997 (CC and CC), and by EPSRC Grant No.","doi":""},{"text":"Castelnovo). C. Castelnovo would like to acknowledge the I2CAM NSF Grant DMR No. 0645461 for travel support, during which part of this work was carried out. GR/R83712/01GR/R83712/01 (C. Castelnovo). C. Castelnovo would like to acknowledge the I2CAM NSF Grant DMR No. 0645461 for travel support, during which part of this work was carried out.","doi":"GR/R83712/01"},{"text":". X.-G Wen, Int. J. Mod. Phys. B. 4239X.-G. Wen, Int. J. Mod. Phys. B 4, 239 (1990);","doi":""},{"text":". Adv. in Phys. 44405Adv. in Phys. 44, 405 (1995);","doi":""},{"text":". Phys. Rev. B. 65165113Phys. Rev. B 65, 165113 (2002).","doi":""},{"text":". F D M Haldane, E H Rezayi, Phys. Rev. B. 312529F. D. M. Haldane, and E. H. Rezayi, Phys. Rev. B 31, 2529 (1985).","doi":""},{"text":". X.-G Wen, Q Niu, Phys. Rev. B. 419377X.-G. Wen, and Q. Niu, Phys. Rev. B 41, 9377 (1990).","doi":""},{"text":". D Arovas, J R Schrieffer, F Wilczek, Phys. Rev. Lett. 53722D. Arovas, J. R. Schrieffer, and F. Wilczek, Phys. Rev. Lett. 53, 722 (1984).","doi":""},{"text":". A Y Kitaev, Ann. Phys. (N.Y.). 303A. Y. Kitaev, Ann. Phys. (N.Y.) 303, 2 (2003).","doi":""},{"text":". M Levin, X.-G Wen, Phys. Rev. B. 7575116M. Levin, and X.-G. Wen, Phys. Rev. B 75, 075116 (2007).","doi":""},{"text":". A Hamma, D A Lidar, arXiv:quant-ph/0607145v4Phys. Rev. Lett.). A. Hamma and D. A. Lidar, (accepted for publication on Phys. Rev. Lett.), arXiv:quant-ph/0607145v4 (2006).","doi":"arXiv:quant-ph/0607145v4"},{"text":". S Trebst, P Werner, M Troyer, K Shtengel, C Nayak, Phys. Rev. Lett. 9870602S. Trebst, P. Werner, M. Troyer, K. Shtengel, and C. Nayak, Phys. Rev. Lett. 98, 070602 (2007).","doi":""},{"text":". A Hamma, W Zhang, S Haas, D A Lidar, arXiv:0705.0026v1A. Hamma, W. Zhang, S. Haas, and D. A. Lidar, arXiv:0705.0026v1 (2007).","doi":"arXiv:0705.0026v1"},{"text":". M Levin, X.-G Wen, Phys. Rev. Lett. 96110405M. Levin, and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006).","doi":""},{"text":". A Y Kitaev, J Preskill, Phys. Rev. Lett. 96110404A. Y. Kitaev, and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006).","doi":""},{"text":". A Hamma, R Ionicioiu, P Zanardi, Phys. Rev. A. 7122315A. Hamma, R. Ionicioiu, and P. Zanardi, Phys. Rev. A 71, 022315 (2005).","doi":""},{"text":". C Castelnovo, C Chamon, C Mudry, P Pujol, Ann. Phys. (N.Y.). 318316C. Castelnovo, C. Chamon, C. Mudry, and P. Pujol, Ann. Phys. (N.Y.) 318, 316 (2005).","doi":""},{"text":". E Fradkin, J E Moore, Phys. Rev. Lett. 9750404E. Fradkin, and J. E. Moore, Phys. Rev. Lett. 97, 050404 (2006).","doi":""},{"text":". C Castelnovo, C Chamon, Phys. Rev. B. 76174416C. Castelnovo and C. Chamon, Phys. Rev. B 76, 174416 (2007).","doi":""},{"text":". A Bugrij, V N Shadura, arXiv:hep-th/9601106JETP Lett. 63384A. Bugrij, and V. N. Shadura, JETP Lett. 63, 384 (1996); arXiv:hep-th/9601106.","doi":"arXiv:hep-th/9601106"},{"text":"This condition is actually immaterial, and can be avoided using an approach similar to the one by Furukawa and Misguich. Phys. Rev. B. 75214407This condition is actually immaterial, and can be avoided us- ing an approach similar to the one by Furukawa and Misguich [Phys. Rev. B 75, 214407 (2007)].","doi":""},{"text":". E Ardonne, P Fendley, E Fradkin, Ann. Phys. (N.Y.). 310493E. Ardonne, P. Fendley, and E. Fradkin, Ann. Phys. (N.Y.) 310, 493 (2004).","doi":""},{"text":"R Baxter, Exactly Solved Models in Statistical Mechanics. LondonAcademic PressR. Baxter, Exactly Solved Models in Statistical Mechanics (Aca- demic Press, London, 1982).","doi":""},{"text":"and references therein. The behavior of fidelity across a topological phase transition in the Kitaev model in presence of a magnetic field was studied numerically in Ref. P Zanardi, L Venuti, P Giorda, Phys. Rev. 7662318P. Zanardi, L. Campos Venuti, P. Giorda, Phys. Rev. A76, 062318 (2007), and references therein. The behavior of fidelity across a topological phase transition in the Kitaev model in presence of a magnetic field was studied numerically in Ref. 9.","doi":""},{"text":"the lack of a local order parameter but display of local non-analyticities -was recently argued by Z. Nussinov and G. Ortiz to characterize as well a temperature-driven topological phase. arXiv:cond-mat/0702377A similar behavior -i.e.. transition in the Kitaev modelA similar behavior -i.e., the lack of a local order param- eter but display of local non-analyticities -was recently ar- gued by Z. Nussinov and G. Ortiz to characterize as well a temperature-driven topological phase transition in the Kitaev model [arXiv:cond-mat/0702377].","doi":"arXiv:cond-mat/0702377"},{"text":". S Papanikolaou, K Raman, E Fradkin, Phys. Rev. B. 76224421S. Papanikolaou, K. Raman, and E. Fradkin, Phys. Rev. B 76, 224421 (2007).","doi":""}],"string":"[\n {\n \"text\": \"DMR-0403997 (CC and CC), and by EPSRC Grant No. DMR-0403997 (CC and CC), and by EPSRC Grant No.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Castelnovo). C. Castelnovo would like to acknowledge the I2CAM NSF Grant DMR No. 0645461 for travel support, during which part of this work was carried out. GR/R83712/01GR/R83712/01 (C. Castelnovo). C. Castelnovo would like to acknowledge the I2CAM NSF Grant DMR No. 0645461 for travel support, during which part of this work was carried out.\",\n \"doi\": \"GR/R83712/01\"\n },\n {\n \"text\": \". X.-G Wen, Int. J. Mod. Phys. B. 4239X.-G. Wen, Int. J. Mod. Phys. B 4, 239 (1990);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Adv. in Phys. 44405Adv. in Phys. 44, 405 (1995);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Phys. Rev. B. 65165113Phys. Rev. B 65, 165113 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F D M Haldane, E H Rezayi, Phys. Rev. B. 312529F. D. M. Haldane, and E. H. Rezayi, Phys. Rev. B 31, 2529 (1985).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". X.-G Wen, Q Niu, Phys. Rev. B. 419377X.-G. Wen, and Q. Niu, Phys. Rev. B 41, 9377 (1990).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Arovas, J R Schrieffer, F Wilczek, Phys. Rev. Lett. 53722D. Arovas, J. R. Schrieffer, and F. Wilczek, Phys. Rev. Lett. 53, 722 (1984).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Y Kitaev, Ann. Phys. (N.Y.). 303A. Y. Kitaev, Ann. Phys. (N.Y.) 303, 2 (2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Levin, X.-G Wen, Phys. Rev. B. 7575116M. Levin, and X.-G. Wen, Phys. Rev. B 75, 075116 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Hamma, D A Lidar, arXiv:quant-ph/0607145v4Phys. Rev. Lett.). A. Hamma and D. A. Lidar, (accepted for publication on Phys. Rev. Lett.), arXiv:quant-ph/0607145v4 (2006).\",\n \"doi\": \"arXiv:quant-ph/0607145v4\"\n },\n {\n \"text\": \". S Trebst, P Werner, M Troyer, K Shtengel, C Nayak, Phys. Rev. Lett. 9870602S. Trebst, P. Werner, M. Troyer, K. Shtengel, and C. Nayak, Phys. Rev. Lett. 98, 070602 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Hamma, W Zhang, S Haas, D A Lidar, arXiv:0705.0026v1A. Hamma, W. Zhang, S. Haas, and D. A. Lidar, arXiv:0705.0026v1 (2007).\",\n \"doi\": \"arXiv:0705.0026v1\"\n },\n {\n \"text\": \". M Levin, X.-G Wen, Phys. Rev. Lett. 96110405M. Levin, and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Y Kitaev, J Preskill, Phys. Rev. Lett. 96110404A. Y. Kitaev, and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Hamma, R Ionicioiu, P Zanardi, Phys. Rev. A. 7122315A. Hamma, R. Ionicioiu, and P. Zanardi, Phys. Rev. A 71, 022315 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Castelnovo, C Chamon, C Mudry, P Pujol, Ann. Phys. (N.Y.). 318316C. Castelnovo, C. Chamon, C. Mudry, and P. Pujol, Ann. Phys. (N.Y.) 318, 316 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Fradkin, J E Moore, Phys. Rev. Lett. 9750404E. Fradkin, and J. E. Moore, Phys. Rev. Lett. 97, 050404 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Castelnovo, C Chamon, Phys. Rev. B. 76174416C. Castelnovo and C. Chamon, Phys. Rev. B 76, 174416 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Bugrij, V N Shadura, arXiv:hep-th/9601106JETP Lett. 63384A. Bugrij, and V. N. Shadura, JETP Lett. 63, 384 (1996); arXiv:hep-th/9601106.\",\n \"doi\": \"arXiv:hep-th/9601106\"\n },\n {\n \"text\": \"This condition is actually immaterial, and can be avoided using an approach similar to the one by Furukawa and Misguich. Phys. Rev. B. 75214407This condition is actually immaterial, and can be avoided us- ing an approach similar to the one by Furukawa and Misguich [Phys. Rev. B 75, 214407 (2007)].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Ardonne, P Fendley, E Fradkin, Ann. Phys. (N.Y.). 310493E. Ardonne, P. Fendley, and E. Fradkin, Ann. Phys. (N.Y.) 310, 493 (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"R Baxter, Exactly Solved Models in Statistical Mechanics. LondonAcademic PressR. Baxter, Exactly Solved Models in Statistical Mechanics (Aca- demic Press, London, 1982).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"and references therein. The behavior of fidelity across a topological phase transition in the Kitaev model in presence of a magnetic field was studied numerically in Ref. P Zanardi, L Venuti, P Giorda, Phys. Rev. 7662318P. Zanardi, L. Campos Venuti, P. Giorda, Phys. Rev. A76, 062318 (2007), and references therein. The behavior of fidelity across a topological phase transition in the Kitaev model in presence of a magnetic field was studied numerically in Ref. 9.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"the lack of a local order parameter but display of local non-analyticities -was recently argued by Z. Nussinov and G. Ortiz to characterize as well a temperature-driven topological phase. arXiv:cond-mat/0702377A similar behavior -i.e.. transition in the Kitaev modelA similar behavior -i.e., the lack of a local order param- eter but display of local non-analyticities -was recently ar- gued by Z. Nussinov and G. Ortiz to characterize as well a temperature-driven topological phase transition in the Kitaev model [arXiv:cond-mat/0702377].\",\n \"doi\": \"arXiv:cond-mat/0702377\"\n },\n {\n \"text\": \". S Papanikolaou, K Raman, E Fradkin, Phys. Rev. B. 76224421S. Papanikolaou, K. Raman, and E. Fradkin, Phys. Rev. B 76, 224421 (2007).\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":474,"cells":{"text":{"kind":"string","value":"\nJOINT LEARNING OF OBJECT GRAPH AND RELATION GRAPH FOR VISUAL QUESTION ANSWERING\n9 May 2022\n\nHao Li \nSchool of Electronic and Computer Engineering\nPeking University\nChina\n\nXu Li \nCognitive Computing Lab\nBaidu ResearchChina\n\nBelhal Karimi belhalkarimi@baidu.com \nCognitive Computing Lab\nBaidu ResearchChina\n\nJie Chen chenj@pcl.ac.cn \nSchool of Electronic and Computer Engineering\nPeking University\nChina\n\nPeng Cheng Laboratory\nShenzhenChina\n\nMingming Sun sunmingming01@baidu.com \nCognitive Computing Lab\nBaidu ResearchChina\n\nJOINT LEARNING OF OBJECT GRAPH AND RELATION GRAPH FOR VISUAL QUESTION ANSWERING\n9 May 202207A30160BAFBAE96F803B639FF0D0DC5arXiv:2205.04188v1[cs.CV]Scene GraphVisual Question AnswerGraph Neural Network\nModeling visual question answering (VQA) through scene graphs can significantly improve the reasoning accuracy and interpretability.However, existing models answer poorly for complex reasoning questions with attributes or relations, which causes false attribute selection or missing relation in Figure1(a).It is because these models cannot balance all kinds of information in scene graphs, neglecting relation and attribute information.In this paper, we introduce a novel Dual Message-passing enhanced Graph Neural Network (DM-GNN), which can obtain a balanced representation by properly encoding multi-scale scene graph information.Specifically, we (i) transform the scene graph into two graphs with diversified focuses on objects and relations; Then we design a dual structure to encode them, which increases the weights from relations (ii) fuse the encoder output with attribute features, which increases the weights from attributes; (iii) propose a message-passing mechanism to enhance the information transfer between objects, relations and attributes.We conduct extensive experiments on datasets including GQA, VG, motif-VG and achieve new state of the art.\n\nINTRODUCTION\n\nVQA tasks require a model to answer a free-form natural language question using visual information from an image.Scene graph (SG) reasoning is an essential instance of VQA tasks [1].The model extracts objects' names, attributes, and relations from the input images and organizes them into a graph representation to generate the scene graph.\n\nSG representation modeling displays several virtues over classical VQA techniques since the features in SG are presented in plain and free text form [2] and the graph struc- Two key issues of traditional scene graph based models that we address: false attribute selection (top: select attribute \"black\" instead of attribute \"off\") and missing relation (bottom: missing \"Behind\") (b) Overview of our DM-GNN model.F N and F E are the object feature map and the relation feature map.F is the full-scale feature map.\n\ntures of SG have better interpretability [3].In this contribution, two reasoning methods on scene graphs are proposed: (i) consider scene graphs as probabilistic graphs and iteratively update nodes' probabilities using soft instructions extracted from questions [4]; (ii) apply Graph Neural Network (GNN) into scene graphs [5,6] to learn joint representations of nodes and their relations, and then feed these representations into a predictor to get the answer.Scene graph reasoning frameworks are useful in VQA [7].However, there still remain imperfections dealing with complex reasoning questions.First, existing models tend to predict wrong answers for complex reasoning questions with attributes.Consider the .....\nDual Graph Encoders\n\nAttribute Features\n\n\nRelation Features\n\n\nFusion Module\n\n\nAttention Fusion Block\n\n\nQ: \"What's on the kid's head\"\n\nAns: \"Sunglasses\" \"why\" question in Fig. 1(a) as an example, false attribute selection occurs because the model cannot associate \"off\" relation with \"dark monitor\" object.The attribute selections require comprehensive supervision from objects, relations, and attributes, but existing methods focus too much on objects while ignoring attributes.Generally, information from objects and relations connected to them are reconstructed into object features in GNN-based methods [8].However, these encoding methods lack information from objects' attributes.NSM methods [4] use soft instructions to update answer possibilities, but they treat attributes as secondary information.\nScene Graph Generator\nSecond, existing approaches answer poorly for complex questions that require information about relations.For instance, for localization questions, as in the \"where\" type of questions, in Fig. 1(a), we observe that missing relation occurs because the model cannot capture the relation information \"Behind\".Existing models have a strong bias towards object features, while considering relation as references.The unbalanced focus on objects and relations makes the models fail to learn discriminative representations for relations.\n\nTo improve the balance of all kinds of information in scene grpahs, we propose the Dual Message-passing enhanced Graph Neural Network (DM-GNN) for VQA, introducing a novel scene graph reasoning model that extracts balanced feature maps from objects, attributes, and relations information in scene graphs.Concretely, as shown in Fig. 1(b), our DM-GNN model is composed of a scene graph generator, a question encoder, dual graph encoders, and a fusion module.Besides, to balance the importance of objects and relations, we transform scene graphs into a relation-significant modality, where nodes represent relations and edges represent objects, and an object-significant modality, in which nodes represent objects and edges represent relations.After receiving scene graphs in two modalities, dual graph encoders can produce feature maps focusing on relations and objects.\n\nFurthermore, to enhance the information transfer between objects, relations and attributes, we modify the gated graph neural network (GGNN) structure in our DM-GNN by adding the message-passing module.It is a bidirectional GRU that guides the internal information flow.The encoder captures information from nodes, edges, and adjacent nodes that connect to them.In Fig. 1(b), the output feature map of the encoder passes through the fusion module, where the attribute features are explicitly modeled into the feature map to increase the information weight from attributes.Then the feature map passes through multi-head attention layers using question features extracted from the question encoder.Hence, the model dynamically focuses on the critical parts of the questions and uses the most similar part of the scene graph as the most adequate answer.Our main contributions are as follows:\n\n• We analyse that existing models answer imperfectly for complex reasoning questions with attributes or relations due to the unbalance focus on three information types in scene graphs, which contain objects, relations and attributes.\n\n• We propose a novel DM-GNN model containing a dual encoder structure and a message-passing module.Our model can obtain a balanced representation by properly encoding multi-scale scene graph information.\n\n• Experimental results on various datasets show that DM-GNN effectively improves the reasoning accuracy on semantically complicated questions.\n\n\nRELATED WORK\n\nVisual Question Answering.Most VQA approaches use sequential models [9] to encode questions and CNN-based pretrained models [10] to encode images.Then they use attention methods [11,12] to fuse features from images and questions.\n\nTransformer models [22] achieve outstanding performances on VQA tasks, yet they are heavy to train and hard to explain [23].Instead, the scene graph model stands for an alternative that is more lightly and explainable.\n\nScene Graph Generation and Reasoning.Scene graph generation (SGG) methods [21] use object detection methods to extract region proposals from images.Scene graph can promote explainable reasoning for downstream multimodal tasks such as VQA [3].In typical scene graph reasoning models, NSM [4] performs sequential reasoning over the scene graph by iteratively traversing its nodes.Other models [5,6,24] use GGNN [25] based model to encode scene graphs.However, previous works are hard to fully utilize the attribute information and learn the comprehensive representation of SG.\n\nGraph Neural Network.GNN [14] is designed to infer on data described by graphs.[15,16] apply GNN-based models on knowledge graphs, which are similar to scene graphs.However, existing GNN-based models cannot effectively process graphs with node attributes and complicated labels.Our DM-GNN model can learn a comprehensive and balanced representation using full-scale scene graph information from objects, attributes, and relations to overcome these problems.\n\n\nDM-GNN METHODOLOGY\n\nOur proposed architecture is illustrated in Fig. 2. We use the scene graph generator from [19].In the question encoder, semantic questions are first projected into an embedding space using GLOVE pretrained word embedding model [17].Then we use long short-term memory (LSTM) networks to generate questions representation q ∈ R dim , where dim is the dimension of the question representation.We introduce our dual graph encoders and fusion module in following subsections.\n\n\nObject/Relation-Significant Graph\n\nWe organize scene graphs into object-significant graphs and relation-significant graphs.\n\nObject-Significant Graph.We define the object significant graph as G obj , where each node represents an object in the image and each edge represents a relation between two objects.Define N as the node set and E as the edge set.For n i , n j ∈ N, e k ∈ E, < n i -e k -n j > denotes the relation tuple that represents relation e k from object n i to object n j .\n\nRelation-Significant Graph.We define relation significant modality as G rel , where each node represents a relation between objects in the image and each edge represents an object, which is completely opposed to the object-significant modality.For e i , e j ∈ E, n k ∈ N, < e i −n k −e j > represents the relations e i and e j have a shared object n k .\n\nAttribute types.Define L as attribute types.For each node n i ∈ N, we define a set of L + 1 property variables n l i L l=0 , where n 0 i represents n i 's name embedding and n l i represents the embedding of node n i 's l th attribute.to update h t i .\n\n\nDual Encoders\n\nWe apply two GGNN-based encoders for object significant graph and relation significant graph.The encoder for object graph focuses on object features and the encoder for relation graph focuses on relation features.The dual structure can balance the importance of relations and objects.\n\nPrior to encoding, every input scene graph is transformed into an information tuple (N, E, A in , A out ): N and E are collections of node embeddings and edge embeddings.A in and A out are the adjacency matrix of incident and output edges.Let h t i is the hidden state of node n i in the encoder at timestep t, then at t = 0, we initialize h 0 i as the GLOVE embedding of n i with zero padding:\n\nMessage-passing Module.To enhance the information transfer from edges and adjacent nodes to the updating nodes, we use the message-passing module (MP) in Fig. 3(a).MP module comes as a replacement of the fully-connected layers from the original GNN model.Consider a tuple < n i , e k , n j > as the processing sample of the message-passing module.The embedding state, noted e k , of the edge e k and neighbor node n j 's hidden state h j are injected into a bidirectional GRU network as input sequence while the node n i 's hidden state h i is injected as the GRU's initial hidden state.The output of the GRU represents the updating information for hidden state h i , which corresponds to the key information from edge e k and node n j that is related to node n i .The sum of every GRU output is n i 's total information gain from n i 's adjacent nodes and edges.We detail the message-passing module formula as follows:\nM P i (A in ) = <ni,e k ,nj >∈Ain k,j GRU([e k , h j ], h i ) , (1) M P i (A out ) = <nj ,e k ,ni>∈Aout k,j GRU([e k , h j ], h i ) ,(2)\nwhere M P i (A in ) is n i 's incident information gain, and M P i (A out ) is n i 's output information gain.Propagation Module.In Fig. 3(b), at timestep t, the hidden states of all nodes are updated by the following gated propagator module:\nk t i = [M P t i (A in ), M P t i (A out )] ,(3)\nwhere k t i is the node n i 's representation from all its incident edges, output edges and adjacent nodes.\n\nThen, we incorporate information from adjacent nodes and from the previous timestep leading to an update of each node's hidden state:\nc t i = [h (t−1) i , k (t−1) i ]W + b ,(4)z t i = σ(U z c t i ) ,(5)r t i = σ(U r c t i ) ,(6)\nwhere W, U z and U r are referred to as the trainable weight matrices.At timestep t, z t i and r t i are the update and reset gates, respectively.Then we have:\nht i = tanh(U 1 k (t−1) i + U 2 (r t i h (t−1) i )) ,(7)h t i = (1 − z t i ) h (t−1) i + z t i ht i .(8)\nHere, U 1 and U 2 denote the trainable parameters of the linear layers, the operator is the element-wise multiplication.σ is the ReLU function.After T steps, the encoder generates the final hidden state map G of the graph.Finally, we compute the graph embedding g i ∈ G for node n i as follows:\ng i = σ(f(h T i , n i )) ,(9)\nwhere f(h T i , n i ) is multi-layer perceptron (MLP) which receives the concatenation of h T i and n i .\n\n\nFusion Module and Answer Predictor\n\nOnce the dual encoders, embedded in our model, output the node and relation features, we first fuse the attributes into feature maps.For node feature map G N and relation feature map G E , the fusion feature map F N and F E are defined as\nF N i = [g N i , n 0 i ] • • • , [g , n L i ] , F E j = [g E j , ej], F = [F N , F E ],(10)\nwhere F N i indicates the fusion features of node i and g N i is node i's representation from the encoder.n l i is the attribute embedding of node i. F E j corresponds to the fusion feature of edge j. g E j is edge j's representation from the encoder.e j is j-th edge original embedding.The full-scale feature map, noted F , is obtained by concatenating F N and F E .\n\nThen, the question embedding q and the full-scale feature map F are fed into a multi-head attention layer.The reasoning vector, noted r, and which stems from the graph and the question, is computed using a weighted sum of the feature map using the scores output from the attention layer,\nr = Attention(F, q) .(11)\nIn answer predictor module, we adopt a two-layer MLP noted by f (•).This MLP can be viewed as a classifier over the set of candidate answers.The input of the answer predictor is the concatenation vector (q, r).Such a classifier has been applied in many VQA models [4,11].The answer â reads:\nâ = arg max(softmax(f ((q, r)))) .(12) 4. EXPERIMENTS 4.1. Empirical Results\nResults on VG dataset.Table 1 reports the results on the test sets of the VG ground truth dataset and the motif-VG dataset.Compared to the baseline models, we can observe that our DM-GNN model outperforms the others at 3%-4%.In addition, we provide detailed results on the VG dataset and motif-VG dataset with different question types.Compared to the other scene graph based VQA models, our model performs well in \"what\", \"where\", \"who\" and \"why\" types.On the VG dataset, our model has 6.5% accuracy improvement in \"why\" type questions, which highly requires VQA models' ability to jointly exploit objects, relations and attributes.\n\nResults on GQA dataset.We report in Table 2 the detailed results on the test sets of the GQA dataset.Our DM-GNN model outperforms baselines.We also evaluate our model and other baselines across GQA dataset's various metrics, where \"Binary\" and \"Open\" stand for binary-answer and open domain questions.\"Distribution\" corresponds to the distance between prediction distribution and standard answer distribution.For open domain questions which are difficult for reasoning, our model outperforms the others by 9.63%.For distribution metric, our model also achieves 2nd score.However, the \"Binary\" question is challenging for DM-GNN although it achieves SOTA performances.Specifically, the advantage of DM-GNN is to correctly locate key objects, relations and attributes.That is why it works well for the open domain (\"what, why, how\") questions.But for \"yes/no\" questions, whose answers are not explicit in scene graphs, DM-GNN is easy to locate but hard to correctly answer.\n\n\nAblation Study\n\nWe compare several ablated forms of DM-GNN with our complete model on the VG dataset.The accuracy for each variant of DM-GNN are reported in Table 3.We use the raw GGNN network as the Base model.The Base-Obj and Base-Rel model represent the original GGNN network processing object-significant graph and relation-significant graph.The models with +MP contain the message-passing module.The models with +Dual apply the dual encoder structure.Effect of dual encoder structure.We first validate the efficacy of applying dual structure to balance the importance of relations and objects by splitting our DM-GNN into two single models (Base-Obj and Base-Rel).Both single models perform poorly at 35.3% ± 0.1%.This also shows that both relations and objects are vital to VQA performance.Absence of any of those modules leads to severe accuracy recession.Adding dual encoder structure (+Dual) leads to an empirical gain of 32.6% accuracy upward, which shows that the dual structure is significant in balancing relations and objects.\n\nEffect of the message-passing module.We validate the effectiveness of applying message-passing structure to learn a more comprehensive representation for scene graphs than the raw GGNN structure.Comparing with the Base-Obj and +MP model, we note that after adding the messagepassing structure, there is an improvement of 3.9%.Comparing with the Base-Rel and +MP model, we observe an improvement of 3.6%.The +Dual model is DM-GNN without message-passing module.Comparing with DM-GNN, there is a 7.3% improvement after adding the message-passing module.These empirical results show that message-passing structure can successfully improve the representation quality of scene graphs.\n\nEffect of explicit modeling.The w/o attr model and the w/o rela model remove the explicit attribute modeling and relation modeling part.Comparing w/o attr, w/o rela with DM-GNN, removing attribute modeling has a 3.6% decrease in accuracy and removing relation modeling has 0.7% decrease.shows the visualization on \"why\", which need models to jointly exploit objects, relations and attributes to infer answers.With dual encoders and the message-passing module, our DM-GNN achieves 96.1% on \"why\" questions.\n\n\nVisualization\n\n\nCONCLUSION\n\nWe propose DM-GNN, which encodes each scene graph into feature representations via an object encoder and a relation encoder generating a balanced and full-scale feature map using objects, attributes, and relations information, and demon-\n\nFig. 1 .\n1\nFig. 1.(a) Two key issues of traditional scene graph based models that we address: false attribute selection (top: select attribute \"black\" instead of attribute \"off\") and missing relation (bottom: missing \"Behind\") (b) Overview of our DM-GNN model.F N and F E are the object feature map and the relation feature map.F is the full-scale feature map.\n\n\nFig. 2 .\n2\nFig. 2. Model structure of the Dual Message-passing enhanced Graph Neural Networks.MP stands for the message-passing module.Images are transformed into scene graphs by the scene graph generator.The object-significant form and relationsignificant form of the scene graph are injected into the object encoder and the relation encoder.Nodes' representations are generated from the sum of the MP modules.The representations are then fused with question representations to predict answers.\n\n\nFig. 3 .\n3\nFig. 3. Overview of the pipeline at timestep t.(a) The message-passing module generates M P in and M P out as incident and output information gains.(b) The propagator uses information gains and h t−1 i\n\n\nFig. 4 (\n4\na) shows visualization on \"what\" question type.Three \"what\" question examples aimed at retrieving either object, relation or attribute information.Comparing row 1, row 2 with row 3, Obj and Rel models have strong attention bias toward objects and relations, while their combination Obj+Rel, balances the attention on both sides and captures correct answers.Comparing row 3 with row 4, the messagepassing module increases the score of correct answers.Fig. 4 (b)\n\n\nTable 1 .\n1\nPerformance on different question types of VG dataset.\nModelsBinary↑ Open↑ Validity↑ Distribution↓ Acc.↑Human91.2087.4098.90-89.30BottomUp66.6434.8396.185.9849.74MAC71.2338.9196.165.3454.06SK T-Brain77.4243.1096.267.5459.19PVR77.6943.0196.455.8059.27GRN77.5343.3596.186.0659.37Dream77.8443.7296.388.4059.72LXRT77.7644.9796.308.3160.34NSM78.9449.2596.413.7163.17ReGAT83.5762.5892.709.3270.50DM-GNN(ours) 69.7972.2193.803.7871.21\n\nTable 2 .\n2\nPerformance on the GQA dataset.\nModelsAcc.Base35.4Base-Obj35.4+MP39.3 (+3.9)+Dual67.9 (+32.7)Base-Rel35.2+MP38.8 (+3.6)+Dual67.9 (+32.5)DM-GNN(ours) 75.2+ (w/o attr)71.6 (−3.6)+ (w/o rela)74.5 (−0.7)+ (w/o QF)54.9 (−20.3)\n\nTable 3 .\n3\nAblation Study on VG.\n\nThis work is supported in part by the Nature Science Foundation of China (No.61972217, No.62081360152), Natural Science Foundation of Guangdong Province in China (No.2019B1515120049, 2020B1111340056).Question:Why is all the food on the wooden table?\nVg- Dataset, GroundTruth Motif-VG Question type What Where Who Why Overall What Where Who Why Overall Percentage (54%) (17%) (5%) (3%) (100%) (54%) (17%). \n\nScene graph reasoning for visual question answering. M Hildebrandt, H Li, R Koner, V Tresp, S Günnemann, 2020CoRR\n\nUnderstanding the role of scene graphs in visual question answering. V Damodaran, S Chakravarthy, A Kumar, A Umapathy, T Mitamura, Y Nakashima, N Garcia, C Chu, abs/2101.05479CoRR. 2021\n\nAn empirical study on leveraging scene graphs for visual question answering. C Zhang, W Chao, D Xuan, 2019\n\nLearning by abstraction: The neural state machine. D A Hudson, C D Manning, 2019\n\nFrom strings to things: Knowledgeenabled vqa model that can read and reason. Aj, An Singh, Sh Mishra, An Shekhar, Chakraborty, \n\nRelation-aware graph attention network for visual question answering. L Li, Z Gan, Y Cheng, J Liu, 2019\n\nPrior visual relationship reasoning for visual question answering. Z Yang, Z Qin, J Yu, T Wan, 2020\n\nSpatial-aware graph relation network for large-scale object detection. H Xu, C Jiang, X Liang, Z Li, CVPR. 2019\n\nBERT: pre-training of deep bidirectional transformers for language understanding. J Devlin, M Chang, K Lee, K Toutanova, NAACL-HLT. 2019\n\nDifferential attention for visual question answering. N Badri, Patro, P Vinay, Namboodiri, 2018 CVPR\n\nHierarchical question-image co-attention for visual question answering. J Lu, J Yang, D Batra, D Parikh, 2016\n\nCompositional attention networks for machine reasoning. D A Hudson, C D Manning, 2018\n\nInterpretable visual question answering by reasoning on dependency trees. Q Cao, X Liang, B Li, L Lin, IEEE TPAMI. 2021\n\nGraph attention networks. P Velickovic, G Cucurull, A Casanova, A Romero, P Liò, Y Bengio, \n\nTransition-based chinese semantic dependency graph parsing. Y Wang, J Guo, W Che, T Liu, NLP-NABD. 2016\n\nA neural transition-based approach for semantic dependency graph parsing. Y Wang, W Che, J Guo, T Liu, 2018\n\nGloVe: Global vectors for word representation. J P , R Socher, C Manning, 2014\n\nGQA: A new dataset for real-world visual reasoning and compositional question answering. D A Hudson, C D Manning, 2019\n\nSgg codebase in pytorch. K Tang, 2020\n\nZoom-net: Mining deep feature interactions for visual relationship recognition. G Yin, L Sheng, B Liu, N Yu, X Wang, J Shao, Chen C , 2018\n\nUnbiased scene graph generation from biased training. K Tang, Y Niu, J Huang, J Shi, H Zhang, 2020\n\nUnicoder-vl: A universal encoder for vision and language by cross-modal pre-training. G Li, N Duan, Y Fang, G Ming, J Daxin, 2020\n\nPre-trained models: Past, present and future. X Han, Z Zhang, N Ding, J Wen, J Yuan, W Zhao, J Zhu, 2021CoRR\n\nGraphvqa: Languageguided graph neural networks for scene graph question answering. W Liange, Y Jiang, Z Liu, 2021\n\nGated graph sequence neural networks. Y Li, D Tarlow, M Brockschmidt, R S Zemel, 2016\n\nVisual genome: Connecting language and vision using crowdsourced dense image annotations. R Krishna, Y Zhu, O Groth, J Johnson, K Hata, J Kravitz, S Chen, Y Kalantidis, L Li, D A Shamma, M S Bernstein, Li Fei-Fei, Int. J. Comput. Vis. 2017\n"},"corpus_id":{"kind":"number","value":248571502,"string":"248,571,502"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2205.04188\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"conf/icmcs/LiLKCS22\",\n \"doi\": \"10.1109/icme52920.2022.9859766\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/2205.04188v1.pdf\"\n ],\n \"pdf_sha\": \"744ec954802334820e8cd2faa6e53c5e90d28920\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"https://arxiv.org/pdf/2205.04188\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"JOINT LEARNING OF OBJECT GRAPH AND RELATION GRAPH FOR VISUAL QUESTION ANSWERING"},"authors":{"kind":"list like","value":["Hao Li \nSchool of Electronic and Computer Engineering\nPeking University\nChina","Xu Li \nCognitive Computing Lab\nBaidu ResearchChina","Belhal Karimi belhalkarimi@baidu.com \nCognitive Computing Lab\nBaidu ResearchChina","Jie Chen chenj@pcl.ac.cn \nSchool of Electronic and Computer Engineering\nPeking University\nChina\n\nPeng Cheng Laboratory\nShenzhenChina","Mingming Sun sunmingming01@baidu.com \nCognitive Computing Lab\nBaidu ResearchChina"],"string":"[\n \"Hao Li \\nSchool of Electronic and Computer Engineering\\nPeking University\\nChina\",\n \"Xu Li \\nCognitive Computing Lab\\nBaidu ResearchChina\",\n \"Belhal Karimi belhalkarimi@baidu.com \\nCognitive Computing Lab\\nBaidu ResearchChina\",\n \"Jie Chen chenj@pcl.ac.cn \\nSchool of Electronic and Computer Engineering\\nPeking University\\nChina\\n\\nPeng Cheng Laboratory\\nShenzhenChina\",\n \"Mingming Sun sunmingming01@baidu.com \\nCognitive Computing Lab\\nBaidu ResearchChina\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Modeling visual question answering (VQA) through scene graphs can significantly improve the reasoning accuracy and interpretability.However, existing models answer poorly for complex reasoning questions with attributes or relations, which causes false attribute selection or missing relation in Figure1(a).It is because these models cannot balance all kinds of information in scene graphs, neglecting relation and attribute information.In this paper, we introduce a novel Dual Message-passing enhanced Graph Neural Network (DM-GNN), which can obtain a balanced representation by properly encoding multi-scale scene graph information.Specifically, we (i) transform the scene graph into two graphs with diversified focuses on objects and relations; Then we design a dual structure to encode them, which increases the weights from relations (ii) fuse the encoder output with attribute features, which increases the weights from attributes; (iii) propose a message-passing mechanism to enhance the information transfer between objects, relations and attributes.We conduct extensive experiments on datasets including GQA, VG, motif-VG and achieve new state of the art."},"sections":{"kind":"list like","value":[{"title":"INTRODUCTION","content":"VQA tasks require a model to answer a free-form natural language question using visual information from an image.Scene graph (SG) reasoning is an essential instance of VQA tasks [1].The model extracts objects' names, attributes, and relations from the input images and organizes them into a graph representation to generate the scene graph.\nSG representation modeling displays several virtues over classical VQA techniques since the features in SG are presented in plain and free text form [2] and the graph struc- Two key issues of traditional scene graph based models that we address: false attribute selection (top: select attribute \"black\" instead of attribute \"off\") and missing relation (bottom: missing \"Behind\") (b) Overview of our DM-GNN model.F N and F E are the object feature map and the relation feature map.F is the full-scale feature map.\ntures of SG have better interpretability [3].In this contribution, two reasoning methods on scene graphs are proposed: (i) consider scene graphs as probabilistic graphs and iteratively update nodes' probabilities using soft instructions extracted from questions [4]; (ii) apply Graph Neural Network (GNN) into scene graphs [5,6] to learn joint representations of nodes and their relations, and then feed these representations into a predictor to get the answer.Scene graph reasoning frameworks are useful in VQA [7].However, there still remain imperfections dealing with complex reasoning questions.First, existing models tend to predict wrong answers for complex reasoning questions with attributes.Consider the ....."},{"title":"Attribute Features","content":""},{"title":"Relation Features","content":""},{"title":"Fusion Module","content":""},{"title":"Attention Fusion Block","content":""},{"title":"Q: \"What's on the kid's head\"","content":"Ans: \"Sunglasses\" \"why\" question in Fig. 1(a) as an example, false attribute selection occurs because the model cannot associate \"off\" relation with \"dark monitor\" object.The attribute selections require comprehensive supervision from objects, relations, and attributes, but existing methods focus too much on objects while ignoring attributes.Generally, information from objects and relations connected to them are reconstructed into object features in GNN-based methods [8].However, these encoding methods lack information from objects' attributes.NSM methods [4] use soft instructions to update answer possibilities, but they treat attributes as secondary information.\nSecond, existing approaches answer poorly for complex questions that require information about relations.For instance, for localization questions, as in the \"where\" type of questions, in Fig. 1(a), we observe that missing relation occurs because the model cannot capture the relation information \"Behind\".Existing models have a strong bias towards object features, while considering relation as references.The unbalanced focus on objects and relations makes the models fail to learn discriminative representations for relations.\nTo improve the balance of all kinds of information in scene grpahs, we propose the Dual Message-passing enhanced Graph Neural Network (DM-GNN) for VQA, introducing a novel scene graph reasoning model that extracts balanced feature maps from objects, attributes, and relations information in scene graphs.Concretely, as shown in Fig. 1(b), our DM-GNN model is composed of a scene graph generator, a question encoder, dual graph encoders, and a fusion module.Besides, to balance the importance of objects and relations, we transform scene graphs into a relation-significant modality, where nodes represent relations and edges represent objects, and an object-significant modality, in which nodes represent objects and edges represent relations.After receiving scene graphs in two modalities, dual graph encoders can produce feature maps focusing on relations and objects.\nFurthermore, to enhance the information transfer between objects, relations and attributes, we modify the gated graph neural network (GGNN) structure in our DM-GNN by adding the message-passing module.It is a bidirectional GRU that guides the internal information flow.The encoder captures information from nodes, edges, and adjacent nodes that connect to them.In Fig. 1(b), the output feature map of the encoder passes through the fusion module, where the attribute features are explicitly modeled into the feature map to increase the information weight from attributes.Then the feature map passes through multi-head attention layers using question features extracted from the question encoder.Hence, the model dynamically focuses on the critical parts of the questions and uses the most similar part of the scene graph as the most adequate answer.Our main contributions are as follows:\n• We analyse that existing models answer imperfectly for complex reasoning questions with attributes or relations due to the unbalance focus on three information types in scene graphs, which contain objects, relations and attributes.\n• We propose a novel DM-GNN model containing a dual encoder structure and a message-passing module.Our model can obtain a balanced representation by properly encoding multi-scale scene graph information.\n• Experimental results on various datasets show that DM-GNN effectively improves the reasoning accuracy on semantically complicated questions."},{"title":"RELATED WORK","content":"Visual Question Answering.Most VQA approaches use sequential models [9] to encode questions and CNN-based pretrained models [10] to encode images.Then they use attention methods [11,12] to fuse features from images and questions.\nTransformer models [22] achieve outstanding performances on VQA tasks, yet they are heavy to train and hard to explain [23].Instead, the scene graph model stands for an alternative that is more lightly and explainable.\nScene Graph Generation and Reasoning.Scene graph generation (SGG) methods [21] use object detection methods to extract region proposals from images.Scene graph can promote explainable reasoning for downstream multimodal tasks such as VQA [3].In typical scene graph reasoning models, NSM [4] performs sequential reasoning over the scene graph by iteratively traversing its nodes.Other models [5,6,24] use GGNN [25] based model to encode scene graphs.However, previous works are hard to fully utilize the attribute information and learn the comprehensive representation of SG.\nGraph Neural Network.GNN [14] is designed to infer on data described by graphs.[15,16] apply GNN-based models on knowledge graphs, which are similar to scene graphs.However, existing GNN-based models cannot effectively process graphs with node attributes and complicated labels.Our DM-GNN model can learn a comprehensive and balanced representation using full-scale scene graph information from objects, attributes, and relations to overcome these problems."},{"title":"DM-GNN METHODOLOGY","content":"Our proposed architecture is illustrated in Fig. 2. We use the scene graph generator from [19].In the question encoder, semantic questions are first projected into an embedding space using GLOVE pretrained word embedding model [17].Then we use long short-term memory (LSTM) networks to generate questions representation q ∈ R dim , where dim is the dimension of the question representation.We introduce our dual graph encoders and fusion module in following subsections."},{"title":"Object/Relation-Significant Graph","content":"We organize scene graphs into object-significant graphs and relation-significant graphs.\nObject-Significant Graph.We define the object significant graph as G obj , where each node represents an object in the image and each edge represents a relation between two objects.Define N as the node set and E as the edge set.For n i , n j ∈ N, e k ∈ E, < n i -e k -n j > denotes the relation tuple that represents relation e k from object n i to object n j .\nRelation-Significant Graph.We define relation significant modality as G rel , where each node represents a relation between objects in the image and each edge represents an object, which is completely opposed to the object-significant modality.For e i , e j ∈ E, n k ∈ N, < e i −n k −e j > represents the relations e i and e j have a shared object n k .\nAttribute types.Define L as attribute types.For each node n i ∈ N, we define a set of L + 1 property variables n l i L l=0 , where n 0 i represents n i 's name embedding and n l i represents the embedding of node n i 's l th attribute.to update h t i ."},{"title":"Dual Encoders","content":"We apply two GGNN-based encoders for object significant graph and relation significant graph.The encoder for object graph focuses on object features and the encoder for relation graph focuses on relation features.The dual structure can balance the importance of relations and objects.\nPrior to encoding, every input scene graph is transformed into an information tuple (N, E, A in , A out ): N and E are collections of node embeddings and edge embeddings.A in and A out are the adjacency matrix of incident and output edges.Let h t i is the hidden state of node n i in the encoder at timestep t, then at t = 0, we initialize h 0 i as the GLOVE embedding of n i with zero padding:\nMessage-passing Module.To enhance the information transfer from edges and adjacent nodes to the updating nodes, we use the message-passing module (MP) in Fig. 3(a).MP module comes as a replacement of the fully-connected layers from the original GNN model.Consider a tuple < n i , e k , n j > as the processing sample of the message-passing module.The embedding state, noted e k , of the edge e k and neighbor node n j 's hidden state h j are injected into a bidirectional GRU network as input sequence while the node n i 's hidden state h i is injected as the GRU's initial hidden state.The output of the GRU represents the updating information for hidden state h i , which corresponds to the key information from edge e k and node n j that is related to node n i .The sum of every GRU output is n i 's total information gain from n i 's adjacent nodes and edges.We detail the message-passing module formula as follows:\nwhere M P i (A in ) is n i 's incident information gain, and M P i (A out ) is n i 's output information gain.Propagation Module.In Fig. 3(b), at timestep t, the hidden states of all nodes are updated by the following gated propagator module:\nwhere k t i is the node n i 's representation from all its incident edges, output edges and adjacent nodes.\nThen, we incorporate information from adjacent nodes and from the previous timestep leading to an update of each node's hidden state:\nwhere W, U z and U r are referred to as the trainable weight matrices.At timestep t, z t i and r t i are the update and reset gates, respectively.Then we have:\nHere, U 1 and U 2 denote the trainable parameters of the linear layers, the operator is the element-wise multiplication.σ is the ReLU function.After T steps, the encoder generates the final hidden state map G of the graph.Finally, we compute the graph embedding g i ∈ G for node n i as follows:\nwhere f(h T i , n i ) is multi-layer perceptron (MLP) which receives the concatenation of h T i and n i ."},{"title":"Fusion Module and Answer Predictor","content":"Once the dual encoders, embedded in our model, output the node and relation features, we first fuse the attributes into feature maps.For node feature map G N and relation feature map G E , the fusion feature map F N and F E are defined as\nwhere F N i indicates the fusion features of node i and g N i is node i's representation from the encoder.n l i is the attribute embedding of node i. F E j corresponds to the fusion feature of edge j. g E j is edge j's representation from the encoder.e j is j-th edge original embedding.The full-scale feature map, noted F , is obtained by concatenating F N and F E .\nThen, the question embedding q and the full-scale feature map F are fed into a multi-head attention layer.The reasoning vector, noted r, and which stems from the graph and the question, is computed using a weighted sum of the feature map using the scores output from the attention layer,\nIn answer predictor module, we adopt a two-layer MLP noted by f (•).This MLP can be viewed as a classifier over the set of candidate answers.The input of the answer predictor is the concatenation vector (q, r).Such a classifier has been applied in many VQA models [4,11].The answer â reads:\nResults on VG dataset.Table 1 reports the results on the test sets of the VG ground truth dataset and the motif-VG dataset.Compared to the baseline models, we can observe that our DM-GNN model outperforms the others at 3%-4%.In addition, we provide detailed results on the VG dataset and motif-VG dataset with different question types.Compared to the other scene graph based VQA models, our model performs well in \"what\", \"where\", \"who\" and \"why\" types.On the VG dataset, our model has 6.5% accuracy improvement in \"why\" type questions, which highly requires VQA models' ability to jointly exploit objects, relations and attributes.\nResults on GQA dataset.We report in Table 2 the detailed results on the test sets of the GQA dataset.Our DM-GNN model outperforms baselines.We also evaluate our model and other baselines across GQA dataset's various metrics, where \"Binary\" and \"Open\" stand for binary-answer and open domain questions.\"Distribution\" corresponds to the distance between prediction distribution and standard answer distribution.For open domain questions which are difficult for reasoning, our model outperforms the others by 9.63%.For distribution metric, our model also achieves 2nd score.However, the \"Binary\" question is challenging for DM-GNN although it achieves SOTA performances.Specifically, the advantage of DM-GNN is to correctly locate key objects, relations and attributes.That is why it works well for the open domain (\"what, why, how\") questions.But for \"yes/no\" questions, whose answers are not explicit in scene graphs, DM-GNN is easy to locate but hard to correctly answer."},{"title":"Ablation Study","content":"We compare several ablated forms of DM-GNN with our complete model on the VG dataset.The accuracy for each variant of DM-GNN are reported in Table 3.We use the raw GGNN network as the Base model.The Base-Obj and Base-Rel model represent the original GGNN network processing object-significant graph and relation-significant graph.The models with +MP contain the message-passing module.The models with +Dual apply the dual encoder structure.Effect of dual encoder structure.We first validate the efficacy of applying dual structure to balance the importance of relations and objects by splitting our DM-GNN into two single models (Base-Obj and Base-Rel).Both single models perform poorly at 35.3% ± 0.1%.This also shows that both relations and objects are vital to VQA performance.Absence of any of those modules leads to severe accuracy recession.Adding dual encoder structure (+Dual) leads to an empirical gain of 32.6% accuracy upward, which shows that the dual structure is significant in balancing relations and objects.\nEffect of the message-passing module.We validate the effectiveness of applying message-passing structure to learn a more comprehensive representation for scene graphs than the raw GGNN structure.Comparing with the Base-Obj and +MP model, we note that after adding the messagepassing structure, there is an improvement of 3.9%.Comparing with the Base-Rel and +MP model, we observe an improvement of 3.6%.The +Dual model is DM-GNN without message-passing module.Comparing with DM-GNN, there is a 7.3% improvement after adding the message-passing module.These empirical results show that message-passing structure can successfully improve the representation quality of scene graphs.\nEffect of explicit modeling.The w/o attr model and the w/o rela model remove the explicit attribute modeling and relation modeling part.Comparing w/o attr, w/o rela with DM-GNN, removing attribute modeling has a 3.6% decrease in accuracy and removing relation modeling has 0.7% decrease.shows the visualization on \"why\", which need models to jointly exploit objects, relations and attributes to infer answers.With dual encoders and the message-passing module, our DM-GNN achieves 96.1% on \"why\" questions."},{"title":"Visualization","content":""},{"title":"CONCLUSION","content":"We propose DM-GNN, which encodes each scene graph into feature representations via an object encoder and a relation encoder generating a balanced and full-scale feature map using objects, attributes, and relations information, and demon-"},{"title":"Fig. 1 .","content":"Fig. 1.(a) Two key issues of traditional scene graph based models that we address: false attribute selection (top: select attribute \"black\" instead of attribute \"off\") and missing relation (bottom: missing \"Behind\") (b) Overview of our DM-GNN model.F N and F E are the object feature map and the relation feature map.F is the full-scale feature map."},{"title":"Fig. 2 .","content":"Fig. 2. Model structure of the Dual Message-passing enhanced Graph Neural Networks.MP stands for the message-passing module.Images are transformed into scene graphs by the scene graph generator.The object-significant form and relationsignificant form of the scene graph are injected into the object encoder and the relation encoder.Nodes' representations are generated from the sum of the MP modules.The representations are then fused with question representations to predict answers."},{"title":"Fig. 3 .","content":"Fig. 3. Overview of the pipeline at timestep t.(a) The message-passing module generates M P in and M P out as incident and output information gains.(b) The propagator uses information gains and h t−1 i"},{"title":"Fig. 4 (","content":"a) shows visualization on \"what\" question type.Three \"what\" question examples aimed at retrieving either object, relation or attribute information.Comparing row 1, row 2 with row 3, Obj and Rel models have strong attention bias toward objects and relations, while their combination Obj+Rel, balances the attention on both sides and captures correct answers.Comparing row 3 with row 4, the messagepassing module increases the score of correct answers.Fig. 4 (b)"},{"title":"Table 1 .","content":"Performance on different question types of VG dataset."},{"title":"Table 2 .","content":"Performance on the GQA dataset."},{"title":"Table 3 .","content":"Ablation Study on VG."}],"string":"[\n {\n \"title\": \"INTRODUCTION\",\n \"content\": \"VQA tasks require a model to answer a free-form natural language question using visual information from an image.Scene graph (SG) reasoning is an essential instance of VQA tasks [1].The model extracts objects' names, attributes, and relations from the input images and organizes them into a graph representation to generate the scene graph.\\nSG representation modeling displays several virtues over classical VQA techniques since the features in SG are presented in plain and free text form [2] and the graph struc- Two key issues of traditional scene graph based models that we address: false attribute selection (top: select attribute \\\"black\\\" instead of attribute \\\"off\\\") and missing relation (bottom: missing \\\"Behind\\\") (b) Overview of our DM-GNN model.F N and F E are the object feature map and the relation feature map.F is the full-scale feature map.\\ntures of SG have better interpretability [3].In this contribution, two reasoning methods on scene graphs are proposed: (i) consider scene graphs as probabilistic graphs and iteratively update nodes' probabilities using soft instructions extracted from questions [4]; (ii) apply Graph Neural Network (GNN) into scene graphs [5,6] to learn joint representations of nodes and their relations, and then feed these representations into a predictor to get the answer.Scene graph reasoning frameworks are useful in VQA [7].However, there still remain imperfections dealing with complex reasoning questions.First, existing models tend to predict wrong answers for complex reasoning questions with attributes.Consider the .....\"\n },\n {\n \"title\": \"Attribute Features\",\n \"content\": \"\"\n },\n {\n \"title\": \"Relation Features\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fusion Module\",\n \"content\": \"\"\n },\n {\n \"title\": \"Attention Fusion Block\",\n \"content\": \"\"\n },\n {\n \"title\": \"Q: \\\"What's on the kid's head\\\"\",\n \"content\": \"Ans: \\\"Sunglasses\\\" \\\"why\\\" question in Fig. 1(a) as an example, false attribute selection occurs because the model cannot associate \\\"off\\\" relation with \\\"dark monitor\\\" object.The attribute selections require comprehensive supervision from objects, relations, and attributes, but existing methods focus too much on objects while ignoring attributes.Generally, information from objects and relations connected to them are reconstructed into object features in GNN-based methods [8].However, these encoding methods lack information from objects' attributes.NSM methods [4] use soft instructions to update answer possibilities, but they treat attributes as secondary information.\\nSecond, existing approaches answer poorly for complex questions that require information about relations.For instance, for localization questions, as in the \\\"where\\\" type of questions, in Fig. 1(a), we observe that missing relation occurs because the model cannot capture the relation information \\\"Behind\\\".Existing models have a strong bias towards object features, while considering relation as references.The unbalanced focus on objects and relations makes the models fail to learn discriminative representations for relations.\\nTo improve the balance of all kinds of information in scene grpahs, we propose the Dual Message-passing enhanced Graph Neural Network (DM-GNN) for VQA, introducing a novel scene graph reasoning model that extracts balanced feature maps from objects, attributes, and relations information in scene graphs.Concretely, as shown in Fig. 1(b), our DM-GNN model is composed of a scene graph generator, a question encoder, dual graph encoders, and a fusion module.Besides, to balance the importance of objects and relations, we transform scene graphs into a relation-significant modality, where nodes represent relations and edges represent objects, and an object-significant modality, in which nodes represent objects and edges represent relations.After receiving scene graphs in two modalities, dual graph encoders can produce feature maps focusing on relations and objects.\\nFurthermore, to enhance the information transfer between objects, relations and attributes, we modify the gated graph neural network (GGNN) structure in our DM-GNN by adding the message-passing module.It is a bidirectional GRU that guides the internal information flow.The encoder captures information from nodes, edges, and adjacent nodes that connect to them.In Fig. 1(b), the output feature map of the encoder passes through the fusion module, where the attribute features are explicitly modeled into the feature map to increase the information weight from attributes.Then the feature map passes through multi-head attention layers using question features extracted from the question encoder.Hence, the model dynamically focuses on the critical parts of the questions and uses the most similar part of the scene graph as the most adequate answer.Our main contributions are as follows:\\n• We analyse that existing models answer imperfectly for complex reasoning questions with attributes or relations due to the unbalance focus on three information types in scene graphs, which contain objects, relations and attributes.\\n• We propose a novel DM-GNN model containing a dual encoder structure and a message-passing module.Our model can obtain a balanced representation by properly encoding multi-scale scene graph information.\\n• Experimental results on various datasets show that DM-GNN effectively improves the reasoning accuracy on semantically complicated questions.\"\n },\n {\n \"title\": \"RELATED WORK\",\n \"content\": \"Visual Question Answering.Most VQA approaches use sequential models [9] to encode questions and CNN-based pretrained models [10] to encode images.Then they use attention methods [11,12] to fuse features from images and questions.\\nTransformer models [22] achieve outstanding performances on VQA tasks, yet they are heavy to train and hard to explain [23].Instead, the scene graph model stands for an alternative that is more lightly and explainable.\\nScene Graph Generation and Reasoning.Scene graph generation (SGG) methods [21] use object detection methods to extract region proposals from images.Scene graph can promote explainable reasoning for downstream multimodal tasks such as VQA [3].In typical scene graph reasoning models, NSM [4] performs sequential reasoning over the scene graph by iteratively traversing its nodes.Other models [5,6,24] use GGNN [25] based model to encode scene graphs.However, previous works are hard to fully utilize the attribute information and learn the comprehensive representation of SG.\\nGraph Neural Network.GNN [14] is designed to infer on data described by graphs.[15,16] apply GNN-based models on knowledge graphs, which are similar to scene graphs.However, existing GNN-based models cannot effectively process graphs with node attributes and complicated labels.Our DM-GNN model can learn a comprehensive and balanced representation using full-scale scene graph information from objects, attributes, and relations to overcome these problems.\"\n },\n {\n \"title\": \"DM-GNN METHODOLOGY\",\n \"content\": \"Our proposed architecture is illustrated in Fig. 2. We use the scene graph generator from [19].In the question encoder, semantic questions are first projected into an embedding space using GLOVE pretrained word embedding model [17].Then we use long short-term memory (LSTM) networks to generate questions representation q ∈ R dim , where dim is the dimension of the question representation.We introduce our dual graph encoders and fusion module in following subsections.\"\n },\n {\n \"title\": \"Object/Relation-Significant Graph\",\n \"content\": \"We organize scene graphs into object-significant graphs and relation-significant graphs.\\nObject-Significant Graph.We define the object significant graph as G obj , where each node represents an object in the image and each edge represents a relation between two objects.Define N as the node set and E as the edge set.For n i , n j ∈ N, e k ∈ E, < n i -e k -n j > denotes the relation tuple that represents relation e k from object n i to object n j .\\nRelation-Significant Graph.We define relation significant modality as G rel , where each node represents a relation between objects in the image and each edge represents an object, which is completely opposed to the object-significant modality.For e i , e j ∈ E, n k ∈ N, < e i −n k −e j > represents the relations e i and e j have a shared object n k .\\nAttribute types.Define L as attribute types.For each node n i ∈ N, we define a set of L + 1 property variables n l i L l=0 , where n 0 i represents n i 's name embedding and n l i represents the embedding of node n i 's l th attribute.to update h t i .\"\n },\n {\n \"title\": \"Dual Encoders\",\n \"content\": \"We apply two GGNN-based encoders for object significant graph and relation significant graph.The encoder for object graph focuses on object features and the encoder for relation graph focuses on relation features.The dual structure can balance the importance of relations and objects.\\nPrior to encoding, every input scene graph is transformed into an information tuple (N, E, A in , A out ): N and E are collections of node embeddings and edge embeddings.A in and A out are the adjacency matrix of incident and output edges.Let h t i is the hidden state of node n i in the encoder at timestep t, then at t = 0, we initialize h 0 i as the GLOVE embedding of n i with zero padding:\\nMessage-passing Module.To enhance the information transfer from edges and adjacent nodes to the updating nodes, we use the message-passing module (MP) in Fig. 3(a).MP module comes as a replacement of the fully-connected layers from the original GNN model.Consider a tuple < n i , e k , n j > as the processing sample of the message-passing module.The embedding state, noted e k , of the edge e k and neighbor node n j 's hidden state h j are injected into a bidirectional GRU network as input sequence while the node n i 's hidden state h i is injected as the GRU's initial hidden state.The output of the GRU represents the updating information for hidden state h i , which corresponds to the key information from edge e k and node n j that is related to node n i .The sum of every GRU output is n i 's total information gain from n i 's adjacent nodes and edges.We detail the message-passing module formula as follows:\\nwhere M P i (A in ) is n i 's incident information gain, and M P i (A out ) is n i 's output information gain.Propagation Module.In Fig. 3(b), at timestep t, the hidden states of all nodes are updated by the following gated propagator module:\\nwhere k t i is the node n i 's representation from all its incident edges, output edges and adjacent nodes.\\nThen, we incorporate information from adjacent nodes and from the previous timestep leading to an update of each node's hidden state:\\nwhere W, U z and U r are referred to as the trainable weight matrices.At timestep t, z t i and r t i are the update and reset gates, respectively.Then we have:\\nHere, U 1 and U 2 denote the trainable parameters of the linear layers, the operator is the element-wise multiplication.σ is the ReLU function.After T steps, the encoder generates the final hidden state map G of the graph.Finally, we compute the graph embedding g i ∈ G for node n i as follows:\\nwhere f(h T i , n i ) is multi-layer perceptron (MLP) which receives the concatenation of h T i and n i .\"\n },\n {\n \"title\": \"Fusion Module and Answer Predictor\",\n \"content\": \"Once the dual encoders, embedded in our model, output the node and relation features, we first fuse the attributes into feature maps.For node feature map G N and relation feature map G E , the fusion feature map F N and F E are defined as\\nwhere F N i indicates the fusion features of node i and g N i is node i's representation from the encoder.n l i is the attribute embedding of node i. F E j corresponds to the fusion feature of edge j. g E j is edge j's representation from the encoder.e j is j-th edge original embedding.The full-scale feature map, noted F , is obtained by concatenating F N and F E .\\nThen, the question embedding q and the full-scale feature map F are fed into a multi-head attention layer.The reasoning vector, noted r, and which stems from the graph and the question, is computed using a weighted sum of the feature map using the scores output from the attention layer,\\nIn answer predictor module, we adopt a two-layer MLP noted by f (•).This MLP can be viewed as a classifier over the set of candidate answers.The input of the answer predictor is the concatenation vector (q, r).Such a classifier has been applied in many VQA models [4,11].The answer â reads:\\nResults on VG dataset.Table 1 reports the results on the test sets of the VG ground truth dataset and the motif-VG dataset.Compared to the baseline models, we can observe that our DM-GNN model outperforms the others at 3%-4%.In addition, we provide detailed results on the VG dataset and motif-VG dataset with different question types.Compared to the other scene graph based VQA models, our model performs well in \\\"what\\\", \\\"where\\\", \\\"who\\\" and \\\"why\\\" types.On the VG dataset, our model has 6.5% accuracy improvement in \\\"why\\\" type questions, which highly requires VQA models' ability to jointly exploit objects, relations and attributes.\\nResults on GQA dataset.We report in Table 2 the detailed results on the test sets of the GQA dataset.Our DM-GNN model outperforms baselines.We also evaluate our model and other baselines across GQA dataset's various metrics, where \\\"Binary\\\" and \\\"Open\\\" stand for binary-answer and open domain questions.\\\"Distribution\\\" corresponds to the distance between prediction distribution and standard answer distribution.For open domain questions which are difficult for reasoning, our model outperforms the others by 9.63%.For distribution metric, our model also achieves 2nd score.However, the \\\"Binary\\\" question is challenging for DM-GNN although it achieves SOTA performances.Specifically, the advantage of DM-GNN is to correctly locate key objects, relations and attributes.That is why it works well for the open domain (\\\"what, why, how\\\") questions.But for \\\"yes/no\\\" questions, whose answers are not explicit in scene graphs, DM-GNN is easy to locate but hard to correctly answer.\"\n },\n {\n \"title\": \"Ablation Study\",\n \"content\": \"We compare several ablated forms of DM-GNN with our complete model on the VG dataset.The accuracy for each variant of DM-GNN are reported in Table 3.We use the raw GGNN network as the Base model.The Base-Obj and Base-Rel model represent the original GGNN network processing object-significant graph and relation-significant graph.The models with +MP contain the message-passing module.The models with +Dual apply the dual encoder structure.Effect of dual encoder structure.We first validate the efficacy of applying dual structure to balance the importance of relations and objects by splitting our DM-GNN into two single models (Base-Obj and Base-Rel).Both single models perform poorly at 35.3% ± 0.1%.This also shows that both relations and objects are vital to VQA performance.Absence of any of those modules leads to severe accuracy recession.Adding dual encoder structure (+Dual) leads to an empirical gain of 32.6% accuracy upward, which shows that the dual structure is significant in balancing relations and objects.\\nEffect of the message-passing module.We validate the effectiveness of applying message-passing structure to learn a more comprehensive representation for scene graphs than the raw GGNN structure.Comparing with the Base-Obj and +MP model, we note that after adding the messagepassing structure, there is an improvement of 3.9%.Comparing with the Base-Rel and +MP model, we observe an improvement of 3.6%.The +Dual model is DM-GNN without message-passing module.Comparing with DM-GNN, there is a 7.3% improvement after adding the message-passing module.These empirical results show that message-passing structure can successfully improve the representation quality of scene graphs.\\nEffect of explicit modeling.The w/o attr model and the w/o rela model remove the explicit attribute modeling and relation modeling part.Comparing w/o attr, w/o rela with DM-GNN, removing attribute modeling has a 3.6% decrease in accuracy and removing relation modeling has 0.7% decrease.shows the visualization on \\\"why\\\", which need models to jointly exploit objects, relations and attributes to infer answers.With dual encoders and the message-passing module, our DM-GNN achieves 96.1% on \\\"why\\\" questions.\"\n },\n {\n \"title\": \"Visualization\",\n \"content\": \"\"\n },\n {\n \"title\": \"CONCLUSION\",\n \"content\": \"We propose DM-GNN, which encodes each scene graph into feature representations via an object encoder and a relation encoder generating a balanced and full-scale feature map using objects, attributes, and relations information, and demon-\"\n },\n {\n \"title\": \"Fig. 1 .\",\n \"content\": \"Fig. 1.(a) Two key issues of traditional scene graph based models that we address: false attribute selection (top: select attribute \\\"black\\\" instead of attribute \\\"off\\\") and missing relation (bottom: missing \\\"Behind\\\") (b) Overview of our DM-GNN model.F N and F E are the object feature map and the relation feature map.F is the full-scale feature map.\"\n },\n {\n \"title\": \"Fig. 2 .\",\n \"content\": \"Fig. 2. Model structure of the Dual Message-passing enhanced Graph Neural Networks.MP stands for the message-passing module.Images are transformed into scene graphs by the scene graph generator.The object-significant form and relationsignificant form of the scene graph are injected into the object encoder and the relation encoder.Nodes' representations are generated from the sum of the MP modules.The representations are then fused with question representations to predict answers.\"\n },\n {\n \"title\": \"Fig. 3 .\",\n \"content\": \"Fig. 3. Overview of the pipeline at timestep t.(a) The message-passing module generates M P in and M P out as incident and output information gains.(b) The propagator uses information gains and h t−1 i\"\n },\n {\n \"title\": \"Fig. 4 (\",\n \"content\": \"a) shows visualization on \\\"what\\\" question type.Three \\\"what\\\" question examples aimed at retrieving either object, relation or attribute information.Comparing row 1, row 2 with row 3, Obj and Rel models have strong attention bias toward objects and relations, while their combination Obj+Rel, balances the attention on both sides and captures correct answers.Comparing row 3 with row 4, the messagepassing module increases the score of correct answers.Fig. 4 (b)\"\n },\n {\n \"title\": \"Table 1 .\",\n \"content\": \"Performance on different question types of VG dataset.\"\n },\n {\n \"title\": \"Table 2 .\",\n \"content\": \"Performance on the GQA dataset.\"\n },\n {\n \"title\": \"Table 3 .\",\n \"content\": \"Ablation Study on VG.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Fig. 1.(a) Two key issues of traditional scene graph based models that we address: false attribute selection (top: select attribute \"black\" instead of attribute \"off\") and missing relation (bottom: missing \"Behind\") (b) Overview of our DM-GNN model.F N and F E are the object feature map and the relation feature map.F is the full-scale feature map.","content":"Fig. 1 .\n1\nFig. 1.(a) Two key issues of traditional scene graph based models that we address: false attribute selection (top: select attribute \"black\" instead of attribute \"off\") and missing relation (bottom: missing \"Behind\") (b) Overview of our DM-GNN model.F N and F E are the object feature map and the relation feature map.F is the full-scale feature map."},{"id":"","caption":"Fig. 2. Model structure of the Dual Message-passing enhanced Graph Neural Networks.MP stands for the message-passing module.Images are transformed into scene graphs by the scene graph generator.The object-significant form and relationsignificant form of the scene graph are injected into the object encoder and the relation encoder.Nodes' representations are generated from the sum of the MP modules.The representations are then fused with question representations to predict answers.","content":"Fig. 2 .\n2\nFig. 2. Model structure of the Dual Message-passing enhanced Graph Neural Networks.MP stands for the message-passing module.Images are transformed into scene graphs by the scene graph generator.The object-significant form and relationsignificant form of the scene graph are injected into the object encoder and the relation encoder.Nodes' representations are generated from the sum of the MP modules.The representations are then fused with question representations to predict answers."},{"id":"","caption":"Fig. 3. Overview of the pipeline at timestep t.(a) The message-passing module generates M P in and M P out as incident and output information gains.(b) The propagator uses information gains and h t−1 i","content":"Fig. 3 .\n3\nFig. 3. Overview of the pipeline at timestep t.(a) The message-passing module generates M P in and M P out as incident and output information gains.(b) The propagator uses information gains and h t−1 i"},{"id":"","caption":"a) shows visualization on \"what\" question type.Three \"what\" question examples aimed at retrieving either object, relation or attribute information.Comparing row 1, row 2 with row 3, Obj and Rel models have strong attention bias toward objects and relations, while their combination Obj+Rel, balances the attention on both sides and captures correct answers.Comparing row 3 with row 4, the messagepassing module increases the score of correct answers.Fig. 4 (b)","content":"Fig. 4 (\n4\na) shows visualization on \"what\" question type.Three \"what\" question examples aimed at retrieving either object, relation or attribute information.Comparing row 1, row 2 with row 3, Obj and Rel models have strong attention bias toward objects and relations, while their combination Obj+Rel, balances the attention on both sides and captures correct answers.Comparing row 3 with row 4, the messagepassing module increases the score of correct answers.Fig. 4 (b)"},{"id":"","caption":"Performance on different question types of VG dataset.","content":"Table 1 .\n1\nPerformance on different question types of VG dataset.\nModelsBinary↑ Open↑ Validity↑ Distribution↓ Acc.↑Human91.2087.4098.90-89.30BottomUp66.6434.8396.185.9849.74MAC71.2338.9196.165.3454.06SK T-Brain77.4243.1096.267.5459.19PVR77.6943.0196.455.8059.27GRN77.5343.3596.186.0659.37Dream77.8443.7296.388.4059.72LXRT77.7644.9796.308.3160.34NSM78.9449.2596.413.7163.17ReGAT83.5762.5892.709.3270.50DM-GNN(ours) 69.7972.2193.803.7871.21"},{"id":"","caption":"Performance on the GQA dataset.","content":"Table 2 .\n2\nPerformance on the GQA dataset.\nModelsAcc.Base35.4Base-Obj35.4+MP39.3 (+3.9)+Dual67.9 (+32.7)Base-Rel35.2+MP38.8 (+3.6)+Dual67.9 (+32.5)DM-GNN(ours) 75.2+ (w/o attr)71.6 (−3.6)+ (w/o rela)74.5 (−0.7)+ (w/o QF)54.9 (−20.3)"},{"id":"","caption":"Ablation Study on VG.","content":"Table 3 .\n3\nAblation Study on VG."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Fig. 1.(a) Two key issues of traditional scene graph based models that we address: false attribute selection (top: select attribute \\\"black\\\" instead of attribute \\\"off\\\") and missing relation (bottom: missing \\\"Behind\\\") (b) Overview of our DM-GNN model.F N and F E are the object feature map and the relation feature map.F is the full-scale feature map.\",\n \"content\": \"Fig. 1 .\\n1\\nFig. 1.(a) Two key issues of traditional scene graph based models that we address: false attribute selection (top: select attribute \\\"black\\\" instead of attribute \\\"off\\\") and missing relation (bottom: missing \\\"Behind\\\") (b) Overview of our DM-GNN model.F N and F E are the object feature map and the relation feature map.F is the full-scale feature map.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Fig. 2. Model structure of the Dual Message-passing enhanced Graph Neural Networks.MP stands for the message-passing module.Images are transformed into scene graphs by the scene graph generator.The object-significant form and relationsignificant form of the scene graph are injected into the object encoder and the relation encoder.Nodes' representations are generated from the sum of the MP modules.The representations are then fused with question representations to predict answers.\",\n \"content\": \"Fig. 2 .\\n2\\nFig. 2. Model structure of the Dual Message-passing enhanced Graph Neural Networks.MP stands for the message-passing module.Images are transformed into scene graphs by the scene graph generator.The object-significant form and relationsignificant form of the scene graph are injected into the object encoder and the relation encoder.Nodes' representations are generated from the sum of the MP modules.The representations are then fused with question representations to predict answers.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Fig. 3. Overview of the pipeline at timestep t.(a) The message-passing module generates M P in and M P out as incident and output information gains.(b) The propagator uses information gains and h t−1 i\",\n \"content\": \"Fig. 3 .\\n3\\nFig. 3. Overview of the pipeline at timestep t.(a) The message-passing module generates M P in and M P out as incident and output information gains.(b) The propagator uses information gains and h t−1 i\"\n },\n {\n \"id\": \"\",\n \"caption\": \"a) shows visualization on \\\"what\\\" question type.Three \\\"what\\\" question examples aimed at retrieving either object, relation or attribute information.Comparing row 1, row 2 with row 3, Obj and Rel models have strong attention bias toward objects and relations, while their combination Obj+Rel, balances the attention on both sides and captures correct answers.Comparing row 3 with row 4, the messagepassing module increases the score of correct answers.Fig. 4 (b)\",\n \"content\": \"Fig. 4 (\\n4\\na) shows visualization on \\\"what\\\" question type.Three \\\"what\\\" question examples aimed at retrieving either object, relation or attribute information.Comparing row 1, row 2 with row 3, Obj and Rel models have strong attention bias toward objects and relations, while their combination Obj+Rel, balances the attention on both sides and captures correct answers.Comparing row 3 with row 4, the messagepassing module increases the score of correct answers.Fig. 4 (b)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Performance on different question types of VG dataset.\",\n \"content\": \"Table 1 .\\n1\\nPerformance on different question types of VG dataset.\\nModelsBinary↑ Open↑ Validity↑ Distribution↓ Acc.↑Human91.2087.4098.90-89.30BottomUp66.6434.8396.185.9849.74MAC71.2338.9196.165.3454.06SK T-Brain77.4243.1096.267.5459.19PVR77.6943.0196.455.8059.27GRN77.5343.3596.186.0659.37Dream77.8443.7296.388.4059.72LXRT77.7644.9796.308.3160.34NSM78.9449.2596.413.7163.17ReGAT83.5762.5892.709.3270.50DM-GNN(ours) 69.7972.2193.803.7871.21\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Performance on the GQA dataset.\",\n \"content\": \"Table 2 .\\n2\\nPerformance on the GQA dataset.\\nModelsAcc.Base35.4Base-Obj35.4+MP39.3 (+3.9)+Dual67.9 (+32.7)Base-Rel35.2+MP38.8 (+3.6)+Dual67.9 (+32.5)DM-GNN(ours) 75.2+ (w/o attr)71.6 (−3.6)+ (w/o rela)74.5 (−0.7)+ (w/o QF)54.9 (−20.3)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Ablation Study on VG.\",\n \"content\": \"Table 3 .\\n3\\nAblation Study on VG.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Vg- Dataset, GroundTruth Motif-VG Question type What Where Who Why Overall What Where Who Why Overall Percentage (54%) (17%) (5%) (3%) (100%) (54%) (17%).","doi":""},{"text":"Scene graph reasoning for visual question answering. M Hildebrandt, H Li, R Koner, V Tresp, S Günnemann, 2020CoRR","doi":""},{"text":"Understanding the role of scene graphs in visual question answering. V Damodaran, S Chakravarthy, A Kumar, A Umapathy, T Mitamura, Y Nakashima, N Garcia, C Chu, abs/2101.05479CoRR. 2021","doi":""},{"text":"An empirical study on leveraging scene graphs for visual question answering. C Zhang, W Chao, D Xuan, 2019","doi":""},{"text":"Learning by abstraction: The neural state machine. D A Hudson, C D Manning, 2019","doi":""},{"text":"From strings to things: Knowledgeenabled vqa model that can read and reason. Aj, An Singh, Sh Mishra, An Shekhar, Chakraborty,","doi":""},{"text":"Relation-aware graph attention network for visual question answering. L Li, Z Gan, Y Cheng, J Liu, 2019","doi":""},{"text":"Prior visual relationship reasoning for visual question answering. Z Yang, Z Qin, J Yu, T Wan, 2020","doi":""},{"text":"Spatial-aware graph relation network for large-scale object detection. H Xu, C Jiang, X Liang, Z Li, CVPR. 2019","doi":""},{"text":"BERT: pre-training of deep bidirectional transformers for language understanding. J Devlin, M Chang, K Lee, K Toutanova, NAACL-HLT. 2019","doi":""},{"text":"Differential attention for visual question answering. N Badri, Patro, P Vinay, Namboodiri, 2018 CVPR","doi":""},{"text":"Hierarchical question-image co-attention for visual question answering. J Lu, J Yang, D Batra, D Parikh, 2016","doi":""},{"text":"Compositional attention networks for machine reasoning. D A Hudson, C D Manning, 2018","doi":""},{"text":"Interpretable visual question answering by reasoning on dependency trees. Q Cao, X Liang, B Li, L Lin, IEEE TPAMI. 2021","doi":""},{"text":"Graph attention networks. P Velickovic, G Cucurull, A Casanova, A Romero, P Liò, Y Bengio,","doi":""},{"text":"Transition-based chinese semantic dependency graph parsing. Y Wang, J Guo, W Che, T Liu, NLP-NABD. 2016","doi":""},{"text":"A neural transition-based approach for semantic dependency graph parsing. Y Wang, W Che, J Guo, T Liu, 2018","doi":""},{"text":"GloVe: Global vectors for word representation. J P , R Socher, C Manning, 2014","doi":""},{"text":"GQA: A new dataset for real-world visual reasoning and compositional question answering. D A Hudson, C D Manning, 2019","doi":""},{"text":"Sgg codebase in pytorch. K Tang, 2020","doi":""},{"text":"Zoom-net: Mining deep feature interactions for visual relationship recognition. G Yin, L Sheng, B Liu, N Yu, X Wang, J Shao, Chen C , 2018","doi":""},{"text":"Unbiased scene graph generation from biased training. K Tang, Y Niu, J Huang, J Shi, H Zhang, 2020","doi":""},{"text":"Unicoder-vl: A universal encoder for vision and language by cross-modal pre-training. G Li, N Duan, Y Fang, G Ming, J Daxin, 2020","doi":""},{"text":"Pre-trained models: Past, present and future. X Han, Z Zhang, N Ding, J Wen, J Yuan, W Zhao, J Zhu, 2021CoRR","doi":""},{"text":"Graphvqa: Languageguided graph neural networks for scene graph question answering. W Liange, Y Jiang, Z Liu, 2021","doi":""},{"text":"Gated graph sequence neural networks. Y Li, D Tarlow, M Brockschmidt, R S Zemel, 2016","doi":""},{"text":"Visual genome: Connecting language and vision using crowdsourced dense image annotations. R Krishna, Y Zhu, O Groth, J Johnson, K Hata, J Kravitz, S Chen, Y Kalantidis, L Li, D A Shamma, M S Bernstein, Li Fei-Fei, Int. J. Comput. Vis. 2017","doi":""}],"string":"[\n {\n \"text\": \"Vg- Dataset, GroundTruth Motif-VG Question type What Where Who Why Overall What Where Who Why Overall Percentage (54%) (17%) (5%) (3%) (100%) (54%) (17%).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Scene graph reasoning for visual question answering. M Hildebrandt, H Li, R Koner, V Tresp, S Günnemann, 2020CoRR\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Understanding the role of scene graphs in visual question answering. V Damodaran, S Chakravarthy, A Kumar, A Umapathy, T Mitamura, Y Nakashima, N Garcia, C Chu, abs/2101.05479CoRR. 2021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An empirical study on leveraging scene graphs for visual question answering. C Zhang, W Chao, D Xuan, 2019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Learning by abstraction: The neural state machine. D A Hudson, C D Manning, 2019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"From strings to things: Knowledgeenabled vqa model that can read and reason. Aj, An Singh, Sh Mishra, An Shekhar, Chakraborty,\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Relation-aware graph attention network for visual question answering. L Li, Z Gan, Y Cheng, J Liu, 2019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Prior visual relationship reasoning for visual question answering. Z Yang, Z Qin, J Yu, T Wan, 2020\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Spatial-aware graph relation network for large-scale object detection. H Xu, C Jiang, X Liang, Z Li, CVPR. 2019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"BERT: pre-training of deep bidirectional transformers for language understanding. J Devlin, M Chang, K Lee, K Toutanova, NAACL-HLT. 2019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Differential attention for visual question answering. N Badri, Patro, P Vinay, Namboodiri, 2018 CVPR\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hierarchical question-image co-attention for visual question answering. J Lu, J Yang, D Batra, D Parikh, 2016\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Compositional attention networks for machine reasoning. D A Hudson, C D Manning, 2018\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Interpretable visual question answering by reasoning on dependency trees. Q Cao, X Liang, B Li, L Lin, IEEE TPAMI. 2021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Graph attention networks. P Velickovic, G Cucurull, A Casanova, A Romero, P Liò, Y Bengio,\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Transition-based chinese semantic dependency graph parsing. Y Wang, J Guo, W Che, T Liu, NLP-NABD. 2016\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A neural transition-based approach for semantic dependency graph parsing. Y Wang, W Che, J Guo, T Liu, 2018\",\n \"doi\": \"\"\n },\n {\n \"text\": \"GloVe: Global vectors for word representation. J P , R Socher, C Manning, 2014\",\n \"doi\": \"\"\n },\n {\n \"text\": \"GQA: A new dataset for real-world visual reasoning and compositional question answering. D A Hudson, C D Manning, 2019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Sgg codebase in pytorch. K Tang, 2020\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Zoom-net: Mining deep feature interactions for visual relationship recognition. G Yin, L Sheng, B Liu, N Yu, X Wang, J Shao, Chen C , 2018\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Unbiased scene graph generation from biased training. K Tang, Y Niu, J Huang, J Shi, H Zhang, 2020\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Unicoder-vl: A universal encoder for vision and language by cross-modal pre-training. G Li, N Duan, Y Fang, G Ming, J Daxin, 2020\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Pre-trained models: Past, present and future. X Han, Z Zhang, N Ding, J Wen, J Yuan, W Zhao, J Zhu, 2021CoRR\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Graphvqa: Languageguided graph neural networks for scene graph question answering. W Liange, Y Jiang, Z Liu, 2021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Gated graph sequence neural networks. Y Li, D Tarlow, M Brockschmidt, R S Zemel, 2016\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Visual genome: Connecting language and vision using crowdsourced dense image annotations. R Krishna, Y Zhu, O Groth, J Johnson, K Hata, J Kravitz, S Chen, Y Kalantidis, L Li, D A Shamma, M S Bernstein, Li Fei-Fei, Int. J. Comput. Vis. 2017\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":475,"cells":{"text":{"kind":"string","value":"\nExperimental designs for binary data in switching measurements on superconducting Josephson junctions\n2007\n\nJuha Karvanen \nNational Public Health Institute\nHelsinkiFinland\n\nJuha J Vartiainen \nHelsinki University of Technology\nEspooFinland\n\nAndrey Timofeev \nHelsinki University of Technology\nEspooFinland\n\nJukka Pekola \nHelsinki University of Technology\nEspooFinland\n\nExperimental designs for binary data in switching measurements on superconducting Josephson junctions\n\nAppl. Statist\n5622007[Received September 2005. Revised November 2006]0035-9254/07/56167Complementary log-log-linkD-optimalityEscape measurementsLogistic regressionOptimal designQuantum physics\nWe study the optimal design of switching measurements of small Josephson junction circuits which operate in the macroscopic quantum tunnelling regime. In the experiment, sequences of current pulses are applied to the Josephson junction sample, while the voltage over the structure is monitored. The appearance of a voltage pulse to a single applied current pulse, being governed by the laws of quantum mechanics, is purely random. Starting from the D-optimality criterion we derive the optimal design for the estimation of the unknown parameters of the underlying Gumbel-type distribution. As a practical method for the measurements, we propose a sequential design that combines heuristic search for initial estimates and maximum likelihood estimation. The design presented has immediate applications in the area of superconducting electronics, implying faster data acquisition. The experimental results presented confirm the usefulness of the method.\n\nIntroduction\n\nRandomness plays a central role in quantum physics. On the fundamental level, randomness is not caused only by unknown covariates or measurement error but is understood to be an intrinsic property of nature itself (Ballentine, 1998). From the statistical point of view, this leads to fascinating problems where the distributional assumptions arise directly from the fundamental laws of physics.\n\nJosephson junctions (JJs) are important non-linear components of superconducting electronics. A typical junction involves two superconducting electrodes which are separated by an insulator gap (Josephson, 1962). In practice, miniature JJ circuits are manufactured by lithographic means and operated at cryogenic temperatures, i.e. below 4.2 K. The strong dependence of the physical parameters of JJ circuits as a function of changes in environmental variables, e.g. temperature, electric noise and magnetic fields, gives JJ circuits several applications as ultrasensitive sensors (Pekola et al., 2005). Moreover, certain JJ circuits are promising candidates for realization of quantum computation (Makhlin et al., 2001). For a thorough review of JJ circuits see, for instance, Grabert and Devoret (1992).\nV 5µm I (a) (b) (c) (d)\n\nFig. 1.\n\nIllustration of a switching measurement: (a) scanning electron micrograph of a typical two-junction JJ sample; (b) sketch of applied current I pulses and resulting voltage V pulses; (c) associated switching probability as a function of the height of the current pulses; (d) quantum mechanical process leading to switching; the phase 'particle' is initially localized in a local minimum of the potential-when the current pulse is applied, the phase particle tunnels through the potential barrier and escapes downhill An experiment called switching measurement (Fig. 1) is a common way to probe the properties of a JJ circuit sample. In the experiment, sequences of current pulses are applied to the sample, while the voltage over the structure is monitored. The superconducting electrodes of the sample are described by a complex-valued quantum mechanical wave function and the physical state of the JJ is described by the phase difference over the junction. The dynamics of the phase difference are analogous to those of a massive particle in a tilted cosine potential; see Fig. 1(d). A sufficiently high current pulse causes the phase difference to escape from the local minimum of the potential to the direction of steepest descent. In the experiment the observed voltage pulse indicates that switching has occurred. The appearance of a voltage pulse from a single applied current pulse, being governed by the laws of quantum mechanics, is purely random. Below, we denote an observed voltage pulse by Y = 1; otherwise Y = 0. The analysis of the statistics of this random variable with respect to current pulse heights yields information about the physical parameters of a JJ circuit.\n\nWe study the switching dynamics of a single JJ. Quantum mechanical arguments that are studied in more detail in Section 2 suggest that the switching rate is of the form\nγ = exp.ax + b/,\nwhere x = x.I/ is a monotonic function depending on the height I of the applied current pulse and a and b are the parameters depending on the geometry and environment of the junction. Moreover, we note that x.I/ must be independent of the parameters a and b. Assuming a constant switching rate during the pulse we obtain the probability of measuring a voltage pulse at an applied current pulse\nP.Y = 1/ = 1 − exp{− exp.ax + b/}:\n. 1/ For simplicity we consider the case x.I/ = I in this paper. In statistics, model (1) is often called a complementary log-log-regression. Assume that a sequence of n current pulses of height I i , i = 1, 2, . . . , n, is applied to the sample. The data obtained consist of covariate values x i , i = 1, 2, . . . , n, and corresponding binary responses y i . Our aim is to choose the covariates such that the parameters a and b can be estimated as well as possible from the measured data. In statistical terms this is an optimal design problem (Atkinson and Donev, 1992;Pukelsheim, 1993;Liski et al., 2002;Berger and Wong, 2005). The literature on optimal designs is concentrated on linear models but work has been done also with non-linear models. Several researchers (Kalish and Rosenberger, 1978;Abdelbasit and Plackett, 1983;Minkin, 1987;Sitter and Wu, 1993;Mathew and Sinha, 2001) have considered optimal designs for binary data in the case of logistic regression\nP.Y = 1/ = 1 1 + exp{−.ax + b/} , .2/\nwhere a and b are unknown parameters and x is a covariate. Compared with logistic regression the complementary log-log-regression (1) is less frequently studied or applied. Complementary log-log-regression is often used as an alternative model for logistic regression (2) if the symmetry assumption does not fit the data, but in our application the complementary log-log-regression arises directly from the physical properties of JJs. The key reference considering optimal design for complementary log-log-regression is the work by Ford et al. (1992). They provided the general framework for the D-and c-optimal designs and the solution for complementary log-log-regression is obtained as a special case of their general results. In this paper we propose a sequential two-point design for estimating the unknown parameters of a JJ circuit. The design is derived by using the D-optimality criterion, which is equivalent to minimizing the generalized variance of the parameters. On the basis of the data that are available, the design yields the optimal current pulse heights for the next measurement in realtime. We are ultimately interested in minimizing the time that is needed for an experiment. The sequential design proposed also takes into account the special requirements of switching measurements, e.g. the time that is needed for the reconfiguration of the signal generator.\n\nThe rest of the paper is organized as follows. In Section 2, a more detailed description of physical background of the switching measurements is given. In Section 3, the optimal design for the complementary log-log-regression that is defined in equation (1) is derived. In Section 4, we propose a sequential design for switching measurements. In Section 5, the sequential design is applied to switching measurements of a JJ circuit. Results from the real data are compared with simulation results. Finally, Section 6 concludes the paper.\n\nThe measurement data and the related program code can be obtained from http://www.blackwellpublishing.com/rss\n\n\nPhysical background\n\nIn this section we study the quantum mechanical arguments leading to model (1). The switching dynamics are described by a Poisson process where the switching rate γ is constant over the current pulse which has duration Δt. The probability of switching is then obtained as\nP.Y = 1/ = 1 − exp.−Δtγ/: .3/\nThe duration Δt of the current pulse is very short and remains fixed in the experiment. Thus, we consider Δt as a part of the switching rate and do not write it explicitly in the subsequent equations. The relationship between the switching rate γ and the height of the current pulse I Switching rate γ as a function of the relative bias current I=I C ( , switching rate γ MQT in the macroscopic quantum tunnelling; -------, switching rate γ TA under the thermal activation model at temperature 1 K; ---, γ TA at temperature 0.1 K); the JJ parameters are set to have typical values, Josephson energy E J D 10 21 J and capacitance C D 50 fF; γ MQT is calculated on the high Q limit; the relationship between γ and the height of the current pulse I is nearly linear on the logarithmic scale, which supports the use of the approximation γ.I / D exp.aI C b/ can be written in the form γ.I/ = exp{a x.I/ + b}, .4/ where x.I/ is a monotonic function of the bias current I and a and b are constants depending on the physical parameters of the JJ circuit. A commonly used approximation is to assume that the activation threshold amplitude is linearly proportional to current I. This leads to the switching rate γ.I/ = exp.aI + b/: .5/\n\nAlthough not based on microscopic theory, the switching rate model (5) describes the results of typical switching measurements quite well (Fig. 2).\n\nTo obtain the functional form of x.I/ in the switching rate model (4) we need a more accurate physical model to describe the system that is under consideration. Theoretically, the superconducting electrodes of the JJ sample are modelled by a complex-valued quantum mechanical wave function and the physical state of the JJ is described by the phase difference over the junction. The dynamics of the phase difference φ are analogous to those of a massive particle in a tilted cosine potential U under gravitational force,\nU = −E J I I C φ + cos.φ/ ,.6/\nwhere E J is so-called Josephson energy, I is the magnitude of the bias current applied and I C is the critical current for which the potential changes into a monotonic function. The potential is depicted during a pulse in Fig. 1(d). The applied current changes the shape of the potential landscape of a JJ circuit in a controlled way. The phase difference φ initially set into one of the local minima becomes a metastable state for finite current I; the state may decay by escape of the phase from the local minimum of the potential to the direction of steepest descent. This escape event is called switching. In the experiment the voltage over the JJ sample is monitored and the observed voltage pulse indicates that switching has occurred. Depending on the temperature, model (1) is explained by the thermal activation model of classical physics or by quantum mechanical analysis. In an experiment at temperatures above 0.2 K, there will be thermal fluctuations in the bias current I as the JJ is connected to the measurement wiring. These fluctuations make the phase particle absorb energy and eventually escape from the potential well. One of the central results in statistical physics is that the probability for the system to be in energy state E at temperature T is proportional to exp.−E=k B T/, where k B ≈ 1:381 × 10 −23 J K −1 is the Boltzmann constant. To escape from the well the phase particle needs to have more energy than the barrier height ΔU. On the basis of these principles the average escape rate for the potential of equation (6) can be shown to follow Kramer's formula\nγ TA .I/ = ω P .I/ 2π exp − ΔU.I/ k B T , .7/\nwhere ΔU is the height of the potential barrier\nΔU.I/ = 4 √ 2E J 3 1 − I I C 3=2 , .8/\nand ω P is the characteristic oscillation frequency\nω P .I/ = E J C 1=2ē 2 1 − I I C 1=4 , .9/\nwhere C is the capacitance of the JJ,ē = 1:602 × 10 −19 C is the elementary charge and ≈ 1:055 × 10 −34 J s is the reduced Planck constant. In these equations E J and C are the unknown quantities-I C is a function of E J . It is easy to see that the switching rate (7) can be written in the form γ TA .I/ = exp{a x.I/ + b}. When a JJ is cooled to tens of millikelvins we can still observe the phase particle escaping from the quasi-stationary potential (Voss and Webb, 1981;Jackel et al., 1981). Since the thermal activation, equation (7), yields a vanishing switching rate for that temperature range, the microscopic mechanism must be different from that described above. This process is called macroscopic quantum tunnelling (Caldeira and Leggett, 1980) and to explain it a detailed quantum mechanical analysis is required. The analysis leads to switching rate (Grabert et al., 1985) γ MQT .I/ = A.I/ exp{−B.I/}:\n\n. 10/\n\nThe prefactor in equation (10) is\nA.I/ = ω P .I/ 2π ΔU.I/ ω P .I/ 1=2 χ 1 2Q , .11/ where χ.α/ = 12 √ .6π/{1 + cα + O.α 2 /}, with c ≈ 2:86\n, and Q is the quality factor of the oscillator describing the effective environment of the junction. The exponent reads\nB.I/ = ΔU.I/ ω P .I/ s 1 2Q , .12/ where s.α/ = 36 5 1 + 45ζ.3/ π 3 α + O.α 2 /\nand ζ.3/ ≈ 1:202 is the value of the Riemann ζ-function at 3. Again, the switching rate (10) can be written in the form γ MQT .I/ = exp{ax.I/ + b}.\n\n\nD-optimal designs for complementary log-log-regression\n\nAssume that a binary random variable Y is distributed as\nP.Y = 1/ = 1 − exp{− exp.ax + b/}, P.Y = 0/ = exp{− exp.ax + b/}, .13/\nwhere a and b are unknown parameters and x is a covariate. The cumulative distribution function\nF.z/ = 1 − exp{− exp.z/},.14/\nwhere z = ax + b, is known in the literature by the names Gumbel distribution, Gompertz distribution, log-Weibull distribution, Fisher-Tippett distribution and extreme value distribution, whereas the inverse of equation (14),\nF −1 .u/ = ax + b = log{− log.1 − u/}, .15/\nis called the complementary log-log-link and is one of the standard link functions in generalized linear models. The problem is to select such covariate values x 1 , x 2 , . . . , x n that the parameters a and b can be estimated from the observed data .x i , y i /, i = 1, 2, . . . , n, in optimal manner. Here we define the optimality as D-optimality, which is equivalent to maximizing the determinant of the information matrix, and estimate the parameters by maximum likelihood. Starting from definition (13) we obtain the log-likelihood of the data .x i , y i /, i = 1, 2, . . . , n:\nL.a, b/ = n i=1 .y i log[1 − exp{− exp.ax i + b/}] − .1 − y i / exp.ax i + b//: .16/ The information matrix J = −E ⎛ ⎜ ⎝ @ 2 L @a 2 @ 2 L @a @b @ 2 L @b @a @ 2 L @b 2 ⎞ ⎟ ⎠ .17/ takes the form J = ⎛ ⎜ ⎝ n i=1 g.z i /x 2 i n i=1 g.z i /x i n i=1 g.z i /x i n i=1 g.z i /, ⎞ ⎟ ⎠, .18/ where z i = ax i + b and g.z/ = exp.2z/ exp{exp.z/} − 1 :\n. 19/ The D-optimality is equivalent to finding such x 1 , x 2 , . . . , x n that\ndet.J/ = n i=1 g.z i /x 2 i n i=1 g.z i / − n i=1 g.z i /x i 2 = n j=i+1 n−1 i=1 g.z i / g.z j /.x i − x j / 2 .20/\nis maximized. It is assumed in the maximization that the true values of a and b are known (see Minkin (1987) and Sitter (1992) for a discussion). First we find a locally optimal two-point design n = 2. This simplifies equation (20) to\ndet.J/ = 1 a 2 g.z 1 / g.z 2 /.z 1 − z 2 / 2 : . 21/\nThe contour plot of det.J/ as a function z 1 and z 2 is depicted in Fig. 3. The values z 1 and z 2 that maximize det.J/,\nz Å 1 ≈ 0:979 633, z Å 2 ≈ −1:33774,.22/\ncan be solved numerically from equation (21). The corresponding response probabilities are\nF.z Å 1 / ≈ 0:930 295, F.z Å 2 / ≈ 0:230 826,\n.23/ indicating that the design is asymmetric with respect to the middle point of the response curve.\n\nSince the function is symmetric with respect to exchanging the variables, the mirror points provide the same maximum value also. To show that the two-point maximum (22) is a local maximum also when n > 2, we define functions v.z 1 , z 1 / = g.z 1 / g.z 2 /.z 1 − z 2 / 2 , .24/ v .z 1 , z 2 / = @ @z 1 v.z 1 , z 2 /: .25/ By inspecting the functional form of v. Because v .z Å 1 , z Å 2 / = 0, a solution for equation (29) is obtained if\nz i , z j / we observe that v.z Å 1 , z Å 2 / = v.z Å 2 , z Å 1 / ⇒ local maximum of v,.26/z i ∈ {z Å 1 , z Å 2 } for all i: .30/\nFrom equation (26) we conclude that a local maximum is obtained, for instance, when\nz i = z Å 1 if i is odd, z Å 2 if i is even.\n.31/\n\nFor the complete proof that the solution (22) is the global maximum, we refer to the general results by Ford et al. (1992). Using a theorem by Kiefer and Wolfowitz (1960), Ford et al. (1992) showed for a class of response curves that the D-optimal design is a two-point design. The proof for the complementary log-log-regression is obtained as a special case of the general results.\n\n\nSequential designs for switching measurements\n\nOur goal is to design a fast automated estimation procedure for switching measurements. A fast estimation procedure allows, for example, the tracking of changes in the parameters as a function of changes in environmental variables. The speed of the procedure is measured in terms of the total time used for measuring and estimation. Sequential designs are a natural choice because it is easy to control the current during the experiment. In sequential designs (also known as two-stage and multistage designs), we start with initial estimates, collect additional data, obtain new estimates and use them as the initial estimates for the next stage.\n\nIn switching measurements, the number of stages can be high: this is a significant difference compared with medical dose-response experiments where practical reasons usually limit the number of stages. However, it is not necessarily advisable to choose the number of stages as high as possible, i.e. update the estimates after every two observations. This is because adding two new observations would have a very small effect on the estimates (especially if the number of observations is already large) but the estimation itself would spend time that could be used for measuring. Additionally, whenever the height of the current pulse is changed, the reconfiguration of the signal generator also takes a time that is considerably longer than the time that is needed for measuring the response for a single pulse. The optimal updating interval depends also on the computational resources and facilities, and our experiences suggest that it is always reasonable to measure at least 20 times using the same pulse height. Our recommendation is only the lower limit and based on empirically experience on the specific measurement instruments that we are using. In the experiment that is presented in Section 5, we measured 50 times using the same pulse height at the beginning of the experiment and increased this number by 10% at every stage so that at stage 50 we measured 5379 times using the same pulse height.\n\nObtaining good initial estimates for the parameters is a challenging problem for switching measurements. Naturally, the results from the previous experiments can be used to obtain initial estimates for the parameters but there is no guarantee that this would lead to good initial estimates. The problem of poor initial estimates in the case of logistic regression was considered for example by King and Wang (2000) and Sitter (1992). They proposed sophisticated minimax approaches that have good optimality properties even if the initial estimates are poor. These approaches are probably applicable also to our problem but, because of their computational complexity, we choose a simpler approach that especially suits switching measurements. It is usually possible to specify an interval from where the covariates should be taken but the problem is that this interval might be very wide compared with the interval where the probability of response 1 increases, for instance, from 0.01 to 0.99. This implies that initial covariates that are chosen randomly from the interval may be very inefficient. In the worst case, poorly chosen initial covariates may lead to the situation where maximum likelihood estimates (MLEs) do not exist for the initial data. The existence of MLEs for model (13) requires that (Albert and Anderson, 1984) max.x i |y i = 0/ > min.x i |y i = 1/, max.x i |y i = 1/ > min.x i |y i = 0/:\n\n.32/ This condition is fulfilled, for instance, if we have two points where both 0s and 1s are measured as responses. Our approach for the initial estimation aims at finding two such points as quickly as possible. This heuristic procedure can be presented as follows. x u − x l /=4. The sign of \" is determined according to the measured response: sgn.\"/ = sgn.0:5 −ȳ/, whereȳ is the average response for x. If 0:5 −ȳ = 0 the sign of \" is chosen randomly. (d) If MLEs exist for the data measured so far, proceed to the maximum likelihood estimation. Otherwise, divide \" by 2 and return to step (c).\n\nStep (b) requires that we measure at least two times at each point. In practice, it is preferable to measure at least tens of times at each point because of the time that is needed for the reconfiguration of the signal generator. When the binary search in step (b) converges we have found one of the two points that are required for the maximum likelihood estimation. The other point that is needed is then found in the neighbourhood of point x using again binary search. Owing to use of binary search in steps (b) and (c), the procedure works reliably and efficiently. At every step of the binary search, the length of the current interval is divided by 2. Consequently, the procedure converges exponentially fast regardless of the choice of the initial interval.\n\nAfter the heuristic procedure has converged the experiment continues sequentially: the current values ofâ andb are used to determine the optimal covariates for the next stage, measurements are performed and finally maximum likelihood estimation is applied to update values ofâ and b. In switching measurements, we are also interested in parameters that directly describe the location and the scale of the response curve in addition to parameters a and b. Let us define the middle point θ and width of the curve λ, as follows: . 36/ It is often practical to define the stopping rule for the data acquisition by using the variances of θ and λ. Special attention should be paid to the stability of the measurement environment. Instability, such as a change in the temperature during the experiment, causes an increase in the variances. This emphasizes the need for fast and reliable experimental design.\nθ = 1 a {F −\n\nExamples with real and simulated data\n\nWe measured a sample consisting of an aluminium-aluminium oxide-aluminium JJ circuit in a dilution refrigerator at 20 mK temperature. The sample parameters a and b depend on the area of the JJ and the thickness of the oxide layer. The sample fabricated involves a JJ whose area is 1 μm × 2 μm and the oxide layer is 3 nm thick. The sample was connected to computer-controlled measurement electronics to apply the current pulses and to record the resulting voltage pulses. The resistance of the sample at room temperature suggests that a pulse of 300 nA always causes a switching (response 1). This gives the upper limit for the initial estimation. The lower limit for the initial estimation, 200 nA, can be roughly estimated from the dimensions of the JJ. The sequential design that was presented in Section 4 was used to construct the applied current pulse sequences in realtime. Each stage consisted of a measuring phase and an estimation phase. Since the reconfiguration of our signal generator takes a relatively long time compared with the length of a single pulse, we used sequences of tens or hundreds of pulses to collect data at each stage. In the set-up the pulse lengths are of the order of milliseconds. To keep the time that is needed for computations much below the actual measurement time, the length of the pulse sequence should grow exponentially. We started with a sequence of 50 pulses per point and let the number of pulses grow by 10% at each stage. In the test experiments, the practically sufficient accuracy of the estimates was reached after 20-30 stages, implying that 6000-17 000 pulses were needed in total. In our set-up, the experiment (50 stages) took 8 min 16 s from which 1 min 30 s were used for estimation. Table 1 and Fig. 4 illustrate the estimation of the unknown parameters in the experiment. In Fig. 4(b) the functional form of the response curve is verified empirically. The response curve is compared with individual response probabilities that are independently estimated using 2000 pulses per point. The results confirm that the response curve is given by the Gumbel distribution as suggested by the theory.\n\nThe performance of the sequential design is also studied in a simulation example. The simulation aims to imitate the real data example. The response variable Y is generated from model (13) with parameters a = 0:24 and b = −61 which are motivated by the actual experiment that was presented above. Our goal is to estimate these parameters when we initially know only that at point x min = 200 the response is 0 with high probability and at point x max = 300 the response is 1 with high probability. The estimation starts by using the heuristic procedure described in Section 4 and continues as maximum likelihood estimation. In the initial estimation, we measure 25 times at each point. In the sequential maximum likelihood estimation, the parameter estimates are updated when the total number of observations has increased by 20% from the previous update. The simulation is repeated 500 times and MLEsâ andb are recorded for different numbers of observations n = 50, n = 100, . . . , n = 20 000. The sequential design is compared with the theoretically D-optimal design where the optimal covariate values x Å 1 = .z Å 1 + 61/=0:24 = 258:25 and x Å 2 = .z Å 2 + 61/=0:24 = 248:59 are assumed to be known already at the beginning of the experiment. As a performance measure we calculate the mean-squared errors (MSEs) between the estimates and the true parameter values. Table 2 summarizes the MLEs obtained and their standard deviations. The numbers reported are means from 500 simulations. The MSEs between the estimates and the true parameter values are reported in Table 3. The MSEs are calculated only from those simulation runs where MLEs exist. Naturally, the theoretically optimal design results in smaller MSE values than the sequential design but it can also be seen that for large values of n the performance of the sequential design approaches the performance of the theoretically optimal design. The results demonstrate the practical importance of good experimental design and the significance of reliable estimation of initial parameters. We conclude that the sequential design proposed performed reasonably well compared with the theoretically D-optimal design when n > 200. In switching measurements, the number of observations is always in thousands. A comparison of estimates and their standard deviations in Table 1 and Table 2 does not reveal any major differences between the experimental data and simulated data. This supports the practical usability of the sequential design.\n\n\nConclusion\n\nWe have considered optimal designs for switching measurements of superconducting JJ circuits. The D-optimality criterion suggests that the measurements at points where ax 1 + b = 0:979 633 and ax 2 + b = −1:337 74 will yield maximal information about the estimated parameters assuming that the probability distribution of the measurement values are given by equation (1). As a practical method for switching measurements, we propose a sequential design that combines heuristic search for initial estimates and maximum likelihood estimation. The fast data acquisition allows tracking of fast environmental changes. The experimental design presented is now in daily use in the Low Temperature Laboratory at Helsinki University of Technology.\n\nThe current work opens several directions for future research. Besides D-optimality we may also consider other optimality criteria. These criteria will probably lead to designs that are different from the D-optimal design but because of the complicated functional forms it may be difficult to derive analytical results.\n\nThe problem of choosing the number of measurements per stage was approached empirically in this paper. A more systematic approach would determine the optimal number of measurements per stage as a function of the observed data. It seems reasonable to assume that each measurement takes a certain time and the change of the measurement point takes additional fixed time. The problem is to choose the number of measurements per stage in such a way that the expected total time that is needed to achieve the accuracy required is minimized. This is a non-trivial problem and probably only approximate solutions are achievable.\n\nThe time that is used for the experiment could be further reduced if the estimation and the measuring were done simultaneously. This requires good co-ordination of data processing and knowledge on the time that is needed for each operation. In sequential designs, the simultaneous estimation and measuring implies that the current estimates are not based on all data measured so far but all data processed so far.\n\nAlthough the complementary log-log-regression model presented describes well some JJ circuits involving several junctions, we probably need more flexible models for complicated structures of JJs. The modelling can be approached from two different directions: we may derive the exact physical model and try to find the optimal design for it, or we may start with a flexible parametric model for which the optimal design is known and estimate the model parameters from the measured data.\n\nStatistical methods can be also applied to the modelling of JJ parameters as a function of environmental variables, such as temperature. This direction of research leads to adaptive sequential designs where the challenge is not only to estimate the parameters but also to track their changes in the optimal way.\n\nFig. 2 .\n2Fig. 2. Switching rate γ as a function of the relative bias current I=I C ( , switching rate γ MQT in the macroscopic quantum tunnelling; -------, switching rate γ TA under the thermal activation model at temperature 1 K; ---, γ TA at temperature 0.1 K); the JJ parameters are set to have typical values, Josephson energy E J D 10 21 J and capacitance C D 50 fF; γ MQT is calculated on the high Q limit; the relationship between γ and the height of the current pulse I is nearly linear on the logarithmic scale, which supports the use of the approximation γ.I / D exp.aI C b/\n\nFig. 3 .\n3Contours of the ratio 100 det{J.z 1 , z 2 /}= det{J.z Å 1 , z Å 2 /}, where det{J.z 1 , z 2 /} is the determinant of the information matrix for the two-point design .z 1 , z 2 /: the maximum of det.J/ is achieved at the point .\n\n( a )\naUse previous knowledge to construct such an interval [x min , x max ] that we can be sure that F.x max ; a, b/ − F.x min ; a, b/ is close to 1. Constructing this kind of interval is usually possible even if we have very little knowledge about the parameters.(b) Use binary search to find such a point x from the interval [x min , x max ] that both 0s and 1s are measured as responses. In binary search, we measure the responses at the middle point of the current interval. If only 0s are measured, the middle point is taken as the new starting-point of the interval. If only 1s are measured, the middle point is taken as the new end point of the interval. Let x be found as the middle point of the interval [x l , x u ], i.e. x = .x l + x u /=2. (c) Measure at the point x + \", where |\"| = .\n\nFig. 4 .\n4parameters are frequently used in switching measurements to characterize the response curve. The covariance matrix of .θ λ/ T is given by AJ −1 A T , Estimated response curve for the measured JJ: (a) estimated response curve with data used in the estimation (pulse heights I and corresponding empirical probabilities P ); (b) estimated response curve and verification points obtained as an average of responses of 2000 independent pulses\n\nTable 1 .\n1Estimates and their standard deviations for the measured JJStage \nnâ \nstd(â)b \nstd(b) \n\n1 \n100 \nNo MLE \nNo MLE \n2 \n210 \nNo MLE \nNo MLE \n3 \n332 \n0.3800 \n0.1933 \n−95.5966 \n48.4418 \n4 \n466 \n0.2585 \n0.0392 \n−65.1942 \n9.8369 \n5 \n614 \n0.2301 \n0.0224 \n−58.1248 \n5.6501 \n6 \n776 \n0.2520 \n0.0177 \n−63.6371 \n4.4625 \n7 \n954 \n0.2474 \n0.0148 \n−62.5090 \n3.7268 \n8 \n1150 \n0.2368 \n0.0126 \n−59.8039 \n3.1855 \n9 \n1366 \n0.2366 \n0.0112 \n−59.7514 \n2.8222 \n10 \n1604 \n0.2426 \n0.0102 \n−61.2916 \n2.5744 \n15 \n3200 \n0.2350 \n0.0067 \n−59.3476 \n1.7022 \n20 \n5762 \n0.2427 \n0.0050 \n−61.2986 \n1.2637 \n25 \n9890 \n0.2433 \n0.0038 \n−61.4761 \n0.9572 \n30 \n16550 \n0.2424 \n0.0029 \n−61.2533 \n0.7359 \n40 \n44578 \n0.2410 \n0.0017 \n−60.8778 \n0.4430 \n50 \n117288 \n0.2402 \n0.0011 \n−60.6282 \n0.2708 \n\nθ \nstd(θ)λ \nstd(λ) \n\n1 \n100 \nNo MLE \nNo MLE \n2 \n210 \nNo MLE \nNo MLE \n3 \n332 \n250.5738 \n0.2519 \n8.1158 \n4.1282 \n4 \n466 \n250.8324 \n0.3037 \n11.9342 \n1.8080 \n5 \n614 \n251.0506 \n0.2984 \n13.4066 \n1.3070 \n6 \n776 \n251.0632 \n0.2519 \n12.2392 \n0.8591 \n7 \n954 \n251.1471 \n0.2327 \n12.4655 \n0.7432 \n8 \n1150 \n251.0128 \n0.2219 \n13.0259 \n0.6933 \n9 \n1366 \n250.9891 \n0.2058 \n13.0362 \n0.6148 \n10 \n1604 \n251.1269 \n0.1874 \n12.7136 \n0.5330 \n15 \n3200 \n250.9385 \n0.1386 \n13.1228 \n0.3751 \n20 \n5762 \n251.1056 \n0.1019 \n12.7110 \n0.2610 \n25 \n9890 \n251.1342 \n0.0780 \n12.6756 \n0.1965 \n30 \n16550 \n251.1604 \n0.0605 \n12.7233 \n0.1522 \n40 \n44578 \n251.1016 \n0.0372 \n12.7992 \n0.0927 \n50 \n117288 \n250.8531 \n0.0231 \n12.8395 \n0.0571 \n\n\n\nTable 2 .\n2Means of estimates and their standard deviations from 500 simulation runsn \nSequential design \nOptimal design \nSequential design \nOptimal design \n\na \nstd(â)â \nstd(â)b \nstd(b)b \nstd(b) \n\n50 \n0.4342 \n0.3953 \n0.2454 \n0.0659 \n−109.6947 \n99.0491 \n−62.4331 \n16.8447 \n100 \n0.2786 \n0.1151 \n0.2476 \n0.0387 \n−70.7275 \n28.8748 \n−62.9339 \n9.8681 \n200 \n0.2482 \n0.0373 \n0.2420 \n0.0262 \n−63.0732 \n9.4241 \n−61.4907 \n6.6879 \n300 \n0.2448 \n0.0254 \n0.2418 \n0.0213 \n−62.2302 \n6.4475 \n−61.4518 \n5.4436 \n400 \n0.2433 \n0.0203 \n0.2413 \n0.0184 \n−61.8373 \n5.1676 \n−61.3297 \n4.6997 \n500 \n0.2425 \n0.0176 \n0.2405 \n0.0164 \n−61.6395 \n4.4892 \n−61.1274 \n4.1928 \n700 \n0.2415 \n0.0145 \n0.2408 \n0.0139 \n−61.3900 \n3.6946 \n−61.2015 \n3.5441 \n1000 \n0.2418 \n0.0120 \n0.2409 \n0.0116 \n−61.4646 \n3.0482 \n−61.2335 \n2.9653 \n1500 \n0.2410 \n0.0096 \n0.2406 \n0.0095 \n−61.2491 \n2.4610 \n−61.1591 \n2.4188 \n2000 \n0.2411 \n0.0083 \n0.2408 \n0.0082 \n−61.2892 \n2.1225 \n−61.2076 \n2.0953 \n3000 \n0.2408 \n0.0068 \n0.2405 \n0.0067 \n−61.2001 \n1.7240 \n−61.1375 \n1.7089 \n4000 \n0.2406 \n0.0058 \n0.2405 \n0.0058 \n−61.1488 \n1.4894 \n−61.1361 \n1.4799 \n5000 \n0.2405 \n0.0052 \n0.2402 \n0.0052 \n−61.1338 \n1.3305 \n−61.0467 \n1.3223 \n7000 \n0.2404 \n0.0044 \n0.2401 \n0.0044 \n−61.0939 \n1.1225 \n−61.0178 \n1.1170 \n10000 \n0.2402 \n0.0037 \n0.2401 \n0.0037 \n−61.0621 \n0.9378 \n−61.0184 \n0.9344 \n15000 \n0.2402 \n0.0030 \n0.2400 \n0.0030 \n−61.0505 \n0.7648 \n−61.0077 \n0.7629 \n20000 \n0.2401 \n0.0026 \n0.2401 \n0.0026 \n−61.0296 \n0.6619 \n−61.0128 \n0.6607 \n\nθ \nstd(θ )θ \nstd(θ )λ \nstd(λ)λ \nstd(λ) \n\n50 \n254.3894 \n7.2977 \n252.7209 \n1.3866 \n21.0825 \n38.7785 \n13.1512 \n3.6519 \n100 \n253.1132 \n1.7524 \n252.6023 \n0.8089 \n14.3413 \n7.8859 \n12.7611 \n2.0261 \n200 \n252.7223 \n0.5869 \n252.5675 \n0.5638 \n12.7674 \n1.9482 \n12.8914 \n1.4084 \n300 \n252.6707 \n0.4449 \n252.5894 \n0.4589 \n12.7352 \n1.3232 \n12.8599 \n1.1416 \n400 \n252.6603 \n0.3783 \n252.5911 \n0.3972 \n12.7684 \n1.0674 \n12.8631 \n0.9864 \n500 \n252.6529 \n0.3393 \n252.5970 \n0.3554 \n12.7853 \n0.9307 \n12.8892 \n0.8837 \n700 \n252.6479 \n0.2892 \n252.6178 \n0.2994 \n12.8159 \n0.7700 \n12.8514 \n0.7427 \n1000 \n252.6554 \n0.2433 \n252.6364 \n0.2501 \n12.7847 \n0.6323 \n12.8315 \n0.6196 \n1500 \n252.6535 \n0.2003 \n252.6472 \n0.2041 \n12.8202 \n0.5134 \n12.8361 \n0.5058 \n2000 \n252.6544 \n0.1741 \n252.6507 \n0.1766 \n12.8081 \n0.4419 \n12.8224 \n0.4373 \n3000 \n252.6413 \n0.1428 \n252.6444 \n0.1443 \n12.8203 \n0.3597 \n12.8329 \n0.3573 \n4000 \n252.6417 \n0.1240 \n252.6472 \n0.1249 \n12.8284 \n0.3111 \n12.8305 \n0.3093 \n5000 \n252.6426 \n0.1110 \n252.6423 \n0.1118 \n12.8301 \n0.2780 \n12.8474 \n0.2771 \n7000 \n252.6454 \n0.0940 \n252.6379 \n0.0945 \n12.8368 \n0.2348 \n12.8517 \n0.2342 \n10000 \n252.6435 \n0.0788 \n252.6366 \n0.0791 \n12.8420 \n0.1963 \n12.8504 \n0.1959 \n15000 \n252.6401 \n0.0644 \n252.6376 \n0.0646 \n12.8432 \n0.1601 \n12.8518 \n0.1600 \n20000 \n252.6391 \n0.0558 \n252.6391 \n0.0559 \n12.8470 \n0.1387 \n12.8503 \n0.1385 \n\n\n\nTable 3 .\n3MSEs for parameter estimatesâ andb as functions of the number of observations n † 399793 †The sequential design proposed and the theoretically D-optimal design are compared. Reported MSEs are means from 500 simulation runs.n \nMSE ofâ \nMSE ofb \n\nSequential \nOptimal \nSequential \nOptimal \n\n50 \n0.206843 \n0.002715 \n12983.570197 \n179.184180 \n100 \n0.021449 \n0.001546 \n1348.770300 \n101.345741 \n200 \n0.001771 \n0.000667 \n112.423833 \n43.457153 \n300 \n0.000686 \n0.000477 \n44.089057 \n31.041668 \n400 \n0.000439 \n0.000373 \n28.286722 \n24.314540 \n500 \n0.000316 \n0.000297 \n20.389242 \n19.364127 \n700 \n0.000211 \n0.000196 \n13.672440 \n12.766761 \n1000 \n0.000139 \n0.000132 \n9.035246 \n8.643545 \n1500 \n0.000093 \n0.000080 \n6.046665 \n5.201665 \n2000 \n0.000077 \n0.000063 \n4.993876 \n4.132176 \n3000 \n0.000049 \n0.000045 \n3.198680 \n2.937851 \n4000 \n0.000037 \n0.000032 \n2.390940 \n2.100386 \n5000 \n0.000030 \n0.000024 \n1.946899 \n1.584166 \n7000 \n0.000021 \n0.000017 \n1.379387 \n1.116941 \n10000 \n0.000015 \n0.000012 \n0.961786 \n0.811790 \n15000 \n0.000010 \n0.000008 \n0.634116 \n0.523743 \n20000 \n0.000007 \n0.000006 \n0.473926 \n0.\n\nExperimental design for binary data. K M Abdelbasit, R L Plackett, J. Am. Statist. Ass. 78Abdelbasit, K. M. and Plackett, R. L. (1983) Experimental design for binary data. J. Am. Statist. Ass., 78, 90-98.\n\nOn the existence of maximum likelihood estimates in logistic regression models. A Albert, J A Anderson, Biometrika. 71Albert, A. and Anderson, J. A. (1984) On the existence of maximum likelihood estimates in logistic regression models. Biometrika, 71, 1-10.\n\nOptimum Experimental Design. A C Atkinson, A N Donev, Oxford University PressOxfordAtkinson, A. C. and Donev, A. N. (1992) Optimum Experimental Design. Oxford: Oxford University Press.\n\nQuantum Mechanics: a Modern Development. L E Ballentine, World Scientific PressSingaporeBallentine, L. E. (1998) Quantum Mechanics: a Modern Development. Singapore: World Scientific Press.\n\nApplied Optimal Design. M P F Berger, Wong, W.-K.WileyNew YorkBerger, M. P. F. and Wong, W.-K. (eds) (2005) Applied Optimal Design. New York: Wiley.\n\nInfluence of dissipation on quantum tunneling in macroscopic systems. A O Caldeira, A J Leggett, Phys. Rev. Lett. 46Caldeira, A. O. and Leggett, A. J. (1980) Influence of dissipation on quantum tunneling in macroscopic systems. Phys. Rev. Lett., 46, 211-214.\n\nThe use of a canonical form in the construction of locally optimal designs for non-linear problems. I Ford, B Torsney, C F J Wu, J. R. Statist. Soc. B. 54Ford, I., Torsney, B. and Wu, C. F. J. (1992) The use of a canonical form in the construction of locally optimal designs for non-linear problems. J. R. Statist. Soc. B, 54, 569-583.\n\nSingle charge tunneling. H Grabert, Devoret, NATO Adv. Study Inst. B. M. H.294Grabert, H. and Devoret, M. H. (eds) (1992) Single charge tunneling. NATO Adv. Study Inst. B, 294.\n\nTemperature dependence of quantum decay rates in dissipative systems. H Grabert, P Olschowski, U Weiss, Phys. Rev. B. 32Grabert, H., Olschowski, P. and Weiss, U. (1985) Temperature dependence of quantum decay rates in dissipative systems. Phys. Rev. B, 32, 3348-3350.\n\nDecay of the zero-voltage state in small-area, high-current-density Josephson junctions. L D Jackel, J P Gordon, E L Hu, R E Howard, L A Fetter, D M Tennant, R W Epworth, J Kurkijarvi, Phys. Rev. Lett. 47Jackel, L. D., Gordon, J. P., Hu, E. L., Howard, R. E., Fetter, L. A., Tennant, D. M., Epworth, R. W. and Kurkijarvi, J. (1981) Decay of the zero-voltage state in small-area, high-current-density Josephson junctions. Phys. Rev. Lett., 47, 697-700.\n\nOptimal designs for the estimation of the logistic regression. B D Josephson, L A Kalish, J L Rosenberger, 33Phys. Lett. 1251Pennsylvania State UniversityTechnical ReportPossible new effects in superconductive tunnellingJosephson, B. D. (1962) Possible new effects in superconductive tunnelling. Phys. Lett., 1, 251. Kalish, L. A. and Rosenberger, J. L. (1978) Optimal designs for the estimation of the logistic regression. Technical Report 33. Pennsylvania State University, Philadelphia.\n\nThe equivalence of two extremum problems. J Kiefer, J Wolfowitz, Can. J. Math. 12Kiefer, J. and Wolfowitz, J. (1960) The equivalence of two extremum problems. Can. J. Math., 12, 363-366.\n\nMinimax D-optimal designs for the logistic model. J King, W.-K Wong, Biometrics. 56King, J. and Wong, W.-K. (2000) Minimax D-optimal designs for the logistic model. Biometrics, 56, 1263-1267.\n\nTopics in Optimal Design. E P Liski, N K Mandal, K R Shah, B K Sinha, SpringerNew YorkLiski, E. P., Mandal, N. K., Shah, K. R. and Sinha, B. K. (2002) Topics in Optimal Design. New York: Springer.\n\nQuantum-state engineering with Josephson-junction devices. Y Makhlin, G Schön, A Shnirman, Rev. Mod. Phys. 73357Makhlin, Y., Schön, G. and Shnirman, A. (2001) Quantum-state engineering with Josephson-junction devices. Rev. Mod. Phys., 73, 357.\n\nOptimal designs for binary data under logistic regression. T Mathew, B K Sinha, J. Statist. Planng Inf. 93Mathew, T. and Sinha, B. K. (2001) Optimal designs for binary data under logistic regression. J. Statist. Planng Inf., 93, 295-307.\n\nOptimal designs for binary data. S Minkin, J. Am. Statist. Ass. 82Minkin, S. (1987) Optimal designs for binary data. J. Am. Statist. Ass., 82, 1098-1103.\n\nShotnoise-driven escape in hysteretic Josephson junctions. J P Pekola, T E Nieminen, M Meschke, J M Kivioja, A O Niskanen, J J Vartiainen, F Pukelsheim, Phys. Rev. Lett. 95WileyBiometricsPekola, J. P., Nieminen, T. E., Meschke, M., Kivioja, J. M., Niskanen, A. O. and Vartiainen, J. J. (2005) Shot- noise-driven escape in hysteretic Josephson junctions. Phys. Rev. Lett., 95, 197004. Pukelsheim, F. (1993) Optimal Design of Experiments. New York: Wiley. Sitter, R. R. (1992) Robust designs for binary data. Biometrics, 48, 1145-1155.\n\nOptimal designs for binary response experiments: Fieller, D and A criteria. R R Sitter, C F J Wu, Scand. J. Statist. 20Sitter, R. R. and Wu, C. F. J. (1993) Optimal designs for binary response experiments: Fieller, D and A criteria. Scand. J. Statist., 20, 329-341.\n\nMacroscopic quantum tunneling in 1-μm Nb Josephson junctions. R F Voss, R A Webb, Phys. Rev. Lett. 47Voss, R. F. and Webb, R. A. (1981) Macroscopic quantum tunneling in 1-μm Nb Josephson junctions. Phys. Rev. Lett., 47, 265-268.\n"},"corpus_id":{"kind":"number","value":16899178,"string":"16,899,178"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"cond-mat/0610507\",\n \"mag\": 1975915405,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1111/j.1467-9876.2007.00572.x\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [],\n \"pdf_sha\": \"69ac5b6bb2dd1fb450fc0f5f2eba30ef41e7a0fe\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/cond-mat/0610507v1.pdf\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Experimental designs for binary data in switching measurements on superconducting Josephson junctions"},"authors":{"kind":"list like","value":["Juha Karvanen \nNational Public Health Institute\nHelsinkiFinland","Juha J Vartiainen \nHelsinki University of Technology\nEspooFinland","Andrey Timofeev \nHelsinki University of Technology\nEspooFinland","Jukka Pekola \nHelsinki University of Technology\nEspooFinland"],"string":"[\n \"Juha Karvanen \\nNational Public Health Institute\\nHelsinkiFinland\",\n \"Juha J Vartiainen \\nHelsinki University of Technology\\nEspooFinland\",\n \"Andrey Timofeev \\nHelsinki University of Technology\\nEspooFinland\",\n \"Jukka Pekola \\nHelsinki University of Technology\\nEspooFinland\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We study the optimal design of switching measurements of small Josephson junction circuits which operate in the macroscopic quantum tunnelling regime. In the experiment, sequences of current pulses are applied to the Josephson junction sample, while the voltage over the structure is monitored. The appearance of a voltage pulse to a single applied current pulse, being governed by the laws of quantum mechanics, is purely random. Starting from the D-optimality criterion we derive the optimal design for the estimation of the unknown parameters of the underlying Gumbel-type distribution. As a practical method for the measurements, we propose a sequential design that combines heuristic search for initial estimates and maximum likelihood estimation. The design presented has immediate applications in the area of superconducting electronics, implying faster data acquisition. The experimental results presented confirm the usefulness of the method."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Randomness plays a central role in quantum physics. On the fundamental level, randomness is not caused only by unknown covariates or measurement error but is understood to be an intrinsic property of nature itself (Ballentine, 1998). From the statistical point of view, this leads to fascinating problems where the distributional assumptions arise directly from the fundamental laws of physics.\nJosephson junctions (JJs) are important non-linear components of superconducting electronics. A typical junction involves two superconducting electrodes which are separated by an insulator gap (Josephson, 1962). In practice, miniature JJ circuits are manufactured by lithographic means and operated at cryogenic temperatures, i.e. below 4.2 K. The strong dependence of the physical parameters of JJ circuits as a function of changes in environmental variables, e.g. temperature, electric noise and magnetic fields, gives JJ circuits several applications as ultrasensitive sensors (Pekola et al., 2005). Moreover, certain JJ circuits are promising candidates for realization of quantum computation (Makhlin et al., 2001). For a thorough review of JJ circuits see, for instance, Grabert and Devoret (1992)."},{"title":"Fig. 1.","content":"Illustration of a switching measurement: (a) scanning electron micrograph of a typical two-junction JJ sample; (b) sketch of applied current I pulses and resulting voltage V pulses; (c) associated switching probability as a function of the height of the current pulses; (d) quantum mechanical process leading to switching; the phase 'particle' is initially localized in a local minimum of the potential-when the current pulse is applied, the phase particle tunnels through the potential barrier and escapes downhill An experiment called switching measurement (Fig. 1) is a common way to probe the properties of a JJ circuit sample. In the experiment, sequences of current pulses are applied to the sample, while the voltage over the structure is monitored. The superconducting electrodes of the sample are described by a complex-valued quantum mechanical wave function and the physical state of the JJ is described by the phase difference over the junction. The dynamics of the phase difference are analogous to those of a massive particle in a tilted cosine potential; see Fig. 1(d). A sufficiently high current pulse causes the phase difference to escape from the local minimum of the potential to the direction of steepest descent. In the experiment the observed voltage pulse indicates that switching has occurred. The appearance of a voltage pulse from a single applied current pulse, being governed by the laws of quantum mechanics, is purely random. Below, we denote an observed voltage pulse by Y = 1; otherwise Y = 0. The analysis of the statistics of this random variable with respect to current pulse heights yields information about the physical parameters of a JJ circuit.\nWe study the switching dynamics of a single JJ. Quantum mechanical arguments that are studied in more detail in Section 2 suggest that the switching rate is of the form\nwhere x = x.I/ is a monotonic function depending on the height I of the applied current pulse and a and b are the parameters depending on the geometry and environment of the junction. Moreover, we note that x.I/ must be independent of the parameters a and b. Assuming a constant switching rate during the pulse we obtain the probability of measuring a voltage pulse at an applied current pulse\n. 1/ For simplicity we consider the case x.I/ = I in this paper. In statistics, model (1) is often called a complementary log-log-regression. Assume that a sequence of n current pulses of height I i , i = 1, 2, . . . , n, is applied to the sample. The data obtained consist of covariate values x i , i = 1, 2, . . . , n, and corresponding binary responses y i . Our aim is to choose the covariates such that the parameters a and b can be estimated as well as possible from the measured data. In statistical terms this is an optimal design problem (Atkinson and Donev, 1992;Pukelsheim, 1993;Liski et al., 2002;Berger and Wong, 2005). The literature on optimal designs is concentrated on linear models but work has been done also with non-linear models. Several researchers (Kalish and Rosenberger, 1978;Abdelbasit and Plackett, 1983;Minkin, 1987;Sitter and Wu, 1993;Mathew and Sinha, 2001) have considered optimal designs for binary data in the case of logistic regression\nwhere a and b are unknown parameters and x is a covariate. Compared with logistic regression the complementary log-log-regression (1) is less frequently studied or applied. Complementary log-log-regression is often used as an alternative model for logistic regression (2) if the symmetry assumption does not fit the data, but in our application the complementary log-log-regression arises directly from the physical properties of JJs. The key reference considering optimal design for complementary log-log-regression is the work by Ford et al. (1992). They provided the general framework for the D-and c-optimal designs and the solution for complementary log-log-regression is obtained as a special case of their general results. In this paper we propose a sequential two-point design for estimating the unknown parameters of a JJ circuit. The design is derived by using the D-optimality criterion, which is equivalent to minimizing the generalized variance of the parameters. On the basis of the data that are available, the design yields the optimal current pulse heights for the next measurement in realtime. We are ultimately interested in minimizing the time that is needed for an experiment. The sequential design proposed also takes into account the special requirements of switching measurements, e.g. the time that is needed for the reconfiguration of the signal generator.\nThe rest of the paper is organized as follows. In Section 2, a more detailed description of physical background of the switching measurements is given. In Section 3, the optimal design for the complementary log-log-regression that is defined in equation (1) is derived. In Section 4, we propose a sequential design for switching measurements. In Section 5, the sequential design is applied to switching measurements of a JJ circuit. Results from the real data are compared with simulation results. Finally, Section 6 concludes the paper.\nThe measurement data and the related program code can be obtained from http://www.blackwellpublishing.com/rss"},{"title":"Physical background","content":"In this section we study the quantum mechanical arguments leading to model (1). The switching dynamics are described by a Poisson process where the switching rate γ is constant over the current pulse which has duration Δt. The probability of switching is then obtained as\nThe duration Δt of the current pulse is very short and remains fixed in the experiment. Thus, we consider Δt as a part of the switching rate and do not write it explicitly in the subsequent equations. The relationship between the switching rate γ and the height of the current pulse I Switching rate γ as a function of the relative bias current I=I C ( , switching rate γ MQT in the macroscopic quantum tunnelling; -------, switching rate γ TA under the thermal activation model at temperature 1 K; ---, γ TA at temperature 0.1 K); the JJ parameters are set to have typical values, Josephson energy E J D 10 21 J and capacitance C D 50 fF; γ MQT is calculated on the high Q limit; the relationship between γ and the height of the current pulse I is nearly linear on the logarithmic scale, which supports the use of the approximation γ.I / D exp.aI C b/ can be written in the form γ.I/ = exp{a x.I/ + b}, .4/ where x.I/ is a monotonic function of the bias current I and a and b are constants depending on the physical parameters of the JJ circuit. A commonly used approximation is to assume that the activation threshold amplitude is linearly proportional to current I. This leads to the switching rate γ.I/ = exp.aI + b/: .5/\nAlthough not based on microscopic theory, the switching rate model (5) describes the results of typical switching measurements quite well (Fig. 2).\nTo obtain the functional form of x.I/ in the switching rate model (4) we need a more accurate physical model to describe the system that is under consideration. Theoretically, the superconducting electrodes of the JJ sample are modelled by a complex-valued quantum mechanical wave function and the physical state of the JJ is described by the phase difference over the junction. The dynamics of the phase difference φ are analogous to those of a massive particle in a tilted cosine potential U under gravitational force,\nwhere E J is so-called Josephson energy, I is the magnitude of the bias current applied and I C is the critical current for which the potential changes into a monotonic function. The potential is depicted during a pulse in Fig. 1(d). The applied current changes the shape of the potential landscape of a JJ circuit in a controlled way. The phase difference φ initially set into one of the local minima becomes a metastable state for finite current I; the state may decay by escape of the phase from the local minimum of the potential to the direction of steepest descent. This escape event is called switching. In the experiment the voltage over the JJ sample is monitored and the observed voltage pulse indicates that switching has occurred. Depending on the temperature, model (1) is explained by the thermal activation model of classical physics or by quantum mechanical analysis. In an experiment at temperatures above 0.2 K, there will be thermal fluctuations in the bias current I as the JJ is connected to the measurement wiring. These fluctuations make the phase particle absorb energy and eventually escape from the potential well. One of the central results in statistical physics is that the probability for the system to be in energy state E at temperature T is proportional to exp.−E=k B T/, where k B ≈ 1:381 × 10 −23 J K −1 is the Boltzmann constant. To escape from the well the phase particle needs to have more energy than the barrier height ΔU. On the basis of these principles the average escape rate for the potential of equation (6) can be shown to follow Kramer's formula\nwhere ΔU is the height of the potential barrier\nand ω P is the characteristic oscillation frequency\nwhere C is the capacitance of the JJ,ē = 1:602 × 10 −19 C is the elementary charge and ≈ 1:055 × 10 −34 J s is the reduced Planck constant. In these equations E J and C are the unknown quantities-I C is a function of E J . It is easy to see that the switching rate (7) can be written in the form γ TA .I/ = exp{a x.I/ + b}. When a JJ is cooled to tens of millikelvins we can still observe the phase particle escaping from the quasi-stationary potential (Voss and Webb, 1981;Jackel et al., 1981). Since the thermal activation, equation (7), yields a vanishing switching rate for that temperature range, the microscopic mechanism must be different from that described above. This process is called macroscopic quantum tunnelling (Caldeira and Leggett, 1980) and to explain it a detailed quantum mechanical analysis is required. The analysis leads to switching rate (Grabert et al., 1985) γ MQT .I/ = A.I/ exp{−B.I/}:\n. 10/\nThe prefactor in equation (10) is\n, and Q is the quality factor of the oscillator describing the effective environment of the junction. The exponent reads\nand ζ.3/ ≈ 1:202 is the value of the Riemann ζ-function at 3. Again, the switching rate (10) can be written in the form γ MQT .I/ = exp{ax.I/ + b}."},{"title":"D-optimal designs for complementary log-log-regression","content":"Assume that a binary random variable Y is distributed as\nwhere a and b are unknown parameters and x is a covariate. The cumulative distribution function\nwhere z = ax + b, is known in the literature by the names Gumbel distribution, Gompertz distribution, log-Weibull distribution, Fisher-Tippett distribution and extreme value distribution, whereas the inverse of equation (14),\nis called the complementary log-log-link and is one of the standard link functions in generalized linear models. The problem is to select such covariate values x 1 , x 2 , . . . , x n that the parameters a and b can be estimated from the observed data .x i , y i /, i = 1, 2, . . . , n, in optimal manner. Here we define the optimality as D-optimality, which is equivalent to maximizing the determinant of the information matrix, and estimate the parameters by maximum likelihood. Starting from definition (13) we obtain the log-likelihood of the data .x i , y i /, i = 1, 2, . . . , n:\n. 19/ The D-optimality is equivalent to finding such x 1 , x 2 , . . . , x n that\nis maximized. It is assumed in the maximization that the true values of a and b are known (see Minkin (1987) and Sitter (1992) for a discussion). First we find a locally optimal two-point design n = 2. This simplifies equation (20) to\nThe contour plot of det.J/ as a function z 1 and z 2 is depicted in Fig. 3. The values z 1 and z 2 that maximize det.J/,\ncan be solved numerically from equation (21). The corresponding response probabilities are\n.23/ indicating that the design is asymmetric with respect to the middle point of the response curve.\nSince the function is symmetric with respect to exchanging the variables, the mirror points provide the same maximum value also. To show that the two-point maximum (22) is a local maximum also when n > 2, we define functions v.z 1 , z 1 / = g.z 1 / g.z 2 /.z 1 − z 2 / 2 , .24/ v .z 1 , z 2 / = @ @z 1 v.z 1 , z 2 /: .25/ By inspecting the functional form of v. Because v .z Å 1 , z Å 2 / = 0, a solution for equation (29) is obtained if\nFrom equation (26) we conclude that a local maximum is obtained, for instance, when\n.31/\nFor the complete proof that the solution (22) is the global maximum, we refer to the general results by Ford et al. (1992). Using a theorem by Kiefer and Wolfowitz (1960), Ford et al. (1992) showed for a class of response curves that the D-optimal design is a two-point design. The proof for the complementary log-log-regression is obtained as a special case of the general results."},{"title":"Sequential designs for switching measurements","content":"Our goal is to design a fast automated estimation procedure for switching measurements. A fast estimation procedure allows, for example, the tracking of changes in the parameters as a function of changes in environmental variables. The speed of the procedure is measured in terms of the total time used for measuring and estimation. Sequential designs are a natural choice because it is easy to control the current during the experiment. In sequential designs (also known as two-stage and multistage designs), we start with initial estimates, collect additional data, obtain new estimates and use them as the initial estimates for the next stage.\nIn switching measurements, the number of stages can be high: this is a significant difference compared with medical dose-response experiments where practical reasons usually limit the number of stages. However, it is not necessarily advisable to choose the number of stages as high as possible, i.e. update the estimates after every two observations. This is because adding two new observations would have a very small effect on the estimates (especially if the number of observations is already large) but the estimation itself would spend time that could be used for measuring. Additionally, whenever the height of the current pulse is changed, the reconfiguration of the signal generator also takes a time that is considerably longer than the time that is needed for measuring the response for a single pulse. The optimal updating interval depends also on the computational resources and facilities, and our experiences suggest that it is always reasonable to measure at least 20 times using the same pulse height. Our recommendation is only the lower limit and based on empirically experience on the specific measurement instruments that we are using. In the experiment that is presented in Section 5, we measured 50 times using the same pulse height at the beginning of the experiment and increased this number by 10% at every stage so that at stage 50 we measured 5379 times using the same pulse height.\nObtaining good initial estimates for the parameters is a challenging problem for switching measurements. Naturally, the results from the previous experiments can be used to obtain initial estimates for the parameters but there is no guarantee that this would lead to good initial estimates. The problem of poor initial estimates in the case of logistic regression was considered for example by King and Wang (2000) and Sitter (1992). They proposed sophisticated minimax approaches that have good optimality properties even if the initial estimates are poor. These approaches are probably applicable also to our problem but, because of their computational complexity, we choose a simpler approach that especially suits switching measurements. It is usually possible to specify an interval from where the covariates should be taken but the problem is that this interval might be very wide compared with the interval where the probability of response 1 increases, for instance, from 0.01 to 0.99. This implies that initial covariates that are chosen randomly from the interval may be very inefficient. In the worst case, poorly chosen initial covariates may lead to the situation where maximum likelihood estimates (MLEs) do not exist for the initial data. The existence of MLEs for model (13) requires that (Albert and Anderson, 1984) max.x i |y i = 0/ > min.x i |y i = 1/, max.x i |y i = 1/ > min.x i |y i = 0/:\n.32/ This condition is fulfilled, for instance, if we have two points where both 0s and 1s are measured as responses. Our approach for the initial estimation aims at finding two such points as quickly as possible. This heuristic procedure can be presented as follows. x u − x l /=4. The sign of \" is determined according to the measured response: sgn.\"/ = sgn.0:5 −ȳ/, whereȳ is the average response for x. If 0:5 −ȳ = 0 the sign of \" is chosen randomly. (d) If MLEs exist for the data measured so far, proceed to the maximum likelihood estimation. Otherwise, divide \" by 2 and return to step (c).\nStep (b) requires that we measure at least two times at each point. In practice, it is preferable to measure at least tens of times at each point because of the time that is needed for the reconfiguration of the signal generator. When the binary search in step (b) converges we have found one of the two points that are required for the maximum likelihood estimation. The other point that is needed is then found in the neighbourhood of point x using again binary search. Owing to use of binary search in steps (b) and (c), the procedure works reliably and efficiently. At every step of the binary search, the length of the current interval is divided by 2. Consequently, the procedure converges exponentially fast regardless of the choice of the initial interval.\nAfter the heuristic procedure has converged the experiment continues sequentially: the current values ofâ andb are used to determine the optimal covariates for the next stage, measurements are performed and finally maximum likelihood estimation is applied to update values ofâ and b. In switching measurements, we are also interested in parameters that directly describe the location and the scale of the response curve in addition to parameters a and b. Let us define the middle point θ and width of the curve λ, as follows: . 36/ It is often practical to define the stopping rule for the data acquisition by using the variances of θ and λ. Special attention should be paid to the stability of the measurement environment. Instability, such as a change in the temperature during the experiment, causes an increase in the variances. This emphasizes the need for fast and reliable experimental design."},{"title":"Examples with real and simulated data","content":"We measured a sample consisting of an aluminium-aluminium oxide-aluminium JJ circuit in a dilution refrigerator at 20 mK temperature. The sample parameters a and b depend on the area of the JJ and the thickness of the oxide layer. The sample fabricated involves a JJ whose area is 1 μm × 2 μm and the oxide layer is 3 nm thick. The sample was connected to computer-controlled measurement electronics to apply the current pulses and to record the resulting voltage pulses. The resistance of the sample at room temperature suggests that a pulse of 300 nA always causes a switching (response 1). This gives the upper limit for the initial estimation. The lower limit for the initial estimation, 200 nA, can be roughly estimated from the dimensions of the JJ. The sequential design that was presented in Section 4 was used to construct the applied current pulse sequences in realtime. Each stage consisted of a measuring phase and an estimation phase. Since the reconfiguration of our signal generator takes a relatively long time compared with the length of a single pulse, we used sequences of tens or hundreds of pulses to collect data at each stage. In the set-up the pulse lengths are of the order of milliseconds. To keep the time that is needed for computations much below the actual measurement time, the length of the pulse sequence should grow exponentially. We started with a sequence of 50 pulses per point and let the number of pulses grow by 10% at each stage. In the test experiments, the practically sufficient accuracy of the estimates was reached after 20-30 stages, implying that 6000-17 000 pulses were needed in total. In our set-up, the experiment (50 stages) took 8 min 16 s from which 1 min 30 s were used for estimation. Table 1 and Fig. 4 illustrate the estimation of the unknown parameters in the experiment. In Fig. 4(b) the functional form of the response curve is verified empirically. The response curve is compared with individual response probabilities that are independently estimated using 2000 pulses per point. The results confirm that the response curve is given by the Gumbel distribution as suggested by the theory.\nThe performance of the sequential design is also studied in a simulation example. The simulation aims to imitate the real data example. The response variable Y is generated from model (13) with parameters a = 0:24 and b = −61 which are motivated by the actual experiment that was presented above. Our goal is to estimate these parameters when we initially know only that at point x min = 200 the response is 0 with high probability and at point x max = 300 the response is 1 with high probability. The estimation starts by using the heuristic procedure described in Section 4 and continues as maximum likelihood estimation. In the initial estimation, we measure 25 times at each point. In the sequential maximum likelihood estimation, the parameter estimates are updated when the total number of observations has increased by 20% from the previous update. The simulation is repeated 500 times and MLEsâ andb are recorded for different numbers of observations n = 50, n = 100, . . . , n = 20 000. The sequential design is compared with the theoretically D-optimal design where the optimal covariate values x Å 1 = .z Å 1 + 61/=0:24 = 258:25 and x Å 2 = .z Å 2 + 61/=0:24 = 248:59 are assumed to be known already at the beginning of the experiment. As a performance measure we calculate the mean-squared errors (MSEs) between the estimates and the true parameter values. Table 2 summarizes the MLEs obtained and their standard deviations. The numbers reported are means from 500 simulations. The MSEs between the estimates and the true parameter values are reported in Table 3. The MSEs are calculated only from those simulation runs where MLEs exist. Naturally, the theoretically optimal design results in smaller MSE values than the sequential design but it can also be seen that for large values of n the performance of the sequential design approaches the performance of the theoretically optimal design. The results demonstrate the practical importance of good experimental design and the significance of reliable estimation of initial parameters. We conclude that the sequential design proposed performed reasonably well compared with the theoretically D-optimal design when n > 200. In switching measurements, the number of observations is always in thousands. A comparison of estimates and their standard deviations in Table 1 and Table 2 does not reveal any major differences between the experimental data and simulated data. This supports the practical usability of the sequential design."},{"title":"Conclusion","content":"We have considered optimal designs for switching measurements of superconducting JJ circuits. The D-optimality criterion suggests that the measurements at points where ax 1 + b = 0:979 633 and ax 2 + b = −1:337 74 will yield maximal information about the estimated parameters assuming that the probability distribution of the measurement values are given by equation (1). As a practical method for switching measurements, we propose a sequential design that combines heuristic search for initial estimates and maximum likelihood estimation. The fast data acquisition allows tracking of fast environmental changes. The experimental design presented is now in daily use in the Low Temperature Laboratory at Helsinki University of Technology.\nThe current work opens several directions for future research. Besides D-optimality we may also consider other optimality criteria. These criteria will probably lead to designs that are different from the D-optimal design but because of the complicated functional forms it may be difficult to derive analytical results.\nThe problem of choosing the number of measurements per stage was approached empirically in this paper. A more systematic approach would determine the optimal number of measurements per stage as a function of the observed data. It seems reasonable to assume that each measurement takes a certain time and the change of the measurement point takes additional fixed time. The problem is to choose the number of measurements per stage in such a way that the expected total time that is needed to achieve the accuracy required is minimized. This is a non-trivial problem and probably only approximate solutions are achievable.\nThe time that is used for the experiment could be further reduced if the estimation and the measuring were done simultaneously. This requires good co-ordination of data processing and knowledge on the time that is needed for each operation. In sequential designs, the simultaneous estimation and measuring implies that the current estimates are not based on all data measured so far but all data processed so far.\nAlthough the complementary log-log-regression model presented describes well some JJ circuits involving several junctions, we probably need more flexible models for complicated structures of JJs. The modelling can be approached from two different directions: we may derive the exact physical model and try to find the optimal design for it, or we may start with a flexible parametric model for which the optimal design is known and estimate the model parameters from the measured data.\nStatistical methods can be also applied to the modelling of JJ parameters as a function of environmental variables, such as temperature. This direction of research leads to adaptive sequential designs where the challenge is not only to estimate the parameters but also to track their changes in the optimal way."},{"title":"Fig. 2 .","content":""},{"title":"Fig. 3 .","content":""},{"title":"( a )","content":""},{"title":"Fig. 4 .","content":""},{"title":"Table 1 .","content":""},{"title":"Table 2 .","content":""},{"title":"Table 3 .","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Randomness plays a central role in quantum physics. On the fundamental level, randomness is not caused only by unknown covariates or measurement error but is understood to be an intrinsic property of nature itself (Ballentine, 1998). From the statistical point of view, this leads to fascinating problems where the distributional assumptions arise directly from the fundamental laws of physics.\\nJosephson junctions (JJs) are important non-linear components of superconducting electronics. A typical junction involves two superconducting electrodes which are separated by an insulator gap (Josephson, 1962). In practice, miniature JJ circuits are manufactured by lithographic means and operated at cryogenic temperatures, i.e. below 4.2 K. The strong dependence of the physical parameters of JJ circuits as a function of changes in environmental variables, e.g. temperature, electric noise and magnetic fields, gives JJ circuits several applications as ultrasensitive sensors (Pekola et al., 2005). Moreover, certain JJ circuits are promising candidates for realization of quantum computation (Makhlin et al., 2001). For a thorough review of JJ circuits see, for instance, Grabert and Devoret (1992).\"\n },\n {\n \"title\": \"Fig. 1.\",\n \"content\": \"Illustration of a switching measurement: (a) scanning electron micrograph of a typical two-junction JJ sample; (b) sketch of applied current I pulses and resulting voltage V pulses; (c) associated switching probability as a function of the height of the current pulses; (d) quantum mechanical process leading to switching; the phase 'particle' is initially localized in a local minimum of the potential-when the current pulse is applied, the phase particle tunnels through the potential barrier and escapes downhill An experiment called switching measurement (Fig. 1) is a common way to probe the properties of a JJ circuit sample. In the experiment, sequences of current pulses are applied to the sample, while the voltage over the structure is monitored. The superconducting electrodes of the sample are described by a complex-valued quantum mechanical wave function and the physical state of the JJ is described by the phase difference over the junction. The dynamics of the phase difference are analogous to those of a massive particle in a tilted cosine potential; see Fig. 1(d). A sufficiently high current pulse causes the phase difference to escape from the local minimum of the potential to the direction of steepest descent. In the experiment the observed voltage pulse indicates that switching has occurred. The appearance of a voltage pulse from a single applied current pulse, being governed by the laws of quantum mechanics, is purely random. Below, we denote an observed voltage pulse by Y = 1; otherwise Y = 0. The analysis of the statistics of this random variable with respect to current pulse heights yields information about the physical parameters of a JJ circuit.\\nWe study the switching dynamics of a single JJ. Quantum mechanical arguments that are studied in more detail in Section 2 suggest that the switching rate is of the form\\nwhere x = x.I/ is a monotonic function depending on the height I of the applied current pulse and a and b are the parameters depending on the geometry and environment of the junction. Moreover, we note that x.I/ must be independent of the parameters a and b. Assuming a constant switching rate during the pulse we obtain the probability of measuring a voltage pulse at an applied current pulse\\n. 1/ For simplicity we consider the case x.I/ = I in this paper. In statistics, model (1) is often called a complementary log-log-regression. Assume that a sequence of n current pulses of height I i , i = 1, 2, . . . , n, is applied to the sample. The data obtained consist of covariate values x i , i = 1, 2, . . . , n, and corresponding binary responses y i . Our aim is to choose the covariates such that the parameters a and b can be estimated as well as possible from the measured data. In statistical terms this is an optimal design problem (Atkinson and Donev, 1992;Pukelsheim, 1993;Liski et al., 2002;Berger and Wong, 2005). The literature on optimal designs is concentrated on linear models but work has been done also with non-linear models. Several researchers (Kalish and Rosenberger, 1978;Abdelbasit and Plackett, 1983;Minkin, 1987;Sitter and Wu, 1993;Mathew and Sinha, 2001) have considered optimal designs for binary data in the case of logistic regression\\nwhere a and b are unknown parameters and x is a covariate. Compared with logistic regression the complementary log-log-regression (1) is less frequently studied or applied. Complementary log-log-regression is often used as an alternative model for logistic regression (2) if the symmetry assumption does not fit the data, but in our application the complementary log-log-regression arises directly from the physical properties of JJs. The key reference considering optimal design for complementary log-log-regression is the work by Ford et al. (1992). They provided the general framework for the D-and c-optimal designs and the solution for complementary log-log-regression is obtained as a special case of their general results. In this paper we propose a sequential two-point design for estimating the unknown parameters of a JJ circuit. The design is derived by using the D-optimality criterion, which is equivalent to minimizing the generalized variance of the parameters. On the basis of the data that are available, the design yields the optimal current pulse heights for the next measurement in realtime. We are ultimately interested in minimizing the time that is needed for an experiment. The sequential design proposed also takes into account the special requirements of switching measurements, e.g. the time that is needed for the reconfiguration of the signal generator.\\nThe rest of the paper is organized as follows. In Section 2, a more detailed description of physical background of the switching measurements is given. In Section 3, the optimal design for the complementary log-log-regression that is defined in equation (1) is derived. In Section 4, we propose a sequential design for switching measurements. In Section 5, the sequential design is applied to switching measurements of a JJ circuit. Results from the real data are compared with simulation results. Finally, Section 6 concludes the paper.\\nThe measurement data and the related program code can be obtained from http://www.blackwellpublishing.com/rss\"\n },\n {\n \"title\": \"Physical background\",\n \"content\": \"In this section we study the quantum mechanical arguments leading to model (1). The switching dynamics are described by a Poisson process where the switching rate γ is constant over the current pulse which has duration Δt. The probability of switching is then obtained as\\nThe duration Δt of the current pulse is very short and remains fixed in the experiment. Thus, we consider Δt as a part of the switching rate and do not write it explicitly in the subsequent equations. The relationship between the switching rate γ and the height of the current pulse I Switching rate γ as a function of the relative bias current I=I C ( , switching rate γ MQT in the macroscopic quantum tunnelling; -------, switching rate γ TA under the thermal activation model at temperature 1 K; ---, γ TA at temperature 0.1 K); the JJ parameters are set to have typical values, Josephson energy E J D 10 21 J and capacitance C D 50 fF; γ MQT is calculated on the high Q limit; the relationship between γ and the height of the current pulse I is nearly linear on the logarithmic scale, which supports the use of the approximation γ.I / D exp.aI C b/ can be written in the form γ.I/ = exp{a x.I/ + b}, .4/ where x.I/ is a monotonic function of the bias current I and a and b are constants depending on the physical parameters of the JJ circuit. A commonly used approximation is to assume that the activation threshold amplitude is linearly proportional to current I. This leads to the switching rate γ.I/ = exp.aI + b/: .5/\\nAlthough not based on microscopic theory, the switching rate model (5) describes the results of typical switching measurements quite well (Fig. 2).\\nTo obtain the functional form of x.I/ in the switching rate model (4) we need a more accurate physical model to describe the system that is under consideration. Theoretically, the superconducting electrodes of the JJ sample are modelled by a complex-valued quantum mechanical wave function and the physical state of the JJ is described by the phase difference over the junction. The dynamics of the phase difference φ are analogous to those of a massive particle in a tilted cosine potential U under gravitational force,\\nwhere E J is so-called Josephson energy, I is the magnitude of the bias current applied and I C is the critical current for which the potential changes into a monotonic function. The potential is depicted during a pulse in Fig. 1(d). The applied current changes the shape of the potential landscape of a JJ circuit in a controlled way. The phase difference φ initially set into one of the local minima becomes a metastable state for finite current I; the state may decay by escape of the phase from the local minimum of the potential to the direction of steepest descent. This escape event is called switching. In the experiment the voltage over the JJ sample is monitored and the observed voltage pulse indicates that switching has occurred. Depending on the temperature, model (1) is explained by the thermal activation model of classical physics or by quantum mechanical analysis. In an experiment at temperatures above 0.2 K, there will be thermal fluctuations in the bias current I as the JJ is connected to the measurement wiring. These fluctuations make the phase particle absorb energy and eventually escape from the potential well. One of the central results in statistical physics is that the probability for the system to be in energy state E at temperature T is proportional to exp.−E=k B T/, where k B ≈ 1:381 × 10 −23 J K −1 is the Boltzmann constant. To escape from the well the phase particle needs to have more energy than the barrier height ΔU. On the basis of these principles the average escape rate for the potential of equation (6) can be shown to follow Kramer's formula\\nwhere ΔU is the height of the potential barrier\\nand ω P is the characteristic oscillation frequency\\nwhere C is the capacitance of the JJ,ē = 1:602 × 10 −19 C is the elementary charge and ≈ 1:055 × 10 −34 J s is the reduced Planck constant. In these equations E J and C are the unknown quantities-I C is a function of E J . It is easy to see that the switching rate (7) can be written in the form γ TA .I/ = exp{a x.I/ + b}. When a JJ is cooled to tens of millikelvins we can still observe the phase particle escaping from the quasi-stationary potential (Voss and Webb, 1981;Jackel et al., 1981). Since the thermal activation, equation (7), yields a vanishing switching rate for that temperature range, the microscopic mechanism must be different from that described above. This process is called macroscopic quantum tunnelling (Caldeira and Leggett, 1980) and to explain it a detailed quantum mechanical analysis is required. The analysis leads to switching rate (Grabert et al., 1985) γ MQT .I/ = A.I/ exp{−B.I/}:\\n. 10/\\nThe prefactor in equation (10) is\\n, and Q is the quality factor of the oscillator describing the effective environment of the junction. The exponent reads\\nand ζ.3/ ≈ 1:202 is the value of the Riemann ζ-function at 3. Again, the switching rate (10) can be written in the form γ MQT .I/ = exp{ax.I/ + b}.\"\n },\n {\n \"title\": \"D-optimal designs for complementary log-log-regression\",\n \"content\": \"Assume that a binary random variable Y is distributed as\\nwhere a and b are unknown parameters and x is a covariate. The cumulative distribution function\\nwhere z = ax + b, is known in the literature by the names Gumbel distribution, Gompertz distribution, log-Weibull distribution, Fisher-Tippett distribution and extreme value distribution, whereas the inverse of equation (14),\\nis called the complementary log-log-link and is one of the standard link functions in generalized linear models. The problem is to select such covariate values x 1 , x 2 , . . . , x n that the parameters a and b can be estimated from the observed data .x i , y i /, i = 1, 2, . . . , n, in optimal manner. Here we define the optimality as D-optimality, which is equivalent to maximizing the determinant of the information matrix, and estimate the parameters by maximum likelihood. Starting from definition (13) we obtain the log-likelihood of the data .x i , y i /, i = 1, 2, . . . , n:\\n. 19/ The D-optimality is equivalent to finding such x 1 , x 2 , . . . , x n that\\nis maximized. It is assumed in the maximization that the true values of a and b are known (see Minkin (1987) and Sitter (1992) for a discussion). First we find a locally optimal two-point design n = 2. This simplifies equation (20) to\\nThe contour plot of det.J/ as a function z 1 and z 2 is depicted in Fig. 3. The values z 1 and z 2 that maximize det.J/,\\ncan be solved numerically from equation (21). The corresponding response probabilities are\\n.23/ indicating that the design is asymmetric with respect to the middle point of the response curve.\\nSince the function is symmetric with respect to exchanging the variables, the mirror points provide the same maximum value also. To show that the two-point maximum (22) is a local maximum also when n > 2, we define functions v.z 1 , z 1 / = g.z 1 / g.z 2 /.z 1 − z 2 / 2 , .24/ v .z 1 , z 2 / = @ @z 1 v.z 1 , z 2 /: .25/ By inspecting the functional form of v. Because v .z Å 1 , z Å 2 / = 0, a solution for equation (29) is obtained if\\nFrom equation (26) we conclude that a local maximum is obtained, for instance, when\\n.31/\\nFor the complete proof that the solution (22) is the global maximum, we refer to the general results by Ford et al. (1992). Using a theorem by Kiefer and Wolfowitz (1960), Ford et al. (1992) showed for a class of response curves that the D-optimal design is a two-point design. The proof for the complementary log-log-regression is obtained as a special case of the general results.\"\n },\n {\n \"title\": \"Sequential designs for switching measurements\",\n \"content\": \"Our goal is to design a fast automated estimation procedure for switching measurements. A fast estimation procedure allows, for example, the tracking of changes in the parameters as a function of changes in environmental variables. The speed of the procedure is measured in terms of the total time used for measuring and estimation. Sequential designs are a natural choice because it is easy to control the current during the experiment. In sequential designs (also known as two-stage and multistage designs), we start with initial estimates, collect additional data, obtain new estimates and use them as the initial estimates for the next stage.\\nIn switching measurements, the number of stages can be high: this is a significant difference compared with medical dose-response experiments where practical reasons usually limit the number of stages. However, it is not necessarily advisable to choose the number of stages as high as possible, i.e. update the estimates after every two observations. This is because adding two new observations would have a very small effect on the estimates (especially if the number of observations is already large) but the estimation itself would spend time that could be used for measuring. Additionally, whenever the height of the current pulse is changed, the reconfiguration of the signal generator also takes a time that is considerably longer than the time that is needed for measuring the response for a single pulse. The optimal updating interval depends also on the computational resources and facilities, and our experiences suggest that it is always reasonable to measure at least 20 times using the same pulse height. Our recommendation is only the lower limit and based on empirically experience on the specific measurement instruments that we are using. In the experiment that is presented in Section 5, we measured 50 times using the same pulse height at the beginning of the experiment and increased this number by 10% at every stage so that at stage 50 we measured 5379 times using the same pulse height.\\nObtaining good initial estimates for the parameters is a challenging problem for switching measurements. Naturally, the results from the previous experiments can be used to obtain initial estimates for the parameters but there is no guarantee that this would lead to good initial estimates. The problem of poor initial estimates in the case of logistic regression was considered for example by King and Wang (2000) and Sitter (1992). They proposed sophisticated minimax approaches that have good optimality properties even if the initial estimates are poor. These approaches are probably applicable also to our problem but, because of their computational complexity, we choose a simpler approach that especially suits switching measurements. It is usually possible to specify an interval from where the covariates should be taken but the problem is that this interval might be very wide compared with the interval where the probability of response 1 increases, for instance, from 0.01 to 0.99. This implies that initial covariates that are chosen randomly from the interval may be very inefficient. In the worst case, poorly chosen initial covariates may lead to the situation where maximum likelihood estimates (MLEs) do not exist for the initial data. The existence of MLEs for model (13) requires that (Albert and Anderson, 1984) max.x i |y i = 0/ > min.x i |y i = 1/, max.x i |y i = 1/ > min.x i |y i = 0/:\\n.32/ This condition is fulfilled, for instance, if we have two points where both 0s and 1s are measured as responses. Our approach for the initial estimation aims at finding two such points as quickly as possible. This heuristic procedure can be presented as follows. x u − x l /=4. The sign of \\\" is determined according to the measured response: sgn.\\\"/ = sgn.0:5 −ȳ/, whereȳ is the average response for x. If 0:5 −ȳ = 0 the sign of \\\" is chosen randomly. (d) If MLEs exist for the data measured so far, proceed to the maximum likelihood estimation. Otherwise, divide \\\" by 2 and return to step (c).\\nStep (b) requires that we measure at least two times at each point. In practice, it is preferable to measure at least tens of times at each point because of the time that is needed for the reconfiguration of the signal generator. When the binary search in step (b) converges we have found one of the two points that are required for the maximum likelihood estimation. The other point that is needed is then found in the neighbourhood of point x using again binary search. Owing to use of binary search in steps (b) and (c), the procedure works reliably and efficiently. At every step of the binary search, the length of the current interval is divided by 2. Consequently, the procedure converges exponentially fast regardless of the choice of the initial interval.\\nAfter the heuristic procedure has converged the experiment continues sequentially: the current values ofâ andb are used to determine the optimal covariates for the next stage, measurements are performed and finally maximum likelihood estimation is applied to update values ofâ and b. In switching measurements, we are also interested in parameters that directly describe the location and the scale of the response curve in addition to parameters a and b. Let us define the middle point θ and width of the curve λ, as follows: . 36/ It is often practical to define the stopping rule for the data acquisition by using the variances of θ and λ. Special attention should be paid to the stability of the measurement environment. Instability, such as a change in the temperature during the experiment, causes an increase in the variances. This emphasizes the need for fast and reliable experimental design.\"\n },\n {\n \"title\": \"Examples with real and simulated data\",\n \"content\": \"We measured a sample consisting of an aluminium-aluminium oxide-aluminium JJ circuit in a dilution refrigerator at 20 mK temperature. The sample parameters a and b depend on the area of the JJ and the thickness of the oxide layer. The sample fabricated involves a JJ whose area is 1 μm × 2 μm and the oxide layer is 3 nm thick. The sample was connected to computer-controlled measurement electronics to apply the current pulses and to record the resulting voltage pulses. The resistance of the sample at room temperature suggests that a pulse of 300 nA always causes a switching (response 1). This gives the upper limit for the initial estimation. The lower limit for the initial estimation, 200 nA, can be roughly estimated from the dimensions of the JJ. The sequential design that was presented in Section 4 was used to construct the applied current pulse sequences in realtime. Each stage consisted of a measuring phase and an estimation phase. Since the reconfiguration of our signal generator takes a relatively long time compared with the length of a single pulse, we used sequences of tens or hundreds of pulses to collect data at each stage. In the set-up the pulse lengths are of the order of milliseconds. To keep the time that is needed for computations much below the actual measurement time, the length of the pulse sequence should grow exponentially. We started with a sequence of 50 pulses per point and let the number of pulses grow by 10% at each stage. In the test experiments, the practically sufficient accuracy of the estimates was reached after 20-30 stages, implying that 6000-17 000 pulses were needed in total. In our set-up, the experiment (50 stages) took 8 min 16 s from which 1 min 30 s were used for estimation. Table 1 and Fig. 4 illustrate the estimation of the unknown parameters in the experiment. In Fig. 4(b) the functional form of the response curve is verified empirically. The response curve is compared with individual response probabilities that are independently estimated using 2000 pulses per point. The results confirm that the response curve is given by the Gumbel distribution as suggested by the theory.\\nThe performance of the sequential design is also studied in a simulation example. The simulation aims to imitate the real data example. The response variable Y is generated from model (13) with parameters a = 0:24 and b = −61 which are motivated by the actual experiment that was presented above. Our goal is to estimate these parameters when we initially know only that at point x min = 200 the response is 0 with high probability and at point x max = 300 the response is 1 with high probability. The estimation starts by using the heuristic procedure described in Section 4 and continues as maximum likelihood estimation. In the initial estimation, we measure 25 times at each point. In the sequential maximum likelihood estimation, the parameter estimates are updated when the total number of observations has increased by 20% from the previous update. The simulation is repeated 500 times and MLEsâ andb are recorded for different numbers of observations n = 50, n = 100, . . . , n = 20 000. The sequential design is compared with the theoretically D-optimal design where the optimal covariate values x Å 1 = .z Å 1 + 61/=0:24 = 258:25 and x Å 2 = .z Å 2 + 61/=0:24 = 248:59 are assumed to be known already at the beginning of the experiment. As a performance measure we calculate the mean-squared errors (MSEs) between the estimates and the true parameter values. Table 2 summarizes the MLEs obtained and their standard deviations. The numbers reported are means from 500 simulations. The MSEs between the estimates and the true parameter values are reported in Table 3. The MSEs are calculated only from those simulation runs where MLEs exist. Naturally, the theoretically optimal design results in smaller MSE values than the sequential design but it can also be seen that for large values of n the performance of the sequential design approaches the performance of the theoretically optimal design. The results demonstrate the practical importance of good experimental design and the significance of reliable estimation of initial parameters. We conclude that the sequential design proposed performed reasonably well compared with the theoretically D-optimal design when n > 200. In switching measurements, the number of observations is always in thousands. A comparison of estimates and their standard deviations in Table 1 and Table 2 does not reveal any major differences between the experimental data and simulated data. This supports the practical usability of the sequential design.\"\n },\n {\n \"title\": \"Conclusion\",\n \"content\": \"We have considered optimal designs for switching measurements of superconducting JJ circuits. The D-optimality criterion suggests that the measurements at points where ax 1 + b = 0:979 633 and ax 2 + b = −1:337 74 will yield maximal information about the estimated parameters assuming that the probability distribution of the measurement values are given by equation (1). As a practical method for switching measurements, we propose a sequential design that combines heuristic search for initial estimates and maximum likelihood estimation. The fast data acquisition allows tracking of fast environmental changes. The experimental design presented is now in daily use in the Low Temperature Laboratory at Helsinki University of Technology.\\nThe current work opens several directions for future research. Besides D-optimality we may also consider other optimality criteria. These criteria will probably lead to designs that are different from the D-optimal design but because of the complicated functional forms it may be difficult to derive analytical results.\\nThe problem of choosing the number of measurements per stage was approached empirically in this paper. A more systematic approach would determine the optimal number of measurements per stage as a function of the observed data. It seems reasonable to assume that each measurement takes a certain time and the change of the measurement point takes additional fixed time. The problem is to choose the number of measurements per stage in such a way that the expected total time that is needed to achieve the accuracy required is minimized. This is a non-trivial problem and probably only approximate solutions are achievable.\\nThe time that is used for the experiment could be further reduced if the estimation and the measuring were done simultaneously. This requires good co-ordination of data processing and knowledge on the time that is needed for each operation. In sequential designs, the simultaneous estimation and measuring implies that the current estimates are not based on all data measured so far but all data processed so far.\\nAlthough the complementary log-log-regression model presented describes well some JJ circuits involving several junctions, we probably need more flexible models for complicated structures of JJs. The modelling can be approached from two different directions: we may derive the exact physical model and try to find the optimal design for it, or we may start with a flexible parametric model for which the optimal design is known and estimate the model parameters from the measured data.\\nStatistical methods can be also applied to the modelling of JJ parameters as a function of environmental variables, such as temperature. This direction of research leads to adaptive sequential designs where the challenge is not only to estimate the parameters but also to track their changes in the optimal way.\"\n },\n {\n \"title\": \"Fig. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"( a )\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 3 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Fig. 2. Switching rate γ as a function of the relative bias current I=I C ( , switching rate γ MQT in the macroscopic quantum tunnelling; -------, switching rate γ TA under the thermal activation model at temperature 1 K; ---, γ TA at temperature 0.1 K); the JJ parameters are set to have typical values, Josephson energy E J D 10 21 J and capacitance C D 50 fF; γ MQT is calculated on the high Q limit; the relationship between γ and the height of the current pulse I is nearly linear on the logarithmic scale, which supports the use of the approximation γ.I / D exp.aI C b/","content":"Fig. 2 .\n2Fig. 2. Switching rate γ as a function of the relative bias current I=I C ( , switching rate γ MQT in the macroscopic quantum tunnelling; -------, switching rate γ TA under the thermal activation model at temperature 1 K; ---, γ TA at temperature 0.1 K); the JJ parameters are set to have typical values, Josephson energy E J D 10 21 J and capacitance C D 50 fF; γ MQT is calculated on the high Q limit; the relationship between γ and the height of the current pulse I is nearly linear on the logarithmic scale, which supports the use of the approximation γ.I / D exp.aI C b/"},{"id":"","caption":"Contours of the ratio 100 det{J.z 1 , z 2 /}= det{J.z Å 1 , z Å 2 /}, where det{J.z 1 , z 2 /} is the determinant of the information matrix for the two-point design .z 1 , z 2 /: the maximum of det.J/ is achieved at the point .","content":"Fig. 3 .\n3Contours of the ratio 100 det{J.z 1 , z 2 /}= det{J.z Å 1 , z Å 2 /}, where det{J.z 1 , z 2 /} is the determinant of the information matrix for the two-point design .z 1 , z 2 /: the maximum of det.J/ is achieved at the point ."},{"id":"","caption":"Use previous knowledge to construct such an interval [x min , x max ] that we can be sure that F.x max ; a, b/ − F.x min ; a, b/ is close to 1. Constructing this kind of interval is usually possible even if we have very little knowledge about the parameters.(b) Use binary search to find such a point x from the interval [x min , x max ] that both 0s and 1s are measured as responses. In binary search, we measure the responses at the middle point of the current interval. If only 0s are measured, the middle point is taken as the new starting-point of the interval. If only 1s are measured, the middle point is taken as the new end point of the interval. Let x be found as the middle point of the interval [x l , x u ], i.e. x = .x l + x u /=2. (c) Measure at the point x + \", where |\"| = .","content":"( a )\naUse previous knowledge to construct such an interval [x min , x max ] that we can be sure that F.x max ; a, b/ − F.x min ; a, b/ is close to 1. Constructing this kind of interval is usually possible even if we have very little knowledge about the parameters.(b) Use binary search to find such a point x from the interval [x min , x max ] that both 0s and 1s are measured as responses. In binary search, we measure the responses at the middle point of the current interval. If only 0s are measured, the middle point is taken as the new starting-point of the interval. If only 1s are measured, the middle point is taken as the new end point of the interval. Let x be found as the middle point of the interval [x l , x u ], i.e. x = .x l + x u /=2. (c) Measure at the point x + \", where |\"| = ."},{"id":"","caption":"parameters are frequently used in switching measurements to characterize the response curve. The covariance matrix of .θ λ/ T is given by AJ −1 A T , Estimated response curve for the measured JJ: (a) estimated response curve with data used in the estimation (pulse heights I and corresponding empirical probabilities P ); (b) estimated response curve and verification points obtained as an average of responses of 2000 independent pulses","content":"Fig. 4 .\n4parameters are frequently used in switching measurements to characterize the response curve. The covariance matrix of .θ λ/ T is given by AJ −1 A T , Estimated response curve for the measured JJ: (a) estimated response curve with data used in the estimation (pulse heights I and corresponding empirical probabilities P ); (b) estimated response curve and verification points obtained as an average of responses of 2000 independent pulses"},{"id":"","caption":"Estimates and their standard deviations for the measured JJ","content":"Table 1 .\n1Estimates and their standard deviations for the measured JJStage \nnâ \nstd(â)b \nstd(b) \n\n1 \n100 \nNo MLE \nNo MLE \n2 \n210 \nNo MLE \nNo MLE \n3 \n332 \n0.3800 \n0.1933 \n−95.5966 \n48.4418 \n4 \n466 \n0.2585 \n0.0392 \n−65.1942 \n9.8369 \n5 \n614 \n0.2301 \n0.0224 \n−58.1248 \n5.6501 \n6 \n776 \n0.2520 \n0.0177 \n−63.6371 \n4.4625 \n7 \n954 \n0.2474 \n0.0148 \n−62.5090 \n3.7268 \n8 \n1150 \n0.2368 \n0.0126 \n−59.8039 \n3.1855 \n9 \n1366 \n0.2366 \n0.0112 \n−59.7514 \n2.8222 \n10 \n1604 \n0.2426 \n0.0102 \n−61.2916 \n2.5744 \n15 \n3200 \n0.2350 \n0.0067 \n−59.3476 \n1.7022 \n20 \n5762 \n0.2427 \n0.0050 \n−61.2986 \n1.2637 \n25 \n9890 \n0.2433 \n0.0038 \n−61.4761 \n0.9572 \n30 \n16550 \n0.2424 \n0.0029 \n−61.2533 \n0.7359 \n40 \n44578 \n0.2410 \n0.0017 \n−60.8778 \n0.4430 \n50 \n117288 \n0.2402 \n0.0011 \n−60.6282 \n0.2708 \n\nθ \nstd(θ)λ \nstd(λ) \n\n1 \n100 \nNo MLE \nNo MLE \n2 \n210 \nNo MLE \nNo MLE \n3 \n332 \n250.5738 \n0.2519 \n8.1158 \n4.1282 \n4 \n466 \n250.8324 \n0.3037 \n11.9342 \n1.8080 \n5 \n614 \n251.0506 \n0.2984 \n13.4066 \n1.3070 \n6 \n776 \n251.0632 \n0.2519 \n12.2392 \n0.8591 \n7 \n954 \n251.1471 \n0.2327 \n12.4655 \n0.7432 \n8 \n1150 \n251.0128 \n0.2219 \n13.0259 \n0.6933 \n9 \n1366 \n250.9891 \n0.2058 \n13.0362 \n0.6148 \n10 \n1604 \n251.1269 \n0.1874 \n12.7136 \n0.5330 \n15 \n3200 \n250.9385 \n0.1386 \n13.1228 \n0.3751 \n20 \n5762 \n251.1056 \n0.1019 \n12.7110 \n0.2610 \n25 \n9890 \n251.1342 \n0.0780 \n12.6756 \n0.1965 \n30 \n16550 \n251.1604 \n0.0605 \n12.7233 \n0.1522 \n40 \n44578 \n251.1016 \n0.0372 \n12.7992 \n0.0927 \n50 \n117288 \n250.8531 \n0.0231 \n12.8395 \n0.0571"},{"id":"","caption":"Means of estimates and their standard deviations from 500 simulation runs","content":"Table 2 .\n2Means of estimates and their standard deviations from 500 simulation runsn \nSequential design \nOptimal design \nSequential design \nOptimal design \n\na \nstd(â)â \nstd(â)b \nstd(b)b \nstd(b) \n\n50 \n0.4342 \n0.3953 \n0.2454 \n0.0659 \n−109.6947 \n99.0491 \n−62.4331 \n16.8447 \n100 \n0.2786 \n0.1151 \n0.2476 \n0.0387 \n−70.7275 \n28.8748 \n−62.9339 \n9.8681 \n200 \n0.2482 \n0.0373 \n0.2420 \n0.0262 \n−63.0732 \n9.4241 \n−61.4907 \n6.6879 \n300 \n0.2448 \n0.0254 \n0.2418 \n0.0213 \n−62.2302 \n6.4475 \n−61.4518 \n5.4436 \n400 \n0.2433 \n0.0203 \n0.2413 \n0.0184 \n−61.8373 \n5.1676 \n−61.3297 \n4.6997 \n500 \n0.2425 \n0.0176 \n0.2405 \n0.0164 \n−61.6395 \n4.4892 \n−61.1274 \n4.1928 \n700 \n0.2415 \n0.0145 \n0.2408 \n0.0139 \n−61.3900 \n3.6946 \n−61.2015 \n3.5441 \n1000 \n0.2418 \n0.0120 \n0.2409 \n0.0116 \n−61.4646 \n3.0482 \n−61.2335 \n2.9653 \n1500 \n0.2410 \n0.0096 \n0.2406 \n0.0095 \n−61.2491 \n2.4610 \n−61.1591 \n2.4188 \n2000 \n0.2411 \n0.0083 \n0.2408 \n0.0082 \n−61.2892 \n2.1225 \n−61.2076 \n2.0953 \n3000 \n0.2408 \n0.0068 \n0.2405 \n0.0067 \n−61.2001 \n1.7240 \n−61.1375 \n1.7089 \n4000 \n0.2406 \n0.0058 \n0.2405 \n0.0058 \n−61.1488 \n1.4894 \n−61.1361 \n1.4799 \n5000 \n0.2405 \n0.0052 \n0.2402 \n0.0052 \n−61.1338 \n1.3305 \n−61.0467 \n1.3223 \n7000 \n0.2404 \n0.0044 \n0.2401 \n0.0044 \n−61.0939 \n1.1225 \n−61.0178 \n1.1170 \n10000 \n0.2402 \n0.0037 \n0.2401 \n0.0037 \n−61.0621 \n0.9378 \n−61.0184 \n0.9344 \n15000 \n0.2402 \n0.0030 \n0.2400 \n0.0030 \n−61.0505 \n0.7648 \n−61.0077 \n0.7629 \n20000 \n0.2401 \n0.0026 \n0.2401 \n0.0026 \n−61.0296 \n0.6619 \n−61.0128 \n0.6607 \n\nθ \nstd(θ )θ \nstd(θ )λ \nstd(λ)λ \nstd(λ) \n\n50 \n254.3894 \n7.2977 \n252.7209 \n1.3866 \n21.0825 \n38.7785 \n13.1512 \n3.6519 \n100 \n253.1132 \n1.7524 \n252.6023 \n0.8089 \n14.3413 \n7.8859 \n12.7611 \n2.0261 \n200 \n252.7223 \n0.5869 \n252.5675 \n0.5638 \n12.7674 \n1.9482 \n12.8914 \n1.4084 \n300 \n252.6707 \n0.4449 \n252.5894 \n0.4589 \n12.7352 \n1.3232 \n12.8599 \n1.1416 \n400 \n252.6603 \n0.3783 \n252.5911 \n0.3972 \n12.7684 \n1.0674 \n12.8631 \n0.9864 \n500 \n252.6529 \n0.3393 \n252.5970 \n0.3554 \n12.7853 \n0.9307 \n12.8892 \n0.8837 \n700 \n252.6479 \n0.2892 \n252.6178 \n0.2994 \n12.8159 \n0.7700 \n12.8514 \n0.7427 \n1000 \n252.6554 \n0.2433 \n252.6364 \n0.2501 \n12.7847 \n0.6323 \n12.8315 \n0.6196 \n1500 \n252.6535 \n0.2003 \n252.6472 \n0.2041 \n12.8202 \n0.5134 \n12.8361 \n0.5058 \n2000 \n252.6544 \n0.1741 \n252.6507 \n0.1766 \n12.8081 \n0.4419 \n12.8224 \n0.4373 \n3000 \n252.6413 \n0.1428 \n252.6444 \n0.1443 \n12.8203 \n0.3597 \n12.8329 \n0.3573 \n4000 \n252.6417 \n0.1240 \n252.6472 \n0.1249 \n12.8284 \n0.3111 \n12.8305 \n0.3093 \n5000 \n252.6426 \n0.1110 \n252.6423 \n0.1118 \n12.8301 \n0.2780 \n12.8474 \n0.2771 \n7000 \n252.6454 \n0.0940 \n252.6379 \n0.0945 \n12.8368 \n0.2348 \n12.8517 \n0.2342 \n10000 \n252.6435 \n0.0788 \n252.6366 \n0.0791 \n12.8420 \n0.1963 \n12.8504 \n0.1959 \n15000 \n252.6401 \n0.0644 \n252.6376 \n0.0646 \n12.8432 \n0.1601 \n12.8518 \n0.1600 \n20000 \n252.6391 \n0.0558 \n252.6391 \n0.0559 \n12.8470 \n0.1387 \n12.8503 \n0.1385"},{"id":"","caption":"MSEs for parameter estimatesâ andb as functions of the number of observations n † 399793 †The sequential design proposed and the theoretically D-optimal design are compared. Reported MSEs are means from 500 simulation runs.","content":"Table 3 .\n3MSEs for parameter estimatesâ andb as functions of the number of observations n † 399793 †The sequential design proposed and the theoretically D-optimal design are compared. Reported MSEs are means from 500 simulation runs.n \nMSE ofâ \nMSE ofb \n\nSequential \nOptimal \nSequential \nOptimal \n\n50 \n0.206843 \n0.002715 \n12983.570197 \n179.184180 \n100 \n0.021449 \n0.001546 \n1348.770300 \n101.345741 \n200 \n0.001771 \n0.000667 \n112.423833 \n43.457153 \n300 \n0.000686 \n0.000477 \n44.089057 \n31.041668 \n400 \n0.000439 \n0.000373 \n28.286722 \n24.314540 \n500 \n0.000316 \n0.000297 \n20.389242 \n19.364127 \n700 \n0.000211 \n0.000196 \n13.672440 \n12.766761 \n1000 \n0.000139 \n0.000132 \n9.035246 \n8.643545 \n1500 \n0.000093 \n0.000080 \n6.046665 \n5.201665 \n2000 \n0.000077 \n0.000063 \n4.993876 \n4.132176 \n3000 \n0.000049 \n0.000045 \n3.198680 \n2.937851 \n4000 \n0.000037 \n0.000032 \n2.390940 \n2.100386 \n5000 \n0.000030 \n0.000024 \n1.946899 \n1.584166 \n7000 \n0.000021 \n0.000017 \n1.379387 \n1.116941 \n10000 \n0.000015 \n0.000012 \n0.961786 \n0.811790 \n15000 \n0.000010 \n0.000008 \n0.634116 \n0.523743 \n20000 \n0.000007 \n0.000006 \n0.473926 \n0."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Fig. 2. Switching rate γ as a function of the relative bias current I=I C ( , switching rate γ MQT in the macroscopic quantum tunnelling; -------, switching rate γ TA under the thermal activation model at temperature 1 K; ---, γ TA at temperature 0.1 K); the JJ parameters are set to have typical values, Josephson energy E J D 10 21 J and capacitance C D 50 fF; γ MQT is calculated on the high Q limit; the relationship between γ and the height of the current pulse I is nearly linear on the logarithmic scale, which supports the use of the approximation γ.I / D exp.aI C b/\",\n \"content\": \"Fig. 2 .\\n2Fig. 2. Switching rate γ as a function of the relative bias current I=I C ( , switching rate γ MQT in the macroscopic quantum tunnelling; -------, switching rate γ TA under the thermal activation model at temperature 1 K; ---, γ TA at temperature 0.1 K); the JJ parameters are set to have typical values, Josephson energy E J D 10 21 J and capacitance C D 50 fF; γ MQT is calculated on the high Q limit; the relationship between γ and the height of the current pulse I is nearly linear on the logarithmic scale, which supports the use of the approximation γ.I / D exp.aI C b/\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Contours of the ratio 100 det{J.z 1 , z 2 /}= det{J.z Å 1 , z Å 2 /}, where det{J.z 1 , z 2 /} is the determinant of the information matrix for the two-point design .z 1 , z 2 /: the maximum of det.J/ is achieved at the point .\",\n \"content\": \"Fig. 3 .\\n3Contours of the ratio 100 det{J.z 1 , z 2 /}= det{J.z Å 1 , z Å 2 /}, where det{J.z 1 , z 2 /} is the determinant of the information matrix for the two-point design .z 1 , z 2 /: the maximum of det.J/ is achieved at the point .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Use previous knowledge to construct such an interval [x min , x max ] that we can be sure that F.x max ; a, b/ − F.x min ; a, b/ is close to 1. Constructing this kind of interval is usually possible even if we have very little knowledge about the parameters.(b) Use binary search to find such a point x from the interval [x min , x max ] that both 0s and 1s are measured as responses. In binary search, we measure the responses at the middle point of the current interval. If only 0s are measured, the middle point is taken as the new starting-point of the interval. If only 1s are measured, the middle point is taken as the new end point of the interval. Let x be found as the middle point of the interval [x l , x u ], i.e. x = .x l + x u /=2. (c) Measure at the point x + \\\", where |\\\"| = .\",\n \"content\": \"( a )\\naUse previous knowledge to construct such an interval [x min , x max ] that we can be sure that F.x max ; a, b/ − F.x min ; a, b/ is close to 1. Constructing this kind of interval is usually possible even if we have very little knowledge about the parameters.(b) Use binary search to find such a point x from the interval [x min , x max ] that both 0s and 1s are measured as responses. In binary search, we measure the responses at the middle point of the current interval. If only 0s are measured, the middle point is taken as the new starting-point of the interval. If only 1s are measured, the middle point is taken as the new end point of the interval. Let x be found as the middle point of the interval [x l , x u ], i.e. x = .x l + x u /=2. (c) Measure at the point x + \\\", where |\\\"| = .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"parameters are frequently used in switching measurements to characterize the response curve. The covariance matrix of .θ λ/ T is given by AJ −1 A T , Estimated response curve for the measured JJ: (a) estimated response curve with data used in the estimation (pulse heights I and corresponding empirical probabilities P ); (b) estimated response curve and verification points obtained as an average of responses of 2000 independent pulses\",\n \"content\": \"Fig. 4 .\\n4parameters are frequently used in switching measurements to characterize the response curve. The covariance matrix of .θ λ/ T is given by AJ −1 A T , Estimated response curve for the measured JJ: (a) estimated response curve with data used in the estimation (pulse heights I and corresponding empirical probabilities P ); (b) estimated response curve and verification points obtained as an average of responses of 2000 independent pulses\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Estimates and their standard deviations for the measured JJ\",\n \"content\": \"Table 1 .\\n1Estimates and their standard deviations for the measured JJStage \\nnâ \\nstd(â)b \\nstd(b) \\n\\n1 \\n100 \\nNo MLE \\nNo MLE \\n2 \\n210 \\nNo MLE \\nNo MLE \\n3 \\n332 \\n0.3800 \\n0.1933 \\n−95.5966 \\n48.4418 \\n4 \\n466 \\n0.2585 \\n0.0392 \\n−65.1942 \\n9.8369 \\n5 \\n614 \\n0.2301 \\n0.0224 \\n−58.1248 \\n5.6501 \\n6 \\n776 \\n0.2520 \\n0.0177 \\n−63.6371 \\n4.4625 \\n7 \\n954 \\n0.2474 \\n0.0148 \\n−62.5090 \\n3.7268 \\n8 \\n1150 \\n0.2368 \\n0.0126 \\n−59.8039 \\n3.1855 \\n9 \\n1366 \\n0.2366 \\n0.0112 \\n−59.7514 \\n2.8222 \\n10 \\n1604 \\n0.2426 \\n0.0102 \\n−61.2916 \\n2.5744 \\n15 \\n3200 \\n0.2350 \\n0.0067 \\n−59.3476 \\n1.7022 \\n20 \\n5762 \\n0.2427 \\n0.0050 \\n−61.2986 \\n1.2637 \\n25 \\n9890 \\n0.2433 \\n0.0038 \\n−61.4761 \\n0.9572 \\n30 \\n16550 \\n0.2424 \\n0.0029 \\n−61.2533 \\n0.7359 \\n40 \\n44578 \\n0.2410 \\n0.0017 \\n−60.8778 \\n0.4430 \\n50 \\n117288 \\n0.2402 \\n0.0011 \\n−60.6282 \\n0.2708 \\n\\nθ \\nstd(θ)λ \\nstd(λ) \\n\\n1 \\n100 \\nNo MLE \\nNo MLE \\n2 \\n210 \\nNo MLE \\nNo MLE \\n3 \\n332 \\n250.5738 \\n0.2519 \\n8.1158 \\n4.1282 \\n4 \\n466 \\n250.8324 \\n0.3037 \\n11.9342 \\n1.8080 \\n5 \\n614 \\n251.0506 \\n0.2984 \\n13.4066 \\n1.3070 \\n6 \\n776 \\n251.0632 \\n0.2519 \\n12.2392 \\n0.8591 \\n7 \\n954 \\n251.1471 \\n0.2327 \\n12.4655 \\n0.7432 \\n8 \\n1150 \\n251.0128 \\n0.2219 \\n13.0259 \\n0.6933 \\n9 \\n1366 \\n250.9891 \\n0.2058 \\n13.0362 \\n0.6148 \\n10 \\n1604 \\n251.1269 \\n0.1874 \\n12.7136 \\n0.5330 \\n15 \\n3200 \\n250.9385 \\n0.1386 \\n13.1228 \\n0.3751 \\n20 \\n5762 \\n251.1056 \\n0.1019 \\n12.7110 \\n0.2610 \\n25 \\n9890 \\n251.1342 \\n0.0780 \\n12.6756 \\n0.1965 \\n30 \\n16550 \\n251.1604 \\n0.0605 \\n12.7233 \\n0.1522 \\n40 \\n44578 \\n251.1016 \\n0.0372 \\n12.7992 \\n0.0927 \\n50 \\n117288 \\n250.8531 \\n0.0231 \\n12.8395 \\n0.0571\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Means of estimates and their standard deviations from 500 simulation runs\",\n \"content\": \"Table 2 .\\n2Means of estimates and their standard deviations from 500 simulation runsn \\nSequential design \\nOptimal design \\nSequential design \\nOptimal design \\n\\na \\nstd(â)â \\nstd(â)b \\nstd(b)b \\nstd(b) \\n\\n50 \\n0.4342 \\n0.3953 \\n0.2454 \\n0.0659 \\n−109.6947 \\n99.0491 \\n−62.4331 \\n16.8447 \\n100 \\n0.2786 \\n0.1151 \\n0.2476 \\n0.0387 \\n−70.7275 \\n28.8748 \\n−62.9339 \\n9.8681 \\n200 \\n0.2482 \\n0.0373 \\n0.2420 \\n0.0262 \\n−63.0732 \\n9.4241 \\n−61.4907 \\n6.6879 \\n300 \\n0.2448 \\n0.0254 \\n0.2418 \\n0.0213 \\n−62.2302 \\n6.4475 \\n−61.4518 \\n5.4436 \\n400 \\n0.2433 \\n0.0203 \\n0.2413 \\n0.0184 \\n−61.8373 \\n5.1676 \\n−61.3297 \\n4.6997 \\n500 \\n0.2425 \\n0.0176 \\n0.2405 \\n0.0164 \\n−61.6395 \\n4.4892 \\n−61.1274 \\n4.1928 \\n700 \\n0.2415 \\n0.0145 \\n0.2408 \\n0.0139 \\n−61.3900 \\n3.6946 \\n−61.2015 \\n3.5441 \\n1000 \\n0.2418 \\n0.0120 \\n0.2409 \\n0.0116 \\n−61.4646 \\n3.0482 \\n−61.2335 \\n2.9653 \\n1500 \\n0.2410 \\n0.0096 \\n0.2406 \\n0.0095 \\n−61.2491 \\n2.4610 \\n−61.1591 \\n2.4188 \\n2000 \\n0.2411 \\n0.0083 \\n0.2408 \\n0.0082 \\n−61.2892 \\n2.1225 \\n−61.2076 \\n2.0953 \\n3000 \\n0.2408 \\n0.0068 \\n0.2405 \\n0.0067 \\n−61.2001 \\n1.7240 \\n−61.1375 \\n1.7089 \\n4000 \\n0.2406 \\n0.0058 \\n0.2405 \\n0.0058 \\n−61.1488 \\n1.4894 \\n−61.1361 \\n1.4799 \\n5000 \\n0.2405 \\n0.0052 \\n0.2402 \\n0.0052 \\n−61.1338 \\n1.3305 \\n−61.0467 \\n1.3223 \\n7000 \\n0.2404 \\n0.0044 \\n0.2401 \\n0.0044 \\n−61.0939 \\n1.1225 \\n−61.0178 \\n1.1170 \\n10000 \\n0.2402 \\n0.0037 \\n0.2401 \\n0.0037 \\n−61.0621 \\n0.9378 \\n−61.0184 \\n0.9344 \\n15000 \\n0.2402 \\n0.0030 \\n0.2400 \\n0.0030 \\n−61.0505 \\n0.7648 \\n−61.0077 \\n0.7629 \\n20000 \\n0.2401 \\n0.0026 \\n0.2401 \\n0.0026 \\n−61.0296 \\n0.6619 \\n−61.0128 \\n0.6607 \\n\\nθ \\nstd(θ )θ \\nstd(θ )λ \\nstd(λ)λ \\nstd(λ) \\n\\n50 \\n254.3894 \\n7.2977 \\n252.7209 \\n1.3866 \\n21.0825 \\n38.7785 \\n13.1512 \\n3.6519 \\n100 \\n253.1132 \\n1.7524 \\n252.6023 \\n0.8089 \\n14.3413 \\n7.8859 \\n12.7611 \\n2.0261 \\n200 \\n252.7223 \\n0.5869 \\n252.5675 \\n0.5638 \\n12.7674 \\n1.9482 \\n12.8914 \\n1.4084 \\n300 \\n252.6707 \\n0.4449 \\n252.5894 \\n0.4589 \\n12.7352 \\n1.3232 \\n12.8599 \\n1.1416 \\n400 \\n252.6603 \\n0.3783 \\n252.5911 \\n0.3972 \\n12.7684 \\n1.0674 \\n12.8631 \\n0.9864 \\n500 \\n252.6529 \\n0.3393 \\n252.5970 \\n0.3554 \\n12.7853 \\n0.9307 \\n12.8892 \\n0.8837 \\n700 \\n252.6479 \\n0.2892 \\n252.6178 \\n0.2994 \\n12.8159 \\n0.7700 \\n12.8514 \\n0.7427 \\n1000 \\n252.6554 \\n0.2433 \\n252.6364 \\n0.2501 \\n12.7847 \\n0.6323 \\n12.8315 \\n0.6196 \\n1500 \\n252.6535 \\n0.2003 \\n252.6472 \\n0.2041 \\n12.8202 \\n0.5134 \\n12.8361 \\n0.5058 \\n2000 \\n252.6544 \\n0.1741 \\n252.6507 \\n0.1766 \\n12.8081 \\n0.4419 \\n12.8224 \\n0.4373 \\n3000 \\n252.6413 \\n0.1428 \\n252.6444 \\n0.1443 \\n12.8203 \\n0.3597 \\n12.8329 \\n0.3573 \\n4000 \\n252.6417 \\n0.1240 \\n252.6472 \\n0.1249 \\n12.8284 \\n0.3111 \\n12.8305 \\n0.3093 \\n5000 \\n252.6426 \\n0.1110 \\n252.6423 \\n0.1118 \\n12.8301 \\n0.2780 \\n12.8474 \\n0.2771 \\n7000 \\n252.6454 \\n0.0940 \\n252.6379 \\n0.0945 \\n12.8368 \\n0.2348 \\n12.8517 \\n0.2342 \\n10000 \\n252.6435 \\n0.0788 \\n252.6366 \\n0.0791 \\n12.8420 \\n0.1963 \\n12.8504 \\n0.1959 \\n15000 \\n252.6401 \\n0.0644 \\n252.6376 \\n0.0646 \\n12.8432 \\n0.1601 \\n12.8518 \\n0.1600 \\n20000 \\n252.6391 \\n0.0558 \\n252.6391 \\n0.0559 \\n12.8470 \\n0.1387 \\n12.8503 \\n0.1385\"\n },\n {\n \"id\": \"\",\n \"caption\": \"MSEs for parameter estimatesâ andb as functions of the number of observations n † 399793 †The sequential design proposed and the theoretically D-optimal design are compared. Reported MSEs are means from 500 simulation runs.\",\n \"content\": \"Table 3 .\\n3MSEs for parameter estimatesâ andb as functions of the number of observations n † 399793 †The sequential design proposed and the theoretically D-optimal design are compared. Reported MSEs are means from 500 simulation runs.n \\nMSE ofâ \\nMSE ofb \\n\\nSequential \\nOptimal \\nSequential \\nOptimal \\n\\n50 \\n0.206843 \\n0.002715 \\n12983.570197 \\n179.184180 \\n100 \\n0.021449 \\n0.001546 \\n1348.770300 \\n101.345741 \\n200 \\n0.001771 \\n0.000667 \\n112.423833 \\n43.457153 \\n300 \\n0.000686 \\n0.000477 \\n44.089057 \\n31.041668 \\n400 \\n0.000439 \\n0.000373 \\n28.286722 \\n24.314540 \\n500 \\n0.000316 \\n0.000297 \\n20.389242 \\n19.364127 \\n700 \\n0.000211 \\n0.000196 \\n13.672440 \\n12.766761 \\n1000 \\n0.000139 \\n0.000132 \\n9.035246 \\n8.643545 \\n1500 \\n0.000093 \\n0.000080 \\n6.046665 \\n5.201665 \\n2000 \\n0.000077 \\n0.000063 \\n4.993876 \\n4.132176 \\n3000 \\n0.000049 \\n0.000045 \\n3.198680 \\n2.937851 \\n4000 \\n0.000037 \\n0.000032 \\n2.390940 \\n2.100386 \\n5000 \\n0.000030 \\n0.000024 \\n1.946899 \\n1.584166 \\n7000 \\n0.000021 \\n0.000017 \\n1.379387 \\n1.116941 \\n10000 \\n0.000015 \\n0.000012 \\n0.961786 \\n0.811790 \\n15000 \\n0.000010 \\n0.000008 \\n0.634116 \\n0.523743 \\n20000 \\n0.000007 \\n0.000006 \\n0.473926 \\n0.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Experimental design for binary data. K M Abdelbasit, R L Plackett, J. Am. Statist. Ass. 78Abdelbasit, K. M. and Plackett, R. L. (1983) Experimental design for binary data. J. Am. Statist. Ass., 78, 90-98.","doi":""},{"text":"On the existence of maximum likelihood estimates in logistic regression models. A Albert, J A Anderson, Biometrika. 71Albert, A. and Anderson, J. A. (1984) On the existence of maximum likelihood estimates in logistic regression models. Biometrika, 71, 1-10.","doi":""},{"text":"Optimum Experimental Design. A C Atkinson, A N Donev, Oxford University PressOxfordAtkinson, A. C. and Donev, A. N. (1992) Optimum Experimental Design. Oxford: Oxford University Press.","doi":""},{"text":"Quantum Mechanics: a Modern Development. L E Ballentine, World Scientific PressSingaporeBallentine, L. E. (1998) Quantum Mechanics: a Modern Development. Singapore: World Scientific Press.","doi":""},{"text":"Applied Optimal Design. M P F Berger, Wong, W.-K.WileyNew YorkBerger, M. P. F. and Wong, W.-K. (eds) (2005) Applied Optimal Design. New York: Wiley.","doi":""},{"text":"Influence of dissipation on quantum tunneling in macroscopic systems. A O Caldeira, A J Leggett, Phys. Rev. Lett. 46Caldeira, A. O. and Leggett, A. J. (1980) Influence of dissipation on quantum tunneling in macroscopic systems. Phys. Rev. Lett., 46, 211-214.","doi":""},{"text":"The use of a canonical form in the construction of locally optimal designs for non-linear problems. I Ford, B Torsney, C F J Wu, J. R. Statist. Soc. B. 54Ford, I., Torsney, B. and Wu, C. F. J. (1992) The use of a canonical form in the construction of locally optimal designs for non-linear problems. J. R. Statist. Soc. B, 54, 569-583.","doi":""},{"text":"Single charge tunneling. H Grabert, Devoret, NATO Adv. Study Inst. B. M. H.294Grabert, H. and Devoret, M. H. (eds) (1992) Single charge tunneling. NATO Adv. Study Inst. B, 294.","doi":""},{"text":"Temperature dependence of quantum decay rates in dissipative systems. H Grabert, P Olschowski, U Weiss, Phys. Rev. B. 32Grabert, H., Olschowski, P. and Weiss, U. (1985) Temperature dependence of quantum decay rates in dissipative systems. Phys. Rev. B, 32, 3348-3350.","doi":""},{"text":"Decay of the zero-voltage state in small-area, high-current-density Josephson junctions. L D Jackel, J P Gordon, E L Hu, R E Howard, L A Fetter, D M Tennant, R W Epworth, J Kurkijarvi, Phys. Rev. Lett. 47Jackel, L. D., Gordon, J. P., Hu, E. L., Howard, R. E., Fetter, L. A., Tennant, D. M., Epworth, R. W. and Kurkijarvi, J. (1981) Decay of the zero-voltage state in small-area, high-current-density Josephson junctions. Phys. Rev. Lett., 47, 697-700.","doi":""},{"text":"Optimal designs for the estimation of the logistic regression. B D Josephson, L A Kalish, J L Rosenberger, 33Phys. Lett. 1251Pennsylvania State UniversityTechnical ReportPossible new effects in superconductive tunnellingJosephson, B. D. (1962) Possible new effects in superconductive tunnelling. Phys. Lett., 1, 251. Kalish, L. A. and Rosenberger, J. L. (1978) Optimal designs for the estimation of the logistic regression. Technical Report 33. Pennsylvania State University, Philadelphia.","doi":"33"},{"text":"The equivalence of two extremum problems. J Kiefer, J Wolfowitz, Can. J. Math. 12Kiefer, J. and Wolfowitz, J. (1960) The equivalence of two extremum problems. Can. J. Math., 12, 363-366.","doi":""},{"text":"Minimax D-optimal designs for the logistic model. J King, W.-K Wong, Biometrics. 56King, J. and Wong, W.-K. (2000) Minimax D-optimal designs for the logistic model. Biometrics, 56, 1263-1267.","doi":""},{"text":"Topics in Optimal Design. E P Liski, N K Mandal, K R Shah, B K Sinha, SpringerNew YorkLiski, E. P., Mandal, N. K., Shah, K. R. and Sinha, B. K. (2002) Topics in Optimal Design. New York: Springer.","doi":""},{"text":"Quantum-state engineering with Josephson-junction devices. Y Makhlin, G Schön, A Shnirman, Rev. Mod. Phys. 73357Makhlin, Y., Schön, G. and Shnirman, A. (2001) Quantum-state engineering with Josephson-junction devices. Rev. Mod. Phys., 73, 357.","doi":""},{"text":"Optimal designs for binary data under logistic regression. T Mathew, B K Sinha, J. Statist. Planng Inf. 93Mathew, T. and Sinha, B. K. (2001) Optimal designs for binary data under logistic regression. J. Statist. Planng Inf., 93, 295-307.","doi":""},{"text":"Optimal designs for binary data. S Minkin, J. Am. Statist. Ass. 82Minkin, S. (1987) Optimal designs for binary data. J. Am. Statist. Ass., 82, 1098-1103.","doi":""},{"text":"Shotnoise-driven escape in hysteretic Josephson junctions. J P Pekola, T E Nieminen, M Meschke, J M Kivioja, A O Niskanen, J J Vartiainen, F Pukelsheim, Phys. Rev. Lett. 95WileyBiometricsPekola, J. P., Nieminen, T. E., Meschke, M., Kivioja, J. M., Niskanen, A. O. and Vartiainen, J. J. (2005) Shot- noise-driven escape in hysteretic Josephson junctions. Phys. Rev. Lett., 95, 197004. Pukelsheim, F. (1993) Optimal Design of Experiments. New York: Wiley. Sitter, R. R. (1992) Robust designs for binary data. Biometrics, 48, 1145-1155.","doi":""},{"text":"Optimal designs for binary response experiments: Fieller, D and A criteria. R R Sitter, C F J Wu, Scand. J. Statist. 20Sitter, R. R. and Wu, C. F. J. (1993) Optimal designs for binary response experiments: Fieller, D and A criteria. Scand. J. Statist., 20, 329-341.","doi":""},{"text":"Macroscopic quantum tunneling in 1-μm Nb Josephson junctions. R F Voss, R A Webb, Phys. Rev. Lett. 47Voss, R. F. and Webb, R. A. (1981) Macroscopic quantum tunneling in 1-μm Nb Josephson junctions. Phys. Rev. Lett., 47, 265-268.","doi":""}],"string":"[\n {\n \"text\": \"Experimental design for binary data. K M Abdelbasit, R L Plackett, J. Am. Statist. Ass. 78Abdelbasit, K. M. and Plackett, R. L. (1983) Experimental design for binary data. J. Am. Statist. Ass., 78, 90-98.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the existence of maximum likelihood estimates in logistic regression models. A Albert, J A Anderson, Biometrika. 71Albert, A. and Anderson, J. A. (1984) On the existence of maximum likelihood estimates in logistic regression models. Biometrika, 71, 1-10.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optimum Experimental Design. A C Atkinson, A N Donev, Oxford University PressOxfordAtkinson, A. C. and Donev, A. N. (1992) Optimum Experimental Design. Oxford: Oxford University Press.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quantum Mechanics: a Modern Development. L E Ballentine, World Scientific PressSingaporeBallentine, L. E. (1998) Quantum Mechanics: a Modern Development. Singapore: World Scientific Press.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Applied Optimal Design. M P F Berger, Wong, W.-K.WileyNew YorkBerger, M. P. F. and Wong, W.-K. (eds) (2005) Applied Optimal Design. New York: Wiley.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Influence of dissipation on quantum tunneling in macroscopic systems. A O Caldeira, A J Leggett, Phys. Rev. Lett. 46Caldeira, A. O. and Leggett, A. J. (1980) Influence of dissipation on quantum tunneling in macroscopic systems. Phys. Rev. Lett., 46, 211-214.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The use of a canonical form in the construction of locally optimal designs for non-linear problems. I Ford, B Torsney, C F J Wu, J. R. Statist. Soc. B. 54Ford, I., Torsney, B. and Wu, C. F. J. (1992) The use of a canonical form in the construction of locally optimal designs for non-linear problems. J. R. Statist. Soc. B, 54, 569-583.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Single charge tunneling. H Grabert, Devoret, NATO Adv. Study Inst. B. M. H.294Grabert, H. and Devoret, M. H. (eds) (1992) Single charge tunneling. NATO Adv. Study Inst. B, 294.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Temperature dependence of quantum decay rates in dissipative systems. H Grabert, P Olschowski, U Weiss, Phys. Rev. B. 32Grabert, H., Olschowski, P. and Weiss, U. (1985) Temperature dependence of quantum decay rates in dissipative systems. Phys. Rev. B, 32, 3348-3350.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Decay of the zero-voltage state in small-area, high-current-density Josephson junctions. L D Jackel, J P Gordon, E L Hu, R E Howard, L A Fetter, D M Tennant, R W Epworth, J Kurkijarvi, Phys. Rev. Lett. 47Jackel, L. D., Gordon, J. P., Hu, E. L., Howard, R. E., Fetter, L. A., Tennant, D. M., Epworth, R. W. and Kurkijarvi, J. (1981) Decay of the zero-voltage state in small-area, high-current-density Josephson junctions. Phys. Rev. Lett., 47, 697-700.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optimal designs for the estimation of the logistic regression. B D Josephson, L A Kalish, J L Rosenberger, 33Phys. Lett. 1251Pennsylvania State UniversityTechnical ReportPossible new effects in superconductive tunnellingJosephson, B. D. (1962) Possible new effects in superconductive tunnelling. Phys. Lett., 1, 251. Kalish, L. A. and Rosenberger, J. L. (1978) Optimal designs for the estimation of the logistic regression. Technical Report 33. Pennsylvania State University, Philadelphia.\",\n \"doi\": \"33\"\n },\n {\n \"text\": \"The equivalence of two extremum problems. J Kiefer, J Wolfowitz, Can. J. Math. 12Kiefer, J. and Wolfowitz, J. (1960) The equivalence of two extremum problems. Can. J. Math., 12, 363-366.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Minimax D-optimal designs for the logistic model. J King, W.-K Wong, Biometrics. 56King, J. and Wong, W.-K. (2000) Minimax D-optimal designs for the logistic model. Biometrics, 56, 1263-1267.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Topics in Optimal Design. E P Liski, N K Mandal, K R Shah, B K Sinha, SpringerNew YorkLiski, E. P., Mandal, N. K., Shah, K. R. and Sinha, B. K. (2002) Topics in Optimal Design. New York: Springer.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quantum-state engineering with Josephson-junction devices. Y Makhlin, G Schön, A Shnirman, Rev. Mod. Phys. 73357Makhlin, Y., Schön, G. and Shnirman, A. (2001) Quantum-state engineering with Josephson-junction devices. Rev. Mod. Phys., 73, 357.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optimal designs for binary data under logistic regression. T Mathew, B K Sinha, J. Statist. Planng Inf. 93Mathew, T. and Sinha, B. K. (2001) Optimal designs for binary data under logistic regression. J. Statist. Planng Inf., 93, 295-307.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optimal designs for binary data. S Minkin, J. Am. Statist. Ass. 82Minkin, S. (1987) Optimal designs for binary data. J. Am. Statist. Ass., 82, 1098-1103.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Shotnoise-driven escape in hysteretic Josephson junctions. J P Pekola, T E Nieminen, M Meschke, J M Kivioja, A O Niskanen, J J Vartiainen, F Pukelsheim, Phys. Rev. Lett. 95WileyBiometricsPekola, J. P., Nieminen, T. E., Meschke, M., Kivioja, J. M., Niskanen, A. O. and Vartiainen, J. J. (2005) Shot- noise-driven escape in hysteretic Josephson junctions. Phys. Rev. Lett., 95, 197004. Pukelsheim, F. (1993) Optimal Design of Experiments. New York: Wiley. Sitter, R. R. (1992) Robust designs for binary data. Biometrics, 48, 1145-1155.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optimal designs for binary response experiments: Fieller, D and A criteria. R R Sitter, C F J Wu, Scand. J. Statist. 20Sitter, R. R. and Wu, C. F. J. (1993) Optimal designs for binary response experiments: Fieller, D and A criteria. Scand. J. Statist., 20, 329-341.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Macroscopic quantum tunneling in 1-μm Nb Josephson junctions. R F Voss, R A Webb, Phys. Rev. Lett. 47Voss, R. F. and Webb, R. A. (1981) Macroscopic quantum tunneling in 1-μm Nb Josephson junctions. Phys. Rev. Lett., 47, 265-268.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":476,"cells":{"text":{"kind":"string","value":"\nBoxSup: Exploiting Bounding Boxes to Supervise Convolutional Networks for Semantic Segmentation\n\n\nJifeng Dai jifdai@microsoft.com \nMicrosoft Research\n\n\nKaiming He kahe@microsoft.com \nMicrosoft Research\n\n\nJian Sun jiansun@microsoft.com \nMicrosoft Research\n\n\nBoxSup: Exploiting Bounding Boxes to Supervise Convolutional Networks for Semantic Segmentation\n\nRecent leading approaches to semantic segmentation rely on deep convolutional networks trained with humanannotated, pixel-level segmentation masks. Such pixelaccurate supervision demands expensive labeling effort and limits the performance of deep networks that usually benefit from more training data. In this paper, we propose a method that achieves competitive accuracy but only requires easily obtained bounding box annotations. The basic idea is to iterate between automatically generating region proposals and training convolutional networks. These two steps gradually recover segmentation masks for improving the networks, and vise versa. Our method, called \"BoxSup\", produces competitive results (e.g., 62.0% mAP for validation) supervised by boxes only, on par with strong baselines (e.g., 63.8% mAP) fully supervised by masks under the same setting. By leveraging a large amount of bounding boxes, BoxSup further unleashes the power of deep convolutional networks and yields state-of-the-art results on PAS-CAL VOC 2012 and PASCAL-CONTEXT[24].\n\nIntroduction\n\nIn the past few months, tremendous progress has been made in the field of semantic segmentation [12,22,13,6,5,23]. Deep convolutional neural networks (CNNs) [19,18] that play as rich hierarchical feature extractors are a key to these methods. These networks are trained on large-scale datasets [7,27] as classifiers, and transferred to the semantic segmentation tasks based on the annotated segmentation masks as supervision.\n\nBut pixel-level mask annotations are time-consuming, frustrating, and in the end commercially expensive to obtain. According to the annotation report of the large-scale Microsoft COCO dataset [21], the workload of labeling segmentation masks is more than 15 times heavier than that of spotting object locations. Further, the crowdsourcing annotators need to be specially trained for the tedious and diffi-cult task of labeling per-pixel masks. These facts limit the amount of available segmentation mask annotations, and thus hinder the performance of CNNs that in general desire large-scale data for training. On the contrary, bounding box annotations are more economical than masks. There have already existed a large number of available box-level annotations in datasets like PASCAL VOC 2007 1 [8] and ImageNet [27]. Though these box-level annotations are less precise than pixel-level masks, their amount may help improve training deep networks for semantic segmentation.\n\nIn addition, current leading approaches have not fully utilized the detailed pixel-level annotations. For example, in the Convolutional Feature Masking (CFM) method [6], the fine-resolution masks are used to generate very lowresolution (e.g., 6 × 6) masks on the feature maps. In the Fully Convolutional Network (FCN) method [22], the network predictions are regressed to the ground-truth masks using a large stride (e.g., 8 pixels). These methods yield competitive results without explicitly harnessing the finer masks. If we consider the box-level annotations as very coarse masks, can we still retain comparably good results without using the segmentation masks?\n\nIn this work, we investigate bounding box annotations as an alternative or extra source of supervision to train convolutional networks for semantic segmentation 2 . We resort to unsupervised region proposal methods [31,2] to generate candidate segmentation masks. The convolutional network is trained under the supervision of these approximate masks. The updated network in turn improves the estimated masks used for training. This process is iterated. Although the masks are coarse at the beginning, they are gradually improved and then provide useful information for network training. Fig. 1 illustrates our training algorithm.\n\nWe extensively evaluate our method, called \"BoxSup\", on the PASCAL segmentation benchmarks [8,24] box-supervised (i.e., using bounding box annotations) method shows a graceful degradation compared with its mask-supervised (i.e., using mask annotations) counterpart. As such, our method waives the requirement of pixel-level masks for training. Further, our semi-supervised variant in which 9/10 mask annotations are replaced with bounding box annotations yields comparable accuracy with the fully mask-supervised counterpart. This suggests that we may save expensive labeling effort by using bounding box annotations dominantly. Moreover, our method makes it possible to harness the large number of available box annotations to improve the mask-supervised results. Using the limited provided mask annotations and extra large-scale bounding box annotations, our method achieves state-of-the-art results on both PASCAL VOC 2012 and PASCAL-CONTEXT [24] benchmarks. Why can a large amount of bounding boxes help improve convolutional networks? Our error analysis reveals that a BoxSup model trained with a large set of boxes effectively increases the object recognition accuracy (the accuracy in the middle of an object), and its improvement on object boundaries is secondary. Though a box is too coarse to contain detailed segmentation information, it provides an instance for learning to distinguish object categories. The large-scale object instances improve the feature quality of the learned convolutional networks, and thus impact the overall performance for semantic segmentation.\n\n\nRelated Work\n\nDeep convolutional networks in general have better accuracy with the growing size of training data, as is evidenced in [18,34]. The ImageNet classification dataset [27] is one of the largest datasets with quality labels, but the current available datasets for object detection, semantic segmentation, and many other vision tasks mostly have orders of magnitudes fewer labeled samples. The milestone work of R-CNN [9] proposes to pre-train deep networks as classifiers on the large-scale ImageNet dataset and go on training (fine-tuning) them for other tasks that have limited number of training data. This transfer learning strategy is widely adopted for object detection [9,14,30], semantic segmentation [12,22,13,6,5,23], visual tracking [32], and other visual recognition tasks. With the continuously improving deep convolutional models [34,28,4,14,29,30,15], the accuracy of these vision tasks also improves thanks to the more powerful generic features learned from large-scale datasets.\n\nAlthough pre-training partially relieves the problem of limited data, the amount of the task-specific data for finetuning still matters. In [1], it has been found that augmenting the object detection training set by combining the VOC 2007 and VOC 2012 sets improves object detection accuracy compared with using VOC 2007 only. In [20], the training set for object detection is augmented by visual tracking results obtained from videos and improves detection accuracy. These experiments demonstrate the importance of dataset sizes for task-specific network training.\n\nFor semantic segmentation, there have been existing papers [33,10] that investigate exploiting bounding box annotations instead of masks. But the box-level annotations have not been used to supervised deep convolutional networks in those works.\n\n\nBaseline\n\nOur BoxSup method is in general applicable for many existing CNN-based mask-supervised semantic segmentation methods, such as FCN [22], improvements on FCN [5,35], and others [13,6,23]. In this paper, we adopt our implementation of the FCN method [22] refined by CRF [5] as the mask-supervised baseline, which we briefly introduce as follows.\n\nThe network training of FCN [22] is formulated as a perpixel regression problem to the ground-truth segmentation masks. Formally, the objective function can be written as:\nE(θ) = p e(X θ (p), l(p)),(1)\nwhere p is a pixel index, l(p) is the ground-truth semantic label at a pixel, and X θ (p) is the per-pixel labeling produced by the fully convolutional network with parameters θ. e(X θ (p), l(p)) is the per-pixel loss function. The network parameters θ are updated by back-propagation and stochastic gradient descent (SGD). A CRF is used to post-process the FCN results [5]. The objective function in Eqn.(1) demands pixel-level segmentation masks l(p) as supervision. It is not directly applicable if only bounding box annotations are given as supervision. Next we introduce our method for addressing this problem.\n\n\nApproach\n\n\nUnsupervised Segmentation for Supervised Training\n\nTo harness the bounding boxes annotations, it is desired to estimate segmentation masks from them. This is a widely studied supervised image segmentation problem, and can be addressed by, e.g., GrabCut [26]. But GrabCut can only generate one or a few samples from one box, which may be insufficient for deep network training.\n\nWe propose to generate a set of candidate segments using unsupervised region proposal methods (e.g., Selective Search [31]) due to their nice properties. First, region proposal methods have high recall rates [2] of having a good candidate in the proposal pool. Second, region proposal methods generate candidates of greater variance, which provide a kind of data augmentation [18] for network training. We will show by experiments the improvements of these properties.\n\nThe candidate segments are used to update the deep convolutional network. The semantic features learned by the network are then used to pick better candidates. This procedure is iterated. We formulate this procedure as an objective function as we will describe below.\n\nIt is worth noticing that the region proposal is only used for networking training. For inference, the trained FCN is directly applied on the image and produces pixel-wise predictions. So our usage of region proposals does not impact the test-time efficiency.\n\n\nFormulation\n\nAs a pre-processing, we use a region proposal method to generate segmentation masks. We adopt Multiscale Combinatorial Grouping (MCG) [2] by default, while other methods [31,17] are also evaluated. The proposal candidate masks are fixed throughout the training procedure. But during training, each candidate mask will be assigned a label which can be a semantic category or background. The labels assigned to the masks will be updated.\n\nWith a ground-truth bounding box annotation, we expect it to pick out a candidate mask that overlaps the box as much as possible. Formally, we define an overlapping objective function E o as:\nE o = 1 N S (1 − IoU(B, S))δ(l B , l S ).(2)\nHere S represents a candidate segment mask, and B represents a ground-truth bounding box annotation. IoU(B, S) ∈ [0, 1] is the intersection-over-union ratio computed from the ground-truth box B and the tight bounding box of the segment S. The function δ is equal to one if the semantic label l S assigned to segment S is the same as the ground-truth label l B of the bounding box B, and zero otherwise. Minimizing E o favors higher IoU scores when the semantic labels are consistent. This objective function is normalized by the number of candidate segments N . Figure 3: Update of segmentation masks during training. Here we show the masks in epoch #1, epoch #5, and epoch #20.\n\nEach segmentation mask will be used as the supervision for the next epoch.\n\nWith the candidate masks and their estimated semantic labels, we can supervise the deep convolutional network as in Eqn. (1). Formally, we consider the following regression objective function E r :\nE r = p e(X θ (p), l S (p)).(3)\nHere l S is the estimated semantic label used as supervision for the network training. This objective function is the same as Eqn.\n\n(1) except that its regression target is the estimated candidate segment. We minimize an objective function that combines the above two terms:\nmin θ,{l S } i (E o + λE r )(4)\nHere the summation i runs over the training images, and λ = 3 is a fixed weighting parameter. The variables to be optimized are the network parameters θ and the labeling {l S } of all candidate segments {S}. If only the term E o exists, the optimization problem in Eqn.(4) trivially finds a candidate segment that has the largest IoU score with the box; if only the term E r exists, the optimization problem in Eqn.(4) is equivalent to FCN. Our formulation simultaneously considers both cases.\n\n\nTraining Algorithm\n\nThe objective function in Eqn.(4) involves a problem of assigning labels to the candidate segments. Next we propose a greedy iterative solution to find a local optimum.\n\nWith the network parameters θ fixed, we update the semantic labeling {l S } for all candidate segments. In our implementation, we only consider the case in which one ground-truth bounding box can \"activate\" (i.e., assign a non-background label to) one and only one candidate. As such, we can simply update the semantic labeling by selecting a single candidate segment for each ground-truth bounding box, such that its cost E o + λE r is the smallest among all candidates. The selected segment is assigned the groundtruth semantic label associated with that bounding box. All other pixels are assigned the background label.\n\nThe above winner-takes-all selection tends to repeatedly use the same or very similar candidate segments, and the optimization procedure may be trapped in poor local optima. To increase the sample variance for better stochastic training, we further adopt a random sampling method to select the candidate segment for each ground-truth bounding box. Instead of selecting the single segment with the largest cost E o + λE r , we randomly sample a segment from the first k segments with the largest costs. In this paper we use k = 5. This random sampling strategy improves the accuracy by about 2% on the validation set.\n\nWith the semantic labeling {l S } of all candidate segments fixed, we update the network parameters θ. In this case, the problem becomes the FCN problem [22] as in Eqn. (1). This problem is minimized by SGD.\n\nWe iteratively perform the above two steps, fixing one set of variables and solving for the other set. For each iteration, we update the network parameters using one training epoch (i.e., all training images are visited once), and after that we update the segment labeling of all images. Fig.3 shows the gradually updated segmentation masks during training. The network is initialized by the model pre-trained in the Ima-geNet classification dataset, and our algorithm starts from the step of updating segment labels.\n\nOur method is applicable for the semi-supervised case (the ground-truth annotations are mixtures of segmentation masks and bounding boxes). The labeling l(p) is given by candidate proposals as above if a sample only has groundtruth boxes, and is simply assigned as the true label if a sample has ground-truth masks.\n\nIn the SGD training of updating the network, we use a mini-batch size of 20, following [22]. The learning rate is initialized to be 0.001 and divided by 10 after every 15 epochs. The training is terminated after 45 epochs.\n\n\nExperiments\n\nIn all our experiments, we use the publicly released VGG-16 model 3 [29] that is pre-trained on ImageNet [27]. The VGG model is also used by all competitors [22, 13, 6, \n\n\nExperiments on PASCAL VOC 2012\n\nWe first evaluate our method on the PASCAL VOC 2012 semantic segmentation benchmark [8]. This dataset involves 20 semantic categories of objects. We use the \"comp6\" evaluation protocol. The accuracy is evaluated by mean IoU scores. The original training data has 1,464 images. Following [11], the training data with ground-truth segmentation masks are augmented to 10,582 images. The validation and test sets have 1,449 and 1,456 images respectively. When evaluating the validation set or the test set, we only use the training set for training. A held-out 100 random validation images are used for cross-validation to set hyper-parameters. Table 1 compares the results of using different strategies of supervision on the validation set. When all ground-truth masks are used as supervision, the result is our implementation of the baseline DeepLab-CRF [5]. Our reproduction has a score of 63.8 (Table 1, \"mask only\"), which is very close to 63.74 reported in [5] under the same setting. So we believe that our reproduced baseline is convincing.\n\n\nComparisons of Supervision Strategies\n\nWhen all 10,582 training samples are replaced with bounding box annotations, our method yields a score of 62.0 ( Table 1, \"box only\"). Though the supervision information is substantially weakened, our method shows a graceful degradation (1.8%) compared with the strongly supervised baseline of 63.8. This indicates that in practice we can avoid the expensive mask labeling effort by using only bounding boxes, with small accuracy loss. Table 1 also shows the semi-supervised result of our method. This result uses the ground-truth masks of the original 1,464 training images and the bounding box annotations of the rest 9k images. The score is 63.5 (Table 1, \"semi\"), on par with the strongly supervised baseline. Such semi-supervision replaces 9/10 of the segmentation mask annotations with bounding box annotations. This means that we can greatly reduce the labeling effort by dominantly using bounding box annotations.\n\nAs a proof of concept, we further evaluate using a sub- stantially larger set of boxes. We use the Microsoft COCO dataset [21] that has 123,287 images with available groundtruth segmentation masks. This dataset has 80 semantic categories, and we only use the 20 categories that also present in PASCAL VOC. For our mask-supervised baseline, the result is a score of 68.1 (Table 1). Then we replace the ground-truth segmentation masks in COCO with their tight bounding boxes. Our semi-supervised result is 68.2 (Table 1), on par with the strongly supervised baseline. Fig. 5 shows some visual results in the validation set. The semi-supervised result (68.2) that uses VOC+COCO is considerably better than the strongly supervised result (63.8) that uses VOC only. The 4.4% gain is contributed by the extra large-scale bounding boxes in the 123k COCO images. This comparison suggests a promising strategywe may make use of the larger amount of existing bounding boxes annotations to improve the overall semantic segmentation results, as further analyzed below.\n\n\nError Analysis\n\nWhy can a large set of bounding boxes help improve convolutional networks? The error in semantic segmentation can be roughly thought of as two types: (i) recognition error that is due to confusions of recognizing object categories, and (ii) boundary error that is due to misalignments of pixel-level labels on object boundaries. Although the bounding box annotations have no information about the object boundaries, they provide extra object instances for recognizing them. We may expect that the large amount of boxes mainly improve the recognition accuracy.\n\nTo analyze the error, we separately evaluate the performance on the boundary regions and interior regions. Following [16,5], we generate a \"trimap\" near the ground-truth boundaries (Fig. 4, top). We evaluate mean IoU scores inside/outside the bands, referred to as boundary/interior regions. Fig. 4 (bottom) shows the results of using different band widths for the trimaps.\n\nFor the interior region, the accuracy of using the extra COCO boxes (red solid line, Fig. 4) is considerably higher than that of using VOC masks only (blue solid line). On the contrary, the improvement on the boundary regions is relatively smaller (red dash line vs. blue dash line). Note that correctly recognizing the interior may also help improve the boundaries (e.g., due to the CRF post-processing). So the improvement of the extra boxes on the boundary regions is secondary.\n\nBecause the accuracy in the interior region is mainly determined by correctly recognizing objects, this analysis suggests that the large amount of boxes improve the feature quality of a learned BoxSup model for better recognition.\n\n\nComparisons of Estimated Masks for Supervision\n\nIn Table 2 we evaluate different methods of estimating masks from bounding boxes for supervision. As a naïve baseline, we fill each bounding box with its semantic label, and consider it as a rectangular mask (Fig. 2(c)). Using these rectangular masks as the supervision throughout training, the score is 52.3 on the validation set. We also use GrabCut [26] to generate segmentation masks from boxes ( Fig. 2(d) Table 4: Results on PASCAL VOC 2012 test set. In the supervision (\"sup\") column, \"mask\" means all training samples are with segmentation mask annotations, \"box\" means all training samples are with bounding box annotations, and \"semi\" means mixtures. \"V\" denotes the VOC data, \"C\" denotes the COCO data, and \"V 07 \" denotes the VOC 2007 data which only has bounding boxes available.\n\nthroughout training, the score is 55.2. In both cases, the masks are not updated by the network feedbacks. Our method has a score 62.0 (Table 2) using the same set of bounding box annotations. This is a considerable gain over the baseline using fixed GrabCut masks. This indicates the importance of the mask quality for supervision. Fig. 3 shows that our method iteratively updates the masks by the network, which in turn improves the network training.\n\nWe also evaluate a variant of our method where each time the updated mask is the candidate with the largest cost, instead of randomly sampled from the first k candidates (see Sec. 4.3). This variant has a lower score of 59.7 ( Table 2). The random sampling strategy, which is data augmentation and increases sample variances, is beneficial for training. Table 2 also shows the result of the concurrent method WSSL [5] under the same evaluation setting. Its results is 58.5. This result suggests that our method estimates more accurate masks than [5] for supervision.\n\n\nComparisons of Region Proposals\n\nOur method resorts to unsupervised region proposals for training. In Table 3, we compare the effects of various region proposals on our method: Selective Search (SS) [31], Geodesic Object Proposals (GOP) [17], and MCG [2]. Table 3 shows that MCG [2] has the best accuracy, which is consistent with its segmentation quality evaluated by other metrics in [2]. Note that at test-time our method does not need region proposals. So the better accuracy of using MCG implies that our method effectively makes use of the higher quality segmentation masks to train a better network.\n\n\nComparisons on the Test Set\n\nNext we compare with the state-of-the-art methods on the PASCAL VOC 2012 test set. In Table 4, the methods are based on the same FCN baseline and thus fair comparisons are made to evaluate the impact of mask/box/semisupervision. As shown in Table 4, our box-supervised result that only uses VOC bounding boxes is 64.6. This compares favorably with the WSSL [25] counterpart (60.4) under the same setting. On the other hand, our box-supervised result has a graceful degradation (1.8%) compared with the masksupervised DeepLab-CRF (66.4 [5]) using the VOC training data. Moreover, our semi-supervised variant which replaces 9/10 segmentation mask annotations with bounding boxes has a score of 66.2. This is on par with the mask-supervised counterpart of DeepLab-CRF, but the supervision information used by our method is much weaker.\n\nIn the WSSL paper [25], by using all segmentation mask annotations in VOC and COCO, the strongly masksupervised result is 70.4. Our semi-supervised method shows a higher score of 71.0. Remarkably, our result uses the bounding box annotations from the 123k COCO images. So our method has a more accurate result but uses much weaker annotations than [25].\n\nOn the other hand, compared with the DeepLab-CRF result (66.4), our method has a 4.6% gain enjoyed from exploiting the bounding box annotations of the COCO dataset. This comparison demonstrates the power of our method that exploits large-scale bounding box annotations to improve accuracy.\n\n\nExploiting Boxes in PASCAL VOC 2007\n\nTo further demonstrate the effect of BoxSup, we exploit the bounding boxes in the PASCAL VOC 2007 dataset [8]. This dataset has no mask annotations. It is a de facto dataset which mask-supervised methods are not able to use.\n\nWe exploit all 10k images in the VOC 2007 trainval and test sets. We train a BoxSup model using the union set of VOC 2007 boxes, COCO boxes, and the augmented VOC 2012 training set. The score improves from 71.0 to 73.1 (Table 4) because of the extra box training data. It is reasonable for us to expect further improvement if more bounding box annotations are available.\n\n\nBaseline Improvement\n\nAlthough our focus is mainly on exploiting boxes as supervision, it is worth noticing that our method may also benefit from other improvements on the mask-sup baseline (FCN in our case). Concurrent with our work, there are a series of improvements [35,5] made on FCN, which achieve excellent results using strong mask-supervision from VOC and COCO data.\n\nTo show the potential of our BoxSup method in parallel with improvements on the baseline, we use a simple testtime augmentation to boost our results. Instead of comput- Table 5: Results on PASCAL-CONTEXT [24] validation. Our baseline is our implementation of FCN+CRF. \"V\" denotes the VOC data, and \"C\" denotes the COCO data.\n\ning pixel-wise predictions on a single scale, we compute the score maps from two extra scales (±20% of the original image size) and bilinearly re-scale the score maps to the original size. The scores from three scales are averaged. This simple modification boosts our result from 73.1 to 75.2 (BoxSup+, Table 4) in the VOC 2012 test set. This result is on par with the latest results using strong masksupervision from both VOC and COCO, but in our case the COCO dataset only provides bounding boxes.\n\n\nExperiments on PASCAL-CONTEXT\n\nWe further perform experiments on the recently labeled PASCAL-CONTEXT dataset [24]. This dataset provides ground-truth semantic labels for the whole scene, including object and stuff (e.g., grass, sky, water). Following the protocol in [24,6,22], the semantic segmentation is performed on the most frequent 59 categories (identified by [24]) plus a background category. The accuracy is measured by mean IoU scores. The training and evaluation are performed on the training and validation sets that have 4,998 and 5,105 images respectively.\n\nTo train a BoxSup model for this dataset, we first use the box annotations from all 80 object categories in the COCO dataset to train the FCN (using VGG-16). This network ends with an 81-way (with an extra one for background) layer. Then we remove this last layer and add a new 60way layer for the 59 categories of PASCAL-CONTEXT. We fine-tune this model in the 5k training images of PASCAL-CONTEXT. A CRF for post-processing is also used. We do no use the test-time scale augmentation. Table 5 shows the results in PASCAL-CONTEXT. The methods of CFM [6] and FCN [22] are both based on the VGG-16 model. Our baseline method, which is our implementation of FCN+CRF, has a score of 35.7 using masks of the 5k training images. Using our BoxSup model pretrained using the COCO boxes, the result is improved to 40.5. The 4.8% gain is solely because of the bounding box annotations in COCO that improve our network training. Fig. 6 shows some examples of our results for joint object and stuff segmentation.\n\n\nConclusion\n\nThe proposed BoxSup method can effectively harness bounding box annotations to train deep networks for semantic segmentation. Our BoxSup method that uses 133k bounding boxes and 10k masks achieves state-of-the-art results. Our error analysis suggests that semantic segmentation accuracy is hampered by the failure of recognizing objects, which large-scale data may help with.\n\nFigure 1 :\n1Overview of our training approach supervised by bounding boxes.\n\nFigure 2 :\n2Segmentation masks used as supervision. (a) A training image. (b) Ground-truth. (c) Each box is naïvely considered as a rectangle mask. (d) A segmentation mask is generated by GrabCut [26]. (e) For our method, the supervision is estimated from region proposals (MCG [2]) by considering bounding box annotations and network feedbacks.\n\nFigure 4 :\n4Error analysis on the validation set. Top: (from left to right) image, ground-truth, boundary regions marked as white, interior regions marked as white). Bottom: boundary and interior mean IoU, using VOC masks only (blue) and using extra COCO boxes (red).\n\nFigure 5 :\n5Example semantic segmentation results on PASCAL VOC 2012 validation using our method. (a) Images. (b) Supervised by masks in VOC. (c) Supervised by boxes in VOC. (d) Supervised by masks in VOC and boxes in COCO.\n\nFigure 6 :\n6Example results on PASCAL-CONTEXT validation. (a) Images. (b) Results of our baseline (35.7 mean IoU), trained using VOC masks. (c) Results of BoxSup (40.5 mean IoU), trained using VOC masks and COCO boxes.\n\n\n. Ourtrain image \nwith gt boxes \n\ndog \n\nperson \n\n… \n\ncandidate masks \n\nupdate \nnetwork \n\nupdate \nmasks \n\nBoxSup training \n\nfeedback \niteration \n\nestimated masks \n\nepoch \n#5 \n\nepoch \n#1 \n\nepoch \n#20 \n\nnetwork for \nsegmentation \n\n\n\nTable 2 :\n2Comparisons of estimated masks for supervision \nin PASCAL VOC 2012 validation. All methods only use \n10,582 bounding boxes as annotations, with no ground-\ntruth segmentation mask used. \n\nSS \nGOP \nMCG \n\nmean IoU \n59.5 \n60.4 \n62.0 \n\n\n\nTable 3 :\n3Comparisons of the effects of region proposal \nmethods on our method in PASCAL VOC 2012 validation. \nAll methods only use 10,582 bounding boxes as annota-\ntions, with no ground-truth segmentation mask used. \n\n\n\n\n). With the GrabCut masks as the supervisionmethod \nsup. \nmask # \nbox # mIoU \n\nFCN [22] \nmask \nV 10k \n-62.2 \nDeepLabCRF [5] mask \nV 10k \n-66.4 \nWSSL [25] \nbox \n-\nV 10k 60.4 \nBoxSup \nbox \n-\nV 10k 64.6 \nBoxSup \nsemi \nV 1.4k \nV 9k 66.2 \nWSSL [25] \nmask V+C 133k \n-70.4 \nBoxSup \nsemi \nV 10k \nC 123k 71.0 \nBoxSup \nsemi \nV 10k V07+C 133k 73.1 \nBoxSup+ \nsemi \nV 10k V07+C 133k 75.2 \n\n\nThe PASCAL VOC 2007 dataset only has bounding box annotations.2 The idea of using bounding box annotations for CNN-based semantic segmentation is developed concurrently and independently in[25]. We also compare with the results of[25].\nwww.robots.ox.ac.uk/˜vgg/research/very_deep/\n\nAnalyzing the performance of multilayer neural networks for object recognition. P Agrawal, R Girshick, J Malik, ECCV. P. Agrawal, R. Girshick, and J. Malik. Analyzing the perfor- mance of multilayer neural networks for object recognition. In ECCV, 2014.\n\nMultiscale combinatorial grouping. P Arbeláez, J Pont-Tuset, J T Barron, F Marques, J Malik, CVPR. P. Arbeláez, J. Pont-Tuset, J. T. Barron, F. Marques, and J. Malik. Multiscale combinatorial grouping. In CVPR, 2014.\n\nSemantic segmentation with second-order pooling. J Carreira, R Caseiro, J Batista, C Sminchisescu, ECCV. J. Carreira, R. Caseiro, J. Batista, and C. Sminchisescu. Se- mantic segmentation with second-order pooling. In ECCV. 2012.\n\nReturn of the devil in the details: Delving deep into convolutional nets. K Chatfield, K Simonyan, A Vedaldi, A Zisserman, BMVC. K. Chatfield, K. Simonyan, A. Vedaldi, and A. Zisserman. Return of the devil in the details: Delving deep into convo- lutional nets. In BMVC, 2014.\n\nSemantic image segmentation with deep convolutional nets and fully connected crfs. L.-C Chen, G Papandreou, I Kokkinos, K Murphy, A L Yuille, ICLR. L.-C. Chen, G. Papandreou, I. Kokkinos, K. Murphy, and A. L. Yuille. Semantic image segmentation with deep con- volutional nets and fully connected crfs. In ICLR, 2015.\n\nConvolutional feature masking for joint object and stuff segmentation. J Dai, K He, J Sun, CVPR. J. Dai, K. He, and J. Sun. Convolutional feature masking for joint object and stuff segmentation. In CVPR, 2015.\n\nImagenet: A large-scale hierarchical image database. J Deng, W Dong, R Socher, L.-J Li, K Li, L Fei-Fei, CVPR. J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei- Fei. Imagenet: A large-scale hierarchical image database. In CVPR, 2009.\n\nM Everingham, L Van Gool, C K Williams, J Winn, A Zisserman, The PASCAL Visual Object Classes (VOC) Challenge. IJCV. M. Everingham, L. Van Gool, C. K. Williams, J. Winn, and A. Zisserman. The PASCAL Visual Object Classes (VOC) Challenge. IJCV, 2010.\n\nRich feature hierarchies for accurate object detection and semantic segmentation. R Girshick, J Donahue, T Darrell, J Malik, R. Girshick, J. Donahue, T. Darrell, and J. Malik. Rich fea- ture hierarchies for accurate object detection and semantic segmentation. 2014.\n\nImagenet autoannotation with segmentation propagation. IJCV. M Guillaumin, D Küttel, V Ferrari, M. Guillaumin, D. Küttel, and V. Ferrari. Imagenet auto- annotation with segmentation propagation. IJCV, 2014.\n\nSemantic contours from inverse detectors. B Hariharan, P Arbeláez, L Bourdev, S Maji, J Malik, ICCV. B. Hariharan, P. Arbeláez, L. Bourdev, S. Maji, and J. Malik. Semantic contours from inverse detectors. In ICCV, 2011.\n\nSimultaneous detection and segmentation. B Hariharan, P Arbeláez, R Girshick, J Malik, ECCV. B. Hariharan, P. Arbeláez, R. Girshick, and J. Malik. Simul- taneous detection and segmentation. In ECCV. 2014.\n\nHypercolumns for object segmentation and fine-grained localization. B Hariharan, P Arbeláez, R Girshick, J Malik, CVPR. B. Hariharan, P. Arbeláez, R. Girshick, and J. Malik. Hyper- columns for object segmentation and fine-grained localiza- tion. In CVPR, 2015.\n\nSpatial pyramid pooling in deep convolutional networks for visual recognition. K He, X Zhang, S Ren, J Sun, ECCV. K. He, X. Zhang, S. Ren, and J. Sun. Spatial pyramid pooling in deep convolutional networks for visual recognition. In ECCV, 2014.\n\nDelving deep into rectifiers: Surpassing human-level performance on imagenet classification. K He, X Zhang, S Ren, J Sun, arXiv:1502.01852K. He, X. Zhang, S. Ren, and J. Sun. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. arXiv:1502.01852, 2015.\n\nRobust higher order potentials for enforcing label consistency. IJCV. P Kohli, P H Torr, P. Kohli, P. H. Torr, et al. Robust higher order potentials for enforcing label consistency. IJCV, pages 302-324, 2009.\n\nGeodesic object proposals. P Krähenbühl, V Koltun, ECCV. P. Krähenbühl and V. Koltun. Geodesic object proposals. In ECCV, 2014.\n\nImagenet classification with deep convolutional neural networks. A Krizhevsky, I Sutskever, G E Hinton, NIPS. A. Krizhevsky, I. Sutskever, and G. E. Hinton. Imagenet classification with deep convolutional neural networks. In NIPS, 2012.\n\nBackpropagation applied to handwritten zip code recognition. Y Lecun, B Boser, J S Denker, D Henderson, R E Howard, W Hubbard, L D , Neural computationY. LeCun, B. Boser, J. S. Denker, D. Henderson, R. E. Howard, W. Hubbard, and L. D. Jackel. Backpropagation applied to handwritten zip code recognition. Neural compu- tation, 1989.\n\nX Liang, S Liu, Y Wei, L Liu, L Lin, S Yan, arXiv:1411.2861Computational baby learning. X. Liang, S. Liu, Y. Wei, L. Liu, L. Lin, and S. Yan. Compu- tational baby learning. arXiv:1411.2861, 2014.\n\nMicrosoft COCO: Common objects in context. T.-Y Lin, M Maire, S Belongie, J Hays, P Perona, D Ramanan, P Dollár, C L Zitnick, ECCV. T.-Y. Lin, M. Maire, S. Belongie, J. Hays, P. Perona, D. Ra- manan, P. Dollár, and C. L. Zitnick. Microsoft COCO: Com- mon objects in context. In ECCV. 2014.\n\nFully convolutional networks for semantic segmentation. J Long, E Shelhamer, T Darrell, CVPR. J. Long, E. Shelhamer, and T. Darrell. Fully convolutional networks for semantic segmentation. In CVPR, 2015.\n\nFeedforward semantic segmentation with zoom-out features. M Mostajabi, P Yadollahpour, G Shakhnarovich, arXiv:1412.0774arXiv preprintM. Mostajabi, P. Yadollahpour, and G. Shakhnarovich. Feedforward semantic segmentation with zoom-out features. arXiv preprint arXiv:1412.0774, 2014.\n\nThe role of context for object detection and semantic segmentation in the wild. R Mottaghi, X Chen, X Liu, N.-G Cho, S.-W Lee, S Fidler, R Urtasun, A Yuille, CVPR. R. Mottaghi, X. Chen, X. Liu, N.-G. Cho, S.-W. Lee, S. Fi- dler, R. Urtasun, and A. Yuille. The role of context for object detection and semantic segmentation in the wild. In CVPR. 2014.\n\nWeakly-and semi-supervised learning of a dcnn for semantic image segmentation. G Papandreou, L.-C Chen, K Murphy, A L Yuille, arXiv:1502.02734arXiv preprintG. Papandreou, L.-C. Chen, K. Murphy, and A. L. Yuille. Weakly-and semi-supervised learning of a dcnn for seman- tic image segmentation. arXiv preprint arXiv:1502.02734, 2015.\n\nGrabcut: Interactive foreground extraction using iterated graph cuts. C Rother, V Kolmogorov, A Blake, ACM Transactions on Graphics. C. Rother, V. Kolmogorov, and A. Blake. Grabcut: Interac- tive foreground extraction using iterated graph cuts. ACM Transactions on Graphics, 2004.\n\n. O Russakovsky, J Deng, H Su, J Krause, S Satheesh, S Ma, Z Huang, A Karpathy, A Khosla, M Bernstein, arXiv:1409.0575Imagenet large scale visual recognition challengeO. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. Bernstein, et al. Imagenet large scale visual recognition challenge. arXiv:1409.0575, 2014.\n\nOverfeat: Integrated recognition, localization and detection using convolutional networks. P Sermanet, D Eigen, X Zhang, M Mathieu, R Fergus, Y Lecun, ICLR. P. Sermanet, D. Eigen, X. Zhang, M. Mathieu, R. Fergus, and Y. LeCun. Overfeat: Integrated recognition, localization and detection using convolutional networks. In ICLR, 2014.\n\nVery deep convolutional networks for large-scale image recognition. K Simonyan, A Zisserman, ICLR. K. Simonyan and A. Zisserman. Very deep convolutional networks for large-scale image recognition. In ICLR, 2015.\n\nGoing deeper with convolutions. C Szegedy, W Liu, Y Jia, P Sermanet, S Reed, D Anguelov, D Erhan, V Vanhoucke, A Rabinovich, CVPR. C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich. Going deeper with convolutions. In CVPR, 2015.\n\nSelective search for object recognition. J R Uijlings, K E Van De Sande, T Gevers, A W Smeulders, IJCVJ. R. Uijlings, K. E. van de Sande, T. Gevers, and A. W. Smeulders. Selective search for object recognition. IJCV, 2013.\n\nN Wang, S Li, A Gupta, D.-Y Yeung, arXiv:1501.04587Transferring rich feature hierarchies for robust visual tracking. N. Wang, S. Li, A. Gupta, and D.-Y. Yeung. Trans- ferring rich feature hierarchies for robust visual tracking. arXiv:1501.04587, 2015.\n\nSemantic segmentation without annotating segments. W Xia, C Domokos, J Dong, L.-F Cheong, S Yan, ICCV. W. Xia, C. Domokos, J. Dong, L.-F. Cheong, and S. Yan. Se- mantic segmentation without annotating segments. In ICCV, 2013.\n\nVisualizing and understanding convolutional neural networks. M D Zeiler, R Fergus, ECCV. M. D. Zeiler and R. Fergus. Visualizing and understanding convolutional neural networks. In ECCV, 2014.\n\nS Zheng, S Jayasumana, B Romera-Paredes, V Vineet, Z Su, D Du, C Huang, P Torr, arXiv:1502.03240Conditional random fields as recurrent neural networks. S. Zheng, S. Jayasumana, B. Romera-Paredes, V. Vineet, Z. Su, D. Du, C. Huang, and P. Torr. Conditional random fields as recurrent neural networks. arXiv:1502.03240, 2015.\n"},"corpus_id":{"kind":"number","value":1613420,"string":"1,613,420"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1503.01640\",\n \"mag\": 2949086864,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"conf/iccv/DaiHS15\",\n \"doi\": \"10.1109/iccv.2015.191\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1503.01640v2.pdf\"\n ],\n \"pdf_sha\": \"fc882ab8af22e25eacd5909256ed480417aa61bc\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1503.01640\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"BoxSup: Exploiting Bounding Boxes to Supervise Convolutional Networks for Semantic Segmentation"},"authors":{"kind":"list like","value":["Jifeng Dai jifdai@microsoft.com \nMicrosoft Research","Kaiming He kahe@microsoft.com \nMicrosoft Research","Jian Sun jiansun@microsoft.com \nMicrosoft Research"],"string":"[\n \"Jifeng Dai jifdai@microsoft.com \\nMicrosoft Research\",\n \"Kaiming He kahe@microsoft.com \\nMicrosoft Research\",\n \"Jian Sun jiansun@microsoft.com \\nMicrosoft Research\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Recent leading approaches to semantic segmentation rely on deep convolutional networks trained with humanannotated, pixel-level segmentation masks. Such pixelaccurate supervision demands expensive labeling effort and limits the performance of deep networks that usually benefit from more training data. In this paper, we propose a method that achieves competitive accuracy but only requires easily obtained bounding box annotations. The basic idea is to iterate between automatically generating region proposals and training convolutional networks. These two steps gradually recover segmentation masks for improving the networks, and vise versa. Our method, called \"BoxSup\", produces competitive results (e.g., 62.0% mAP for validation) supervised by boxes only, on par with strong baselines (e.g., 63.8% mAP) fully supervised by masks under the same setting. By leveraging a large amount of bounding boxes, BoxSup further unleashes the power of deep convolutional networks and yields state-of-the-art results on PAS-CAL VOC 2012 and PASCAL-CONTEXT[24]."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"In the past few months, tremendous progress has been made in the field of semantic segmentation [12,22,13,6,5,23]. Deep convolutional neural networks (CNNs) [19,18] that play as rich hierarchical feature extractors are a key to these methods. These networks are trained on large-scale datasets [7,27] as classifiers, and transferred to the semantic segmentation tasks based on the annotated segmentation masks as supervision.\nBut pixel-level mask annotations are time-consuming, frustrating, and in the end commercially expensive to obtain. According to the annotation report of the large-scale Microsoft COCO dataset [21], the workload of labeling segmentation masks is more than 15 times heavier than that of spotting object locations. Further, the crowdsourcing annotators need to be specially trained for the tedious and diffi-cult task of labeling per-pixel masks. These facts limit the amount of available segmentation mask annotations, and thus hinder the performance of CNNs that in general desire large-scale data for training. On the contrary, bounding box annotations are more economical than masks. There have already existed a large number of available box-level annotations in datasets like PASCAL VOC 2007 1 [8] and ImageNet [27]. Though these box-level annotations are less precise than pixel-level masks, their amount may help improve training deep networks for semantic segmentation.\nIn addition, current leading approaches have not fully utilized the detailed pixel-level annotations. For example, in the Convolutional Feature Masking (CFM) method [6], the fine-resolution masks are used to generate very lowresolution (e.g., 6 × 6) masks on the feature maps. In the Fully Convolutional Network (FCN) method [22], the network predictions are regressed to the ground-truth masks using a large stride (e.g., 8 pixels). These methods yield competitive results without explicitly harnessing the finer masks. If we consider the box-level annotations as very coarse masks, can we still retain comparably good results without using the segmentation masks?\nIn this work, we investigate bounding box annotations as an alternative or extra source of supervision to train convolutional networks for semantic segmentation 2 . We resort to unsupervised region proposal methods [31,2] to generate candidate segmentation masks. The convolutional network is trained under the supervision of these approximate masks. The updated network in turn improves the estimated masks used for training. This process is iterated. Although the masks are coarse at the beginning, they are gradually improved and then provide useful information for network training. Fig. 1 illustrates our training algorithm.\nWe extensively evaluate our method, called \"BoxSup\", on the PASCAL segmentation benchmarks [8,24] box-supervised (i.e., using bounding box annotations) method shows a graceful degradation compared with its mask-supervised (i.e., using mask annotations) counterpart. As such, our method waives the requirement of pixel-level masks for training. Further, our semi-supervised variant in which 9/10 mask annotations are replaced with bounding box annotations yields comparable accuracy with the fully mask-supervised counterpart. This suggests that we may save expensive labeling effort by using bounding box annotations dominantly. Moreover, our method makes it possible to harness the large number of available box annotations to improve the mask-supervised results. Using the limited provided mask annotations and extra large-scale bounding box annotations, our method achieves state-of-the-art results on both PASCAL VOC 2012 and PASCAL-CONTEXT [24] benchmarks. Why can a large amount of bounding boxes help improve convolutional networks? Our error analysis reveals that a BoxSup model trained with a large set of boxes effectively increases the object recognition accuracy (the accuracy in the middle of an object), and its improvement on object boundaries is secondary. Though a box is too coarse to contain detailed segmentation information, it provides an instance for learning to distinguish object categories. The large-scale object instances improve the feature quality of the learned convolutional networks, and thus impact the overall performance for semantic segmentation."},{"title":"Related Work","content":"Deep convolutional networks in general have better accuracy with the growing size of training data, as is evidenced in [18,34]. The ImageNet classification dataset [27] is one of the largest datasets with quality labels, but the current available datasets for object detection, semantic segmentation, and many other vision tasks mostly have orders of magnitudes fewer labeled samples. The milestone work of R-CNN [9] proposes to pre-train deep networks as classifiers on the large-scale ImageNet dataset and go on training (fine-tuning) them for other tasks that have limited number of training data. This transfer learning strategy is widely adopted for object detection [9,14,30], semantic segmentation [12,22,13,6,5,23], visual tracking [32], and other visual recognition tasks. With the continuously improving deep convolutional models [34,28,4,14,29,30,15], the accuracy of these vision tasks also improves thanks to the more powerful generic features learned from large-scale datasets.\nAlthough pre-training partially relieves the problem of limited data, the amount of the task-specific data for finetuning still matters. In [1], it has been found that augmenting the object detection training set by combining the VOC 2007 and VOC 2012 sets improves object detection accuracy compared with using VOC 2007 only. In [20], the training set for object detection is augmented by visual tracking results obtained from videos and improves detection accuracy. These experiments demonstrate the importance of dataset sizes for task-specific network training.\nFor semantic segmentation, there have been existing papers [33,10] that investigate exploiting bounding box annotations instead of masks. But the box-level annotations have not been used to supervised deep convolutional networks in those works."},{"title":"Baseline","content":"Our BoxSup method is in general applicable for many existing CNN-based mask-supervised semantic segmentation methods, such as FCN [22], improvements on FCN [5,35], and others [13,6,23]. In this paper, we adopt our implementation of the FCN method [22] refined by CRF [5] as the mask-supervised baseline, which we briefly introduce as follows.\nThe network training of FCN [22] is formulated as a perpixel regression problem to the ground-truth segmentation masks. Formally, the objective function can be written as:\nwhere p is a pixel index, l(p) is the ground-truth semantic label at a pixel, and X θ (p) is the per-pixel labeling produced by the fully convolutional network with parameters θ. e(X θ (p), l(p)) is the per-pixel loss function. The network parameters θ are updated by back-propagation and stochastic gradient descent (SGD). A CRF is used to post-process the FCN results [5]. The objective function in Eqn.(1) demands pixel-level segmentation masks l(p) as supervision. It is not directly applicable if only bounding box annotations are given as supervision. Next we introduce our method for addressing this problem."},{"title":"Approach","content":""},{"title":"Unsupervised Segmentation for Supervised Training","content":"To harness the bounding boxes annotations, it is desired to estimate segmentation masks from them. This is a widely studied supervised image segmentation problem, and can be addressed by, e.g., GrabCut [26]. But GrabCut can only generate one or a few samples from one box, which may be insufficient for deep network training.\nWe propose to generate a set of candidate segments using unsupervised region proposal methods (e.g., Selective Search [31]) due to their nice properties. First, region proposal methods have high recall rates [2] of having a good candidate in the proposal pool. Second, region proposal methods generate candidates of greater variance, which provide a kind of data augmentation [18] for network training. We will show by experiments the improvements of these properties.\nThe candidate segments are used to update the deep convolutional network. The semantic features learned by the network are then used to pick better candidates. This procedure is iterated. We formulate this procedure as an objective function as we will describe below.\nIt is worth noticing that the region proposal is only used for networking training. For inference, the trained FCN is directly applied on the image and produces pixel-wise predictions. So our usage of region proposals does not impact the test-time efficiency."},{"title":"Formulation","content":"As a pre-processing, we use a region proposal method to generate segmentation masks. We adopt Multiscale Combinatorial Grouping (MCG) [2] by default, while other methods [31,17] are also evaluated. The proposal candidate masks are fixed throughout the training procedure. But during training, each candidate mask will be assigned a label which can be a semantic category or background. The labels assigned to the masks will be updated.\nWith a ground-truth bounding box annotation, we expect it to pick out a candidate mask that overlaps the box as much as possible. Formally, we define an overlapping objective function E o as:\nHere S represents a candidate segment mask, and B represents a ground-truth bounding box annotation. IoU(B, S) ∈ [0, 1] is the intersection-over-union ratio computed from the ground-truth box B and the tight bounding box of the segment S. The function δ is equal to one if the semantic label l S assigned to segment S is the same as the ground-truth label l B of the bounding box B, and zero otherwise. Minimizing E o favors higher IoU scores when the semantic labels are consistent. This objective function is normalized by the number of candidate segments N . Figure 3: Update of segmentation masks during training. Here we show the masks in epoch #1, epoch #5, and epoch #20.\nEach segmentation mask will be used as the supervision for the next epoch.\nWith the candidate masks and their estimated semantic labels, we can supervise the deep convolutional network as in Eqn. (1). Formally, we consider the following regression objective function E r :\nHere l S is the estimated semantic label used as supervision for the network training. This objective function is the same as Eqn.\n(1) except that its regression target is the estimated candidate segment. We minimize an objective function that combines the above two terms:\nHere the summation i runs over the training images, and λ = 3 is a fixed weighting parameter. The variables to be optimized are the network parameters θ and the labeling {l S } of all candidate segments {S}. If only the term E o exists, the optimization problem in Eqn.(4) trivially finds a candidate segment that has the largest IoU score with the box; if only the term E r exists, the optimization problem in Eqn.(4) is equivalent to FCN. Our formulation simultaneously considers both cases."},{"title":"Training Algorithm","content":"The objective function in Eqn.(4) involves a problem of assigning labels to the candidate segments. Next we propose a greedy iterative solution to find a local optimum.\nWith the network parameters θ fixed, we update the semantic labeling {l S } for all candidate segments. In our implementation, we only consider the case in which one ground-truth bounding box can \"activate\" (i.e., assign a non-background label to) one and only one candidate. As such, we can simply update the semantic labeling by selecting a single candidate segment for each ground-truth bounding box, such that its cost E o + λE r is the smallest among all candidates. The selected segment is assigned the groundtruth semantic label associated with that bounding box. All other pixels are assigned the background label.\nThe above winner-takes-all selection tends to repeatedly use the same or very similar candidate segments, and the optimization procedure may be trapped in poor local optima. To increase the sample variance for better stochastic training, we further adopt a random sampling method to select the candidate segment for each ground-truth bounding box. Instead of selecting the single segment with the largest cost E o + λE r , we randomly sample a segment from the first k segments with the largest costs. In this paper we use k = 5. This random sampling strategy improves the accuracy by about 2% on the validation set.\nWith the semantic labeling {l S } of all candidate segments fixed, we update the network parameters θ. In this case, the problem becomes the FCN problem [22] as in Eqn. (1). This problem is minimized by SGD.\nWe iteratively perform the above two steps, fixing one set of variables and solving for the other set. For each iteration, we update the network parameters using one training epoch (i.e., all training images are visited once), and after that we update the segment labeling of all images. Fig.3 shows the gradually updated segmentation masks during training. The network is initialized by the model pre-trained in the Ima-geNet classification dataset, and our algorithm starts from the step of updating segment labels.\nOur method is applicable for the semi-supervised case (the ground-truth annotations are mixtures of segmentation masks and bounding boxes). The labeling l(p) is given by candidate proposals as above if a sample only has groundtruth boxes, and is simply assigned as the true label if a sample has ground-truth masks.\nIn the SGD training of updating the network, we use a mini-batch size of 20, following [22]. The learning rate is initialized to be 0.001 and divided by 10 after every 15 epochs. The training is terminated after 45 epochs."},{"title":"Experiments","content":"In all our experiments, we use the publicly released VGG-16 model 3 [29] that is pre-trained on ImageNet [27]. The VGG model is also used by all competitors [22, 13, 6,"},{"title":"Experiments on PASCAL VOC 2012","content":"We first evaluate our method on the PASCAL VOC 2012 semantic segmentation benchmark [8]. This dataset involves 20 semantic categories of objects. We use the \"comp6\" evaluation protocol. The accuracy is evaluated by mean IoU scores. The original training data has 1,464 images. Following [11], the training data with ground-truth segmentation masks are augmented to 10,582 images. The validation and test sets have 1,449 and 1,456 images respectively. When evaluating the validation set or the test set, we only use the training set for training. A held-out 100 random validation images are used for cross-validation to set hyper-parameters. Table 1 compares the results of using different strategies of supervision on the validation set. When all ground-truth masks are used as supervision, the result is our implementation of the baseline DeepLab-CRF [5]. Our reproduction has a score of 63.8 (Table 1, \"mask only\"), which is very close to 63.74 reported in [5] under the same setting. So we believe that our reproduced baseline is convincing."},{"title":"Comparisons of Supervision Strategies","content":"When all 10,582 training samples are replaced with bounding box annotations, our method yields a score of 62.0 ( Table 1, \"box only\"). Though the supervision information is substantially weakened, our method shows a graceful degradation (1.8%) compared with the strongly supervised baseline of 63.8. This indicates that in practice we can avoid the expensive mask labeling effort by using only bounding boxes, with small accuracy loss. Table 1 also shows the semi-supervised result of our method. This result uses the ground-truth masks of the original 1,464 training images and the bounding box annotations of the rest 9k images. The score is 63.5 (Table 1, \"semi\"), on par with the strongly supervised baseline. Such semi-supervision replaces 9/10 of the segmentation mask annotations with bounding box annotations. This means that we can greatly reduce the labeling effort by dominantly using bounding box annotations.\nAs a proof of concept, we further evaluate using a sub- stantially larger set of boxes. We use the Microsoft COCO dataset [21] that has 123,287 images with available groundtruth segmentation masks. This dataset has 80 semantic categories, and we only use the 20 categories that also present in PASCAL VOC. For our mask-supervised baseline, the result is a score of 68.1 (Table 1). Then we replace the ground-truth segmentation masks in COCO with their tight bounding boxes. Our semi-supervised result is 68.2 (Table 1), on par with the strongly supervised baseline. Fig. 5 shows some visual results in the validation set. The semi-supervised result (68.2) that uses VOC+COCO is considerably better than the strongly supervised result (63.8) that uses VOC only. The 4.4% gain is contributed by the extra large-scale bounding boxes in the 123k COCO images. This comparison suggests a promising strategywe may make use of the larger amount of existing bounding boxes annotations to improve the overall semantic segmentation results, as further analyzed below."},{"title":"Error Analysis","content":"Why can a large set of bounding boxes help improve convolutional networks? The error in semantic segmentation can be roughly thought of as two types: (i) recognition error that is due to confusions of recognizing object categories, and (ii) boundary error that is due to misalignments of pixel-level labels on object boundaries. Although the bounding box annotations have no information about the object boundaries, they provide extra object instances for recognizing them. We may expect that the large amount of boxes mainly improve the recognition accuracy.\nTo analyze the error, we separately evaluate the performance on the boundary regions and interior regions. Following [16,5], we generate a \"trimap\" near the ground-truth boundaries (Fig. 4, top). We evaluate mean IoU scores inside/outside the bands, referred to as boundary/interior regions. Fig. 4 (bottom) shows the results of using different band widths for the trimaps.\nFor the interior region, the accuracy of using the extra COCO boxes (red solid line, Fig. 4) is considerably higher than that of using VOC masks only (blue solid line). On the contrary, the improvement on the boundary regions is relatively smaller (red dash line vs. blue dash line). Note that correctly recognizing the interior may also help improve the boundaries (e.g., due to the CRF post-processing). So the improvement of the extra boxes on the boundary regions is secondary.\nBecause the accuracy in the interior region is mainly determined by correctly recognizing objects, this analysis suggests that the large amount of boxes improve the feature quality of a learned BoxSup model for better recognition."},{"title":"Comparisons of Estimated Masks for Supervision","content":"In Table 2 we evaluate different methods of estimating masks from bounding boxes for supervision. As a naïve baseline, we fill each bounding box with its semantic label, and consider it as a rectangular mask (Fig. 2(c)). Using these rectangular masks as the supervision throughout training, the score is 52.3 on the validation set. We also use GrabCut [26] to generate segmentation masks from boxes ( Fig. 2(d) Table 4: Results on PASCAL VOC 2012 test set. In the supervision (\"sup\") column, \"mask\" means all training samples are with segmentation mask annotations, \"box\" means all training samples are with bounding box annotations, and \"semi\" means mixtures. \"V\" denotes the VOC data, \"C\" denotes the COCO data, and \"V 07 \" denotes the VOC 2007 data which only has bounding boxes available.\nthroughout training, the score is 55.2. In both cases, the masks are not updated by the network feedbacks. Our method has a score 62.0 (Table 2) using the same set of bounding box annotations. This is a considerable gain over the baseline using fixed GrabCut masks. This indicates the importance of the mask quality for supervision. Fig. 3 shows that our method iteratively updates the masks by the network, which in turn improves the network training.\nWe also evaluate a variant of our method where each time the updated mask is the candidate with the largest cost, instead of randomly sampled from the first k candidates (see Sec. 4.3). This variant has a lower score of 59.7 ( Table 2). The random sampling strategy, which is data augmentation and increases sample variances, is beneficial for training. Table 2 also shows the result of the concurrent method WSSL [5] under the same evaluation setting. Its results is 58.5. This result suggests that our method estimates more accurate masks than [5] for supervision."},{"title":"Comparisons of Region Proposals","content":"Our method resorts to unsupervised region proposals for training. In Table 3, we compare the effects of various region proposals on our method: Selective Search (SS) [31], Geodesic Object Proposals (GOP) [17], and MCG [2]. Table 3 shows that MCG [2] has the best accuracy, which is consistent with its segmentation quality evaluated by other metrics in [2]. Note that at test-time our method does not need region proposals. So the better accuracy of using MCG implies that our method effectively makes use of the higher quality segmentation masks to train a better network."},{"title":"Comparisons on the Test Set","content":"Next we compare with the state-of-the-art methods on the PASCAL VOC 2012 test set. In Table 4, the methods are based on the same FCN baseline and thus fair comparisons are made to evaluate the impact of mask/box/semisupervision. As shown in Table 4, our box-supervised result that only uses VOC bounding boxes is 64.6. This compares favorably with the WSSL [25] counterpart (60.4) under the same setting. On the other hand, our box-supervised result has a graceful degradation (1.8%) compared with the masksupervised DeepLab-CRF (66.4 [5]) using the VOC training data. Moreover, our semi-supervised variant which replaces 9/10 segmentation mask annotations with bounding boxes has a score of 66.2. This is on par with the mask-supervised counterpart of DeepLab-CRF, but the supervision information used by our method is much weaker.\nIn the WSSL paper [25], by using all segmentation mask annotations in VOC and COCO, the strongly masksupervised result is 70.4. Our semi-supervised method shows a higher score of 71.0. Remarkably, our result uses the bounding box annotations from the 123k COCO images. So our method has a more accurate result but uses much weaker annotations than [25].\nOn the other hand, compared with the DeepLab-CRF result (66.4), our method has a 4.6% gain enjoyed from exploiting the bounding box annotations of the COCO dataset. This comparison demonstrates the power of our method that exploits large-scale bounding box annotations to improve accuracy."},{"title":"Exploiting Boxes in PASCAL VOC 2007","content":"To further demonstrate the effect of BoxSup, we exploit the bounding boxes in the PASCAL VOC 2007 dataset [8]. This dataset has no mask annotations. It is a de facto dataset which mask-supervised methods are not able to use.\nWe exploit all 10k images in the VOC 2007 trainval and test sets. We train a BoxSup model using the union set of VOC 2007 boxes, COCO boxes, and the augmented VOC 2012 training set. The score improves from 71.0 to 73.1 (Table 4) because of the extra box training data. It is reasonable for us to expect further improvement if more bounding box annotations are available."},{"title":"Baseline Improvement","content":"Although our focus is mainly on exploiting boxes as supervision, it is worth noticing that our method may also benefit from other improvements on the mask-sup baseline (FCN in our case). Concurrent with our work, there are a series of improvements [35,5] made on FCN, which achieve excellent results using strong mask-supervision from VOC and COCO data.\nTo show the potential of our BoxSup method in parallel with improvements on the baseline, we use a simple testtime augmentation to boost our results. Instead of comput- Table 5: Results on PASCAL-CONTEXT [24] validation. Our baseline is our implementation of FCN+CRF. \"V\" denotes the VOC data, and \"C\" denotes the COCO data.\ning pixel-wise predictions on a single scale, we compute the score maps from two extra scales (±20% of the original image size) and bilinearly re-scale the score maps to the original size. The scores from three scales are averaged. This simple modification boosts our result from 73.1 to 75.2 (BoxSup+, Table 4) in the VOC 2012 test set. This result is on par with the latest results using strong masksupervision from both VOC and COCO, but in our case the COCO dataset only provides bounding boxes."},{"title":"Experiments on PASCAL-CONTEXT","content":"We further perform experiments on the recently labeled PASCAL-CONTEXT dataset [24]. This dataset provides ground-truth semantic labels for the whole scene, including object and stuff (e.g., grass, sky, water). Following the protocol in [24,6,22], the semantic segmentation is performed on the most frequent 59 categories (identified by [24]) plus a background category. The accuracy is measured by mean IoU scores. The training and evaluation are performed on the training and validation sets that have 4,998 and 5,105 images respectively.\nTo train a BoxSup model for this dataset, we first use the box annotations from all 80 object categories in the COCO dataset to train the FCN (using VGG-16). This network ends with an 81-way (with an extra one for background) layer. Then we remove this last layer and add a new 60way layer for the 59 categories of PASCAL-CONTEXT. We fine-tune this model in the 5k training images of PASCAL-CONTEXT. A CRF for post-processing is also used. We do no use the test-time scale augmentation. Table 5 shows the results in PASCAL-CONTEXT. The methods of CFM [6] and FCN [22] are both based on the VGG-16 model. Our baseline method, which is our implementation of FCN+CRF, has a score of 35.7 using masks of the 5k training images. Using our BoxSup model pretrained using the COCO boxes, the result is improved to 40.5. The 4.8% gain is solely because of the bounding box annotations in COCO that improve our network training. Fig. 6 shows some examples of our results for joint object and stuff segmentation."},{"title":"Conclusion","content":"The proposed BoxSup method can effectively harness bounding box annotations to train deep networks for semantic segmentation. Our BoxSup method that uses 133k bounding boxes and 10k masks achieves state-of-the-art results. Our error analysis suggests that semantic segmentation accuracy is hampered by the failure of recognizing objects, which large-scale data may help with."},{"title":"Figure 1 :","content":""},{"title":"Figure 2 :","content":""},{"title":"Figure 4 :","content":""},{"title":"Figure 5 :","content":""},{"title":"Figure 6 :","content":""},{"title":"Table 2 :","content":""},{"title":"Table 3 :","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"In the past few months, tremendous progress has been made in the field of semantic segmentation [12,22,13,6,5,23]. Deep convolutional neural networks (CNNs) [19,18] that play as rich hierarchical feature extractors are a key to these methods. These networks are trained on large-scale datasets [7,27] as classifiers, and transferred to the semantic segmentation tasks based on the annotated segmentation masks as supervision.\\nBut pixel-level mask annotations are time-consuming, frustrating, and in the end commercially expensive to obtain. According to the annotation report of the large-scale Microsoft COCO dataset [21], the workload of labeling segmentation masks is more than 15 times heavier than that of spotting object locations. Further, the crowdsourcing annotators need to be specially trained for the tedious and diffi-cult task of labeling per-pixel masks. These facts limit the amount of available segmentation mask annotations, and thus hinder the performance of CNNs that in general desire large-scale data for training. On the contrary, bounding box annotations are more economical than masks. There have already existed a large number of available box-level annotations in datasets like PASCAL VOC 2007 1 [8] and ImageNet [27]. Though these box-level annotations are less precise than pixel-level masks, their amount may help improve training deep networks for semantic segmentation.\\nIn addition, current leading approaches have not fully utilized the detailed pixel-level annotations. For example, in the Convolutional Feature Masking (CFM) method [6], the fine-resolution masks are used to generate very lowresolution (e.g., 6 × 6) masks on the feature maps. In the Fully Convolutional Network (FCN) method [22], the network predictions are regressed to the ground-truth masks using a large stride (e.g., 8 pixels). These methods yield competitive results without explicitly harnessing the finer masks. If we consider the box-level annotations as very coarse masks, can we still retain comparably good results without using the segmentation masks?\\nIn this work, we investigate bounding box annotations as an alternative or extra source of supervision to train convolutional networks for semantic segmentation 2 . We resort to unsupervised region proposal methods [31,2] to generate candidate segmentation masks. The convolutional network is trained under the supervision of these approximate masks. The updated network in turn improves the estimated masks used for training. This process is iterated. Although the masks are coarse at the beginning, they are gradually improved and then provide useful information for network training. Fig. 1 illustrates our training algorithm.\\nWe extensively evaluate our method, called \\\"BoxSup\\\", on the PASCAL segmentation benchmarks [8,24] box-supervised (i.e., using bounding box annotations) method shows a graceful degradation compared with its mask-supervised (i.e., using mask annotations) counterpart. As such, our method waives the requirement of pixel-level masks for training. Further, our semi-supervised variant in which 9/10 mask annotations are replaced with bounding box annotations yields comparable accuracy with the fully mask-supervised counterpart. This suggests that we may save expensive labeling effort by using bounding box annotations dominantly. Moreover, our method makes it possible to harness the large number of available box annotations to improve the mask-supervised results. Using the limited provided mask annotations and extra large-scale bounding box annotations, our method achieves state-of-the-art results on both PASCAL VOC 2012 and PASCAL-CONTEXT [24] benchmarks. Why can a large amount of bounding boxes help improve convolutional networks? Our error analysis reveals that a BoxSup model trained with a large set of boxes effectively increases the object recognition accuracy (the accuracy in the middle of an object), and its improvement on object boundaries is secondary. Though a box is too coarse to contain detailed segmentation information, it provides an instance for learning to distinguish object categories. The large-scale object instances improve the feature quality of the learned convolutional networks, and thus impact the overall performance for semantic segmentation.\"\n },\n {\n \"title\": \"Related Work\",\n \"content\": \"Deep convolutional networks in general have better accuracy with the growing size of training data, as is evidenced in [18,34]. The ImageNet classification dataset [27] is one of the largest datasets with quality labels, but the current available datasets for object detection, semantic segmentation, and many other vision tasks mostly have orders of magnitudes fewer labeled samples. The milestone work of R-CNN [9] proposes to pre-train deep networks as classifiers on the large-scale ImageNet dataset and go on training (fine-tuning) them for other tasks that have limited number of training data. This transfer learning strategy is widely adopted for object detection [9,14,30], semantic segmentation [12,22,13,6,5,23], visual tracking [32], and other visual recognition tasks. With the continuously improving deep convolutional models [34,28,4,14,29,30,15], the accuracy of these vision tasks also improves thanks to the more powerful generic features learned from large-scale datasets.\\nAlthough pre-training partially relieves the problem of limited data, the amount of the task-specific data for finetuning still matters. In [1], it has been found that augmenting the object detection training set by combining the VOC 2007 and VOC 2012 sets improves object detection accuracy compared with using VOC 2007 only. In [20], the training set for object detection is augmented by visual tracking results obtained from videos and improves detection accuracy. These experiments demonstrate the importance of dataset sizes for task-specific network training.\\nFor semantic segmentation, there have been existing papers [33,10] that investigate exploiting bounding box annotations instead of masks. But the box-level annotations have not been used to supervised deep convolutional networks in those works.\"\n },\n {\n \"title\": \"Baseline\",\n \"content\": \"Our BoxSup method is in general applicable for many existing CNN-based mask-supervised semantic segmentation methods, such as FCN [22], improvements on FCN [5,35], and others [13,6,23]. In this paper, we adopt our implementation of the FCN method [22] refined by CRF [5] as the mask-supervised baseline, which we briefly introduce as follows.\\nThe network training of FCN [22] is formulated as a perpixel regression problem to the ground-truth segmentation masks. Formally, the objective function can be written as:\\nwhere p is a pixel index, l(p) is the ground-truth semantic label at a pixel, and X θ (p) is the per-pixel labeling produced by the fully convolutional network with parameters θ. e(X θ (p), l(p)) is the per-pixel loss function. The network parameters θ are updated by back-propagation and stochastic gradient descent (SGD). A CRF is used to post-process the FCN results [5]. The objective function in Eqn.(1) demands pixel-level segmentation masks l(p) as supervision. It is not directly applicable if only bounding box annotations are given as supervision. Next we introduce our method for addressing this problem.\"\n },\n {\n \"title\": \"Approach\",\n \"content\": \"\"\n },\n {\n \"title\": \"Unsupervised Segmentation for Supervised Training\",\n \"content\": \"To harness the bounding boxes annotations, it is desired to estimate segmentation masks from them. This is a widely studied supervised image segmentation problem, and can be addressed by, e.g., GrabCut [26]. But GrabCut can only generate one or a few samples from one box, which may be insufficient for deep network training.\\nWe propose to generate a set of candidate segments using unsupervised region proposal methods (e.g., Selective Search [31]) due to their nice properties. First, region proposal methods have high recall rates [2] of having a good candidate in the proposal pool. Second, region proposal methods generate candidates of greater variance, which provide a kind of data augmentation [18] for network training. We will show by experiments the improvements of these properties.\\nThe candidate segments are used to update the deep convolutional network. The semantic features learned by the network are then used to pick better candidates. This procedure is iterated. We formulate this procedure as an objective function as we will describe below.\\nIt is worth noticing that the region proposal is only used for networking training. For inference, the trained FCN is directly applied on the image and produces pixel-wise predictions. So our usage of region proposals does not impact the test-time efficiency.\"\n },\n {\n \"title\": \"Formulation\",\n \"content\": \"As a pre-processing, we use a region proposal method to generate segmentation masks. We adopt Multiscale Combinatorial Grouping (MCG) [2] by default, while other methods [31,17] are also evaluated. The proposal candidate masks are fixed throughout the training procedure. But during training, each candidate mask will be assigned a label which can be a semantic category or background. The labels assigned to the masks will be updated.\\nWith a ground-truth bounding box annotation, we expect it to pick out a candidate mask that overlaps the box as much as possible. Formally, we define an overlapping objective function E o as:\\nHere S represents a candidate segment mask, and B represents a ground-truth bounding box annotation. IoU(B, S) ∈ [0, 1] is the intersection-over-union ratio computed from the ground-truth box B and the tight bounding box of the segment S. The function δ is equal to one if the semantic label l S assigned to segment S is the same as the ground-truth label l B of the bounding box B, and zero otherwise. Minimizing E o favors higher IoU scores when the semantic labels are consistent. This objective function is normalized by the number of candidate segments N . Figure 3: Update of segmentation masks during training. Here we show the masks in epoch #1, epoch #5, and epoch #20.\\nEach segmentation mask will be used as the supervision for the next epoch.\\nWith the candidate masks and their estimated semantic labels, we can supervise the deep convolutional network as in Eqn. (1). Formally, we consider the following regression objective function E r :\\nHere l S is the estimated semantic label used as supervision for the network training. This objective function is the same as Eqn.\\n(1) except that its regression target is the estimated candidate segment. We minimize an objective function that combines the above two terms:\\nHere the summation i runs over the training images, and λ = 3 is a fixed weighting parameter. The variables to be optimized are the network parameters θ and the labeling {l S } of all candidate segments {S}. If only the term E o exists, the optimization problem in Eqn.(4) trivially finds a candidate segment that has the largest IoU score with the box; if only the term E r exists, the optimization problem in Eqn.(4) is equivalent to FCN. Our formulation simultaneously considers both cases.\"\n },\n {\n \"title\": \"Training Algorithm\",\n \"content\": \"The objective function in Eqn.(4) involves a problem of assigning labels to the candidate segments. Next we propose a greedy iterative solution to find a local optimum.\\nWith the network parameters θ fixed, we update the semantic labeling {l S } for all candidate segments. In our implementation, we only consider the case in which one ground-truth bounding box can \\\"activate\\\" (i.e., assign a non-background label to) one and only one candidate. As such, we can simply update the semantic labeling by selecting a single candidate segment for each ground-truth bounding box, such that its cost E o + λE r is the smallest among all candidates. The selected segment is assigned the groundtruth semantic label associated with that bounding box. All other pixels are assigned the background label.\\nThe above winner-takes-all selection tends to repeatedly use the same or very similar candidate segments, and the optimization procedure may be trapped in poor local optima. To increase the sample variance for better stochastic training, we further adopt a random sampling method to select the candidate segment for each ground-truth bounding box. Instead of selecting the single segment with the largest cost E o + λE r , we randomly sample a segment from the first k segments with the largest costs. In this paper we use k = 5. This random sampling strategy improves the accuracy by about 2% on the validation set.\\nWith the semantic labeling {l S } of all candidate segments fixed, we update the network parameters θ. In this case, the problem becomes the FCN problem [22] as in Eqn. (1). This problem is minimized by SGD.\\nWe iteratively perform the above two steps, fixing one set of variables and solving for the other set. For each iteration, we update the network parameters using one training epoch (i.e., all training images are visited once), and after that we update the segment labeling of all images. Fig.3 shows the gradually updated segmentation masks during training. The network is initialized by the model pre-trained in the Ima-geNet classification dataset, and our algorithm starts from the step of updating segment labels.\\nOur method is applicable for the semi-supervised case (the ground-truth annotations are mixtures of segmentation masks and bounding boxes). The labeling l(p) is given by candidate proposals as above if a sample only has groundtruth boxes, and is simply assigned as the true label if a sample has ground-truth masks.\\nIn the SGD training of updating the network, we use a mini-batch size of 20, following [22]. The learning rate is initialized to be 0.001 and divided by 10 after every 15 epochs. The training is terminated after 45 epochs.\"\n },\n {\n \"title\": \"Experiments\",\n \"content\": \"In all our experiments, we use the publicly released VGG-16 model 3 [29] that is pre-trained on ImageNet [27]. The VGG model is also used by all competitors [22, 13, 6,\"\n },\n {\n \"title\": \"Experiments on PASCAL VOC 2012\",\n \"content\": \"We first evaluate our method on the PASCAL VOC 2012 semantic segmentation benchmark [8]. This dataset involves 20 semantic categories of objects. We use the \\\"comp6\\\" evaluation protocol. The accuracy is evaluated by mean IoU scores. The original training data has 1,464 images. Following [11], the training data with ground-truth segmentation masks are augmented to 10,582 images. The validation and test sets have 1,449 and 1,456 images respectively. When evaluating the validation set or the test set, we only use the training set for training. A held-out 100 random validation images are used for cross-validation to set hyper-parameters. Table 1 compares the results of using different strategies of supervision on the validation set. When all ground-truth masks are used as supervision, the result is our implementation of the baseline DeepLab-CRF [5]. Our reproduction has a score of 63.8 (Table 1, \\\"mask only\\\"), which is very close to 63.74 reported in [5] under the same setting. So we believe that our reproduced baseline is convincing.\"\n },\n {\n \"title\": \"Comparisons of Supervision Strategies\",\n \"content\": \"When all 10,582 training samples are replaced with bounding box annotations, our method yields a score of 62.0 ( Table 1, \\\"box only\\\"). Though the supervision information is substantially weakened, our method shows a graceful degradation (1.8%) compared with the strongly supervised baseline of 63.8. This indicates that in practice we can avoid the expensive mask labeling effort by using only bounding boxes, with small accuracy loss. Table 1 also shows the semi-supervised result of our method. This result uses the ground-truth masks of the original 1,464 training images and the bounding box annotations of the rest 9k images. The score is 63.5 (Table 1, \\\"semi\\\"), on par with the strongly supervised baseline. Such semi-supervision replaces 9/10 of the segmentation mask annotations with bounding box annotations. This means that we can greatly reduce the labeling effort by dominantly using bounding box annotations.\\nAs a proof of concept, we further evaluate using a sub- stantially larger set of boxes. We use the Microsoft COCO dataset [21] that has 123,287 images with available groundtruth segmentation masks. This dataset has 80 semantic categories, and we only use the 20 categories that also present in PASCAL VOC. For our mask-supervised baseline, the result is a score of 68.1 (Table 1). Then we replace the ground-truth segmentation masks in COCO with their tight bounding boxes. Our semi-supervised result is 68.2 (Table 1), on par with the strongly supervised baseline. Fig. 5 shows some visual results in the validation set. The semi-supervised result (68.2) that uses VOC+COCO is considerably better than the strongly supervised result (63.8) that uses VOC only. The 4.4% gain is contributed by the extra large-scale bounding boxes in the 123k COCO images. This comparison suggests a promising strategywe may make use of the larger amount of existing bounding boxes annotations to improve the overall semantic segmentation results, as further analyzed below.\"\n },\n {\n \"title\": \"Error Analysis\",\n \"content\": \"Why can a large set of bounding boxes help improve convolutional networks? The error in semantic segmentation can be roughly thought of as two types: (i) recognition error that is due to confusions of recognizing object categories, and (ii) boundary error that is due to misalignments of pixel-level labels on object boundaries. Although the bounding box annotations have no information about the object boundaries, they provide extra object instances for recognizing them. We may expect that the large amount of boxes mainly improve the recognition accuracy.\\nTo analyze the error, we separately evaluate the performance on the boundary regions and interior regions. Following [16,5], we generate a \\\"trimap\\\" near the ground-truth boundaries (Fig. 4, top). We evaluate mean IoU scores inside/outside the bands, referred to as boundary/interior regions. Fig. 4 (bottom) shows the results of using different band widths for the trimaps.\\nFor the interior region, the accuracy of using the extra COCO boxes (red solid line, Fig. 4) is considerably higher than that of using VOC masks only (blue solid line). On the contrary, the improvement on the boundary regions is relatively smaller (red dash line vs. blue dash line). Note that correctly recognizing the interior may also help improve the boundaries (e.g., due to the CRF post-processing). So the improvement of the extra boxes on the boundary regions is secondary.\\nBecause the accuracy in the interior region is mainly determined by correctly recognizing objects, this analysis suggests that the large amount of boxes improve the feature quality of a learned BoxSup model for better recognition.\"\n },\n {\n \"title\": \"Comparisons of Estimated Masks for Supervision\",\n \"content\": \"In Table 2 we evaluate different methods of estimating masks from bounding boxes for supervision. As a naïve baseline, we fill each bounding box with its semantic label, and consider it as a rectangular mask (Fig. 2(c)). Using these rectangular masks as the supervision throughout training, the score is 52.3 on the validation set. We also use GrabCut [26] to generate segmentation masks from boxes ( Fig. 2(d) Table 4: Results on PASCAL VOC 2012 test set. In the supervision (\\\"sup\\\") column, \\\"mask\\\" means all training samples are with segmentation mask annotations, \\\"box\\\" means all training samples are with bounding box annotations, and \\\"semi\\\" means mixtures. \\\"V\\\" denotes the VOC data, \\\"C\\\" denotes the COCO data, and \\\"V 07 \\\" denotes the VOC 2007 data which only has bounding boxes available.\\nthroughout training, the score is 55.2. In both cases, the masks are not updated by the network feedbacks. Our method has a score 62.0 (Table 2) using the same set of bounding box annotations. This is a considerable gain over the baseline using fixed GrabCut masks. This indicates the importance of the mask quality for supervision. Fig. 3 shows that our method iteratively updates the masks by the network, which in turn improves the network training.\\nWe also evaluate a variant of our method where each time the updated mask is the candidate with the largest cost, instead of randomly sampled from the first k candidates (see Sec. 4.3). This variant has a lower score of 59.7 ( Table 2). The random sampling strategy, which is data augmentation and increases sample variances, is beneficial for training. Table 2 also shows the result of the concurrent method WSSL [5] under the same evaluation setting. Its results is 58.5. This result suggests that our method estimates more accurate masks than [5] for supervision.\"\n },\n {\n \"title\": \"Comparisons of Region Proposals\",\n \"content\": \"Our method resorts to unsupervised region proposals for training. In Table 3, we compare the effects of various region proposals on our method: Selective Search (SS) [31], Geodesic Object Proposals (GOP) [17], and MCG [2]. Table 3 shows that MCG [2] has the best accuracy, which is consistent with its segmentation quality evaluated by other metrics in [2]. Note that at test-time our method does not need region proposals. So the better accuracy of using MCG implies that our method effectively makes use of the higher quality segmentation masks to train a better network.\"\n },\n {\n \"title\": \"Comparisons on the Test Set\",\n \"content\": \"Next we compare with the state-of-the-art methods on the PASCAL VOC 2012 test set. In Table 4, the methods are based on the same FCN baseline and thus fair comparisons are made to evaluate the impact of mask/box/semisupervision. As shown in Table 4, our box-supervised result that only uses VOC bounding boxes is 64.6. This compares favorably with the WSSL [25] counterpart (60.4) under the same setting. On the other hand, our box-supervised result has a graceful degradation (1.8%) compared with the masksupervised DeepLab-CRF (66.4 [5]) using the VOC training data. Moreover, our semi-supervised variant which replaces 9/10 segmentation mask annotations with bounding boxes has a score of 66.2. This is on par with the mask-supervised counterpart of DeepLab-CRF, but the supervision information used by our method is much weaker.\\nIn the WSSL paper [25], by using all segmentation mask annotations in VOC and COCO, the strongly masksupervised result is 70.4. Our semi-supervised method shows a higher score of 71.0. Remarkably, our result uses the bounding box annotations from the 123k COCO images. So our method has a more accurate result but uses much weaker annotations than [25].\\nOn the other hand, compared with the DeepLab-CRF result (66.4), our method has a 4.6% gain enjoyed from exploiting the bounding box annotations of the COCO dataset. This comparison demonstrates the power of our method that exploits large-scale bounding box annotations to improve accuracy.\"\n },\n {\n \"title\": \"Exploiting Boxes in PASCAL VOC 2007\",\n \"content\": \"To further demonstrate the effect of BoxSup, we exploit the bounding boxes in the PASCAL VOC 2007 dataset [8]. This dataset has no mask annotations. It is a de facto dataset which mask-supervised methods are not able to use.\\nWe exploit all 10k images in the VOC 2007 trainval and test sets. We train a BoxSup model using the union set of VOC 2007 boxes, COCO boxes, and the augmented VOC 2012 training set. The score improves from 71.0 to 73.1 (Table 4) because of the extra box training data. It is reasonable for us to expect further improvement if more bounding box annotations are available.\"\n },\n {\n \"title\": \"Baseline Improvement\",\n \"content\": \"Although our focus is mainly on exploiting boxes as supervision, it is worth noticing that our method may also benefit from other improvements on the mask-sup baseline (FCN in our case). Concurrent with our work, there are a series of improvements [35,5] made on FCN, which achieve excellent results using strong mask-supervision from VOC and COCO data.\\nTo show the potential of our BoxSup method in parallel with improvements on the baseline, we use a simple testtime augmentation to boost our results. Instead of comput- Table 5: Results on PASCAL-CONTEXT [24] validation. Our baseline is our implementation of FCN+CRF. \\\"V\\\" denotes the VOC data, and \\\"C\\\" denotes the COCO data.\\ning pixel-wise predictions on a single scale, we compute the score maps from two extra scales (±20% of the original image size) and bilinearly re-scale the score maps to the original size. The scores from three scales are averaged. This simple modification boosts our result from 73.1 to 75.2 (BoxSup+, Table 4) in the VOC 2012 test set. This result is on par with the latest results using strong masksupervision from both VOC and COCO, but in our case the COCO dataset only provides bounding boxes.\"\n },\n {\n \"title\": \"Experiments on PASCAL-CONTEXT\",\n \"content\": \"We further perform experiments on the recently labeled PASCAL-CONTEXT dataset [24]. This dataset provides ground-truth semantic labels for the whole scene, including object and stuff (e.g., grass, sky, water). Following the protocol in [24,6,22], the semantic segmentation is performed on the most frequent 59 categories (identified by [24]) plus a background category. The accuracy is measured by mean IoU scores. The training and evaluation are performed on the training and validation sets that have 4,998 and 5,105 images respectively.\\nTo train a BoxSup model for this dataset, we first use the box annotations from all 80 object categories in the COCO dataset to train the FCN (using VGG-16). This network ends with an 81-way (with an extra one for background) layer. Then we remove this last layer and add a new 60way layer for the 59 categories of PASCAL-CONTEXT. We fine-tune this model in the 5k training images of PASCAL-CONTEXT. A CRF for post-processing is also used. We do no use the test-time scale augmentation. Table 5 shows the results in PASCAL-CONTEXT. The methods of CFM [6] and FCN [22] are both based on the VGG-16 model. Our baseline method, which is our implementation of FCN+CRF, has a score of 35.7 using masks of the 5k training images. Using our BoxSup model pretrained using the COCO boxes, the result is improved to 40.5. The 4.8% gain is solely because of the bounding box annotations in COCO that improve our network training. Fig. 6 shows some examples of our results for joint object and stuff segmentation.\"\n },\n {\n \"title\": \"Conclusion\",\n \"content\": \"The proposed BoxSup method can effectively harness bounding box annotations to train deep networks for semantic segmentation. Our BoxSup method that uses 133k bounding boxes and 10k masks achieves state-of-the-art results. Our error analysis suggests that semantic segmentation accuracy is hampered by the failure of recognizing objects, which large-scale data may help with.\"\n },\n {\n \"title\": \"Figure 1 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 2 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 4 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 5 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 6 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 2 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 3 :\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Overview of our training approach supervised by bounding boxes.","content":"Figure 1 :\n1Overview of our training approach supervised by bounding boxes."},{"id":"","caption":"Segmentation masks used as supervision. (a) A training image. (b) Ground-truth. (c) Each box is naïvely considered as a rectangle mask. (d) A segmentation mask is generated by GrabCut [26]. (e) For our method, the supervision is estimated from region proposals (MCG [2]) by considering bounding box annotations and network feedbacks.","content":"Figure 2 :\n2Segmentation masks used as supervision. (a) A training image. (b) Ground-truth. (c) Each box is naïvely considered as a rectangle mask. (d) A segmentation mask is generated by GrabCut [26]. (e) For our method, the supervision is estimated from region proposals (MCG [2]) by considering bounding box annotations and network feedbacks."},{"id":"","caption":"Error analysis on the validation set. Top: (from left to right) image, ground-truth, boundary regions marked as white, interior regions marked as white). Bottom: boundary and interior mean IoU, using VOC masks only (blue) and using extra COCO boxes (red).","content":"Figure 4 :\n4Error analysis on the validation set. Top: (from left to right) image, ground-truth, boundary regions marked as white, interior regions marked as white). Bottom: boundary and interior mean IoU, using VOC masks only (blue) and using extra COCO boxes (red)."},{"id":"","caption":"Example semantic segmentation results on PASCAL VOC 2012 validation using our method. (a) Images. (b) Supervised by masks in VOC. (c) Supervised by boxes in VOC. (d) Supervised by masks in VOC and boxes in COCO.","content":"Figure 5 :\n5Example semantic segmentation results on PASCAL VOC 2012 validation using our method. (a) Images. (b) Supervised by masks in VOC. (c) Supervised by boxes in VOC. (d) Supervised by masks in VOC and boxes in COCO."},{"id":"","caption":"Example results on PASCAL-CONTEXT validation. (a) Images. (b) Results of our baseline (35.7 mean IoU), trained using VOC masks. (c) Results of BoxSup (40.5 mean IoU), trained using VOC masks and COCO boxes.","content":"Figure 6 :\n6Example results on PASCAL-CONTEXT validation. (a) Images. (b) Results of our baseline (35.7 mean IoU), trained using VOC masks. (c) Results of BoxSup (40.5 mean IoU), trained using VOC masks and COCO boxes."},{"id":"","caption":". Our","content":". Ourtrain image \nwith gt boxes \n\ndog \n\nperson \n\n… \n\ncandidate masks \n\nupdate \nnetwork \n\nupdate \nmasks \n\nBoxSup training \n\nfeedback \niteration \n\nestimated masks \n\nepoch \n#5 \n\nepoch \n#1 \n\nepoch \n#20 \n\nnetwork for \nsegmentation"},{"id":"","caption":"). With the GrabCut masks as the supervision","content":"Table 2 :\n2Comparisons of estimated masks for supervision \nin PASCAL VOC 2012 validation. All methods only use \n10,582 bounding boxes as annotations, with no ground-\ntruth segmentation mask used. \n\nSS \nGOP \nMCG \n\nmean IoU \n59.5 \n60.4 \n62.0"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Overview of our training approach supervised by bounding boxes.\",\n \"content\": \"Figure 1 :\\n1Overview of our training approach supervised by bounding boxes.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Segmentation masks used as supervision. (a) A training image. (b) Ground-truth. (c) Each box is naïvely considered as a rectangle mask. (d) A segmentation mask is generated by GrabCut [26]. (e) For our method, the supervision is estimated from region proposals (MCG [2]) by considering bounding box annotations and network feedbacks.\",\n \"content\": \"Figure 2 :\\n2Segmentation masks used as supervision. (a) A training image. (b) Ground-truth. (c) Each box is naïvely considered as a rectangle mask. (d) A segmentation mask is generated by GrabCut [26]. (e) For our method, the supervision is estimated from region proposals (MCG [2]) by considering bounding box annotations and network feedbacks.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Error analysis on the validation set. Top: (from left to right) image, ground-truth, boundary regions marked as white, interior regions marked as white). Bottom: boundary and interior mean IoU, using VOC masks only (blue) and using extra COCO boxes (red).\",\n \"content\": \"Figure 4 :\\n4Error analysis on the validation set. Top: (from left to right) image, ground-truth, boundary regions marked as white, interior regions marked as white). Bottom: boundary and interior mean IoU, using VOC masks only (blue) and using extra COCO boxes (red).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Example semantic segmentation results on PASCAL VOC 2012 validation using our method. (a) Images. (b) Supervised by masks in VOC. (c) Supervised by boxes in VOC. (d) Supervised by masks in VOC and boxes in COCO.\",\n \"content\": \"Figure 5 :\\n5Example semantic segmentation results on PASCAL VOC 2012 validation using our method. (a) Images. (b) Supervised by masks in VOC. (c) Supervised by boxes in VOC. (d) Supervised by masks in VOC and boxes in COCO.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Example results on PASCAL-CONTEXT validation. (a) Images. (b) Results of our baseline (35.7 mean IoU), trained using VOC masks. (c) Results of BoxSup (40.5 mean IoU), trained using VOC masks and COCO boxes.\",\n \"content\": \"Figure 6 :\\n6Example results on PASCAL-CONTEXT validation. (a) Images. (b) Results of our baseline (35.7 mean IoU), trained using VOC masks. (c) Results of BoxSup (40.5 mean IoU), trained using VOC masks and COCO boxes.\"\n },\n {\n \"id\": \"\",\n \"caption\": \". Our\",\n \"content\": \". Ourtrain image \\nwith gt boxes \\n\\ndog \\n\\nperson \\n\\n… \\n\\ncandidate masks \\n\\nupdate \\nnetwork \\n\\nupdate \\nmasks \\n\\nBoxSup training \\n\\nfeedback \\niteration \\n\\nestimated masks \\n\\nepoch \\n#5 \\n\\nepoch \\n#1 \\n\\nepoch \\n#20 \\n\\nnetwork for \\nsegmentation\"\n },\n {\n \"id\": \"\",\n \"caption\": \"). With the GrabCut masks as the supervision\",\n \"content\": \"Table 2 :\\n2Comparisons of estimated masks for supervision \\nin PASCAL VOC 2012 validation. All methods only use \\n10,582 bounding boxes as annotations, with no ground-\\ntruth segmentation mask used. \\n\\nSS \\nGOP \\nMCG \\n\\nmean IoU \\n59.5 \\n60.4 \\n62.0\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Analyzing the performance of multilayer neural networks for object recognition. P Agrawal, R Girshick, J Malik, ECCV. P. Agrawal, R. Girshick, and J. Malik. Analyzing the perfor- mance of multilayer neural networks for object recognition. In ECCV, 2014.","doi":""},{"text":"Multiscale combinatorial grouping. P Arbeláez, J Pont-Tuset, J T Barron, F Marques, J Malik, CVPR. P. Arbeláez, J. Pont-Tuset, J. T. Barron, F. Marques, and J. Malik. Multiscale combinatorial grouping. In CVPR, 2014.","doi":""},{"text":"Semantic segmentation with second-order pooling. J Carreira, R Caseiro, J Batista, C Sminchisescu, ECCV. J. Carreira, R. Caseiro, J. Batista, and C. Sminchisescu. Se- mantic segmentation with second-order pooling. In ECCV. 2012.","doi":""},{"text":"Return of the devil in the details: Delving deep into convolutional nets. K Chatfield, K Simonyan, A Vedaldi, A Zisserman, BMVC. K. Chatfield, K. Simonyan, A. Vedaldi, and A. Zisserman. Return of the devil in the details: Delving deep into convo- lutional nets. In BMVC, 2014.","doi":""},{"text":"Semantic image segmentation with deep convolutional nets and fully connected crfs. L.-C Chen, G Papandreou, I Kokkinos, K Murphy, A L Yuille, ICLR. L.-C. Chen, G. Papandreou, I. Kokkinos, K. Murphy, and A. L. Yuille. Semantic image segmentation with deep con- volutional nets and fully connected crfs. In ICLR, 2015.","doi":""},{"text":"Convolutional feature masking for joint object and stuff segmentation. J Dai, K He, J Sun, CVPR. J. Dai, K. He, and J. Sun. Convolutional feature masking for joint object and stuff segmentation. In CVPR, 2015.","doi":""},{"text":"Imagenet: A large-scale hierarchical image database. J Deng, W Dong, R Socher, L.-J Li, K Li, L Fei-Fei, CVPR. J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei- Fei. Imagenet: A large-scale hierarchical image database. In CVPR, 2009.","doi":""},{"text":"M Everingham, L Van Gool, C K Williams, J Winn, A Zisserman, The PASCAL Visual Object Classes (VOC) Challenge. IJCV. M. Everingham, L. Van Gool, C. K. Williams, J. Winn, and A. Zisserman. The PASCAL Visual Object Classes (VOC) Challenge. IJCV, 2010.","doi":""},{"text":"Rich feature hierarchies for accurate object detection and semantic segmentation. R Girshick, J Donahue, T Darrell, J Malik, R. Girshick, J. Donahue, T. Darrell, and J. Malik. Rich fea- ture hierarchies for accurate object detection and semantic segmentation. 2014.","doi":""},{"text":"Imagenet autoannotation with segmentation propagation. IJCV. M Guillaumin, D Küttel, V Ferrari, M. Guillaumin, D. Küttel, and V. Ferrari. Imagenet auto- annotation with segmentation propagation. IJCV, 2014.","doi":""},{"text":"Semantic contours from inverse detectors. B Hariharan, P Arbeláez, L Bourdev, S Maji, J Malik, ICCV. B. Hariharan, P. Arbeláez, L. Bourdev, S. Maji, and J. Malik. Semantic contours from inverse detectors. In ICCV, 2011.","doi":""},{"text":"Simultaneous detection and segmentation. B Hariharan, P Arbeláez, R Girshick, J Malik, ECCV. B. Hariharan, P. Arbeláez, R. Girshick, and J. Malik. Simul- taneous detection and segmentation. In ECCV. 2014.","doi":""},{"text":"Hypercolumns for object segmentation and fine-grained localization. B Hariharan, P Arbeláez, R Girshick, J Malik, CVPR. B. Hariharan, P. Arbeláez, R. Girshick, and J. Malik. Hyper- columns for object segmentation and fine-grained localiza- tion. In CVPR, 2015.","doi":""},{"text":"Spatial pyramid pooling in deep convolutional networks for visual recognition. K He, X Zhang, S Ren, J Sun, ECCV. K. He, X. Zhang, S. Ren, and J. Sun. Spatial pyramid pooling in deep convolutional networks for visual recognition. In ECCV, 2014.","doi":""},{"text":"Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. K He, X Zhang, S Ren, J Sun, arXiv:1502.01852K. He, X. Zhang, S. Ren, and J. Sun. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. arXiv:1502.01852, 2015.","doi":"arXiv:1502.01852"},{"text":"Robust higher order potentials for enforcing label consistency. IJCV. P Kohli, P H Torr, P. Kohli, P. H. Torr, et al. Robust higher order potentials for enforcing label consistency. IJCV, pages 302-324, 2009.","doi":""},{"text":"Geodesic object proposals. P Krähenbühl, V Koltun, ECCV. P. Krähenbühl and V. Koltun. Geodesic object proposals. In ECCV, 2014.","doi":""},{"text":"Imagenet classification with deep convolutional neural networks. A Krizhevsky, I Sutskever, G E Hinton, NIPS. A. Krizhevsky, I. Sutskever, and G. E. Hinton. Imagenet classification with deep convolutional neural networks. In NIPS, 2012.","doi":""},{"text":"Backpropagation applied to handwritten zip code recognition. Y Lecun, B Boser, J S Denker, D Henderson, R E Howard, W Hubbard, L D , Neural computationY. LeCun, B. Boser, J. S. Denker, D. Henderson, R. E. Howard, W. Hubbard, and L. D. Jackel. Backpropagation applied to handwritten zip code recognition. Neural compu- tation, 1989.","doi":""},{"text":"X Liang, S Liu, Y Wei, L Liu, L Lin, S Yan, arXiv:1411.2861Computational baby learning. X. Liang, S. Liu, Y. Wei, L. Liu, L. Lin, and S. Yan. Compu- tational baby learning. arXiv:1411.2861, 2014.","doi":"arXiv:1411.2861"},{"text":"Microsoft COCO: Common objects in context. T.-Y Lin, M Maire, S Belongie, J Hays, P Perona, D Ramanan, P Dollár, C L Zitnick, ECCV. T.-Y. Lin, M. Maire, S. Belongie, J. Hays, P. Perona, D. Ra- manan, P. Dollár, and C. L. Zitnick. Microsoft COCO: Com- mon objects in context. In ECCV. 2014.","doi":""},{"text":"Fully convolutional networks for semantic segmentation. J Long, E Shelhamer, T Darrell, CVPR. J. Long, E. Shelhamer, and T. Darrell. Fully convolutional networks for semantic segmentation. In CVPR, 2015.","doi":""},{"text":"Feedforward semantic segmentation with zoom-out features. M Mostajabi, P Yadollahpour, G Shakhnarovich, arXiv:1412.0774arXiv preprintM. Mostajabi, P. Yadollahpour, and G. Shakhnarovich. Feedforward semantic segmentation with zoom-out features. arXiv preprint arXiv:1412.0774, 2014.","doi":"arXiv:1412.0774"},{"text":"The role of context for object detection and semantic segmentation in the wild. R Mottaghi, X Chen, X Liu, N.-G Cho, S.-W Lee, S Fidler, R Urtasun, A Yuille, CVPR. R. Mottaghi, X. Chen, X. Liu, N.-G. Cho, S.-W. Lee, S. Fi- dler, R. Urtasun, and A. Yuille. The role of context for object detection and semantic segmentation in the wild. In CVPR. 2014.","doi":""},{"text":"Weakly-and semi-supervised learning of a dcnn for semantic image segmentation. G Papandreou, L.-C Chen, K Murphy, A L Yuille, arXiv:1502.02734arXiv preprintG. Papandreou, L.-C. Chen, K. Murphy, and A. L. Yuille. Weakly-and semi-supervised learning of a dcnn for seman- tic image segmentation. arXiv preprint arXiv:1502.02734, 2015.","doi":"arXiv:1502.02734"},{"text":"Grabcut: Interactive foreground extraction using iterated graph cuts. C Rother, V Kolmogorov, A Blake, ACM Transactions on Graphics. C. Rother, V. Kolmogorov, and A. Blake. Grabcut: Interac- tive foreground extraction using iterated graph cuts. ACM Transactions on Graphics, 2004.","doi":""},{"text":". O Russakovsky, J Deng, H Su, J Krause, S Satheesh, S Ma, Z Huang, A Karpathy, A Khosla, M Bernstein, arXiv:1409.0575Imagenet large scale visual recognition challengeO. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. Bernstein, et al. Imagenet large scale visual recognition challenge. arXiv:1409.0575, 2014.","doi":"arXiv:1409.0575"},{"text":"Overfeat: Integrated recognition, localization and detection using convolutional networks. P Sermanet, D Eigen, X Zhang, M Mathieu, R Fergus, Y Lecun, ICLR. P. Sermanet, D. Eigen, X. Zhang, M. Mathieu, R. Fergus, and Y. LeCun. Overfeat: Integrated recognition, localization and detection using convolutional networks. In ICLR, 2014.","doi":""},{"text":"Very deep convolutional networks for large-scale image recognition. K Simonyan, A Zisserman, ICLR. K. Simonyan and A. Zisserman. Very deep convolutional networks for large-scale image recognition. In ICLR, 2015.","doi":""},{"text":"Going deeper with convolutions. C Szegedy, W Liu, Y Jia, P Sermanet, S Reed, D Anguelov, D Erhan, V Vanhoucke, A Rabinovich, CVPR. C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich. Going deeper with convolutions. In CVPR, 2015.","doi":""},{"text":"Selective search for object recognition. J R Uijlings, K E Van De Sande, T Gevers, A W Smeulders, IJCVJ. R. Uijlings, K. E. van de Sande, T. Gevers, and A. W. Smeulders. Selective search for object recognition. IJCV, 2013.","doi":""},{"text":"N Wang, S Li, A Gupta, D.-Y Yeung, arXiv:1501.04587Transferring rich feature hierarchies for robust visual tracking. N. Wang, S. Li, A. Gupta, and D.-Y. Yeung. Trans- ferring rich feature hierarchies for robust visual tracking. arXiv:1501.04587, 2015.","doi":"arXiv:1501.04587"},{"text":"Semantic segmentation without annotating segments. W Xia, C Domokos, J Dong, L.-F Cheong, S Yan, ICCV. W. Xia, C. Domokos, J. Dong, L.-F. Cheong, and S. Yan. Se- mantic segmentation without annotating segments. In ICCV, 2013.","doi":""},{"text":"Visualizing and understanding convolutional neural networks. M D Zeiler, R Fergus, ECCV. M. D. Zeiler and R. Fergus. Visualizing and understanding convolutional neural networks. In ECCV, 2014.","doi":""},{"text":"S Zheng, S Jayasumana, B Romera-Paredes, V Vineet, Z Su, D Du, C Huang, P Torr, arXiv:1502.03240Conditional random fields as recurrent neural networks. S. Zheng, S. Jayasumana, B. Romera-Paredes, V. Vineet, Z. Su, D. Du, C. Huang, and P. Torr. Conditional random fields as recurrent neural networks. arXiv:1502.03240, 2015.","doi":"arXiv:1502.03240"}],"string":"[\n {\n \"text\": \"Analyzing the performance of multilayer neural networks for object recognition. P Agrawal, R Girshick, J Malik, ECCV. P. Agrawal, R. Girshick, and J. Malik. Analyzing the perfor- mance of multilayer neural networks for object recognition. In ECCV, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Multiscale combinatorial grouping. P Arbeláez, J Pont-Tuset, J T Barron, F Marques, J Malik, CVPR. P. Arbeláez, J. Pont-Tuset, J. T. Barron, F. Marques, and J. Malik. Multiscale combinatorial grouping. In CVPR, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Semantic segmentation with second-order pooling. J Carreira, R Caseiro, J Batista, C Sminchisescu, ECCV. J. Carreira, R. Caseiro, J. Batista, and C. Sminchisescu. Se- mantic segmentation with second-order pooling. In ECCV. 2012.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Return of the devil in the details: Delving deep into convolutional nets. K Chatfield, K Simonyan, A Vedaldi, A Zisserman, BMVC. K. Chatfield, K. Simonyan, A. Vedaldi, and A. Zisserman. Return of the devil in the details: Delving deep into convo- lutional nets. In BMVC, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Semantic image segmentation with deep convolutional nets and fully connected crfs. L.-C Chen, G Papandreou, I Kokkinos, K Murphy, A L Yuille, ICLR. L.-C. Chen, G. Papandreou, I. Kokkinos, K. Murphy, and A. L. Yuille. Semantic image segmentation with deep con- volutional nets and fully connected crfs. In ICLR, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Convolutional feature masking for joint object and stuff segmentation. J Dai, K He, J Sun, CVPR. J. Dai, K. He, and J. Sun. Convolutional feature masking for joint object and stuff segmentation. In CVPR, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Imagenet: A large-scale hierarchical image database. J Deng, W Dong, R Socher, L.-J Li, K Li, L Fei-Fei, CVPR. J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei- Fei. Imagenet: A large-scale hierarchical image database. In CVPR, 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"M Everingham, L Van Gool, C K Williams, J Winn, A Zisserman, The PASCAL Visual Object Classes (VOC) Challenge. IJCV. M. Everingham, L. Van Gool, C. K. Williams, J. Winn, and A. Zisserman. The PASCAL Visual Object Classes (VOC) Challenge. IJCV, 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Rich feature hierarchies for accurate object detection and semantic segmentation. R Girshick, J Donahue, T Darrell, J Malik, R. Girshick, J. Donahue, T. Darrell, and J. Malik. Rich fea- ture hierarchies for accurate object detection and semantic segmentation. 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Imagenet autoannotation with segmentation propagation. IJCV. M Guillaumin, D Küttel, V Ferrari, M. Guillaumin, D. Küttel, and V. Ferrari. Imagenet auto- annotation with segmentation propagation. IJCV, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Semantic contours from inverse detectors. B Hariharan, P Arbeláez, L Bourdev, S Maji, J Malik, ICCV. B. Hariharan, P. Arbeláez, L. Bourdev, S. Maji, and J. Malik. Semantic contours from inverse detectors. In ICCV, 2011.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Simultaneous detection and segmentation. B Hariharan, P Arbeláez, R Girshick, J Malik, ECCV. B. Hariharan, P. Arbeláez, R. Girshick, and J. Malik. Simul- taneous detection and segmentation. In ECCV. 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hypercolumns for object segmentation and fine-grained localization. B Hariharan, P Arbeláez, R Girshick, J Malik, CVPR. B. Hariharan, P. Arbeláez, R. Girshick, and J. Malik. Hyper- columns for object segmentation and fine-grained localiza- tion. In CVPR, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Spatial pyramid pooling in deep convolutional networks for visual recognition. K He, X Zhang, S Ren, J Sun, ECCV. K. He, X. Zhang, S. Ren, and J. Sun. Spatial pyramid pooling in deep convolutional networks for visual recognition. In ECCV, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. K He, X Zhang, S Ren, J Sun, arXiv:1502.01852K. He, X. Zhang, S. Ren, and J. Sun. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. arXiv:1502.01852, 2015.\",\n \"doi\": \"arXiv:1502.01852\"\n },\n {\n \"text\": \"Robust higher order potentials for enforcing label consistency. IJCV. P Kohli, P H Torr, P. Kohli, P. H. Torr, et al. Robust higher order potentials for enforcing label consistency. IJCV, pages 302-324, 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Geodesic object proposals. P Krähenbühl, V Koltun, ECCV. P. Krähenbühl and V. Koltun. Geodesic object proposals. In ECCV, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Imagenet classification with deep convolutional neural networks. A Krizhevsky, I Sutskever, G E Hinton, NIPS. A. Krizhevsky, I. Sutskever, and G. E. Hinton. Imagenet classification with deep convolutional neural networks. In NIPS, 2012.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Backpropagation applied to handwritten zip code recognition. Y Lecun, B Boser, J S Denker, D Henderson, R E Howard, W Hubbard, L D , Neural computationY. LeCun, B. Boser, J. S. Denker, D. Henderson, R. E. Howard, W. Hubbard, and L. D. Jackel. Backpropagation applied to handwritten zip code recognition. Neural compu- tation, 1989.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"X Liang, S Liu, Y Wei, L Liu, L Lin, S Yan, arXiv:1411.2861Computational baby learning. X. Liang, S. Liu, Y. Wei, L. Liu, L. Lin, and S. Yan. Compu- tational baby learning. arXiv:1411.2861, 2014.\",\n \"doi\": \"arXiv:1411.2861\"\n },\n {\n \"text\": \"Microsoft COCO: Common objects in context. T.-Y Lin, M Maire, S Belongie, J Hays, P Perona, D Ramanan, P Dollár, C L Zitnick, ECCV. T.-Y. Lin, M. Maire, S. Belongie, J. Hays, P. Perona, D. Ra- manan, P. Dollár, and C. L. Zitnick. Microsoft COCO: Com- mon objects in context. In ECCV. 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fully convolutional networks for semantic segmentation. J Long, E Shelhamer, T Darrell, CVPR. J. Long, E. Shelhamer, and T. Darrell. Fully convolutional networks for semantic segmentation. In CVPR, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Feedforward semantic segmentation with zoom-out features. M Mostajabi, P Yadollahpour, G Shakhnarovich, arXiv:1412.0774arXiv preprintM. Mostajabi, P. Yadollahpour, and G. Shakhnarovich. Feedforward semantic segmentation with zoom-out features. arXiv preprint arXiv:1412.0774, 2014.\",\n \"doi\": \"arXiv:1412.0774\"\n },\n {\n \"text\": \"The role of context for object detection and semantic segmentation in the wild. R Mottaghi, X Chen, X Liu, N.-G Cho, S.-W Lee, S Fidler, R Urtasun, A Yuille, CVPR. R. Mottaghi, X. Chen, X. Liu, N.-G. Cho, S.-W. Lee, S. Fi- dler, R. Urtasun, and A. Yuille. The role of context for object detection and semantic segmentation in the wild. In CVPR. 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Weakly-and semi-supervised learning of a dcnn for semantic image segmentation. G Papandreou, L.-C Chen, K Murphy, A L Yuille, arXiv:1502.02734arXiv preprintG. Papandreou, L.-C. Chen, K. Murphy, and A. L. Yuille. Weakly-and semi-supervised learning of a dcnn for seman- tic image segmentation. arXiv preprint arXiv:1502.02734, 2015.\",\n \"doi\": \"arXiv:1502.02734\"\n },\n {\n \"text\": \"Grabcut: Interactive foreground extraction using iterated graph cuts. C Rother, V Kolmogorov, A Blake, ACM Transactions on Graphics. C. Rother, V. Kolmogorov, and A. Blake. Grabcut: Interac- tive foreground extraction using iterated graph cuts. ACM Transactions on Graphics, 2004.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". O Russakovsky, J Deng, H Su, J Krause, S Satheesh, S Ma, Z Huang, A Karpathy, A Khosla, M Bernstein, arXiv:1409.0575Imagenet large scale visual recognition challengeO. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. Bernstein, et al. Imagenet large scale visual recognition challenge. arXiv:1409.0575, 2014.\",\n \"doi\": \"arXiv:1409.0575\"\n },\n {\n \"text\": \"Overfeat: Integrated recognition, localization and detection using convolutional networks. P Sermanet, D Eigen, X Zhang, M Mathieu, R Fergus, Y Lecun, ICLR. P. Sermanet, D. Eigen, X. Zhang, M. Mathieu, R. Fergus, and Y. LeCun. Overfeat: Integrated recognition, localization and detection using convolutional networks. In ICLR, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Very deep convolutional networks for large-scale image recognition. K Simonyan, A Zisserman, ICLR. K. Simonyan and A. Zisserman. Very deep convolutional networks for large-scale image recognition. In ICLR, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Going deeper with convolutions. C Szegedy, W Liu, Y Jia, P Sermanet, S Reed, D Anguelov, D Erhan, V Vanhoucke, A Rabinovich, CVPR. C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich. Going deeper with convolutions. In CVPR, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Selective search for object recognition. J R Uijlings, K E Van De Sande, T Gevers, A W Smeulders, IJCVJ. R. Uijlings, K. E. van de Sande, T. Gevers, and A. W. Smeulders. Selective search for object recognition. IJCV, 2013.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"N Wang, S Li, A Gupta, D.-Y Yeung, arXiv:1501.04587Transferring rich feature hierarchies for robust visual tracking. N. Wang, S. Li, A. Gupta, and D.-Y. Yeung. Trans- ferring rich feature hierarchies for robust visual tracking. arXiv:1501.04587, 2015.\",\n \"doi\": \"arXiv:1501.04587\"\n },\n {\n \"text\": \"Semantic segmentation without annotating segments. W Xia, C Domokos, J Dong, L.-F Cheong, S Yan, ICCV. W. Xia, C. Domokos, J. Dong, L.-F. Cheong, and S. Yan. Se- mantic segmentation without annotating segments. In ICCV, 2013.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Visualizing and understanding convolutional neural networks. M D Zeiler, R Fergus, ECCV. M. D. Zeiler and R. Fergus. Visualizing and understanding convolutional neural networks. In ECCV, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"S Zheng, S Jayasumana, B Romera-Paredes, V Vineet, Z Su, D Du, C Huang, P Torr, arXiv:1502.03240Conditional random fields as recurrent neural networks. S. Zheng, S. Jayasumana, B. Romera-Paredes, V. Vineet, Z. Su, D. Du, C. Huang, and P. Torr. Conditional random fields as recurrent neural networks. arXiv:1502.03240, 2015.\",\n \"doi\": \"arXiv:1502.03240\"\n }\n]","isConversational":false}}},{"rowIdx":477,"cells":{"text":{"kind":"string","value":"\nEffect of phase relaxation on quantum superpositions in complex collisions\n28 Mar 2006\n\nL Benet \nCentro de Ciencias Físicas\nNational University of Mexico (UNAM)\nMor62210CuernavacaMexico\n\nS Yu Kun \nCentro de Ciencias Físicas\nNational University of Mexico (UNAM)\nMor62210CuernavacaMexico\n\nNonlinear Physics Center\nThe Australian National University\n0200RSPhysSE, CanberraACTAustralia\n\nDepartment of Theoretical Physics\nThe Australian National University\n0200RSPhysSE, CanberraACTAustralia\n\nWang Qi \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina\n\nEffect of phase relaxation on quantum superpositions in complex collisions\n28 Mar 2006(Dated: April 1, 2022)arXiv:quant-ph/0503046v2numbers: 2570Bc0365-w0365Yz3410+x\nWe study the effect of phase relaxation on coherent superpositions of rotating clockwise and anticlockwise wave packets in the regime of strongly overlapping resonances of the intermediate complex. Such highly excited deformed complexes may be created in binary collisions of heavy ions, molecules and atomic clusters. It is shown that phase relaxation leads to a reduction of the interference fringes, thus mimicking the effect of decoherence. This reduction is crucial for the determination of the phase-relaxation width from the data on the excitation function oscillations in heavy-ion collisions and bimolecular chemical reactions. The difference between the effects of phase relaxation and decoherence is discussed.\n\nI. INTRODUCTION\n\nFor highly excited strongly interacting many-body systems the independent particle picture has very limited validity [1]. For high excitations, the interaction results in a quick decay of single-particle [2,3,4] as well as collective modes [2,4], which are not eigenstates of the Hamiltonian of the system. This decay leads to the formation of complicated many-body configurations. Each of these many-body states is characterized by a uniform occupation of the accessible phase space and sharing the energy between many particles of the system. The characteristic time for the formation of such spatially extended ergodic many-body states, independent of the initial conditions, is given by the inverse spreading width, τ erg =h/Γ spr [4]. Introduced by Wigner [5], Γ spr also characterizes the width of the distribution of the expansion coefficients of the many-body eigenstates over a noninteracting mean-field basis. For sufficiently high excitation energy the decay of single-particle modes creates a linear superposition of a very large number of manybody configurations. The question is whether the phase relations between these individually ergodic many-body states in the superposition may still preserve a memory of the way the system was excited. This question is of fundamental importance for the study of relaxation phenomena in nuclear, atomic, molecular and mesoscopic many-body systems, and for many-qubit quantum computation. In particular, if phase relaxation is longer than energy relaxationh/Γ spr , this can extend the time for quantum computing [6,7] beyond the quantum-chaos border [8,9].\n\nTo answer this question from first principles is difficult due to computational limitations. Indeed, in order to solve the full quantum many-body problem one would require a many-qubit quantum computer. Therefore, the only currently available resort to search for possible manifestations of long phase relaxation is the experiment, and a careful data analysis. Nuclear systems are an ideal laboratory to study many-body systems, since nuclear interactions are so strong that external perturbations can often be neglected. In particular, the analysis of Refs. [6,7,10] of the data on asymmetry around 90 • c.m. in angular distributions of evaporated protons from heavy nuclei in nucleon-induced and photonuclear reactions clearly indicates that phase relaxation time in highly excited strongly interacting many-body systems can be up to eight orders of magnitude [7] longer than energy relaxation timeh/Γ spr . This reveals a new form of matter-thermalized non-equilibrated matter introduced by one of us in Refs. [11,12,13].\n\nA more subtle indication of a slow phase relaxation is found in light heavy-ion scattering [14,15,16,17]. It manifests itself in the formation of rotating wave packets whose spreading time, which is given by the inverse phase-relaxation width, is much longer than the energy relaxation timeh/Γ spr . The approach [15,16,17] treats time-delayed light heavy-ion scattering in terms of the formation and decay of quasimolecular resonances [18]. The highly excited coherently rotating intermediate system has the energy of the intrinsic excitation ≥ 10 MeV. The intermediate system is in the regime of strongly overlapping resonances. Therefore, this coherent rotation does not originate from the discreteness of the spectrum, which is not resolved, but is due to the spin off-diagonal correlations between partial width amplitudes. Indeed, the period of the coherent rotation is much shorter than the inverse level spacing of the intermediate system. This reveals a new root, as compared to Bohr's correspondence principle, for a quantum-classical transition in highly excited many-body systems.\n\nThe width of the rotating wave packets is about 1/d + βt/h ≈ 1/d + βθ/hω, where d is the effective number of partial waves, β is the spin off-diagonal phaserelaxation width, ω is the angular velocity, and t and θ are the time and angle of the rotation, respectively. This seemingly allows us to determine the wave-packet spreading rate from the time power spectra at differ-ent scattering angles, which can be reconstructed, for binary collisions, from the data on energy fluctuations of the cross sections [16,17]. These fluctuations originate from the energy-fluctuating collision amplitude corresponding to a resonance time-delayed process. However, for elastic heavy-ion scattering, such energy fluctuations can be typically observed only at backward angles 140 • ≤ θ ≤ 180 • , since for smaller θ the direct reaction contribution grows exponentially and becomes much greater than the time delayed one. Therefore, since at the initial moment of time the deformed intermediate system is oriented in the forward direction θ ≈ 0 • , the spreading of the wave packets within the backward angular interval 140 • ≤ θ ≤ 180 • can hardly be detected reliably. Moreover, the width of the wave packets at this backwardangle range does not allow to determine β, since the angular dispersion 1/d at t = 0 is also unknown. In this paper we show that both d and β can indeed be determined unambiguously. This results from the strong sensitivity of the interference fringes between the rotating clockwise and anticlockwise wave packets for the backward angle range to the the phase relaxation width: As β grows, the interference fringes are suppressed more strongly. The effect is considered in relation to heavy-ion scattering and bimolecular chemical reactions. The difference between the effects of phase relaxation and decoherence is discussed.\n\n\nII. THE TIME POWER SPECTRUM OF THE TIME-DELAYED COLLISIONS\n\nFollowing Ref. [19], we consider spinless collision partners in the entrance and exit channels. The time and angle dependent intensity of the decay (the time power spectrum), P (t, θ), is given by the modulus square of the Fourier component of the energy-fluctuating collision amplitude. It can be also expressed as the Fourier component of the amplitude energy autocorrelation function. P (t, θ) has been obtained in [19] by summing over a very large number of strongly overlapping resonance levels, Γ/D ≫ 1, where Γ is the total decay width and D is the average level spacing of the intermediate complex. As a result, after changing from summation over the resonance levels to integration, which is a good approximation for t ≪h/D, P (t, θ) takes the form:\nP (t, θ) ∝ H(t) exp(−Γt/h) JJ ′ (2J + 1)(2J ′ + 1)[W (J)W (J ′ )] 1/2 exp[i(Φ − ωt)(J − J ′ ) − β|J − J ′ |t/h]P J (θ)P J ′ (θ). (1)\nHere, H(t) is the Heaviside step function, β is the spin phase-relaxation width, ω is the angular velocity of the coherent rotation, Φ is the deflection given by the total spin J (inh units) derivative of the potential phase shifts (direct reaction), and P J (θ) are the Legendre polynomials. The physical meaning of the inverse spin phase-relaxation width,h/β, is the characteristic time for the angular spreading of the clockwise and anticlockwise rotating wave packets. The partial average reaction probability is taken in the J-window form,\nW (J) = |δS J (E)| 2 ∝ exp[−(J −J) 2 /d 2 ]\n, whereJ is the average spin and d is the J-window width.\n\nIt should be noted that a similar effect of coherent rotation of the nuclear molecule, as described by Eq. (1) with β = 0, was found in Ref. [20] for a scattering of heavy nuclei. However, while Eq. (1) has been obtained for a large number of open channels and strongly overlapping resonances of the intermediate system [19], the time power spectrum in Ref. [20] was derived from one open channel and one resonance pole form of the S-matrix elements. Therefore the latter approach is not applicable for the forthcoming analysis of the 12 C+ 24 Mg system since, for this system, the number of open channels is much greater than one. Furthermore, the one-resonance pole form of the S-matrix elements employed in Ref. [20] results in iso-lated resonances with the energy spacing between them to be abouthω. This is not the case for the 12 C+ 24 Mg elastic scattering analyzed in this paper. Indeed, the data on the excitation function for the 12 C+ 24 Mg elastic scattering at θ = 180 • reveal about 15 local maxima on the energy interval ∆E cm = 13 − 22 MeV [21]. Then the interpretation [20] would mean that average level spacing, ≃ 0.6 MeV in our case, between the local maxima is given byhω. This, in turn, would reflect the quasiperiodic behaviour of the cross section energy autocorrelation functions with the period of about 0.6 MeV. This is inconsistent with the data [21] showing the quasiperiodicity of the cross section with the period of about 2.9 MeV yieldinghω = 1.35 − 1.45 MeV [16,17]. Moreover, the interpretation [20] implies that these local maxima in the excitation function should be approximately equidistant. Instead, the data [21,22] show big fluctuations, from 0.3 MeV to 1.3, for the energy spacing between the nearest-neighboring local maxima in the excitation function. Note that the indication against the interpretation in terms of isolated resonances comes from the statistically insignificant channel-channel correlations for the 12 C+ 24 Mg elastic and inelastic scattering at θ = 180 • [16]. Finally, the effect of coherent rotation [20] does not survive a generalization [23] by taking On the basis of the above arguments we believe that the interpretation of the fine structure in the excitation functions for the 12 C+ 24 Mg scattering at θ = 180 • in terms of overlapping, rather than isolated, resonances of the intermediate system is a justified approach.\n\nFirst, we calculate P (t, θ) for the set of parameters obtained from the description [17] of the experimental cross section energy autocorrelation functions [21] for 12 C+ 24 Mg elastic and inelastic scattering at θ = 180 • [22]. For these collisions the analysis of the oscillations in the cross section energy autocorrelation functions [16,17] In Figs. 1 we plot the quantity AP (t, θ)/ σ(E, θ) at four moments of time, with T = 2π/ω being the period of one complete revolution. Here, σ(E, θ) ∝ ∞ 0 dt P (t, θ) is the energy-averaged differential cross section for the time-delayed collision. In Figs. 1, P (t, θ) is scaled with σ(E, θ) for the reasons discussed in Ref. [19]. The constant A is determined by the condition AP (t = 0, θ = 0)/ σ(E, θ = 0) = 1. Figure 1(a) shows the two slightly overlapping wave packets rotating towards each other in the backward direction. In panels (b) and (c), the wave packets strongly overlap around θ = 180 • , producing interference fringes. Finally, in panel (d) the wave packets have passed each other and move apart rotating in the forward direction.\n\nIn Figs. 1 we also plot AP (t, θ)/ σ(E, θ) with the same constant A and the set of parameters as before, except for d and β. We consider two other cases: d = 4, β = 0.04 MeV, and d = 8, β = 0.075 MeV. One can see that when the wave packets overlap only slightly (panels (a) and (d)) one can hardly distinguish between the three combinations of d and β, even though β changes by almost one order of magnitude. On the other hand, from panels (b) and (c), we observe that a fringe contrast due to the interference of the wave packets is very sensitive to absolute values of β and d. This result is further illustrated in Fig. 2 where we plot P (t, θ) calculated for the three sets of β and d values, as a function of t for θ = 170.6 • . One observes that the smaller β is the deeper minima are, due to the destructive interference between the wave packets at t = T /2. However, for θ = 180 • , P (t, θ) in Fig. 2 is insensitive to the three sets of β and d.\n\nNote that in order to reproduce the width of the wave packets in Figs. 1 and 2 for the maximal possible d = I value, one has to take β = 0.1 MeV. Therefore, even the largest possible β value is more than one order of magnitude smaller than Γ spr . The latter is usually estimated from the width of giant resonances [2,4]: Γ spr ≃ 5 MeV. This indicates that energy relaxation, i.e., the process of formation of ergodic many-body configurations, in highly excited nuclear systems can be much faster than phase relaxation.\n\n\nIII. DETERMINATION OF THE TIME POWER SPECTRUM FROM THE DATA ON THE EXCITATION FUNCTION OSCILLATIONS\n\nWe consider a collision of the spinless reaction partners in the entrance and exit channels. The cross section of the collision is given by σ(E, θ) = |f (E, θ)| 2 . Here E is the energy in the entrance channel and θ is the scattering angle. The reaction amplitude f (E, θ) is a linear combination of S-matrix elements S J (E) corresponding to the different total spin values J. Suppose that, for a fixed energy E 1 , a measurement of detailed angular distribution is performed. Then, using, e.g., the method [24], one can find S J (E 1 ) and, therefore, f (E 1 , θ). Measuring the angular distributions for different energies on the energy interval I = E min − E max with the energy step ∆E and energy resolution smaller or about the minimal characteristic energy scale of a variation of f (E, θ) one obtains a detailed energy and angle dependencies of the collision amplitude. Its Fourier component,\nP(t, θ) ∝ Emax Emin dE exp(−iEt/h)f (E, θ) ≈ ∆E N n=0 exp[−i(E min + n∆E)t/h]f (E min + n∆E, θ),(2)\nis the instantaneous decay amplitude received at a de- (2) may be changed to the integration if we use a linear interpolation for f (E, θ) in between the consequent energies E n and E n+1 . This is justified provided the energy step ∆E is smaller or about the minimal characteristic energy scale of a variation of the cross section. For example, in Ref. [22] the energy step is ∆E cm = 133 keV while the minimal characteristic scale of the variation of σ(E, θ = 180 • ) is Γ = 0.3 MeV [16,17]. Having reconstructed P(t, θ) from the excitation function data one obtains the time power spectra P (t, θ) = |P(t, θ)| 2 for any angle. The later quantity is equivalent to that directly measured using real-time methods of femtochemistry, with the energy bands of I for the pump and probe laser pulses, to monitor chemical reactions dynamics [25,26]. Due to its dependence on θ, P (t, θ) determined by the proposed method could provide even more detailed information on the collision dynamics, as compared with the femtochemistry experiments, since latter measurements often allow to obtain P (t, θ) averaged over θ.\n\nFor the energy interval I being long enough to provide a sufficient time resolutionh/I to resolve interference fringes [27] the proposed experiment would allow to test our theoretical predictions. If these were confirmed by the data, then the experimentally determined P (t, θ) can be compared with our calculations presented in Figs. 1 and 2 with the purpose of evaluation β and d.\n\nIn particular, the energy interval of 10.5 MeV, over which excitation function for the 12 C+ 24 Mg elastic scattering at θ = 180 • was measured [22], should be sufficient to resolve the interference fringes in P (t, θ) [27].\n\nIt should be noted that the method proposed above allows to reconstruct P (t, θ) which includes both potential scattering, i.e. fast processes, and the time delayed mechanism. Since the previous analysis [16,17] demonstrated a presence of a strong potential scattering component we expect that this method will yield a strong potential scattering contribution in P (t, θ). However this contribution is expected to be restricted to relatively short times t ps ≤h/∆E ps , where ∆E ps is a characteristic energy interval for the variation of the potential scattering amplitude. For ∆E ps ≃ 2 − 3 MeV [18], t ps ≃ 3 − 2 × 10 −22 sec. Therefore, this potential scattering contribution is expected to be restricted to t/T ≤ 0.1 in Fig. 2 and should not show up at all in Fig. 1.\n\nThe above method to obtain P (t, θ) from the data is model independent and, therefore, most reliable one. However it is very demanding experimentally since σ(E, θ) must be measured for a very large number of angles [24]. There is another way to extract P (t, θ) from the data even if the excitation function is measured for a single angle. This method can be applied if the direct reaction process provides a major contribution to the cross section [17]. We consider again a case of spinless collision partners in the entrance and exit channels. The main idea is based on the decomposition of the col-lision amplitude, f (E, θ) = f dir (E, θ) + δf (E, θ). Here f dir (E, θ) is the energy smooth amplitude corresponding to potential scattering or direct fast process, and δf (E, θ) is the energy fluctuating amplitude, δf (E, θ) = 0, corresponding to the time-delayed collision. Consider first an idealized case of energy independent amplitude f dir (E, θ) when it coincides with the energy averaged amplitude f (E, θ) . The collision cross section has the form σ(E, θ) = |f (E, θ)\n| 2 = σ dir (E, θ) + σ f l (E, θ) + 2Re[δf (E, θ)f dir (E, θ) * ].\nHere the potential scattering or direct reaction cross section σ dir (θ) = |f dir (E, θ)| 2 is energy-independent and σ f l (E, θ) = |δf (E, θ)| 2 . For a relative contribution of the potential scattering or direct reaction cross section to be about 70% or more of the total cross section, ∆σ(E, θ) = σ(E, θ) − σ(E, θ) ≃ 2Re[δf (E, θ)f dir (E, θ) * ], where σ(E, θ) = σ dir (θ) + σ f l (E, θ) is the energy averaged cross section. This means that rapid energy variations of the cross section originate mostly from interference between the energy fluctuating δf (E, θ) and the energy independent amplitude f dir (E, θ). Suppose that the excitation function is measured on energy interval I = E max − E min with energy step ∆E so that the total number of steps is N + 1, where N = I/∆E. Then, for t > 0, we have [17] Emax Emin\ndE exp(−iEt/h)∆σ(E, θ) ∝ Emax Emin dE exp(−iEt/h)δf (E, θ) ≈ ∆E N n=0 exp[−i(E min + n∆E)t/h]δf (E min + n∆E, θ) ∝ P(t, θ)(3)\nwith P (t, θ) = |P(t, θ)| 2 . In Eq. (3) we have employed a causality condition, P(t < 0, θ) = 0. This condition implies that the molecule cannot decay before it is formed at t = 0. Again the above information on P (t, θ) is equivalent to that obtained directly using real-time methods of femtochemistry to monitor unimolecular chemical reactions, with the energy bands of I = (E max − E min ) for the pump and probe laser pulses [25]. The above consideration can be extended to a case when the potential scattering or direct reaction amplitude f dir (E, θ) depends on energy but this dependence is considerably smoother than the energy dependence of the amplitude δf (E, θ) for the time-delayed collision. We shall again assume that a relative contribution of direct processes into the energy averaged cross section is about 70% or more. We define the characteristic energy interval I d of a variation of f dir (E, θ) with I d ≫ Γ, where Γ is the characteristic energy interval of variation of δf (E, θ). The energy interval I d can be evaluated using, e.g., the trend reduction method [28]. In particular, for the 12 C+ 24 Mg scat-tering [22] analyzed in Sec. II, this interval was evaluated to be about 4 − 5 MeV [21], i.e. much greater than Γ = 0.3 MeV. Then one can find the best polynomial fit of the energy dependence of the cross section\nσ(E, θ) = σ d (E, θ) + σ f l (E, θ) + 2Re[δf (E, θ)f dir (E, θ) * ]\nwith the order of polynomial being [I/I d ]+1, where [I/I d ] is integer part of I/I d The resulting energy smooth cross section is denoted asσ(E, θ) ≃ σ d (E, θ) + σ f l (E, θ) with σ f l (E, θ) being energy averaged, i.e. energy independent, cross section of the time-delayed processes. It can be found using the standard procedure employed for the analysis of Ericson fluctuations (see, e.g., Refs. [21,22]). As a result, one finds energy smooth direct reaction cross section σ d (E, θ). Consider σ(E, θ) − σ(E, θ) ≃ 2Re[δf (E, θ)f dir (E, θ) * ]. Assume first that energy dependence of the phase of f dir (E, θ) is negligible, arg[f dir (E, θ)] = constant, and the smooth energy dependence of σ d (E, θ) is due to the energy dependence of |f dir (E, θ)| = σ d (E, θ) 1/2 . Then applying the same arguments as those used to obtain Eq. (3), for t > 0, we have\nP(t, θ) ∝ N n=0 exp[−i(E min + n∆E)t/h][σ(E min + n∆E, θ) −σ(E min + n∆E, θ)]/σ d (E min + n∆E, θ) 1/2 (4)\nwith P (t, θ) = |P(t, θ)| 2 . In order to generalize Eq. (4) by taking into account energy dependence of arg[f dir (E, θ)] ≡ φ(E, θ) (f dir (E, θ) = |f dir (E, θ)| exp[iφ(E, θ)]) we use the linear approximation φ(E, θ) = φ(Ē, θ) + (E − E)dφ(E, θ)/dE| E=Ē , whereĒ = (E max − E min )/2 and dφ(E, θ)/dE| E=Ē = t dir /h with t dir ≪h/Γ being the time delay due to the potential scattering or direct re-actions [29]. Then, it is easy to see, that, for t > t dir , the r.h.s. of Eq. (4) is changed to P(t − t dir , θ). This manifestly demonstrates that the interference between time-delayed and fast direct processes in the cross section is a precondition for monitoring the time evolution, the fast process switches on the clock at the initial moment of time t d playing the role of the pump pulse. In the absence of direct processes the initial moment of time is not defined.\n\nNote that P (t, θ) can also be expressed as a cosine half-Fourier transform of the cross-section energy autocorrelation function C(ε, θ), provided the relative contribution of potential scattering is larger than 70% [16,17]. In Fig. 3 we plot C(ε, θ)/C(ε = 0, θ) = [ Fig. 3 we observe that the sensitivity of P (t, θ) to different values of β and d is considerably stronger than the sensitivity of C(ε, θ) to these values. Also P (t, θ) obtained using Eqs. (3) and (4) does not acquire statistical errors due to the finite energy interval I. Indeed such a method provides information on P (t, θ) equivalent to that obtained in the femtochemistry experiments. Therefore the only uncertainty of this method is a finite time resolution, ∆t ≃h/I, which is the uncertainty in the femtochemistry experiments with the finite energy band I of the pump and probe laser pulses [25,26]. On the other hand, a reconstruction of P (t, θ) from C(ε, θ) may result in additional errors due to possible statistical uncertainties in C(ε, θ) related to the finite energy range I [30]. Such additional statistical uncertainties may not be excluded even though the effects studied in this paper can clearly not be associated with Ericson fluctuations. Indeed, unlike, e.g., oscillating structures in Fig. 3, Ericson theory predicts angle independent Lorentzian shapes for the C(ε, θ). Therefore, for the reasons described above, we suggest that a reconstruction of P (t, θ) directly from the excitation functions (Eqs. (3) and (4)) is more reliable than from C(ε, θ).\n\n\nIV. FROM HEAVY-ION COLLISIONS TO BIMOLECULAR CHEMICAL REACTIONS\n\nAn important question is whether the rotating wave packets and their interference can occur in colliding systems for which more rigorous approaches can be implemented. The encouraging answer has been given by the calculations of the time evolution of the H + D 2 → HD + D bimolecular chemical reaction [31,32,33]. The evidence for rotational wave packets has been demonstrated (see, e.g., Fig. 3 in [31]) due to the interference of the overlapping resonances [34] of the quasibound complex with J = 15 − 20 total spin values, (i.e., within our approach, d ≃ 5). In this peripheral time-delayed collision the rotational wave packets are stable showing a moderate spreading during half a period of the rotation. Within our description, this reveals a small β value in the time-delayed peripheral reaction mechanism, i.e. πβ/hω ≪ 1/d. Then, in accordance with our prediction, the coherent superposition of the rotating wave packets should produce a strong interference contrast. This is indeed the case as clearly seen in Fig. 3 of Ref. [31]. Similar effects of the stable rotating wave packets and their coherent superpositions with strong interference contrast have also been found in the calculations for F + HD → HF + D bimolecular reaction [35,36].\n\nThe above results should allow to test our suggestion for the determination of the phase relaxation time and, in general, for the time evolution of the collision in bimolecular chemical reactions. For example, in [32] a total time dependent amplitude was calculated for the H + D 2 → HD + D reaction. Then it was splitted into short-time direct reaction component and the timedelayed one. Taking the half-Fourier transform, explicit energy and angle dependencies of both the direct and time-delayed amplitudes were obtained. It was found that in the region of backward angles θ ≤ 70 • and for the c.m. energy interval 1.6-2 eV the direct reaction cross section is considerably bigger than the time-delayed one. Then one can apply our method (Sect. III) to obtain P (t, θ) of the time-delayed reaction mechanism from the energy dependence of the double-differential cross sections. This extracted P (t, θ) can be compared with its actual exact form since the latter is known and was used to generate the energy dependence of the doubledifferential cross sections.\n\nThe calculations [32,33] suggests that in the forward angle range, where the Schrödinger cat states originated from the interference of the rotating towards each other wave packets are observed, contribution of the direct reactions is negligible. Therefore, for this particular H + D 2 → HD + D reaction for the c.m. energy less than 2 eV our method can not be applied. Yet one can still check the sensitivity of the method by artificially adding an energy smooth background amplitude to the calculated one for the time-delayed reaction mechanism. The energy smooth amplitude would play a role of the direct reaction amplitude. This would enable one to directly test the accuracy of the determination of the interference fringe contrast from the data on the double differential cross sections generated with the artificially added energy smooth direct reaction amplitude. Such a test may be important since for the higher energy and/or for some other bimolecular chemical reactions, a strong contribution of direct reactions is possible for the same angle range where the Schrödinger cat states show up.\n\nNote that pure energy resolution measurements of the cross sections of bimolecular chemical reactions, in particular H + D 2 → D + HD [37] and F + HD → HF + D [38] reactions, have recently become possible.\n\nOur results suggest that this should enable experimentalists to extract information on the time dependence, in particular on the rotational coherent dynamics and possible wave packet interference, of the bimolecular chemical reactions which previously could only be studied by the pump probe laser pulses technique to monitor unimolecular chemical reactions [25,26].\n\n\nV. PHASE RELAXATION VERSUS DECOHERENCE\n\nSince phase relaxation results in washing out the interference fringes and eventually, for a sufficiently large β, can destroy the Schrödinger cat states of Figs. 1 and 2, this effect appears to be quite similar to decoherence [39]. Yet, there is an essential difference between the two effects. It is instructive to compare the extreme case of very fast phase relaxation (very large β) with very fast decoherence. Very fast phase relaxation corresponds to the regime of random matrix theory, i.e., to random phase relations between all partial width amplitudes corresponding to different strongly overlapping resonance states [4]. This yields angle independent exponential form for the time power spectrum P (t, θ) and unity for the normalized variance of the fluctuating cross section. Analogous to the fast phase relaxation would be quick decoherence between all many-body quasi-stationary resonance states, similar to the decoherence of Fock states due to the coupling of the system to a phase reservoir [39]. Such a decoherence would destroy interference terms between all the many-body eigenstates, leading to the vanishing of interference between different partial width amplitudes. It can be shown that in this case the time power spectrum would have the same angle independent exponential form as for fast phase relaxation. However, in contrast to fast relaxation, quick decoherence would result in washing out the cross section energy fluctuations, reducing its normalized variance to D/Γ ≪ 1 [40]. It should be noted that the suppression of electron transmission intensity fluctuations through nanostructures due to dephasing, an effect similar to decoherence, can be described on the basis of different arguments allowing to evaluate dephasing rates from the amplitude of conductance fluctuations [41].\n\n\nVI. DISCUSSION AND CONCLUSIONS\n\nOur approach is based on a new idea of slow phase relaxation between partial width amplitudes in a regime of strongly overlapping resonances. Contrary to the conventional idea of random phase relations in this regime [4,42,44], our analysis invokes spin off-diagonal phase correlations and their slow decay for the resonance timedelayed collisions. Though these effects may occur for atomic cluster collisions [19] and are already revealed for the bimolecular chemical reactions [31,32,33,35,36], the present consideration is essentially motivated by the pure energy resolution data on fluctuations in 12 C+ 24 Mg elastic and inelastic scattering at θ = 180 • [21,22], demonstrating both the fine energy structure of ≃ 0.3 MeV and quasiperiodicity with period ≃ 3 MeV [16,17]. It is these peculiar features that led us to the interpretation of the 12 C+ 24 Mg scattering in terms of highly excited quasimolecular states with strongly overlapping resonances, stable rotational wave packets and strong sensitivity of their coherent superpositions to the phase relaxation width. Note that quasiperiodic structures in the excitation functions are present also for other heavy-ion colliding systems [15]. We have also described thoroughly how to extract the time power spectrum from the data on the excitation function oscillations.\n\nIt should be mentioned that a conceptually different optical model description of the 12 C+ 24 Mg elastic scattering at backward angles was presented in Ref. [43], where the quasimolecular structures were ruled out. This conclusion was based on a qualitative description of angular oscillations at backward angles for only three energies within the same energy range which is the subject of our study in this paper. Unfortunately, that analysis is inconsistent with the presence of fine energy structure of ≃ 0.3 MeV at θ = 180 • [21], which unambiguously reveals the presence of time-delayed processes at backward angles. On the contrary, the optical model description relies on the energy average S-matrix and therefore is expected to reproduce broad ≃ 2 − 3 MeV energy structures in the cross section. From our point of view, the fact that the presence of the fine energy structures was not taken into account led to an incorrect qualitative and then quantitative identification of the physical picture of the collision process in Ref. [43].\n\nIn conclusion, we have demonstrated a strong sensitivity of Schrödinger cat states to the phase-relaxation width in complex quantum collisions. This should permit to determine the phase relaxation time from measurements of the excitation functions for 12 C+ 24 Mg elastic scattering at backward angles. Such an experiment would be desirable since the Schrödinger cat states predicted in [19] for 12 C+ 24 Mg scattering involve ∼ 10 3 −10 4 many-body configurations of the highly excited intermediate complex. To the best of our knowledge, the internal interactive complexity of these quantum macroscopic superpositions dramatically exceeds [19] all those previously experimentally realized. The proposed method can also be applied for determination of the phase relaxation time from the data on excitation functions for bimolecular chemical reactions.\n\nFIG. 1 :\n1(Color online) Angular dependence of the decay intensity of the highly excited intermediate complex at different moments of time for different combinations of β and d. Panel (a) corresponds to time t = 3T /8; (b) t = 7T /16; (c) t = T /2; (d) t = 5T /8. Here, T is the period of one complete revolution of the complex. Solid lines (black) correspond to β = 0.01 MeV and d = 3; dashed (red) to β = 0.04 MeV and d = 4; dashed-dotted (blue) to β = 0.075 MeV and d = 8. The bigger β the smaller the interference fringe contrast is.\n\n\ninto account intermediate structure by means of inclusion of many isolated resonances in the S-matrix elements.\n\n\nindicates the formation of stable rotational wave packets, in spite of the strong overlap of resonance levels in the highly excited intermediate molecule. The set of parameters is [17]: Φ = 0, d = 3,J = 14, β = 0.01 MeV,hω =1.45 MeV, and Γ = 0.3 MeV.\n\nFIG. 2 :\n2(Color online) Time dependence of the decay intensity of the highly excited intermediate complex at (a) θ = 180 • and (b) θ = 170.6 • , for the different sets of β and d. Solid lines (black) correspond to β = 0.01 MeV and d = 3; dashed (red) to β = 0.04 MeV and d = 4; dashed-dotted (blue) to β = 0.075 MeV and d = 8. The smaller β the deeper the minimum is, due to the interference between the wave packets for θ = 170.6 • at t = T /2.\n\n\ntector at time t with the time uncertainty of abouth/I, for a collision initiated at t = 0. In the sum in Eq. (2), N = I/∆E and (N + 1) is the number of equidistant energy values for which the detailed angular distributions should be measured. The summation in the r.h.s. of Eq.\n\n\ncos(εt/h)P (t, θ)]/ σ(E, θ) for (a) θ = 180 • and (b) θ = 170.6 • , for the three combinations of β and d. Since the P (t, θ = 180 • ) are close for the different sets of β and d, the corresponding C(ε, θ = 180 • ) can hardly be distinguished reliably. This is the reason that the analysis at θ = 180 • [16, 17] is not sufficient to determine unambiguously the values of β and d. However, C(ε, θ = 170.6 • ) is more sensitive to the different sets of β and d. In particular, for ε ≥ 4 − 5 MeV, the oscillations in C(ε, θ = 170.6 • ) for β = 0.01 MeV, d = 3 and β = 0.075 MeV, d = 8 are out of phase with the absolute value of their difference being up to 0.4. Still, from the comparison of Figs. 1(c), 2(b) and\n\nFIG. 3 :\n3(Color online) Directly measurable cross section energy autocorrelation functions calculated at (a) θ = 180 • and (b) θ = 170.6 • for the different sets of β and d. Solid lines (black) corresponds to β = 0.01 MeV and d = 3; dashed (red) to β = 0.04 MeV and d = 4; dashed-dotted (blue) to β = 0.075 MeV and d = 8.\nAcknowledgmentsWe are grateful to F. Leyvraz and T.H. Seligman for useful discussions and suggestions. One of us (LB) acknowledges financial support from the projects IN-101603 (DGAPA-UNAM) and 43375-E (CONACyT).\nE Wigner, Statistical Properties of Nuclei. J.B. GardNew York-LondonPlenum Press11E. Wigner, in Statistical Properties of Nuclei, editor J.B. Gard (Plenum Press, New York-London, 1972), p. 11.\n\n. O Bohigas, H A Weidenmüller, Annu. Rev. Nucl. Part. Sci. 38421O. Bohigas and H.A. Weidenmüller, Annu. Rev. Nucl. Part. Sci. 38, 421 (1988).\n\n. B L Altshuler, Y Gefen, A Kamenev, L S Levitov, Phys. Rev. Lett. 782803B.L. Altshuler, Y. Gefen, A. Kamenev, and L.S. Levitov, Phys. Rev. Lett. 78, 2803 (1997).\n\n. T Guhr, A Müller-Groeling, H A Weidenmüller, Phys. Rep. 299189T. Guhr, A. Müller-Groeling, and H.A. Weidenmüller, Phys. Rep. 299, 189 (1998).\n\n. E Wigner, Ann. Math. 62203E. Wigner, Ann. Math. 62, 548 (1955); 65, 203 (1957).\n\n. J Flores, S Yu, T H Kun, Seligman, quant-ph/0502050Phys. Rev. E. 7217201J. Flores, S.Yu. Kun, and T.H. Seligman, Phys. Rev. E 72, 017201 (2005); quant-ph/0502050.\n\nM Bienert, J Flores, S Yu, T H Kun, SIGMA (Symmetry, Integrability and Geometry: Methods and Applications. 227M. Bienert, J. Flores, S.Yu. Kun, and T.H. Selig- man, SIGMA (Symmetry, Integrability and Geome- try: Methods and Applications) 2, Paper 027 (2006);\n\n. B Georgeot, D L Shepelyansky, Phys. Rev. E. 623504B. Georgeot and D.L. Shepelyansky, Phys. Rev. E 62, 3504 (2000).\n\nD L Shepelyansky, the Proceedings of Nobel Symposium on Quantum Chaos. 90112D.L. Shepelyansky, in the Proceedings of Nobel Sympo- sium on Quantum Chaos 2000, Phys. Scripta T90, 112 (2001).\n\nExperimental proposal for accurate determination of the phase relaxation time and testing a formation of thermalized nonequilibrated matter in highly excited quantum manybody systems. M Bienert, J Flores, S Yu, Kun, submittedM. Bienert, J. Flores, and S.Yu. Kun, \"Experimental proposal for accurate determination of the phase relax- ation time and testing a formation of thermalized non- equilibrated matter in highly excited quantum many- body systems\", submitted.\n\n. S Yu, Kun, Phys. Lett. B. 31916S.Yu. Kun, Phys. Lett. B 319, 16 (1993).\n\n. S Yu, Kun, Z. Phys. A. 348273S.Yu. Kun, Z. Phys. A 348, 273 (1994).\n\n. S Yu, Kun, Z. Phys. A. 357255S.Yu. Kun, Z. Phys. A 357, 255 (1997).\n\n. S Yu, A V Kun, O K Vagov, Vorov, Phys. Rev. C. 59585S.Yu. Kun, A.V. Vagov, and O.K. Vorov, Phys. Rev. C 59, R585 (1999). (2001).\n\n. S Yu, B A Kun, A V Robson, Vagov, Phys. Rev. Lett. 83504S.Yu. Kun, B.A. Robson, and A.V. Vagov, Phys. Rev. Lett. 83, 504 (1999).\n\n. S Yu, A V Kun, W Vagov, Greiner, Phys. Rev. C. 6314608S.Yu. Kun, A.V. Vagov, and W. Greiner, Phys. Rev. C 63, 014608 (2001).\n\n. S Yu, A V Kun, L T Vagov, W Chadderton, Greiner, Int. J. Mod. Phys. E. 11273S.Yu. Kun, A.V. Vagov, L.T. Chadderton, and W. Greiner, Int. J. Mod. Phys. E 11, 273 (2002).\n\nW Greiner, J Y Park, W Scheid, Nuclear Molecules. SingaporeWorld ScientificW. Greiner, J.Y. Park, and W. Scheid, Nuclear Molecules (World Scientific, Singapore, 1995).\n\n. S Yu, L Kun, L T Benet, W Chadderton, F Greiner, Haas, Phys. Rev. C. 6711604S.Yu. Kun, L. Benet, L.T. Chadderton, W. Greiner, and F. Haas, Phys. Rev. C 67, 011604(R) (2003).\n\n. U Heinz, J Reinhardt, B Müller, W Greiner, W T Pinkston, Z. Phys. A. 316341U. Heinz, J. Reinhardt, B. Müller, W. Greiner, and W.T. Pinkston, Z. Phys. A, 316, 341 (1984).\n\n. B Ghosh, R Singh, Pramana J. Phys. 291555B. Ghosh and R. Singh, Pramana J. Phys. 29, 1555 (1987).\n\n. M C Mermaz, Phys. Rev. C. 241512M.C. Mermaz et al., Phys. Rev. C 24, 1512 (1981).\n\n. U Heinz, U Müller, J Reinhardt, B Müller, W Greiner, Ann. Phys. (N.Y.). 158476U. Heinz, U. Müller, J. Reinhardt, B. Müller, W. Greiner, Ann. Phys. (N.Y.) 158, 476 (1984).\n\n. V Chiste, R Lichtenthäler, A C C Villari, L C Gomes, Phys. Rev. C. 54784V. Chiste, R. Lichtenthäler, A.C.C. Villari, and L.C. Gomes, Phys. Rev. C 54, 784 (1996).\n\nA H Zewail, Femtochemistry. SingaporeWorld Scientific1A.H. Zewail, Femtochemistry (World Scientific, Singa- pore, 1995), Vols. 1 and 2.\n\nNobel Lectures. A H Zewail, Nobel Lectures. SingaporeChemistryA.H. Zewail, Nobel Lectures 1999, in Nobel Lectures, Chemistry 1996-2000, editor I. Grenthe, (World Scien- tific, Singapore 2003).\n\n. L Benet, S Yu, Wang Kun, V Qi, Denisov, Phys. Lett. B. 605101L. Benet, S.Yu. Kun, Wang Qi, and V. Denisov, Phys. Lett. B 605, 101 (2005).\n\n. G Pappalardo, Phys. Lett. B. 13320G. Pappalardo, Phys. Lett. B 13, 320 (1964).\n\n. E P Wigner, Phys. Rev. 98145E.P. Wigner, Phys. Rev. 98, 145 (1955).\n\nA Richter, Nuclear Spectroscopy and Reactions. J. ChernyNew YorkAcademic343A. Richter, in Nuclear Spectroscopy and Reactions, edited by J. Cherny (Academic, New York, 1974), Vol. B, p. 343.\n\n. S C Althorpe, F Fernandez-Alonso, B D Bean, J D Ayers, A E Pomerantz, R N Zare, E Wrede, Nature. 41667S.C. Althorpe, F. Fernandez-Alonso, B.D. Bean, J.D. Ay- ers, A.E. Pomerantz, R.N. Zare, and E. Wrede, Nature (London) 416, 67 (2002).\n\n. S C Althorpe, J. Chem. Phys. 1174623S.C. Althorpe, J. Chem. Phys. 117, 4623 (2002).\n\n. S C Althorpe, J. Chem. Phys. 1211175S.C. Althorpe, J. Chem. Phys. 121, 1175 (2004).\n\n. S D Chao, R T Skodje, Chem. Phys. Lett. 336364S.D. Chao and R.T. Skodje, Chem. Phys. Lett. 336, 364 (2001).\n\n. S C Althorpe, Chem. Phys. Lett. 370443S.C. Althorpe, Chem. Phys. Lett. 370, 443 (2003).\n\n. S C Althorpe, J. Phys. Chem. A. 1077152S.C. Althorpe, J. Phys. Chem. A 107, 7152 (2003).\n\n. D Dai, C C Wang, S A Harich, X Wang, X Yang, S D Chao, R T Skodje, Science. 3001730D. Dai, C.C. Wang, S.A. Harich, X. Wang, X. Yang, S.D. Chao, and R.T. Skodje, Science 300, 1730 (2003).\n\n. R T Skodje, D Skouteris, D E Manolopoulos, Shih-Huang Lee, Feng Dong, Kopin Liu, Phys. Rev. Lett. 851206R.T. Skodje, D. Skouteris, D.E. Manolopoulos, Shih- Huang Lee, Feng Dong, and Kopin Liu, Phys. Rev. Lett. 85, 1206 (2000).\n\n. C J Myatt, Nature. 403269C.J. Myatt et al., Nature (London) 403, 269 (2000).\n\n. S Yu, Kun, Phys. Rev. A. 6534701S.Yu. Kun, Phys. Rev. A 65, 034701 (2002).\n\n. C M Marcus, R M Westervelt, P F Hopkins, A C Gossard, Phys. Rev. B. 482460C.M. Marcus, R.M. Westervelt, P.F. Hopkins, and A.C. Gossard, Phys. Rev. B 48, 2460 (1993);\n\n. E Mccann, I V Lerner, J. Phys.: Condens. Matter. 86719E. McCann and I.V. Lerner, J. Phys.: Condens. Matter 8, 6719 (1996);\n\n. P W Brouwer, C W J Beenakker, Phys. Rev. B. 554695P.W. Brouwer and C.W.J. Beenakker, Phys. Rev. B 55, 4695 (1997);\n\n. J P Bird, J. Phys.: Condens. Matter. 11413J.P. Bird, J. Phys.: Condens. Matter 11, R413 (1999).\n\n. A H Zewail, Faradey Discuss, Chem. Soc. 75315A.H. Zewail, Faradey Discuss. Chem. Soc. 75, 315 (1983).\n\n. A Lépine-Szily, M S Hussein, R Lichtenthäler, J Cseh, G Lévai, Phys. Rev. Lett. 823972A. Lépine-Szily, M.S. Hussein, R. Lichtenthäler, J. Cseh, and G. Lévai, Phys. Rev. Lett. 82, 3972 (1999).\n\nProceedings of the Les Houches Summer School, Session LII. U Smilansky, Chaos and Quantum Physics. M.J. Giannoni, A. Voros, and J. Zinn-JustinNorth-Holland, Amsterdam372and references thereinU. Smilansky, in Chaos and Quantum Physics, Proceed- ings of the Les Houches Summer School, Session LII, edited by M.J. Giannoni, A. Voros, and J. Zinn-Justin (North-Holland, Amsterdam, 1991), p. 372, and refer- ences therein.\n"},"corpus_id":{"kind":"number","value":39198288,"string":"39,198,288"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"quant-ph/0503046\",\n \"mag\": 2060521203,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1103/physrevc.73.064602\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/quant-ph/0503046v2.pdf\"\n ],\n \"pdf_sha\": \"021f5666d73fa04c76b4468f838cb14615b2e5e3\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/quant-ph/0503046\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Effect of phase relaxation on quantum superpositions in complex collisions"},"authors":{"kind":"list like","value":["L Benet \nCentro de Ciencias Físicas\nNational University of Mexico (UNAM)\nMor62210CuernavacaMexico","S Yu Kun \nCentro de Ciencias Físicas\nNational University of Mexico (UNAM)\nMor62210CuernavacaMexico\n\nNonlinear Physics Center\nThe Australian National University\n0200RSPhysSE, CanberraACTAustralia\n\nDepartment of Theoretical Physics\nThe Australian National University\n0200RSPhysSE, CanberraACTAustralia","Wang Qi \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina"],"string":"[\n \"L Benet \\nCentro de Ciencias Físicas\\nNational University of Mexico (UNAM)\\nMor62210CuernavacaMexico\",\n \"S Yu Kun \\nCentro de Ciencias Físicas\\nNational University of Mexico (UNAM)\\nMor62210CuernavacaMexico\\n\\nNonlinear Physics Center\\nThe Australian National University\\n0200RSPhysSE, CanberraACTAustralia\\n\\nDepartment of Theoretical Physics\\nThe Australian National University\\n0200RSPhysSE, CanberraACTAustralia\",\n \"Wang Qi \\nInstitute of Modern Physics\\nChinese Academy of Sciences\\n730000LanzhouChina\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We study the effect of phase relaxation on coherent superpositions of rotating clockwise and anticlockwise wave packets in the regime of strongly overlapping resonances of the intermediate complex. Such highly excited deformed complexes may be created in binary collisions of heavy ions, molecules and atomic clusters. It is shown that phase relaxation leads to a reduction of the interference fringes, thus mimicking the effect of decoherence. This reduction is crucial for the determination of the phase-relaxation width from the data on the excitation function oscillations in heavy-ion collisions and bimolecular chemical reactions. The difference between the effects of phase relaxation and decoherence is discussed."},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":"For highly excited strongly interacting many-body systems the independent particle picture has very limited validity [1]. For high excitations, the interaction results in a quick decay of single-particle [2,3,4] as well as collective modes [2,4], which are not eigenstates of the Hamiltonian of the system. This decay leads to the formation of complicated many-body configurations. Each of these many-body states is characterized by a uniform occupation of the accessible phase space and sharing the energy between many particles of the system. The characteristic time for the formation of such spatially extended ergodic many-body states, independent of the initial conditions, is given by the inverse spreading width, τ erg =h/Γ spr [4]. Introduced by Wigner [5], Γ spr also characterizes the width of the distribution of the expansion coefficients of the many-body eigenstates over a noninteracting mean-field basis. For sufficiently high excitation energy the decay of single-particle modes creates a linear superposition of a very large number of manybody configurations. The question is whether the phase relations between these individually ergodic many-body states in the superposition may still preserve a memory of the way the system was excited. This question is of fundamental importance for the study of relaxation phenomena in nuclear, atomic, molecular and mesoscopic many-body systems, and for many-qubit quantum computation. In particular, if phase relaxation is longer than energy relaxationh/Γ spr , this can extend the time for quantum computing [6,7] beyond the quantum-chaos border [8,9].\nTo answer this question from first principles is difficult due to computational limitations. Indeed, in order to solve the full quantum many-body problem one would require a many-qubit quantum computer. Therefore, the only currently available resort to search for possible manifestations of long phase relaxation is the experiment, and a careful data analysis. Nuclear systems are an ideal laboratory to study many-body systems, since nuclear interactions are so strong that external perturbations can often be neglected. In particular, the analysis of Refs. [6,7,10] of the data on asymmetry around 90 • c.m. in angular distributions of evaporated protons from heavy nuclei in nucleon-induced and photonuclear reactions clearly indicates that phase relaxation time in highly excited strongly interacting many-body systems can be up to eight orders of magnitude [7] longer than energy relaxation timeh/Γ spr . This reveals a new form of matter-thermalized non-equilibrated matter introduced by one of us in Refs. [11,12,13].\nA more subtle indication of a slow phase relaxation is found in light heavy-ion scattering [14,15,16,17]. It manifests itself in the formation of rotating wave packets whose spreading time, which is given by the inverse phase-relaxation width, is much longer than the energy relaxation timeh/Γ spr . The approach [15,16,17] treats time-delayed light heavy-ion scattering in terms of the formation and decay of quasimolecular resonances [18]. The highly excited coherently rotating intermediate system has the energy of the intrinsic excitation ≥ 10 MeV. The intermediate system is in the regime of strongly overlapping resonances. Therefore, this coherent rotation does not originate from the discreteness of the spectrum, which is not resolved, but is due to the spin off-diagonal correlations between partial width amplitudes. Indeed, the period of the coherent rotation is much shorter than the inverse level spacing of the intermediate system. This reveals a new root, as compared to Bohr's correspondence principle, for a quantum-classical transition in highly excited many-body systems.\nThe width of the rotating wave packets is about 1/d + βt/h ≈ 1/d + βθ/hω, where d is the effective number of partial waves, β is the spin off-diagonal phaserelaxation width, ω is the angular velocity, and t and θ are the time and angle of the rotation, respectively. This seemingly allows us to determine the wave-packet spreading rate from the time power spectra at differ-ent scattering angles, which can be reconstructed, for binary collisions, from the data on energy fluctuations of the cross sections [16,17]. These fluctuations originate from the energy-fluctuating collision amplitude corresponding to a resonance time-delayed process. However, for elastic heavy-ion scattering, such energy fluctuations can be typically observed only at backward angles 140 • ≤ θ ≤ 180 • , since for smaller θ the direct reaction contribution grows exponentially and becomes much greater than the time delayed one. Therefore, since at the initial moment of time the deformed intermediate system is oriented in the forward direction θ ≈ 0 • , the spreading of the wave packets within the backward angular interval 140 • ≤ θ ≤ 180 • can hardly be detected reliably. Moreover, the width of the wave packets at this backwardangle range does not allow to determine β, since the angular dispersion 1/d at t = 0 is also unknown. In this paper we show that both d and β can indeed be determined unambiguously. This results from the strong sensitivity of the interference fringes between the rotating clockwise and anticlockwise wave packets for the backward angle range to the the phase relaxation width: As β grows, the interference fringes are suppressed more strongly. The effect is considered in relation to heavy-ion scattering and bimolecular chemical reactions. The difference between the effects of phase relaxation and decoherence is discussed."},{"title":"II. THE TIME POWER SPECTRUM OF THE TIME-DELAYED COLLISIONS","content":"Following Ref. [19], we consider spinless collision partners in the entrance and exit channels. The time and angle dependent intensity of the decay (the time power spectrum), P (t, θ), is given by the modulus square of the Fourier component of the energy-fluctuating collision amplitude. It can be also expressed as the Fourier component of the amplitude energy autocorrelation function. P (t, θ) has been obtained in [19] by summing over a very large number of strongly overlapping resonance levels, Γ/D ≫ 1, where Γ is the total decay width and D is the average level spacing of the intermediate complex. As a result, after changing from summation over the resonance levels to integration, which is a good approximation for t ≪h/D, P (t, θ) takes the form:\nHere, H(t) is the Heaviside step function, β is the spin phase-relaxation width, ω is the angular velocity of the coherent rotation, Φ is the deflection given by the total spin J (inh units) derivative of the potential phase shifts (direct reaction), and P J (θ) are the Legendre polynomials. The physical meaning of the inverse spin phase-relaxation width,h/β, is the characteristic time for the angular spreading of the clockwise and anticlockwise rotating wave packets. The partial average reaction probability is taken in the J-window form,\n, whereJ is the average spin and d is the J-window width.\nIt should be noted that a similar effect of coherent rotation of the nuclear molecule, as described by Eq. (1) with β = 0, was found in Ref. [20] for a scattering of heavy nuclei. However, while Eq. (1) has been obtained for a large number of open channels and strongly overlapping resonances of the intermediate system [19], the time power spectrum in Ref. [20] was derived from one open channel and one resonance pole form of the S-matrix elements. Therefore the latter approach is not applicable for the forthcoming analysis of the 12 C+ 24 Mg system since, for this system, the number of open channels is much greater than one. Furthermore, the one-resonance pole form of the S-matrix elements employed in Ref. [20] results in iso-lated resonances with the energy spacing between them to be abouthω. This is not the case for the 12 C+ 24 Mg elastic scattering analyzed in this paper. Indeed, the data on the excitation function for the 12 C+ 24 Mg elastic scattering at θ = 180 • reveal about 15 local maxima on the energy interval ∆E cm = 13 − 22 MeV [21]. Then the interpretation [20] would mean that average level spacing, ≃ 0.6 MeV in our case, between the local maxima is given byhω. This, in turn, would reflect the quasiperiodic behaviour of the cross section energy autocorrelation functions with the period of about 0.6 MeV. This is inconsistent with the data [21] showing the quasiperiodicity of the cross section with the period of about 2.9 MeV yieldinghω = 1.35 − 1.45 MeV [16,17]. Moreover, the interpretation [20] implies that these local maxima in the excitation function should be approximately equidistant. Instead, the data [21,22] show big fluctuations, from 0.3 MeV to 1.3, for the energy spacing between the nearest-neighboring local maxima in the excitation function. Note that the indication against the interpretation in terms of isolated resonances comes from the statistically insignificant channel-channel correlations for the 12 C+ 24 Mg elastic and inelastic scattering at θ = 180 • [16]. Finally, the effect of coherent rotation [20] does not survive a generalization [23] by taking On the basis of the above arguments we believe that the interpretation of the fine structure in the excitation functions for the 12 C+ 24 Mg scattering at θ = 180 • in terms of overlapping, rather than isolated, resonances of the intermediate system is a justified approach.\nFirst, we calculate P (t, θ) for the set of parameters obtained from the description [17] of the experimental cross section energy autocorrelation functions [21] for 12 C+ 24 Mg elastic and inelastic scattering at θ = 180 • [22]. For these collisions the analysis of the oscillations in the cross section energy autocorrelation functions [16,17] In Figs. 1 we plot the quantity AP (t, θ)/ σ(E, θ) at four moments of time, with T = 2π/ω being the period of one complete revolution. Here, σ(E, θ) ∝ ∞ 0 dt P (t, θ) is the energy-averaged differential cross section for the time-delayed collision. In Figs. 1, P (t, θ) is scaled with σ(E, θ) for the reasons discussed in Ref. [19]. The constant A is determined by the condition AP (t = 0, θ = 0)/ σ(E, θ = 0) = 1. Figure 1(a) shows the two slightly overlapping wave packets rotating towards each other in the backward direction. In panels (b) and (c), the wave packets strongly overlap around θ = 180 • , producing interference fringes. Finally, in panel (d) the wave packets have passed each other and move apart rotating in the forward direction.\nIn Figs. 1 we also plot AP (t, θ)/ σ(E, θ) with the same constant A and the set of parameters as before, except for d and β. We consider two other cases: d = 4, β = 0.04 MeV, and d = 8, β = 0.075 MeV. One can see that when the wave packets overlap only slightly (panels (a) and (d)) one can hardly distinguish between the three combinations of d and β, even though β changes by almost one order of magnitude. On the other hand, from panels (b) and (c), we observe that a fringe contrast due to the interference of the wave packets is very sensitive to absolute values of β and d. This result is further illustrated in Fig. 2 where we plot P (t, θ) calculated for the three sets of β and d values, as a function of t for θ = 170.6 • . One observes that the smaller β is the deeper minima are, due to the destructive interference between the wave packets at t = T /2. However, for θ = 180 • , P (t, θ) in Fig. 2 is insensitive to the three sets of β and d.\nNote that in order to reproduce the width of the wave packets in Figs. 1 and 2 for the maximal possible d = I value, one has to take β = 0.1 MeV. Therefore, even the largest possible β value is more than one order of magnitude smaller than Γ spr . The latter is usually estimated from the width of giant resonances [2,4]: Γ spr ≃ 5 MeV. This indicates that energy relaxation, i.e., the process of formation of ergodic many-body configurations, in highly excited nuclear systems can be much faster than phase relaxation."},{"title":"III. DETERMINATION OF THE TIME POWER SPECTRUM FROM THE DATA ON THE EXCITATION FUNCTION OSCILLATIONS","content":"We consider a collision of the spinless reaction partners in the entrance and exit channels. The cross section of the collision is given by σ(E, θ) = |f (E, θ)| 2 . Here E is the energy in the entrance channel and θ is the scattering angle. The reaction amplitude f (E, θ) is a linear combination of S-matrix elements S J (E) corresponding to the different total spin values J. Suppose that, for a fixed energy E 1 , a measurement of detailed angular distribution is performed. Then, using, e.g., the method [24], one can find S J (E 1 ) and, therefore, f (E 1 , θ). Measuring the angular distributions for different energies on the energy interval I = E min − E max with the energy step ∆E and energy resolution smaller or about the minimal characteristic energy scale of a variation of f (E, θ) one obtains a detailed energy and angle dependencies of the collision amplitude. Its Fourier component,\nis the instantaneous decay amplitude received at a de- (2) may be changed to the integration if we use a linear interpolation for f (E, θ) in between the consequent energies E n and E n+1 . This is justified provided the energy step ∆E is smaller or about the minimal characteristic energy scale of a variation of the cross section. For example, in Ref. [22] the energy step is ∆E cm = 133 keV while the minimal characteristic scale of the variation of σ(E, θ = 180 • ) is Γ = 0.3 MeV [16,17]. Having reconstructed P(t, θ) from the excitation function data one obtains the time power spectra P (t, θ) = |P(t, θ)| 2 for any angle. The later quantity is equivalent to that directly measured using real-time methods of femtochemistry, with the energy bands of I for the pump and probe laser pulses, to monitor chemical reactions dynamics [25,26]. Due to its dependence on θ, P (t, θ) determined by the proposed method could provide even more detailed information on the collision dynamics, as compared with the femtochemistry experiments, since latter measurements often allow to obtain P (t, θ) averaged over θ.\nFor the energy interval I being long enough to provide a sufficient time resolutionh/I to resolve interference fringes [27] the proposed experiment would allow to test our theoretical predictions. If these were confirmed by the data, then the experimentally determined P (t, θ) can be compared with our calculations presented in Figs. 1 and 2 with the purpose of evaluation β and d.\nIn particular, the energy interval of 10.5 MeV, over which excitation function for the 12 C+ 24 Mg elastic scattering at θ = 180 • was measured [22], should be sufficient to resolve the interference fringes in P (t, θ) [27].\nIt should be noted that the method proposed above allows to reconstruct P (t, θ) which includes both potential scattering, i.e. fast processes, and the time delayed mechanism. Since the previous analysis [16,17] demonstrated a presence of a strong potential scattering component we expect that this method will yield a strong potential scattering contribution in P (t, θ). However this contribution is expected to be restricted to relatively short times t ps ≤h/∆E ps , where ∆E ps is a characteristic energy interval for the variation of the potential scattering amplitude. For ∆E ps ≃ 2 − 3 MeV [18], t ps ≃ 3 − 2 × 10 −22 sec. Therefore, this potential scattering contribution is expected to be restricted to t/T ≤ 0.1 in Fig. 2 and should not show up at all in Fig. 1.\nThe above method to obtain P (t, θ) from the data is model independent and, therefore, most reliable one. However it is very demanding experimentally since σ(E, θ) must be measured for a very large number of angles [24]. There is another way to extract P (t, θ) from the data even if the excitation function is measured for a single angle. This method can be applied if the direct reaction process provides a major contribution to the cross section [17]. We consider again a case of spinless collision partners in the entrance and exit channels. The main idea is based on the decomposition of the col-lision amplitude, f (E, θ) = f dir (E, θ) + δf (E, θ). Here f dir (E, θ) is the energy smooth amplitude corresponding to potential scattering or direct fast process, and δf (E, θ) is the energy fluctuating amplitude, δf (E, θ) = 0, corresponding to the time-delayed collision. Consider first an idealized case of energy independent amplitude f dir (E, θ) when it coincides with the energy averaged amplitude f (E, θ) . The collision cross section has the form σ(E, θ) = |f (E, θ)\nHere the potential scattering or direct reaction cross section σ dir (θ) = |f dir (E, θ)| 2 is energy-independent and σ f l (E, θ) = |δf (E, θ)| 2 . For a relative contribution of the potential scattering or direct reaction cross section to be about 70% or more of the total cross section, ∆σ(E, θ) = σ(E, θ) − σ(E, θ) ≃ 2Re[δf (E, θ)f dir (E, θ) * ], where σ(E, θ) = σ dir (θ) + σ f l (E, θ) is the energy averaged cross section. This means that rapid energy variations of the cross section originate mostly from interference between the energy fluctuating δf (E, θ) and the energy independent amplitude f dir (E, θ). Suppose that the excitation function is measured on energy interval I = E max − E min with energy step ∆E so that the total number of steps is N + 1, where N = I/∆E. Then, for t > 0, we have [17] Emax Emin\nwith P (t, θ) = |P(t, θ)| 2 . In Eq. (3) we have employed a causality condition, P(t < 0, θ) = 0. This condition implies that the molecule cannot decay before it is formed at t = 0. Again the above information on P (t, θ) is equivalent to that obtained directly using real-time methods of femtochemistry to monitor unimolecular chemical reactions, with the energy bands of I = (E max − E min ) for the pump and probe laser pulses [25]. The above consideration can be extended to a case when the potential scattering or direct reaction amplitude f dir (E, θ) depends on energy but this dependence is considerably smoother than the energy dependence of the amplitude δf (E, θ) for the time-delayed collision. We shall again assume that a relative contribution of direct processes into the energy averaged cross section is about 70% or more. We define the characteristic energy interval I d of a variation of f dir (E, θ) with I d ≫ Γ, where Γ is the characteristic energy interval of variation of δf (E, θ). The energy interval I d can be evaluated using, e.g., the trend reduction method [28]. In particular, for the 12 C+ 24 Mg scat-tering [22] analyzed in Sec. II, this interval was evaluated to be about 4 − 5 MeV [21], i.e. much greater than Γ = 0.3 MeV. Then one can find the best polynomial fit of the energy dependence of the cross section\nwith the order of polynomial being [I/I d ]+1, where [I/I d ] is integer part of I/I d The resulting energy smooth cross section is denoted asσ(E, θ) ≃ σ d (E, θ) + σ f l (E, θ) with σ f l (E, θ) being energy averaged, i.e. energy independent, cross section of the time-delayed processes. It can be found using the standard procedure employed for the analysis of Ericson fluctuations (see, e.g., Refs. [21,22]). As a result, one finds energy smooth direct reaction cross section σ d (E, θ). Consider σ(E, θ) − σ(E, θ) ≃ 2Re[δf (E, θ)f dir (E, θ) * ]. Assume first that energy dependence of the phase of f dir (E, θ) is negligible, arg[f dir (E, θ)] = constant, and the smooth energy dependence of σ d (E, θ) is due to the energy dependence of |f dir (E, θ)| = σ d (E, θ) 1/2 . Then applying the same arguments as those used to obtain Eq. (3), for t > 0, we have\nwith P (t, θ) = |P(t, θ)| 2 . In order to generalize Eq. (4) by taking into account energy dependence of arg[f dir (E, θ)] ≡ φ(E, θ) (f dir (E, θ) = |f dir (E, θ)| exp[iφ(E, θ)]) we use the linear approximation φ(E, θ) = φ(Ē, θ) + (E − E)dφ(E, θ)/dE| E=Ē , whereĒ = (E max − E min )/2 and dφ(E, θ)/dE| E=Ē = t dir /h with t dir ≪h/Γ being the time delay due to the potential scattering or direct re-actions [29]. Then, it is easy to see, that, for t > t dir , the r.h.s. of Eq. (4) is changed to P(t − t dir , θ). This manifestly demonstrates that the interference between time-delayed and fast direct processes in the cross section is a precondition for monitoring the time evolution, the fast process switches on the clock at the initial moment of time t d playing the role of the pump pulse. In the absence of direct processes the initial moment of time is not defined.\nNote that P (t, θ) can also be expressed as a cosine half-Fourier transform of the cross-section energy autocorrelation function C(ε, θ), provided the relative contribution of potential scattering is larger than 70% [16,17]. In Fig. 3 we plot C(ε, θ)/C(ε = 0, θ) = [ Fig. 3 we observe that the sensitivity of P (t, θ) to different values of β and d is considerably stronger than the sensitivity of C(ε, θ) to these values. Also P (t, θ) obtained using Eqs. (3) and (4) does not acquire statistical errors due to the finite energy interval I. Indeed such a method provides information on P (t, θ) equivalent to that obtained in the femtochemistry experiments. Therefore the only uncertainty of this method is a finite time resolution, ∆t ≃h/I, which is the uncertainty in the femtochemistry experiments with the finite energy band I of the pump and probe laser pulses [25,26]. On the other hand, a reconstruction of P (t, θ) from C(ε, θ) may result in additional errors due to possible statistical uncertainties in C(ε, θ) related to the finite energy range I [30]. Such additional statistical uncertainties may not be excluded even though the effects studied in this paper can clearly not be associated with Ericson fluctuations. Indeed, unlike, e.g., oscillating structures in Fig. 3, Ericson theory predicts angle independent Lorentzian shapes for the C(ε, θ). Therefore, for the reasons described above, we suggest that a reconstruction of P (t, θ) directly from the excitation functions (Eqs. (3) and (4)) is more reliable than from C(ε, θ)."},{"title":"IV. FROM HEAVY-ION COLLISIONS TO BIMOLECULAR CHEMICAL REACTIONS","content":"An important question is whether the rotating wave packets and their interference can occur in colliding systems for which more rigorous approaches can be implemented. The encouraging answer has been given by the calculations of the time evolution of the H + D 2 → HD + D bimolecular chemical reaction [31,32,33]. The evidence for rotational wave packets has been demonstrated (see, e.g., Fig. 3 in [31]) due to the interference of the overlapping resonances [34] of the quasibound complex with J = 15 − 20 total spin values, (i.e., within our approach, d ≃ 5). In this peripheral time-delayed collision the rotational wave packets are stable showing a moderate spreading during half a period of the rotation. Within our description, this reveals a small β value in the time-delayed peripheral reaction mechanism, i.e. πβ/hω ≪ 1/d. Then, in accordance with our prediction, the coherent superposition of the rotating wave packets should produce a strong interference contrast. This is indeed the case as clearly seen in Fig. 3 of Ref. [31]. Similar effects of the stable rotating wave packets and their coherent superpositions with strong interference contrast have also been found in the calculations for F + HD → HF + D bimolecular reaction [35,36].\nThe above results should allow to test our suggestion for the determination of the phase relaxation time and, in general, for the time evolution of the collision in bimolecular chemical reactions. For example, in [32] a total time dependent amplitude was calculated for the H + D 2 → HD + D reaction. Then it was splitted into short-time direct reaction component and the timedelayed one. Taking the half-Fourier transform, explicit energy and angle dependencies of both the direct and time-delayed amplitudes were obtained. It was found that in the region of backward angles θ ≤ 70 • and for the c.m. energy interval 1.6-2 eV the direct reaction cross section is considerably bigger than the time-delayed one. Then one can apply our method (Sect. III) to obtain P (t, θ) of the time-delayed reaction mechanism from the energy dependence of the double-differential cross sections. This extracted P (t, θ) can be compared with its actual exact form since the latter is known and was used to generate the energy dependence of the doubledifferential cross sections.\nThe calculations [32,33] suggests that in the forward angle range, where the Schrödinger cat states originated from the interference of the rotating towards each other wave packets are observed, contribution of the direct reactions is negligible. Therefore, for this particular H + D 2 → HD + D reaction for the c.m. energy less than 2 eV our method can not be applied. Yet one can still check the sensitivity of the method by artificially adding an energy smooth background amplitude to the calculated one for the time-delayed reaction mechanism. The energy smooth amplitude would play a role of the direct reaction amplitude. This would enable one to directly test the accuracy of the determination of the interference fringe contrast from the data on the double differential cross sections generated with the artificially added energy smooth direct reaction amplitude. Such a test may be important since for the higher energy and/or for some other bimolecular chemical reactions, a strong contribution of direct reactions is possible for the same angle range where the Schrödinger cat states show up.\nNote that pure energy resolution measurements of the cross sections of bimolecular chemical reactions, in particular H + D 2 → D + HD [37] and F + HD → HF + D [38] reactions, have recently become possible.\nOur results suggest that this should enable experimentalists to extract information on the time dependence, in particular on the rotational coherent dynamics and possible wave packet interference, of the bimolecular chemical reactions which previously could only be studied by the pump probe laser pulses technique to monitor unimolecular chemical reactions [25,26]."},{"title":"V. PHASE RELAXATION VERSUS DECOHERENCE","content":"Since phase relaxation results in washing out the interference fringes and eventually, for a sufficiently large β, can destroy the Schrödinger cat states of Figs. 1 and 2, this effect appears to be quite similar to decoherence [39]. Yet, there is an essential difference between the two effects. It is instructive to compare the extreme case of very fast phase relaxation (very large β) with very fast decoherence. Very fast phase relaxation corresponds to the regime of random matrix theory, i.e., to random phase relations between all partial width amplitudes corresponding to different strongly overlapping resonance states [4]. This yields angle independent exponential form for the time power spectrum P (t, θ) and unity for the normalized variance of the fluctuating cross section. Analogous to the fast phase relaxation would be quick decoherence between all many-body quasi-stationary resonance states, similar to the decoherence of Fock states due to the coupling of the system to a phase reservoir [39]. Such a decoherence would destroy interference terms between all the many-body eigenstates, leading to the vanishing of interference between different partial width amplitudes. It can be shown that in this case the time power spectrum would have the same angle independent exponential form as for fast phase relaxation. However, in contrast to fast relaxation, quick decoherence would result in washing out the cross section energy fluctuations, reducing its normalized variance to D/Γ ≪ 1 [40]. It should be noted that the suppression of electron transmission intensity fluctuations through nanostructures due to dephasing, an effect similar to decoherence, can be described on the basis of different arguments allowing to evaluate dephasing rates from the amplitude of conductance fluctuations [41]."},{"title":"VI. DISCUSSION AND CONCLUSIONS","content":"Our approach is based on a new idea of slow phase relaxation between partial width amplitudes in a regime of strongly overlapping resonances. Contrary to the conventional idea of random phase relations in this regime [4,42,44], our analysis invokes spin off-diagonal phase correlations and their slow decay for the resonance timedelayed collisions. Though these effects may occur for atomic cluster collisions [19] and are already revealed for the bimolecular chemical reactions [31,32,33,35,36], the present consideration is essentially motivated by the pure energy resolution data on fluctuations in 12 C+ 24 Mg elastic and inelastic scattering at θ = 180 • [21,22], demonstrating both the fine energy structure of ≃ 0.3 MeV and quasiperiodicity with period ≃ 3 MeV [16,17]. It is these peculiar features that led us to the interpretation of the 12 C+ 24 Mg scattering in terms of highly excited quasimolecular states with strongly overlapping resonances, stable rotational wave packets and strong sensitivity of their coherent superpositions to the phase relaxation width. Note that quasiperiodic structures in the excitation functions are present also for other heavy-ion colliding systems [15]. We have also described thoroughly how to extract the time power spectrum from the data on the excitation function oscillations.\nIt should be mentioned that a conceptually different optical model description of the 12 C+ 24 Mg elastic scattering at backward angles was presented in Ref. [43], where the quasimolecular structures were ruled out. This conclusion was based on a qualitative description of angular oscillations at backward angles for only three energies within the same energy range which is the subject of our study in this paper. Unfortunately, that analysis is inconsistent with the presence of fine energy structure of ≃ 0.3 MeV at θ = 180 • [21], which unambiguously reveals the presence of time-delayed processes at backward angles. On the contrary, the optical model description relies on the energy average S-matrix and therefore is expected to reproduce broad ≃ 2 − 3 MeV energy structures in the cross section. From our point of view, the fact that the presence of the fine energy structures was not taken into account led to an incorrect qualitative and then quantitative identification of the physical picture of the collision process in Ref. [43].\nIn conclusion, we have demonstrated a strong sensitivity of Schrödinger cat states to the phase-relaxation width in complex quantum collisions. This should permit to determine the phase relaxation time from measurements of the excitation functions for 12 C+ 24 Mg elastic scattering at backward angles. Such an experiment would be desirable since the Schrödinger cat states predicted in [19] for 12 C+ 24 Mg scattering involve ∼ 10 3 −10 4 many-body configurations of the highly excited intermediate complex. To the best of our knowledge, the internal interactive complexity of these quantum macroscopic superpositions dramatically exceeds [19] all those previously experimentally realized. The proposed method can also be applied for determination of the phase relaxation time from the data on excitation functions for bimolecular chemical reactions."},{"title":"FIG. 1 :","content":""},{"title":"FIG. 2 :","content":""},{"title":"FIG. 3 :","content":""}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"For highly excited strongly interacting many-body systems the independent particle picture has very limited validity [1]. For high excitations, the interaction results in a quick decay of single-particle [2,3,4] as well as collective modes [2,4], which are not eigenstates of the Hamiltonian of the system. This decay leads to the formation of complicated many-body configurations. Each of these many-body states is characterized by a uniform occupation of the accessible phase space and sharing the energy between many particles of the system. The characteristic time for the formation of such spatially extended ergodic many-body states, independent of the initial conditions, is given by the inverse spreading width, τ erg =h/Γ spr [4]. Introduced by Wigner [5], Γ spr also characterizes the width of the distribution of the expansion coefficients of the many-body eigenstates over a noninteracting mean-field basis. For sufficiently high excitation energy the decay of single-particle modes creates a linear superposition of a very large number of manybody configurations. The question is whether the phase relations between these individually ergodic many-body states in the superposition may still preserve a memory of the way the system was excited. This question is of fundamental importance for the study of relaxation phenomena in nuclear, atomic, molecular and mesoscopic many-body systems, and for many-qubit quantum computation. In particular, if phase relaxation is longer than energy relaxationh/Γ spr , this can extend the time for quantum computing [6,7] beyond the quantum-chaos border [8,9].\\nTo answer this question from first principles is difficult due to computational limitations. Indeed, in order to solve the full quantum many-body problem one would require a many-qubit quantum computer. Therefore, the only currently available resort to search for possible manifestations of long phase relaxation is the experiment, and a careful data analysis. Nuclear systems are an ideal laboratory to study many-body systems, since nuclear interactions are so strong that external perturbations can often be neglected. In particular, the analysis of Refs. [6,7,10] of the data on asymmetry around 90 • c.m. in angular distributions of evaporated protons from heavy nuclei in nucleon-induced and photonuclear reactions clearly indicates that phase relaxation time in highly excited strongly interacting many-body systems can be up to eight orders of magnitude [7] longer than energy relaxation timeh/Γ spr . This reveals a new form of matter-thermalized non-equilibrated matter introduced by one of us in Refs. [11,12,13].\\nA more subtle indication of a slow phase relaxation is found in light heavy-ion scattering [14,15,16,17]. It manifests itself in the formation of rotating wave packets whose spreading time, which is given by the inverse phase-relaxation width, is much longer than the energy relaxation timeh/Γ spr . The approach [15,16,17] treats time-delayed light heavy-ion scattering in terms of the formation and decay of quasimolecular resonances [18]. The highly excited coherently rotating intermediate system has the energy of the intrinsic excitation ≥ 10 MeV. The intermediate system is in the regime of strongly overlapping resonances. Therefore, this coherent rotation does not originate from the discreteness of the spectrum, which is not resolved, but is due to the spin off-diagonal correlations between partial width amplitudes. Indeed, the period of the coherent rotation is much shorter than the inverse level spacing of the intermediate system. This reveals a new root, as compared to Bohr's correspondence principle, for a quantum-classical transition in highly excited many-body systems.\\nThe width of the rotating wave packets is about 1/d + βt/h ≈ 1/d + βθ/hω, where d is the effective number of partial waves, β is the spin off-diagonal phaserelaxation width, ω is the angular velocity, and t and θ are the time and angle of the rotation, respectively. This seemingly allows us to determine the wave-packet spreading rate from the time power spectra at differ-ent scattering angles, which can be reconstructed, for binary collisions, from the data on energy fluctuations of the cross sections [16,17]. These fluctuations originate from the energy-fluctuating collision amplitude corresponding to a resonance time-delayed process. However, for elastic heavy-ion scattering, such energy fluctuations can be typically observed only at backward angles 140 • ≤ θ ≤ 180 • , since for smaller θ the direct reaction contribution grows exponentially and becomes much greater than the time delayed one. Therefore, since at the initial moment of time the deformed intermediate system is oriented in the forward direction θ ≈ 0 • , the spreading of the wave packets within the backward angular interval 140 • ≤ θ ≤ 180 • can hardly be detected reliably. Moreover, the width of the wave packets at this backwardangle range does not allow to determine β, since the angular dispersion 1/d at t = 0 is also unknown. In this paper we show that both d and β can indeed be determined unambiguously. This results from the strong sensitivity of the interference fringes between the rotating clockwise and anticlockwise wave packets for the backward angle range to the the phase relaxation width: As β grows, the interference fringes are suppressed more strongly. The effect is considered in relation to heavy-ion scattering and bimolecular chemical reactions. The difference between the effects of phase relaxation and decoherence is discussed.\"\n },\n {\n \"title\": \"II. THE TIME POWER SPECTRUM OF THE TIME-DELAYED COLLISIONS\",\n \"content\": \"Following Ref. [19], we consider spinless collision partners in the entrance and exit channels. The time and angle dependent intensity of the decay (the time power spectrum), P (t, θ), is given by the modulus square of the Fourier component of the energy-fluctuating collision amplitude. It can be also expressed as the Fourier component of the amplitude energy autocorrelation function. P (t, θ) has been obtained in [19] by summing over a very large number of strongly overlapping resonance levels, Γ/D ≫ 1, where Γ is the total decay width and D is the average level spacing of the intermediate complex. As a result, after changing from summation over the resonance levels to integration, which is a good approximation for t ≪h/D, P (t, θ) takes the form:\\nHere, H(t) is the Heaviside step function, β is the spin phase-relaxation width, ω is the angular velocity of the coherent rotation, Φ is the deflection given by the total spin J (inh units) derivative of the potential phase shifts (direct reaction), and P J (θ) are the Legendre polynomials. The physical meaning of the inverse spin phase-relaxation width,h/β, is the characteristic time for the angular spreading of the clockwise and anticlockwise rotating wave packets. The partial average reaction probability is taken in the J-window form,\\n, whereJ is the average spin and d is the J-window width.\\nIt should be noted that a similar effect of coherent rotation of the nuclear molecule, as described by Eq. (1) with β = 0, was found in Ref. [20] for a scattering of heavy nuclei. However, while Eq. (1) has been obtained for a large number of open channels and strongly overlapping resonances of the intermediate system [19], the time power spectrum in Ref. [20] was derived from one open channel and one resonance pole form of the S-matrix elements. Therefore the latter approach is not applicable for the forthcoming analysis of the 12 C+ 24 Mg system since, for this system, the number of open channels is much greater than one. Furthermore, the one-resonance pole form of the S-matrix elements employed in Ref. [20] results in iso-lated resonances with the energy spacing between them to be abouthω. This is not the case for the 12 C+ 24 Mg elastic scattering analyzed in this paper. Indeed, the data on the excitation function for the 12 C+ 24 Mg elastic scattering at θ = 180 • reveal about 15 local maxima on the energy interval ∆E cm = 13 − 22 MeV [21]. Then the interpretation [20] would mean that average level spacing, ≃ 0.6 MeV in our case, between the local maxima is given byhω. This, in turn, would reflect the quasiperiodic behaviour of the cross section energy autocorrelation functions with the period of about 0.6 MeV. This is inconsistent with the data [21] showing the quasiperiodicity of the cross section with the period of about 2.9 MeV yieldinghω = 1.35 − 1.45 MeV [16,17]. Moreover, the interpretation [20] implies that these local maxima in the excitation function should be approximately equidistant. Instead, the data [21,22] show big fluctuations, from 0.3 MeV to 1.3, for the energy spacing between the nearest-neighboring local maxima in the excitation function. Note that the indication against the interpretation in terms of isolated resonances comes from the statistically insignificant channel-channel correlations for the 12 C+ 24 Mg elastic and inelastic scattering at θ = 180 • [16]. Finally, the effect of coherent rotation [20] does not survive a generalization [23] by taking On the basis of the above arguments we believe that the interpretation of the fine structure in the excitation functions for the 12 C+ 24 Mg scattering at θ = 180 • in terms of overlapping, rather than isolated, resonances of the intermediate system is a justified approach.\\nFirst, we calculate P (t, θ) for the set of parameters obtained from the description [17] of the experimental cross section energy autocorrelation functions [21] for 12 C+ 24 Mg elastic and inelastic scattering at θ = 180 • [22]. For these collisions the analysis of the oscillations in the cross section energy autocorrelation functions [16,17] In Figs. 1 we plot the quantity AP (t, θ)/ σ(E, θ) at four moments of time, with T = 2π/ω being the period of one complete revolution. Here, σ(E, θ) ∝ ∞ 0 dt P (t, θ) is the energy-averaged differential cross section for the time-delayed collision. In Figs. 1, P (t, θ) is scaled with σ(E, θ) for the reasons discussed in Ref. [19]. The constant A is determined by the condition AP (t = 0, θ = 0)/ σ(E, θ = 0) = 1. Figure 1(a) shows the two slightly overlapping wave packets rotating towards each other in the backward direction. In panels (b) and (c), the wave packets strongly overlap around θ = 180 • , producing interference fringes. Finally, in panel (d) the wave packets have passed each other and move apart rotating in the forward direction.\\nIn Figs. 1 we also plot AP (t, θ)/ σ(E, θ) with the same constant A and the set of parameters as before, except for d and β. We consider two other cases: d = 4, β = 0.04 MeV, and d = 8, β = 0.075 MeV. One can see that when the wave packets overlap only slightly (panels (a) and (d)) one can hardly distinguish between the three combinations of d and β, even though β changes by almost one order of magnitude. On the other hand, from panels (b) and (c), we observe that a fringe contrast due to the interference of the wave packets is very sensitive to absolute values of β and d. This result is further illustrated in Fig. 2 where we plot P (t, θ) calculated for the three sets of β and d values, as a function of t for θ = 170.6 • . One observes that the smaller β is the deeper minima are, due to the destructive interference between the wave packets at t = T /2. However, for θ = 180 • , P (t, θ) in Fig. 2 is insensitive to the three sets of β and d.\\nNote that in order to reproduce the width of the wave packets in Figs. 1 and 2 for the maximal possible d = I value, one has to take β = 0.1 MeV. Therefore, even the largest possible β value is more than one order of magnitude smaller than Γ spr . The latter is usually estimated from the width of giant resonances [2,4]: Γ spr ≃ 5 MeV. This indicates that energy relaxation, i.e., the process of formation of ergodic many-body configurations, in highly excited nuclear systems can be much faster than phase relaxation.\"\n },\n {\n \"title\": \"III. DETERMINATION OF THE TIME POWER SPECTRUM FROM THE DATA ON THE EXCITATION FUNCTION OSCILLATIONS\",\n \"content\": \"We consider a collision of the spinless reaction partners in the entrance and exit channels. The cross section of the collision is given by σ(E, θ) = |f (E, θ)| 2 . Here E is the energy in the entrance channel and θ is the scattering angle. The reaction amplitude f (E, θ) is a linear combination of S-matrix elements S J (E) corresponding to the different total spin values J. Suppose that, for a fixed energy E 1 , a measurement of detailed angular distribution is performed. Then, using, e.g., the method [24], one can find S J (E 1 ) and, therefore, f (E 1 , θ). Measuring the angular distributions for different energies on the energy interval I = E min − E max with the energy step ∆E and energy resolution smaller or about the minimal characteristic energy scale of a variation of f (E, θ) one obtains a detailed energy and angle dependencies of the collision amplitude. Its Fourier component,\\nis the instantaneous decay amplitude received at a de- (2) may be changed to the integration if we use a linear interpolation for f (E, θ) in between the consequent energies E n and E n+1 . This is justified provided the energy step ∆E is smaller or about the minimal characteristic energy scale of a variation of the cross section. For example, in Ref. [22] the energy step is ∆E cm = 133 keV while the minimal characteristic scale of the variation of σ(E, θ = 180 • ) is Γ = 0.3 MeV [16,17]. Having reconstructed P(t, θ) from the excitation function data one obtains the time power spectra P (t, θ) = |P(t, θ)| 2 for any angle. The later quantity is equivalent to that directly measured using real-time methods of femtochemistry, with the energy bands of I for the pump and probe laser pulses, to monitor chemical reactions dynamics [25,26]. Due to its dependence on θ, P (t, θ) determined by the proposed method could provide even more detailed information on the collision dynamics, as compared with the femtochemistry experiments, since latter measurements often allow to obtain P (t, θ) averaged over θ.\\nFor the energy interval I being long enough to provide a sufficient time resolutionh/I to resolve interference fringes [27] the proposed experiment would allow to test our theoretical predictions. If these were confirmed by the data, then the experimentally determined P (t, θ) can be compared with our calculations presented in Figs. 1 and 2 with the purpose of evaluation β and d.\\nIn particular, the energy interval of 10.5 MeV, over which excitation function for the 12 C+ 24 Mg elastic scattering at θ = 180 • was measured [22], should be sufficient to resolve the interference fringes in P (t, θ) [27].\\nIt should be noted that the method proposed above allows to reconstruct P (t, θ) which includes both potential scattering, i.e. fast processes, and the time delayed mechanism. Since the previous analysis [16,17] demonstrated a presence of a strong potential scattering component we expect that this method will yield a strong potential scattering contribution in P (t, θ). However this contribution is expected to be restricted to relatively short times t ps ≤h/∆E ps , where ∆E ps is a characteristic energy interval for the variation of the potential scattering amplitude. For ∆E ps ≃ 2 − 3 MeV [18], t ps ≃ 3 − 2 × 10 −22 sec. Therefore, this potential scattering contribution is expected to be restricted to t/T ≤ 0.1 in Fig. 2 and should not show up at all in Fig. 1.\\nThe above method to obtain P (t, θ) from the data is model independent and, therefore, most reliable one. However it is very demanding experimentally since σ(E, θ) must be measured for a very large number of angles [24]. There is another way to extract P (t, θ) from the data even if the excitation function is measured for a single angle. This method can be applied if the direct reaction process provides a major contribution to the cross section [17]. We consider again a case of spinless collision partners in the entrance and exit channels. The main idea is based on the decomposition of the col-lision amplitude, f (E, θ) = f dir (E, θ) + δf (E, θ). Here f dir (E, θ) is the energy smooth amplitude corresponding to potential scattering or direct fast process, and δf (E, θ) is the energy fluctuating amplitude, δf (E, θ) = 0, corresponding to the time-delayed collision. Consider first an idealized case of energy independent amplitude f dir (E, θ) when it coincides with the energy averaged amplitude f (E, θ) . The collision cross section has the form σ(E, θ) = |f (E, θ)\\nHere the potential scattering or direct reaction cross section σ dir (θ) = |f dir (E, θ)| 2 is energy-independent and σ f l (E, θ) = |δf (E, θ)| 2 . For a relative contribution of the potential scattering or direct reaction cross section to be about 70% or more of the total cross section, ∆σ(E, θ) = σ(E, θ) − σ(E, θ) ≃ 2Re[δf (E, θ)f dir (E, θ) * ], where σ(E, θ) = σ dir (θ) + σ f l (E, θ) is the energy averaged cross section. This means that rapid energy variations of the cross section originate mostly from interference between the energy fluctuating δf (E, θ) and the energy independent amplitude f dir (E, θ). Suppose that the excitation function is measured on energy interval I = E max − E min with energy step ∆E so that the total number of steps is N + 1, where N = I/∆E. Then, for t > 0, we have [17] Emax Emin\\nwith P (t, θ) = |P(t, θ)| 2 . In Eq. (3) we have employed a causality condition, P(t < 0, θ) = 0. This condition implies that the molecule cannot decay before it is formed at t = 0. Again the above information on P (t, θ) is equivalent to that obtained directly using real-time methods of femtochemistry to monitor unimolecular chemical reactions, with the energy bands of I = (E max − E min ) for the pump and probe laser pulses [25]. The above consideration can be extended to a case when the potential scattering or direct reaction amplitude f dir (E, θ) depends on energy but this dependence is considerably smoother than the energy dependence of the amplitude δf (E, θ) for the time-delayed collision. We shall again assume that a relative contribution of direct processes into the energy averaged cross section is about 70% or more. We define the characteristic energy interval I d of a variation of f dir (E, θ) with I d ≫ Γ, where Γ is the characteristic energy interval of variation of δf (E, θ). The energy interval I d can be evaluated using, e.g., the trend reduction method [28]. In particular, for the 12 C+ 24 Mg scat-tering [22] analyzed in Sec. II, this interval was evaluated to be about 4 − 5 MeV [21], i.e. much greater than Γ = 0.3 MeV. Then one can find the best polynomial fit of the energy dependence of the cross section\\nwith the order of polynomial being [I/I d ]+1, where [I/I d ] is integer part of I/I d The resulting energy smooth cross section is denoted asσ(E, θ) ≃ σ d (E, θ) + σ f l (E, θ) with σ f l (E, θ) being energy averaged, i.e. energy independent, cross section of the time-delayed processes. It can be found using the standard procedure employed for the analysis of Ericson fluctuations (see, e.g., Refs. [21,22]). As a result, one finds energy smooth direct reaction cross section σ d (E, θ). Consider σ(E, θ) − σ(E, θ) ≃ 2Re[δf (E, θ)f dir (E, θ) * ]. Assume first that energy dependence of the phase of f dir (E, θ) is negligible, arg[f dir (E, θ)] = constant, and the smooth energy dependence of σ d (E, θ) is due to the energy dependence of |f dir (E, θ)| = σ d (E, θ) 1/2 . Then applying the same arguments as those used to obtain Eq. (3), for t > 0, we have\\nwith P (t, θ) = |P(t, θ)| 2 . In order to generalize Eq. (4) by taking into account energy dependence of arg[f dir (E, θ)] ≡ φ(E, θ) (f dir (E, θ) = |f dir (E, θ)| exp[iφ(E, θ)]) we use the linear approximation φ(E, θ) = φ(Ē, θ) + (E − E)dφ(E, θ)/dE| E=Ē , whereĒ = (E max − E min )/2 and dφ(E, θ)/dE| E=Ē = t dir /h with t dir ≪h/Γ being the time delay due to the potential scattering or direct re-actions [29]. Then, it is easy to see, that, for t > t dir , the r.h.s. of Eq. (4) is changed to P(t − t dir , θ). This manifestly demonstrates that the interference between time-delayed and fast direct processes in the cross section is a precondition for monitoring the time evolution, the fast process switches on the clock at the initial moment of time t d playing the role of the pump pulse. In the absence of direct processes the initial moment of time is not defined.\\nNote that P (t, θ) can also be expressed as a cosine half-Fourier transform of the cross-section energy autocorrelation function C(ε, θ), provided the relative contribution of potential scattering is larger than 70% [16,17]. In Fig. 3 we plot C(ε, θ)/C(ε = 0, θ) = [ Fig. 3 we observe that the sensitivity of P (t, θ) to different values of β and d is considerably stronger than the sensitivity of C(ε, θ) to these values. Also P (t, θ) obtained using Eqs. (3) and (4) does not acquire statistical errors due to the finite energy interval I. Indeed such a method provides information on P (t, θ) equivalent to that obtained in the femtochemistry experiments. Therefore the only uncertainty of this method is a finite time resolution, ∆t ≃h/I, which is the uncertainty in the femtochemistry experiments with the finite energy band I of the pump and probe laser pulses [25,26]. On the other hand, a reconstruction of P (t, θ) from C(ε, θ) may result in additional errors due to possible statistical uncertainties in C(ε, θ) related to the finite energy range I [30]. Such additional statistical uncertainties may not be excluded even though the effects studied in this paper can clearly not be associated with Ericson fluctuations. Indeed, unlike, e.g., oscillating structures in Fig. 3, Ericson theory predicts angle independent Lorentzian shapes for the C(ε, θ). Therefore, for the reasons described above, we suggest that a reconstruction of P (t, θ) directly from the excitation functions (Eqs. (3) and (4)) is more reliable than from C(ε, θ).\"\n },\n {\n \"title\": \"IV. FROM HEAVY-ION COLLISIONS TO BIMOLECULAR CHEMICAL REACTIONS\",\n \"content\": \"An important question is whether the rotating wave packets and their interference can occur in colliding systems for which more rigorous approaches can be implemented. The encouraging answer has been given by the calculations of the time evolution of the H + D 2 → HD + D bimolecular chemical reaction [31,32,33]. The evidence for rotational wave packets has been demonstrated (see, e.g., Fig. 3 in [31]) due to the interference of the overlapping resonances [34] of the quasibound complex with J = 15 − 20 total spin values, (i.e., within our approach, d ≃ 5). In this peripheral time-delayed collision the rotational wave packets are stable showing a moderate spreading during half a period of the rotation. Within our description, this reveals a small β value in the time-delayed peripheral reaction mechanism, i.e. πβ/hω ≪ 1/d. Then, in accordance with our prediction, the coherent superposition of the rotating wave packets should produce a strong interference contrast. This is indeed the case as clearly seen in Fig. 3 of Ref. [31]. Similar effects of the stable rotating wave packets and their coherent superpositions with strong interference contrast have also been found in the calculations for F + HD → HF + D bimolecular reaction [35,36].\\nThe above results should allow to test our suggestion for the determination of the phase relaxation time and, in general, for the time evolution of the collision in bimolecular chemical reactions. For example, in [32] a total time dependent amplitude was calculated for the H + D 2 → HD + D reaction. Then it was splitted into short-time direct reaction component and the timedelayed one. Taking the half-Fourier transform, explicit energy and angle dependencies of both the direct and time-delayed amplitudes were obtained. It was found that in the region of backward angles θ ≤ 70 • and for the c.m. energy interval 1.6-2 eV the direct reaction cross section is considerably bigger than the time-delayed one. Then one can apply our method (Sect. III) to obtain P (t, θ) of the time-delayed reaction mechanism from the energy dependence of the double-differential cross sections. This extracted P (t, θ) can be compared with its actual exact form since the latter is known and was used to generate the energy dependence of the doubledifferential cross sections.\\nThe calculations [32,33] suggests that in the forward angle range, where the Schrödinger cat states originated from the interference of the rotating towards each other wave packets are observed, contribution of the direct reactions is negligible. Therefore, for this particular H + D 2 → HD + D reaction for the c.m. energy less than 2 eV our method can not be applied. Yet one can still check the sensitivity of the method by artificially adding an energy smooth background amplitude to the calculated one for the time-delayed reaction mechanism. The energy smooth amplitude would play a role of the direct reaction amplitude. This would enable one to directly test the accuracy of the determination of the interference fringe contrast from the data on the double differential cross sections generated with the artificially added energy smooth direct reaction amplitude. Such a test may be important since for the higher energy and/or for some other bimolecular chemical reactions, a strong contribution of direct reactions is possible for the same angle range where the Schrödinger cat states show up.\\nNote that pure energy resolution measurements of the cross sections of bimolecular chemical reactions, in particular H + D 2 → D + HD [37] and F + HD → HF + D [38] reactions, have recently become possible.\\nOur results suggest that this should enable experimentalists to extract information on the time dependence, in particular on the rotational coherent dynamics and possible wave packet interference, of the bimolecular chemical reactions which previously could only be studied by the pump probe laser pulses technique to monitor unimolecular chemical reactions [25,26].\"\n },\n {\n \"title\": \"V. PHASE RELAXATION VERSUS DECOHERENCE\",\n \"content\": \"Since phase relaxation results in washing out the interference fringes and eventually, for a sufficiently large β, can destroy the Schrödinger cat states of Figs. 1 and 2, this effect appears to be quite similar to decoherence [39]. Yet, there is an essential difference between the two effects. It is instructive to compare the extreme case of very fast phase relaxation (very large β) with very fast decoherence. Very fast phase relaxation corresponds to the regime of random matrix theory, i.e., to random phase relations between all partial width amplitudes corresponding to different strongly overlapping resonance states [4]. This yields angle independent exponential form for the time power spectrum P (t, θ) and unity for the normalized variance of the fluctuating cross section. Analogous to the fast phase relaxation would be quick decoherence between all many-body quasi-stationary resonance states, similar to the decoherence of Fock states due to the coupling of the system to a phase reservoir [39]. Such a decoherence would destroy interference terms between all the many-body eigenstates, leading to the vanishing of interference between different partial width amplitudes. It can be shown that in this case the time power spectrum would have the same angle independent exponential form as for fast phase relaxation. However, in contrast to fast relaxation, quick decoherence would result in washing out the cross section energy fluctuations, reducing its normalized variance to D/Γ ≪ 1 [40]. It should be noted that the suppression of electron transmission intensity fluctuations through nanostructures due to dephasing, an effect similar to decoherence, can be described on the basis of different arguments allowing to evaluate dephasing rates from the amplitude of conductance fluctuations [41].\"\n },\n {\n \"title\": \"VI. DISCUSSION AND CONCLUSIONS\",\n \"content\": \"Our approach is based on a new idea of slow phase relaxation between partial width amplitudes in a regime of strongly overlapping resonances. Contrary to the conventional idea of random phase relations in this regime [4,42,44], our analysis invokes spin off-diagonal phase correlations and their slow decay for the resonance timedelayed collisions. Though these effects may occur for atomic cluster collisions [19] and are already revealed for the bimolecular chemical reactions [31,32,33,35,36], the present consideration is essentially motivated by the pure energy resolution data on fluctuations in 12 C+ 24 Mg elastic and inelastic scattering at θ = 180 • [21,22], demonstrating both the fine energy structure of ≃ 0.3 MeV and quasiperiodicity with period ≃ 3 MeV [16,17]. It is these peculiar features that led us to the interpretation of the 12 C+ 24 Mg scattering in terms of highly excited quasimolecular states with strongly overlapping resonances, stable rotational wave packets and strong sensitivity of their coherent superpositions to the phase relaxation width. Note that quasiperiodic structures in the excitation functions are present also for other heavy-ion colliding systems [15]. We have also described thoroughly how to extract the time power spectrum from the data on the excitation function oscillations.\\nIt should be mentioned that a conceptually different optical model description of the 12 C+ 24 Mg elastic scattering at backward angles was presented in Ref. [43], where the quasimolecular structures were ruled out. This conclusion was based on a qualitative description of angular oscillations at backward angles for only three energies within the same energy range which is the subject of our study in this paper. Unfortunately, that analysis is inconsistent with the presence of fine energy structure of ≃ 0.3 MeV at θ = 180 • [21], which unambiguously reveals the presence of time-delayed processes at backward angles. On the contrary, the optical model description relies on the energy average S-matrix and therefore is expected to reproduce broad ≃ 2 − 3 MeV energy structures in the cross section. From our point of view, the fact that the presence of the fine energy structures was not taken into account led to an incorrect qualitative and then quantitative identification of the physical picture of the collision process in Ref. [43].\\nIn conclusion, we have demonstrated a strong sensitivity of Schrödinger cat states to the phase-relaxation width in complex quantum collisions. This should permit to determine the phase relaxation time from measurements of the excitation functions for 12 C+ 24 Mg elastic scattering at backward angles. Such an experiment would be desirable since the Schrödinger cat states predicted in [19] for 12 C+ 24 Mg scattering involve ∼ 10 3 −10 4 many-body configurations of the highly excited intermediate complex. To the best of our knowledge, the internal interactive complexity of these quantum macroscopic superpositions dramatically exceeds [19] all those previously experimentally realized. The proposed method can also be applied for determination of the phase relaxation time from the data on excitation functions for bimolecular chemical reactions.\"\n },\n {\n \"title\": \"FIG. 1 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 2 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 3 :\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"(Color online) Angular dependence of the decay intensity of the highly excited intermediate complex at different moments of time for different combinations of β and d. Panel (a) corresponds to time t = 3T /8; (b) t = 7T /16; (c) t = T /2; (d) t = 5T /8. Here, T is the period of one complete revolution of the complex. Solid lines (black) correspond to β = 0.01 MeV and d = 3; dashed (red) to β = 0.04 MeV and d = 4; dashed-dotted (blue) to β = 0.075 MeV and d = 8. The bigger β the smaller the interference fringe contrast is.","content":"FIG. 1 :\n1(Color online) Angular dependence of the decay intensity of the highly excited intermediate complex at different moments of time for different combinations of β and d. Panel (a) corresponds to time t = 3T /8; (b) t = 7T /16; (c) t = T /2; (d) t = 5T /8. Here, T is the period of one complete revolution of the complex. Solid lines (black) correspond to β = 0.01 MeV and d = 3; dashed (red) to β = 0.04 MeV and d = 4; dashed-dotted (blue) to β = 0.075 MeV and d = 8. The bigger β the smaller the interference fringe contrast is."},{"id":"","caption":"into account intermediate structure by means of inclusion of many isolated resonances in the S-matrix elements.","content":"into account intermediate structure by means of inclusion of many isolated resonances in the S-matrix elements."},{"id":"","caption":"indicates the formation of stable rotational wave packets, in spite of the strong overlap of resonance levels in the highly excited intermediate molecule. The set of parameters is [17]: Φ = 0, d = 3,J = 14, β = 0.01 MeV,hω =1.45 MeV, and Γ = 0.3 MeV.","content":"indicates the formation of stable rotational wave packets, in spite of the strong overlap of resonance levels in the highly excited intermediate molecule. The set of parameters is [17]: Φ = 0, d = 3,J = 14, β = 0.01 MeV,hω =1.45 MeV, and Γ = 0.3 MeV."},{"id":"","caption":"(Color online) Time dependence of the decay intensity of the highly excited intermediate complex at (a) θ = 180 • and (b) θ = 170.6 • , for the different sets of β and d. Solid lines (black) correspond to β = 0.01 MeV and d = 3; dashed (red) to β = 0.04 MeV and d = 4; dashed-dotted (blue) to β = 0.075 MeV and d = 8. The smaller β the deeper the minimum is, due to the interference between the wave packets for θ = 170.6 • at t = T /2.","content":"FIG. 2 :\n2(Color online) Time dependence of the decay intensity of the highly excited intermediate complex at (a) θ = 180 • and (b) θ = 170.6 • , for the different sets of β and d. Solid lines (black) correspond to β = 0.01 MeV and d = 3; dashed (red) to β = 0.04 MeV and d = 4; dashed-dotted (blue) to β = 0.075 MeV and d = 8. The smaller β the deeper the minimum is, due to the interference between the wave packets for θ = 170.6 • at t = T /2."},{"id":"","caption":"tector at time t with the time uncertainty of abouth/I, for a collision initiated at t = 0. In the sum in Eq. (2), N = I/∆E and (N + 1) is the number of equidistant energy values for which the detailed angular distributions should be measured. The summation in the r.h.s. of Eq.","content":"tector at time t with the time uncertainty of abouth/I, for a collision initiated at t = 0. In the sum in Eq. (2), N = I/∆E and (N + 1) is the number of equidistant energy values for which the detailed angular distributions should be measured. The summation in the r.h.s. of Eq."},{"id":"","caption":"cos(εt/h)P (t, θ)]/ σ(E, θ) for (a) θ = 180 • and (b) θ = 170.6 • , for the three combinations of β and d. Since the P (t, θ = 180 • ) are close for the different sets of β and d, the corresponding C(ε, θ = 180 • ) can hardly be distinguished reliably. This is the reason that the analysis at θ = 180 • [16, 17] is not sufficient to determine unambiguously the values of β and d. However, C(ε, θ = 170.6 • ) is more sensitive to the different sets of β and d. In particular, for ε ≥ 4 − 5 MeV, the oscillations in C(ε, θ = 170.6 • ) for β = 0.01 MeV, d = 3 and β = 0.075 MeV, d = 8 are out of phase with the absolute value of their difference being up to 0.4. Still, from the comparison of Figs. 1(c), 2(b) and","content":"cos(εt/h)P (t, θ)]/ σ(E, θ) for (a) θ = 180 • and (b) θ = 170.6 • , for the three combinations of β and d. Since the P (t, θ = 180 • ) are close for the different sets of β and d, the corresponding C(ε, θ = 180 • ) can hardly be distinguished reliably. This is the reason that the analysis at θ = 180 • [16, 17] is not sufficient to determine unambiguously the values of β and d. However, C(ε, θ = 170.6 • ) is more sensitive to the different sets of β and d. In particular, for ε ≥ 4 − 5 MeV, the oscillations in C(ε, θ = 170.6 • ) for β = 0.01 MeV, d = 3 and β = 0.075 MeV, d = 8 are out of phase with the absolute value of their difference being up to 0.4. Still, from the comparison of Figs. 1(c), 2(b) and"},{"id":"","caption":"(Color online) Directly measurable cross section energy autocorrelation functions calculated at (a) θ = 180 • and (b) θ = 170.6 • for the different sets of β and d. Solid lines (black) corresponds to β = 0.01 MeV and d = 3; dashed (red) to β = 0.04 MeV and d = 4; dashed-dotted (blue) to β = 0.075 MeV and d = 8.","content":"FIG. 3 :\n3(Color online) Directly measurable cross section energy autocorrelation functions calculated at (a) θ = 180 • and (b) θ = 170.6 • for the different sets of β and d. Solid lines (black) corresponds to β = 0.01 MeV and d = 3; dashed (red) to β = 0.04 MeV and d = 4; dashed-dotted (blue) to β = 0.075 MeV and d = 8."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"(Color online) Angular dependence of the decay intensity of the highly excited intermediate complex at different moments of time for different combinations of β and d. Panel (a) corresponds to time t = 3T /8; (b) t = 7T /16; (c) t = T /2; (d) t = 5T /8. Here, T is the period of one complete revolution of the complex. Solid lines (black) correspond to β = 0.01 MeV and d = 3; dashed (red) to β = 0.04 MeV and d = 4; dashed-dotted (blue) to β = 0.075 MeV and d = 8. The bigger β the smaller the interference fringe contrast is.\",\n \"content\": \"FIG. 1 :\\n1(Color online) Angular dependence of the decay intensity of the highly excited intermediate complex at different moments of time for different combinations of β and d. Panel (a) corresponds to time t = 3T /8; (b) t = 7T /16; (c) t = T /2; (d) t = 5T /8. Here, T is the period of one complete revolution of the complex. Solid lines (black) correspond to β = 0.01 MeV and d = 3; dashed (red) to β = 0.04 MeV and d = 4; dashed-dotted (blue) to β = 0.075 MeV and d = 8. The bigger β the smaller the interference fringe contrast is.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"into account intermediate structure by means of inclusion of many isolated resonances in the S-matrix elements.\",\n \"content\": \"into account intermediate structure by means of inclusion of many isolated resonances in the S-matrix elements.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"indicates the formation of stable rotational wave packets, in spite of the strong overlap of resonance levels in the highly excited intermediate molecule. The set of parameters is [17]: Φ = 0, d = 3,J = 14, β = 0.01 MeV,hω =1.45 MeV, and Γ = 0.3 MeV.\",\n \"content\": \"indicates the formation of stable rotational wave packets, in spite of the strong overlap of resonance levels in the highly excited intermediate molecule. The set of parameters is [17]: Φ = 0, d = 3,J = 14, β = 0.01 MeV,hω =1.45 MeV, and Γ = 0.3 MeV.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(Color online) Time dependence of the decay intensity of the highly excited intermediate complex at (a) θ = 180 • and (b) θ = 170.6 • , for the different sets of β and d. Solid lines (black) correspond to β = 0.01 MeV and d = 3; dashed (red) to β = 0.04 MeV and d = 4; dashed-dotted (blue) to β = 0.075 MeV and d = 8. The smaller β the deeper the minimum is, due to the interference between the wave packets for θ = 170.6 • at t = T /2.\",\n \"content\": \"FIG. 2 :\\n2(Color online) Time dependence of the decay intensity of the highly excited intermediate complex at (a) θ = 180 • and (b) θ = 170.6 • , for the different sets of β and d. Solid lines (black) correspond to β = 0.01 MeV and d = 3; dashed (red) to β = 0.04 MeV and d = 4; dashed-dotted (blue) to β = 0.075 MeV and d = 8. The smaller β the deeper the minimum is, due to the interference between the wave packets for θ = 170.6 • at t = T /2.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"tector at time t with the time uncertainty of abouth/I, for a collision initiated at t = 0. In the sum in Eq. (2), N = I/∆E and (N + 1) is the number of equidistant energy values for which the detailed angular distributions should be measured. The summation in the r.h.s. of Eq.\",\n \"content\": \"tector at time t with the time uncertainty of abouth/I, for a collision initiated at t = 0. In the sum in Eq. (2), N = I/∆E and (N + 1) is the number of equidistant energy values for which the detailed angular distributions should be measured. The summation in the r.h.s. of Eq.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"cos(εt/h)P (t, θ)]/ σ(E, θ) for (a) θ = 180 • and (b) θ = 170.6 • , for the three combinations of β and d. Since the P (t, θ = 180 • ) are close for the different sets of β and d, the corresponding C(ε, θ = 180 • ) can hardly be distinguished reliably. This is the reason that the analysis at θ = 180 • [16, 17] is not sufficient to determine unambiguously the values of β and d. However, C(ε, θ = 170.6 • ) is more sensitive to the different sets of β and d. In particular, for ε ≥ 4 − 5 MeV, the oscillations in C(ε, θ = 170.6 • ) for β = 0.01 MeV, d = 3 and β = 0.075 MeV, d = 8 are out of phase with the absolute value of their difference being up to 0.4. Still, from the comparison of Figs. 1(c), 2(b) and\",\n \"content\": \"cos(εt/h)P (t, θ)]/ σ(E, θ) for (a) θ = 180 • and (b) θ = 170.6 • , for the three combinations of β and d. Since the P (t, θ = 180 • ) are close for the different sets of β and d, the corresponding C(ε, θ = 180 • ) can hardly be distinguished reliably. This is the reason that the analysis at θ = 180 • [16, 17] is not sufficient to determine unambiguously the values of β and d. However, C(ε, θ = 170.6 • ) is more sensitive to the different sets of β and d. In particular, for ε ≥ 4 − 5 MeV, the oscillations in C(ε, θ = 170.6 • ) for β = 0.01 MeV, d = 3 and β = 0.075 MeV, d = 8 are out of phase with the absolute value of their difference being up to 0.4. Still, from the comparison of Figs. 1(c), 2(b) and\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(Color online) Directly measurable cross section energy autocorrelation functions calculated at (a) θ = 180 • and (b) θ = 170.6 • for the different sets of β and d. Solid lines (black) corresponds to β = 0.01 MeV and d = 3; dashed (red) to β = 0.04 MeV and d = 4; dashed-dotted (blue) to β = 0.075 MeV and d = 8.\",\n \"content\": \"FIG. 3 :\\n3(Color online) Directly measurable cross section energy autocorrelation functions calculated at (a) θ = 180 • and (b) θ = 170.6 • for the different sets of β and d. Solid lines (black) corresponds to β = 0.01 MeV and d = 3; dashed (red) to β = 0.04 MeV and d = 4; dashed-dotted (blue) to β = 0.075 MeV and d = 8.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"E Wigner, Statistical Properties of Nuclei. J.B. GardNew York-LondonPlenum Press11E. Wigner, in Statistical Properties of Nuclei, editor J.B. Gard (Plenum Press, New York-London, 1972), p. 11.","doi":""},{"text":". O Bohigas, H A Weidenmüller, Annu. Rev. Nucl. Part. Sci. 38421O. Bohigas and H.A. Weidenmüller, Annu. Rev. Nucl. Part. Sci. 38, 421 (1988).","doi":""},{"text":". B L Altshuler, Y Gefen, A Kamenev, L S Levitov, Phys. Rev. Lett. 782803B.L. Altshuler, Y. Gefen, A. Kamenev, and L.S. Levitov, Phys. Rev. Lett. 78, 2803 (1997).","doi":""},{"text":". T Guhr, A Müller-Groeling, H A Weidenmüller, Phys. Rep. 299189T. Guhr, A. Müller-Groeling, and H.A. Weidenmüller, Phys. Rep. 299, 189 (1998).","doi":""},{"text":". E Wigner, Ann. Math. 62203E. Wigner, Ann. Math. 62, 548 (1955); 65, 203 (1957).","doi":""},{"text":". J Flores, S Yu, T H Kun, Seligman, quant-ph/0502050Phys. Rev. E. 7217201J. Flores, S.Yu. Kun, and T.H. Seligman, Phys. Rev. E 72, 017201 (2005); quant-ph/0502050.","doi":"quant-ph/0502050"},{"text":"M Bienert, J Flores, S Yu, T H Kun, SIGMA (Symmetry, Integrability and Geometry: Methods and Applications. 227M. Bienert, J. Flores, S.Yu. Kun, and T.H. Selig- man, SIGMA (Symmetry, Integrability and Geome- try: Methods and Applications) 2, Paper 027 (2006);","doi":""},{"text":". B Georgeot, D L Shepelyansky, Phys. Rev. E. 623504B. Georgeot and D.L. Shepelyansky, Phys. Rev. E 62, 3504 (2000).","doi":""},{"text":"D L Shepelyansky, the Proceedings of Nobel Symposium on Quantum Chaos. 90112D.L. Shepelyansky, in the Proceedings of Nobel Sympo- sium on Quantum Chaos 2000, Phys. Scripta T90, 112 (2001).","doi":""},{"text":"Experimental proposal for accurate determination of the phase relaxation time and testing a formation of thermalized nonequilibrated matter in highly excited quantum manybody systems. M Bienert, J Flores, S Yu, Kun, submittedM. Bienert, J. Flores, and S.Yu. Kun, \"Experimental proposal for accurate determination of the phase relax- ation time and testing a formation of thermalized non- equilibrated matter in highly excited quantum many- body systems\", submitted.","doi":""},{"text":". S Yu, Kun, Phys. Lett. B. 31916S.Yu. Kun, Phys. Lett. B 319, 16 (1993).","doi":""},{"text":". S Yu, Kun, Z. Phys. A. 348273S.Yu. Kun, Z. Phys. A 348, 273 (1994).","doi":""},{"text":". S Yu, Kun, Z. Phys. A. 357255S.Yu. Kun, Z. Phys. A 357, 255 (1997).","doi":""},{"text":". S Yu, A V Kun, O K Vagov, Vorov, Phys. Rev. C. 59585S.Yu. Kun, A.V. Vagov, and O.K. Vorov, Phys. Rev. C 59, R585 (1999). (2001).","doi":""},{"text":". S Yu, B A Kun, A V Robson, Vagov, Phys. Rev. Lett. 83504S.Yu. Kun, B.A. Robson, and A.V. Vagov, Phys. Rev. Lett. 83, 504 (1999).","doi":""},{"text":". S Yu, A V Kun, W Vagov, Greiner, Phys. Rev. C. 6314608S.Yu. Kun, A.V. Vagov, and W. Greiner, Phys. Rev. C 63, 014608 (2001).","doi":""},{"text":". S Yu, A V Kun, L T Vagov, W Chadderton, Greiner, Int. J. Mod. Phys. E. 11273S.Yu. Kun, A.V. Vagov, L.T. Chadderton, and W. Greiner, Int. J. Mod. Phys. E 11, 273 (2002).","doi":""},{"text":"W Greiner, J Y Park, W Scheid, Nuclear Molecules. SingaporeWorld ScientificW. Greiner, J.Y. Park, and W. Scheid, Nuclear Molecules (World Scientific, Singapore, 1995).","doi":""},{"text":". S Yu, L Kun, L T Benet, W Chadderton, F Greiner, Haas, Phys. Rev. C. 6711604S.Yu. Kun, L. Benet, L.T. Chadderton, W. Greiner, and F. Haas, Phys. Rev. C 67, 011604(R) (2003).","doi":""},{"text":". U Heinz, J Reinhardt, B Müller, W Greiner, W T Pinkston, Z. Phys. A. 316341U. Heinz, J. Reinhardt, B. Müller, W. Greiner, and W.T. Pinkston, Z. Phys. A, 316, 341 (1984).","doi":""},{"text":". B Ghosh, R Singh, Pramana J. Phys. 291555B. Ghosh and R. Singh, Pramana J. Phys. 29, 1555 (1987).","doi":""},{"text":". M C Mermaz, Phys. Rev. C. 241512M.C. Mermaz et al., Phys. Rev. C 24, 1512 (1981).","doi":""},{"text":". U Heinz, U Müller, J Reinhardt, B Müller, W Greiner, Ann. Phys. (N.Y.). 158476U. Heinz, U. Müller, J. Reinhardt, B. Müller, W. Greiner, Ann. Phys. (N.Y.) 158, 476 (1984).","doi":""},{"text":". V Chiste, R Lichtenthäler, A C C Villari, L C Gomes, Phys. Rev. C. 54784V. Chiste, R. Lichtenthäler, A.C.C. Villari, and L.C. Gomes, Phys. Rev. C 54, 784 (1996).","doi":""},{"text":"A H Zewail, Femtochemistry. SingaporeWorld Scientific1A.H. Zewail, Femtochemistry (World Scientific, Singa- pore, 1995), Vols. 1 and 2.","doi":""},{"text":"Nobel Lectures. A H Zewail, Nobel Lectures. SingaporeChemistryA.H. Zewail, Nobel Lectures 1999, in Nobel Lectures, Chemistry 1996-2000, editor I. Grenthe, (World Scien- tific, Singapore 2003).","doi":""},{"text":". L Benet, S Yu, Wang Kun, V Qi, Denisov, Phys. Lett. B. 605101L. Benet, S.Yu. Kun, Wang Qi, and V. Denisov, Phys. Lett. B 605, 101 (2005).","doi":""},{"text":". G Pappalardo, Phys. Lett. B. 13320G. Pappalardo, Phys. Lett. B 13, 320 (1964).","doi":""},{"text":". E P Wigner, Phys. Rev. 98145E.P. Wigner, Phys. Rev. 98, 145 (1955).","doi":""},{"text":"A Richter, Nuclear Spectroscopy and Reactions. J. ChernyNew YorkAcademic343A. Richter, in Nuclear Spectroscopy and Reactions, edited by J. Cherny (Academic, New York, 1974), Vol. B, p. 343.","doi":""},{"text":". S C Althorpe, F Fernandez-Alonso, B D Bean, J D Ayers, A E Pomerantz, R N Zare, E Wrede, Nature. 41667S.C. Althorpe, F. Fernandez-Alonso, B.D. Bean, J.D. Ay- ers, A.E. Pomerantz, R.N. Zare, and E. Wrede, Nature (London) 416, 67 (2002).","doi":""},{"text":". S C Althorpe, J. Chem. Phys. 1174623S.C. Althorpe, J. Chem. Phys. 117, 4623 (2002).","doi":""},{"text":". S C Althorpe, J. Chem. Phys. 1211175S.C. Althorpe, J. Chem. Phys. 121, 1175 (2004).","doi":""},{"text":". S D Chao, R T Skodje, Chem. Phys. Lett. 336364S.D. Chao and R.T. Skodje, Chem. Phys. Lett. 336, 364 (2001).","doi":""},{"text":". S C Althorpe, Chem. Phys. Lett. 370443S.C. Althorpe, Chem. Phys. Lett. 370, 443 (2003).","doi":""},{"text":". S C Althorpe, J. Phys. Chem. A. 1077152S.C. Althorpe, J. Phys. Chem. A 107, 7152 (2003).","doi":""},{"text":". D Dai, C C Wang, S A Harich, X Wang, X Yang, S D Chao, R T Skodje, Science. 3001730D. Dai, C.C. Wang, S.A. Harich, X. Wang, X. Yang, S.D. Chao, and R.T. Skodje, Science 300, 1730 (2003).","doi":""},{"text":". R T Skodje, D Skouteris, D E Manolopoulos, Shih-Huang Lee, Feng Dong, Kopin Liu, Phys. Rev. Lett. 851206R.T. Skodje, D. Skouteris, D.E. Manolopoulos, Shih- Huang Lee, Feng Dong, and Kopin Liu, Phys. Rev. Lett. 85, 1206 (2000).","doi":""},{"text":". C J Myatt, Nature. 403269C.J. Myatt et al., Nature (London) 403, 269 (2000).","doi":""},{"text":". S Yu, Kun, Phys. Rev. A. 6534701S.Yu. Kun, Phys. Rev. A 65, 034701 (2002).","doi":""},{"text":". C M Marcus, R M Westervelt, P F Hopkins, A C Gossard, Phys. Rev. B. 482460C.M. Marcus, R.M. Westervelt, P.F. Hopkins, and A.C. Gossard, Phys. Rev. B 48, 2460 (1993);","doi":""},{"text":". E Mccann, I V Lerner, J. Phys.: Condens. Matter. 86719E. McCann and I.V. Lerner, J. Phys.: Condens. Matter 8, 6719 (1996);","doi":""},{"text":". P W Brouwer, C W J Beenakker, Phys. Rev. B. 554695P.W. Brouwer and C.W.J. Beenakker, Phys. Rev. B 55, 4695 (1997);","doi":""},{"text":". J P Bird, J. Phys.: Condens. Matter. 11413J.P. Bird, J. Phys.: Condens. Matter 11, R413 (1999).","doi":""},{"text":". A H Zewail, Faradey Discuss, Chem. Soc. 75315A.H. Zewail, Faradey Discuss. Chem. Soc. 75, 315 (1983).","doi":""},{"text":". A Lépine-Szily, M S Hussein, R Lichtenthäler, J Cseh, G Lévai, Phys. Rev. Lett. 823972A. Lépine-Szily, M.S. Hussein, R. Lichtenthäler, J. Cseh, and G. Lévai, Phys. Rev. Lett. 82, 3972 (1999).","doi":""},{"text":"Proceedings of the Les Houches Summer School, Session LII. U Smilansky, Chaos and Quantum Physics. M.J. Giannoni, A. Voros, and J. Zinn-JustinNorth-Holland, Amsterdam372and references thereinU. Smilansky, in Chaos and Quantum Physics, Proceed- ings of the Les Houches Summer School, Session LII, edited by M.J. Giannoni, A. Voros, and J. Zinn-Justin (North-Holland, Amsterdam, 1991), p. 372, and refer- ences therein.","doi":""}],"string":"[\n {\n \"text\": \"E Wigner, Statistical Properties of Nuclei. J.B. GardNew York-LondonPlenum Press11E. Wigner, in Statistical Properties of Nuclei, editor J.B. Gard (Plenum Press, New York-London, 1972), p. 11.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". O Bohigas, H A Weidenmüller, Annu. Rev. Nucl. Part. Sci. 38421O. Bohigas and H.A. Weidenmüller, Annu. Rev. Nucl. Part. Sci. 38, 421 (1988).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B L Altshuler, Y Gefen, A Kamenev, L S Levitov, Phys. Rev. Lett. 782803B.L. Altshuler, Y. Gefen, A. Kamenev, and L.S. Levitov, Phys. Rev. Lett. 78, 2803 (1997).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Guhr, A Müller-Groeling, H A Weidenmüller, Phys. Rep. 299189T. Guhr, A. Müller-Groeling, and H.A. Weidenmüller, Phys. Rep. 299, 189 (1998).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Wigner, Ann. Math. 62203E. Wigner, Ann. Math. 62, 548 (1955); 65, 203 (1957).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Flores, S Yu, T H Kun, Seligman, quant-ph/0502050Phys. Rev. E. 7217201J. Flores, S.Yu. Kun, and T.H. Seligman, Phys. Rev. E 72, 017201 (2005); quant-ph/0502050.\",\n \"doi\": \"quant-ph/0502050\"\n },\n {\n \"text\": \"M Bienert, J Flores, S Yu, T H Kun, SIGMA (Symmetry, Integrability and Geometry: Methods and Applications. 227M. Bienert, J. Flores, S.Yu. Kun, and T.H. Selig- man, SIGMA (Symmetry, Integrability and Geome- try: Methods and Applications) 2, Paper 027 (2006);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Georgeot, D L Shepelyansky, Phys. Rev. E. 623504B. Georgeot and D.L. Shepelyansky, Phys. Rev. E 62, 3504 (2000).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"D L Shepelyansky, the Proceedings of Nobel Symposium on Quantum Chaos. 90112D.L. Shepelyansky, in the Proceedings of Nobel Sympo- sium on Quantum Chaos 2000, Phys. Scripta T90, 112 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Experimental proposal for accurate determination of the phase relaxation time and testing a formation of thermalized nonequilibrated matter in highly excited quantum manybody systems. M Bienert, J Flores, S Yu, Kun, submittedM. Bienert, J. Flores, and S.Yu. Kun, \\\"Experimental proposal for accurate determination of the phase relax- ation time and testing a formation of thermalized non- equilibrated matter in highly excited quantum many- body systems\\\", submitted.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Yu, Kun, Phys. Lett. B. 31916S.Yu. Kun, Phys. Lett. B 319, 16 (1993).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Yu, Kun, Z. Phys. A. 348273S.Yu. Kun, Z. Phys. A 348, 273 (1994).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Yu, Kun, Z. Phys. A. 357255S.Yu. Kun, Z. Phys. A 357, 255 (1997).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Yu, A V Kun, O K Vagov, Vorov, Phys. Rev. C. 59585S.Yu. Kun, A.V. Vagov, and O.K. Vorov, Phys. Rev. C 59, R585 (1999). (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Yu, B A Kun, A V Robson, Vagov, Phys. Rev. Lett. 83504S.Yu. Kun, B.A. Robson, and A.V. Vagov, Phys. Rev. Lett. 83, 504 (1999).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Yu, A V Kun, W Vagov, Greiner, Phys. Rev. C. 6314608S.Yu. Kun, A.V. Vagov, and W. Greiner, Phys. Rev. C 63, 014608 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Yu, A V Kun, L T Vagov, W Chadderton, Greiner, Int. J. Mod. Phys. E. 11273S.Yu. Kun, A.V. Vagov, L.T. Chadderton, and W. Greiner, Int. J. Mod. Phys. E 11, 273 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"W Greiner, J Y Park, W Scheid, Nuclear Molecules. SingaporeWorld ScientificW. Greiner, J.Y. Park, and W. Scheid, Nuclear Molecules (World Scientific, Singapore, 1995).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Yu, L Kun, L T Benet, W Chadderton, F Greiner, Haas, Phys. Rev. C. 6711604S.Yu. Kun, L. Benet, L.T. Chadderton, W. Greiner, and F. Haas, Phys. Rev. C 67, 011604(R) (2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". U Heinz, J Reinhardt, B Müller, W Greiner, W T Pinkston, Z. Phys. A. 316341U. Heinz, J. Reinhardt, B. Müller, W. Greiner, and W.T. Pinkston, Z. Phys. A, 316, 341 (1984).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Ghosh, R Singh, Pramana J. Phys. 291555B. Ghosh and R. Singh, Pramana J. Phys. 29, 1555 (1987).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M C Mermaz, Phys. Rev. C. 241512M.C. Mermaz et al., Phys. Rev. C 24, 1512 (1981).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". U Heinz, U Müller, J Reinhardt, B Müller, W Greiner, Ann. Phys. (N.Y.). 158476U. Heinz, U. Müller, J. Reinhardt, B. Müller, W. Greiner, Ann. Phys. (N.Y.) 158, 476 (1984).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Chiste, R Lichtenthäler, A C C Villari, L C Gomes, Phys. Rev. C. 54784V. Chiste, R. Lichtenthäler, A.C.C. Villari, and L.C. Gomes, Phys. Rev. C 54, 784 (1996).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A H Zewail, Femtochemistry. SingaporeWorld Scientific1A.H. Zewail, Femtochemistry (World Scientific, Singa- pore, 1995), Vols. 1 and 2.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nobel Lectures. A H Zewail, Nobel Lectures. SingaporeChemistryA.H. Zewail, Nobel Lectures 1999, in Nobel Lectures, Chemistry 1996-2000, editor I. Grenthe, (World Scien- tific, Singapore 2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Benet, S Yu, Wang Kun, V Qi, Denisov, Phys. Lett. B. 605101L. Benet, S.Yu. Kun, Wang Qi, and V. Denisov, Phys. Lett. B 605, 101 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Pappalardo, Phys. Lett. B. 13320G. Pappalardo, Phys. Lett. B 13, 320 (1964).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E P Wigner, Phys. Rev. 98145E.P. Wigner, Phys. Rev. 98, 145 (1955).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Richter, Nuclear Spectroscopy and Reactions. J. ChernyNew YorkAcademic343A. Richter, in Nuclear Spectroscopy and Reactions, edited by J. Cherny (Academic, New York, 1974), Vol. B, p. 343.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S C Althorpe, F Fernandez-Alonso, B D Bean, J D Ayers, A E Pomerantz, R N Zare, E Wrede, Nature. 41667S.C. Althorpe, F. Fernandez-Alonso, B.D. Bean, J.D. Ay- ers, A.E. Pomerantz, R.N. Zare, and E. Wrede, Nature (London) 416, 67 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S C Althorpe, J. Chem. Phys. 1174623S.C. Althorpe, J. Chem. Phys. 117, 4623 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S C Althorpe, J. Chem. Phys. 1211175S.C. Althorpe, J. Chem. Phys. 121, 1175 (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S D Chao, R T Skodje, Chem. Phys. Lett. 336364S.D. Chao and R.T. Skodje, Chem. Phys. Lett. 336, 364 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S C Althorpe, Chem. Phys. Lett. 370443S.C. Althorpe, Chem. Phys. Lett. 370, 443 (2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S C Althorpe, J. Phys. Chem. A. 1077152S.C. Althorpe, J. Phys. Chem. A 107, 7152 (2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Dai, C C Wang, S A Harich, X Wang, X Yang, S D Chao, R T Skodje, Science. 3001730D. Dai, C.C. Wang, S.A. Harich, X. Wang, X. Yang, S.D. Chao, and R.T. Skodje, Science 300, 1730 (2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R T Skodje, D Skouteris, D E Manolopoulos, Shih-Huang Lee, Feng Dong, Kopin Liu, Phys. Rev. Lett. 851206R.T. Skodje, D. Skouteris, D.E. Manolopoulos, Shih- Huang Lee, Feng Dong, and Kopin Liu, Phys. Rev. Lett. 85, 1206 (2000).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C J Myatt, Nature. 403269C.J. Myatt et al., Nature (London) 403, 269 (2000).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Yu, Kun, Phys. Rev. A. 6534701S.Yu. Kun, Phys. Rev. A 65, 034701 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C M Marcus, R M Westervelt, P F Hopkins, A C Gossard, Phys. Rev. B. 482460C.M. Marcus, R.M. Westervelt, P.F. Hopkins, and A.C. Gossard, Phys. Rev. B 48, 2460 (1993);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Mccann, I V Lerner, J. Phys.: Condens. Matter. 86719E. McCann and I.V. Lerner, J. Phys.: Condens. Matter 8, 6719 (1996);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P W Brouwer, C W J Beenakker, Phys. Rev. B. 554695P.W. Brouwer and C.W.J. Beenakker, Phys. Rev. B 55, 4695 (1997);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J P Bird, J. Phys.: Condens. Matter. 11413J.P. Bird, J. Phys.: Condens. Matter 11, R413 (1999).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A H Zewail, Faradey Discuss, Chem. Soc. 75315A.H. Zewail, Faradey Discuss. Chem. Soc. 75, 315 (1983).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Lépine-Szily, M S Hussein, R Lichtenthäler, J Cseh, G Lévai, Phys. Rev. Lett. 823972A. Lépine-Szily, M.S. Hussein, R. Lichtenthäler, J. Cseh, and G. Lévai, Phys. Rev. Lett. 82, 3972 (1999).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Proceedings of the Les Houches Summer School, Session LII. U Smilansky, Chaos and Quantum Physics. M.J. Giannoni, A. Voros, and J. Zinn-JustinNorth-Holland, Amsterdam372and references thereinU. Smilansky, in Chaos and Quantum Physics, Proceed- ings of the Les Houches Summer School, Session LII, edited by M.J. Giannoni, A. Voros, and J. Zinn-Justin (North-Holland, Amsterdam, 1991), p. 372, and refer- ences therein.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":478,"cells":{"text":{"kind":"string","value":"\nSIGNATURES OF PHOTON-AXION CONVERSION IN THE THERMAL SPECTRA AND POLARIZATION OF NEUTRON STARS\n25 Jan 2012 December 21, 2013\n\nRosalba Perna \nWynn C G Ho \nLicia Verde \nMatthew Van Adelsberg \nRaul Jimenez \nSIGNATURES OF PHOTON-AXION CONVERSION IN THE THERMAL SPECTRA AND POLARIZATION OF NEUTRON STARS\n25 Jan 2012 December 21, 2013DRAFT VERSION DECEMBER 21, 2013 Preprint typeset using L A T E X style emulateapj v. 11/26/04 Draft versionSubject headings: stars: neutron -X-rays: stars -cosmology: miscellaneous\nConversion of photons into axions under the presence of a strong magnetic field can dim the radiation from magnetized astrophysical objects. Here we perform a detailed calculation aimed at quantifying the signatures of photon-axion conversion in the spectra, light curves, and polarization of neutron stars (NSs). We take into account the energy and angle-dependence of the conversion probability and the surface thermal emission from NSs. The latter is computed from magnetized atmosphere models that include the effect of photon polarization mode conversion due to vacuum polarization. The resulting spectral models, inclusive of the general-relativistic effects of gravitational redshift and light deflection, allow us to make realistic predictions for the effects of photon to axion conversion on observed NS spectra, light curves, and polarization signals. We identify unique signatures of the conversion, such as an increase of the effective area of a hot spot as it rotates away from the observer line of sight. For a star emitting from the entire surface, the conversion produces apparent radii that are either larger or smaller (depending on axion mass and coupling strength) than the limits set by NS equations of state. For an emission region that is observed phase-on, photon-axion conversion results in an inversion of the plane of polarization with respect to the no-conversion case. While the quantitative details of the features that we identify depend on NS properties (magnetic field strength, temperature) and axion parameters, the spectral and polarization signatures induced by photon-axion conversion are distinctive enough to make NSs very interesting and promising probes of axion physics.\n\nINTRODUCTION\n\nAxions with low mass (below 1 meV) are a direct prediction of the solution for the strong CP violation problem. Peccei & Quinn (1977) proposed axions as Pseudo-Goldstone bosons arising from the spontaneous breakdown of the U(1) symmetry. The Peccei-Quinn axion occupies a narrow region in the axion parameter space of mass and coupling strength. However, axions (or pseudo-scalar particles) can exist with more generic values for their mass and coupling strength, and are one type of dark matter candidate (e.g. Arvanitaki et al. 2009). This motivates efforts to constrain axion properties even if the parameters do not reach into the Peccei-Quinn regime.\n\nSeveral experimental avenues have been used to detect axions and constrain their properties. These fall into two categories: ground-based experiments and cosmologicalastrophysical efforts. Current limits on the axion mass (10 −6 < m a /eV < 10 −3 ) come from the latter approach. The cooling rate observed in supernova 1987A implies a direct axion mass upper limit: if axions are more massive, then cooling would be predominantly due to axions rather than neutrinos (Eidelman et al. 2004). The lower limit comes from cosmological considerations: if axions constitute dark matter, then they must be cold and not overclose the Universe (Preskill et al. 1983). Constraints on the coupling constant can be obtained with both astrophysical observations and ground-based experiments. The existence of horizontal branch stars with extended morphology, that is, stars with a range of surface tem-peratures, constrains the coupling constant of axion-photon interaction to be g a < 10 −10 GeV −1 (Raffelt 2008). Groundbased experiments also place similar constraints, and future experiments are anticipated that will obtain stronger limits (Andriamonje et al. 1983;De Panlis et al 1987;Wuensch et al. 1989;Asztalos et al. 2004Asztalos et al. , 2010.\n\nRecent developments in understanding radiation dimming (Lai & Heyl 2006;Jimenez et al. 2011) and modification of the polarization pattern (Gill & Heyl 2011) have put further constraints on the m a − g a plane. Limits on cosmological radiation dimming have also been used to place constraints on much lighter axion masses (Avgoustidis et al. 2010).\n\nThe potential use of neutron stars (NSs) as probes of axion parameters has been noted by a number of authors (e.g., Lai & Heyl 2006;Chelouche et al. 2009;Jimenez et al. 2011). In particular, occultation and eclipsing or transiting of a companion and dimming of the spectrum can potentially produce observable signatures. However, the phenomenon of occultation (when a background object passes through the influence of a NS magnetic field) was deemed to have too low a probability to be astrophysically relevant. In addition, there are no known binary systems involving a NS that are detached enough to yield a clean constraint (Jimenez et al. 2011). On the other hand, Lai & Heyl (2006) showed how the thermal X-ray spectrum of a NS (modeled as simple blackbody emission) can be modified in a significant way by the presence of axions. Spectral features due to axion-photon oscillations have been discussed by Chelouche et al. (2009), and shown to be relevant for highly magnetized NSs in the sub-mm wavelength range for an observationally interesting range of axion parameters.\n\nIn this paper, we focus on the soft X-ray band in which NSs are routinely observed. The goal of our work is to perform a detailed calculation of the observable effects that the presence of axions would have on NS spectra, light curves, and polar-ization. Our analysis goes beyond previous computations in the literature for the subject by coupling the calculation of photon-axion conversion probabilities with detailed models of magnetized atmospheres and general relativistic models for emission from the NS surface. For the local surface brightness of the NS, we use accurate magnetized atmosphere models, which yield the angular and energy-dependent photon intensities in the two photon polarization modes. These models include the effect of mode conversion due to the vacuum polarization. The local intensity is then imported into a code for computing spectra and light curves from finite emission regions on the NS surface. The analysis incorporates the effects of gravitational redshift and gravitational light deflection and allows for arbitrary viewing geometries. The resulting spectra are then modified by the photon-axion conversion process, accounting for both the energy and angular dependencies of the conversion probability. Note that we solve the conversion between photon modes and the conversion between axions and photons independently, which is shown by Lai & Heyl (2006) to be a very good approximation for the regimes we explore in this paper. With our calculations, we are able to produce realistic models for the spectra, light curves, and polarization signal of NSs, and use these to identify specific features that carry the telltale signs of photon-axion conversion. We discuss these signs in the context of NS observations in §5.2.\n\nOur paper is organized as follows: In §2, we discuss the formation of the NS thermal spectrum within a magnetized atmosphere, placing particular emphasis on the emergence of two photon polarization modes. In §3, we summarize the calculation of the photon-axion conversion probability for a magnetized object. The computation of the 'observed' thermal spectrum, light curves, and polarization with and without photon-axion conversion is described in §4, and the results are presented in §5. We summarize our findings in §6.\n\n\nPHOTON PROPAGATION IN THE ATMOSPHERE OF\n\n\nMAGNETIZED NEUTRON STARS\n\nIn this section, we describe the main properties of magnetized NS atmospheres and the processes that influence photon propagation in them. Axions do not directly couple to the processes described in this section. However, the details presented below are important for calculating the polarized radiation emerging from the NS atmosphere. As discussed in §3, axions couple only to photons that are linearly polarized in the plane formed by the directions of propagation and the magnetic field; the physics of photon propagation in the NS atmosphere determines the relative fraction of such photons.\n\nRadiation emerging from the hot surfaces of magnetized NSs is the result of radiative transfer through a layer of ionized plasma. Due to the strong gravitational field of the NS, heavy elements settle quickly (Alcock & Illarionov 1980;Brown et al. 2002), and this atmospheric layer is composed of light elements, unless nuclear burning on the NS surface consumes the light elements (Chang et al. 2010, and references therein). Only recently have self-consistent magnetic atmosphere models for partially ionized hydrogen and mid-Z elements been constructed (see Mori & Ho 2007;Ho et al. 2008, and references therein for details; see also Suleimanov et al. 2009). For simplicity, we focus our calculations on fully ionized hydrogen atmospheres (see below). In this case, the scale height is\nH ρ = 2kT s m p g cosδ = 1.65 cos δ T s 10 6 K g * 10 14 cm s −1 −1 cm, (1)\nwhere T s is the effective temperature of the star, g * is the gravitational acceleration, and δ is the angle that the ray makes with the surface normal.\n\nIn a NS atmosphere, radiation propagates in two distinct polarization modes: the extraordinary (X) mode and ordinary (O) mode, which are very nearly linearly polarized perpendicular and parallel, respectively, to the plane formed by the directions of propagation and the magnetic field. The X mode absorption opacity is reduced by a factor of (E Be /E) 2 relative to the O mode opacity, where the electron cyclotron energy is E Be = 115.8B 13 keV and B 13 = B/10 13 G. Thus the X mode radiation decouples from deeper, hotter layers in the NS atmosphere than the O mode radiation. For rays propagating at intermediate angles θ p relative to the magnetic field, the net emission is significantly polarized and dominated by the X mode.\n\nTo produce surface emissivities for the X and O photon modes, we use the model of van Adelsberg & Lai (2006), which quantitatively incorporates the effects of vacuum polarization on the radiative transfer for fully ionized atmospheres with an external magnetic field parallel to the axis of symmetry. This model is reasonably accurate for temperatures T s 10 6 K. At lower temperatures, partial ionization of atomic species becomes important, and partially ionized models are needed (see Ho et al. 2008, and references therein). Therefore, we use the fully ionized model only in the high temperature regime. It should be noted, however, that this is the effective temperature of the star, and that the measured value at the observer is lower due to gravitational redshift. Colder stars are less interesting for the purpose of this study: First, being much dimmer, their measured spectra are generally of lower statistical significance for detailed spectral studies. Second, and more importantly, since the fraction of Omode photons increases with temperature, spectra of colder stars are less sensitive to axion effects (see the discussion below).\n\nFor magnetic field strengths B ∼ 4 × 10 13 G, there is a significant contribution from vacuum polarization to the dielectric properties of the medium (Adler 1971;Tsai & Erber 1975). A ray traversing the density gradient of a magnetized NS atmosphere will eventually encounter a layer of the medium in which the plasma and vacuum contributions to the dielectric tensor are of the same magnitude, leading to a resonance. For the models studied in this paper, the resonance occurs at lower density than the X and O mode decoupling depths (where the optical depth τ ν ≈ 1) for most photon energies and propagation angles. In addition, the integrated opacity across the vacuum resonance is negligible at these field strengths. A discussion of vacuum resonance effects on the mode opacities is given by Ho & Lai (2003). As shown by Lai & Ho (2002) and Lai & Ho (2003a), there is coherent mixing of the modes at the resonance, analogous to the Mikheyev-Smirnov-Wolfenstein (MSW) effect for neutrino oscillations (see, e.g., Bahcall 1989;Haxton 1995). Using the geometric optics approximation, and neglecting damping terms in the dielectric tensor (which only affect the width of the resonance), the amplitudes of the two modes, A O and A X , evolve according to\ni d dz A O A X = ω 2 2 + σ 11 σ 12 σ 21 2 + σ 22 A O A X ,(2)\nwhere (e.g. Lai & Heyl 2006) \nσ 11 = [ξ − v e ] sin 2 θ p − v e 1 − u e cos 2 θ p ,(3)σ 22 = −χ sin 2 θ p − v e 1 − u e ,(4)σ 21 = −σ 21 = i v e u 1/2 e 1 − u e cos θ p .(5)\nIn equations (3)-(5), θ p is the angle between the magnetic field and the photon direction (defined to be the z direction), ω is the photon frequency, v e = (ω pe /ω) 2 and u e = (ω ce /ω) 2 , where ω pe and ω ce are the electron plasma and cyclotron frequencies, respectively. The B-field dependent functions ξ(b) and χ(b) are (Potekhin et al. 2004)\nξ(b) = 4α f b 2 (1 + 1.2b) 45π [1 + 1.33b + 0.56b 2 ] ,(6)\nand\nχ(b) = 7α f b 2 45π [1 + 0.72b 5/4 + (4/15)b 2 ] ,(7)\nwhere α f is the fine structure constant and b is the magnetic field strength in units of the critical field B c = m 2 e c 3 /(eh) = 4.414 × 10 13 G. The equations above assume that v e ≪ 1 and u e ≫ 1. Since v e ∼ 10 −3 (ρ/1 g cm −3 ) and u e ∼ 10 4 (B/10 13 G) at 1 keV, these constraints are easily satisfied throughout the soft X-ray band in which NS thermal spectra typically peak.\n\nEquation (2) determines the evolution of the photon mode amplitudes through the atmosphere (see Ho & Lai 2003;Lai & Ho 2003, for details). The amplitudes are related to the specific intensities of each mode by I X ,O ∝ |A X ,O | 2 . As the photons encounter the resonance, the polarization mode ellipticities, K X ,O (∝ E x /E y ), experience a discontinuity at the resonance. As discussed in Ho & Lai (2003), an alternative description of the polarization state uses ellipticities K ± , which vary continuously through the vacuum resonance. If the variation of the atmosphere density profile is gradual enough, an adiabatic condition is satisfied (see below), and the mode properties are fixed along the curves described by K ± . Before the resonance, a K − photon corresponds to an X mode polarized photon, while a K + photon corresponds to an O mode photon. After the resonance, the correspondence between K ± and K X ,O switches, so that under adiabatic conditions the character of each modes changes. This is analogous to the correspondence between flavor and mass eigenstates in the MSW mechanism for neutrino mixing. The adiabatic condition is set by evaluating the coefficient matrix in Eq. (2) at the resonance and setting the magnitude of the diagonal term equal to the off-diagonal term (Lai & Ho, 2002;2003a). In the non-adiabatic regime, the diagonal terms have a much larger magnitude than the off-diagonal terms, leading to a set of decoupled equations for A X ,O . In the adiabatic regime, the offdiagonal terms dominate over the diagonal ones, leading to a coupled set of equations for A X ,O and mixing of the polarization states.\n\nThe conversion probability P c is derived from Eq. (2) by setting the radiation completely in one mode (e.g., A O = 1, A X = 0) and evolving the equations to z → ∞. If the atmosphere density profile is linear in z, then v e (∝ ρ) is a linear function of z, and the conversion fraction is given by the well known Landau-Zener formula (see Lai & Ho, 2002;2003a, and the references therein):\nP C = 1 − exp −π E/E ad 3 /2 .(8)\nThe adiabatic energy E ad is defined by\nE ad ≈ 2.52 f B tan θ p 1 − E Bi /E 2 2/3 H −1/3 ρ ,(9)\nwhere E Bi is the ion cyclotron energy, and f −2 B is a slowly varying function of the magnetic field (see van Adelsberg & Lai, 2006, and the references therein). Because the vacuum resonance is very narrow in energy (and hence density, for a given photon energy and propagation angle), the linear approximation for the density profile is always satisfied for the atmosphere models considered in van Adelsberg & Lai (2006). In addition, the conversion occurs on a distance scale much less than the scale height of the atmosphere; thus the conversion process can be regarded as occurring exactly at the resonance density, and the asymptotic solution to Eq. (2), in the form of the conversion probability Eq. (8), can be used to treat mode conversion effects. This was shown to be an accurate approximation in van Adelsberg & Lai (2006), and we adopt it here in our calculations of mode conversion.\n\nA photon with energy E encounters the vacuum resonance at a density\nρ V ≈ 0.96 E 1 keV 2 B 10 14 G 2 f −2 B g cm −3 .(10)\nFor a given energy, ρ V is the density at which the vacuum and plasma contributions are of equal magnitude (i.e., when σ 11 = σ 22 ). If E > E ad , then conversion between the modes at this depth is effective. If I X and I O are the values of the mode specific intensities at the resonance, then after conversion they become:\nI ′ X = (1 − P C ) I X + P C I O I ′ O = P C I X + (1 − P C ) I O .(11)\nThis process is automatically incorporated into the models with B = 10 13 G and B = 4 × 10 13 G (T s = 5 × 10 6 K) using the code of van Adelsberg & Lai (2006). For models with B = 4 × 10 12 G, we use the code of Ho & Lai (2001). This code does not include partial mode conversion, but we have included the effect according to the following prescription: The code of Ho & Lai (2001) produces atmosphere profiles ρ d and T d at a set of discrete points in Thomson optical depth, τ d , where d = 1, . . . , D. Since the conversion process occurs at lower densities than both the X and O mode decoupling depths, it does not affect the atmosphere structure and can be treated as a modification operating on the emergent specific intensities. For a given photon energy and propagation angle, we calculate the vacuum resonance density using eq. (10), and then locate a grid point i, such that ρ i ≤ ρ V ≤ ρ i+1 . We then compute a linear approximation to the resonance Thomson depth τ V , using the formula:\nτ V ≈ τ i + ∆τ (ρ V − ρ i ) /∆ρ(12)\nwhere ∆τ = τ i+1 − τ , and ∆ρ = ρ i+1 − ρ i . We relate the resonance temperature T V to the Thomson depth and density in a hydrostatic atmosphere using\nT V ≈ 1.52 g * 10 14 cm 2 s −2 τ V ρ V ,(13)\nand calculate the atmosphere scale height at the vacuum resonance according to Eq. (1). Finally, the adiabatic energy and mode conversion probabilities are calculated according to Eqs. (8) and (9). The emergent intensities are then mixed using Eq. (11). When the surface magnetic field exceeds the quantum critical field B c ∼ 4 × 10 13 G, there are significant vacuum contributions to the dielectric properties of the medium. At the resonance, if the radiation energy is much greater than the adiabatic energy [Eq. (9)], there is a high probability of conversion between the X and O modes. For magnetic field strengths B 7 × 10 13 G, the vacuum resonance lies between the photospheres of the X and O modes (where the mode opacities τ ν ≈ 1). The modes mix at the resonance with a fraction of Xmode photons converting into O-mode photons and vice-versa according to the probability in Eq. (8). The O-mode photons, after converting into X-mode photons at the resonance, decouple from the atmosphere. The X-mode photons that con-vert to the O-mode, subsequently interact strongly with the atmosphere, as they now experience the large opacity for O mode polarization. Any X-mode photons that do not convert encounter a large integrated opacity at the resonance (under typical conditions; see Ho & Lai, 2003). Thus, one of the net effects of vacuum polarization is to shift the location of the X-mode photosphere to the resonance density, which is at lower density and temperature than the original X-mode photosphere. As a result, the high energy spectrum is softer, and the relative fraction of X to O mode photons decreases. At magnetic field strengths B ≪ 7 × 10 13 G, radiation in both modes decouples from the NS atmosphere before encountering the vacuum resonance. In this case, mode conversion es- sentially switches the photospheres of the two modes, with the O-mode photons now being produced in the deeper (hotter, denser) layers of the star. Therefore, if the emission is mostly polarized in the X mode, for E > E ad , the plane of polarization switches to the O mode after resonant conversion (Lai & Ho 2003a, 2003b. The above considerations are particularly relevant for the subsequent mixing between O-mode photons and axions, as emission which would otherwise have a small fraction of O mode radiation now predominantly consists of O mode photons, depending on the magnetic field strength and NS geometry (see van Adelsberg & Perna 2009, for more discussion). Figure 1 shows the fractional contribution to the emer-gent flux of the X and O modes for three values of the field strength, B = 4 × 10 13 G, 10 13 G, and 4 × 10 12 G. The effective temperature is T s = 5 × 10 6 K in all cases. Each panel shows the results for a separate emission angle δ relative to the magnetic field. In general, the X mode dominates the emergent flux at low energies, except around the ion cyclotron feature E Bi . According to the formula for E ad [eq. (9)], there is an angle-dependent point at which mode conversion becomes efficient and the radiation is composed mostly of O mode photons. Since E ad ∝ tan 2/3 θ p = tan 2/3 δ for a magnetic field perpendicular to the surface, at larger values of δ, mode conversion is efficient only at high energies. Note that as θ p → 0, the properties of the two modes become similar, and the dif-ference in the opacities decreases. Figure 2 shows the fractional contributions to the emergent flux of the X mode and O modes, for a fixed magnetic field strength, B = 4 × 10 12 G, at three effective temperatures T s = 5 × 10 6 K, 2.5 × 10 6 K, and 10 6 K. The general trend is an increase in the O/X flux ratio as the temperature increases. This is straightforward to understand: since a higher temperature atmosphere produces more photons with higher E, the average ratio E Be /E decreases. Thus, the X-mode opacity becomes more similar to the O-mode, leading to an increase in the O/X ratio.\n\n\nPHOTON-AXION CONVERSION IN THE MAGNETIC FIELD OF NEUTRON STARS\n\nIn the following we discuss the effect of photon-axion mixing as a photon beam propagates through the dipolar field of a magnetized NS. Firstly, we note that an important assumption of our analysis is that the conversions between the O and X modes and between the O mode and axion can be treated separately (each as a two-state mixing) rather than as a coupled three-state mixing. This assumption is valid as long as the vacuum resonance and the photon-axion resonance are well separated. This was shown to be the case by Lai & Heyl (2006) for highly magnetized NSs at soft X-ray energies and axion masses m a 10 −3 eV. Thus, the assumption of decoupled mixing is satisfactory for our regime of interest, and it will be adopted in our analysis.\n\nOur formalism for photon-axion conversion in a magnetized medium follows that of Raffelt & Stodolski (1988). To be consistent with the notation adopted in the previous section for the two photon polarization states, we denote the amplitudes of photons polarized parallel and perpendicular to the magnetic field as A O and A X , respectively. Under the assumption that the length scale over which the magnetic field varies is much larger than the photon and axion wavelengths, the evolution of the photon and axion fields in the z direction, with frequency ω, is given by\ni d dz A X A O a = ∆ ⊥ 0 0 0 ω + ∆ ∆ M 0 ∆ M ω + ∆ a A X A O a ,(14)\nin units withh = c = 1, where a is the axion field. The diagonal matrix elements are given by\n∆ ⊥ = ω ξ(b) sin θ 2 p 2 ,(15)∆ = ξ(b) ω sin 2 θ p 2 , ∆ a = − m 2 a 2ω ,(16)\nwhere m a is the axion mass and ξ(b) is defined by Eq. (6). The off-diagonal component is given by\n∆ M = 1 2 g a B(r) sin θ p(17)\nwhere g a describes the coupling strength between the O-mode photons and the axions, and r is the distance from the stellar surface. Since mixing occurs only between the axion field and the O-mode photon field, we neglect the evolution of the X-mode field in the discussion that follows. The mixing strength can be measured in terms of the ratio between the off-diagonal term and the difference between the diagonal terms in Eq. (14); a useful parametrization can be made in terms of the mixing angle θ m defined by\nθ m = 1 2 arctan 2 ∆ M ∆ a − ∆ .(18)\nWe begin our analysis by considering a region of space over which the magnetic field can be approximated as homogeneous. Following Raffelt & Stodolsky (1988), the discussion is greatly simplified if we define phases relative to the unmixed component, and neglect a common phase. The solution can be found by first performing a matrix rotation to an eigenstate basis where the propagation matrix is diagonal, propagating the two eigenstates independently, and then rotating back to the photon-axion basis. This yields an evolution equation for the mixing components:\nA O (z) a(z) = M(z) A O (0) a(0) ,(19)\nwhere\nM(z) = cos θ m − sin θ m sin θ m cos θ m M D cos θ m sin θ m − sin θ m cos θ m(20)\nand\nM D = e −i(∆ ′ −∆ )z 0 0 e −i(∆ ′ a −∆ )z ,(21)∆ ′ = ∆ + ∆ a 2 ± ∆ − ∆ a 2 cos 2θ m ,(22)\nand the plus and minus sign indicates ∆ ′ and ∆ ′ a , respectively.\n\nThe solution provided by Eq. (19) assumes that the field is homogeneous. The generalization for photon propagation within the dipolar field around the NS, with B ∝ r −3 , can be made by assuming that the total evolution operator is the product of evolution operators within spatial slices of width ∆z j wherein the B field can be approximated as constant.\n\nWithin each shell, we use Eq. (19) to evolve the photon and axion amplitudes from A O (z j ) and a(z j ) in shell j to A O (z i ) and a(z i ), where z i = (z j + ∆z j ) is the subsequent shell. We tested the code for convergence and found that, for ∆z j ≤ 0.07 km, the amplitudes at infinity converge for the range of B, m a , and g a of interest. Fig. 3 shows the conversion probability for combinations of the axion parameters m a , g a in the soft X-ray band of interest for NS observations. The probability is plotted as a function of photon energy and propagation angle θ p . There is a range in the parameter space of axion mass, coupling strength, and NS properties (such as surface magnetic field strength) where the photon-axion conversion probability is non-negligible. This implies that photon-axion conversion can leave distinctive signatures in NS spectra, making them interesting sources for constraining axion physics. In the next section, we perform a detailed analysis of such signatures. The observed spectrum of a NS is obtained by summing the local emissivities over the emission region. For simplicity, we consider an emitting region that has a constant temperature and magnetic field strength, with the field direction along the surface normal (e.g., a hot magnetic polar cap). Such a region allows us to directly examine the spectral dependencies of photon-axion conversion on these physical quantities. It is often the case that the emission is dominated by a region smaller than the entire star surface (i.e., a hot spot). The spectrum and lightcurves will then be functions of the viewing angle α between the observer and spot axes. As we will show FIG. 3.-Photon-axion conversion probabilities for the magnetic field strength B = 10 13 G and various combinations of the coupling strength, ga, and axion mass, ma. The probability is shown as a function of photon energy and angle θp between B and the direction of photon propagation. In the left panels, the dependence on the coupling strength is explored for fixed axion mass ma = 2 × 10 −6 eV (from top to bottom: ga = 10 −8 , 10 −9 , 10 −10 GeV −1 ). The right panels show the dependence of the conversion probability on the axion mass for fixed coupling strength ga = 10 −8 GeV −1 (from top to bottom: ma = 7 × 10 −7 , 6 × 10 −6 , 1.7 × 10 −5 eV). The dependence of the probability on B is weak for the range considered here (4 × 10 12 − 4 × 10 13 G), so we only display results at the intermediate field strength.\n\nin § 5, phase dependent methods add considerable diagnostic power for detecting the presence of axions.\n\nThe calculation of phase-dependent emission from an extended region on the NS surface follows the formalism developed by Pechenick et al (1983), with generalizations by Perna & Gotthelf (2008; see also Bernardini et al. 2011, for a similar geometry). We define the time-dependent rotational phase γ(t) as the azimuthal angle subtended by the magnetic dipole vector µ around the axis of rotation. It is related to the modulus of the NS angular velocity, Ω(t), by γ(t) = Ω(t)t. We choose the coordinate system so that the observer is located along the z axis; the inclination angle of the rotation axis,Ω, with respect to the line of sight is denoted by α R , while the angle between the magnetic dipole vector and the rotation an-gle is denoted by α M . Thus, the angle between µ and the line of sight is given by\ncos α(t) = cos α R cos α M + sin α R sin α M cos γ(t). (23)\nThe emission region is assumed to be circular, with opening angle β, and is centered around the magnetic dipole axis. Hence, α(t) also indicates the time-dependent angular separation between the spot axis and the line of sight as the star rotates. We describe points on the NS surface by means of the polar angle θ, and the azimuthal angle, ϕ, in spherical polar coordinates (see Fig. 1 in Perna & Gotthelf 2008 for a visual representation of this geometry).\n\nThe emission region is restricted to θ ≤ β for α = 0, while for α = 0 it is identified by the conditions\nα − β ≤ θ ≤ α + β 2π − ϕ ⋆ ≤ ϕ ≤ ϕ ⋆ if β ≤ α (24) with ϕ ⋆ = arccos cos β − cosα cos θ sin α sin θ ,(25)\nand by the condition\nθ ≤ θ * (α, ϕ, β), i f β > α .(26)\nIn the latter case, the outer boundary of the spot, θ ⋆ (α, ϕ, β), must be determined numerically from the expression cos β = sin α sin θ ⋆ cos ϕ + cosα cos θ ⋆ .\n\nDue to the intense gravitational field of the NS, photons emitted at the NS surface are substantially deflected as they travel to the observer. A photon emitted from colatitude θ will reach the observer (at infinity) if emitted at an angle δ with respect to the surface normal; the relation between the two angles is given by the ray-tracing function\nθ(δ) = Rs/2R 0 du x 1 − R s R R s 2R 2 − (1 − 2u)u 2 x 2 −1/2 ,(28)\nwhere x ≡ sin δ, R and M are the NS radius and mass, respectively, and R s ≡ 2GM/c 2 is the Schwarzchild radius.\n\nThe spectrum at the observer is obtained, as a function of the viewing angle, α(γ), by integrating the local emission over the observable surface; this procedure yields the flux seen by an observer at a distance D ≫ R. Accounting for the gravitational redshift of the radiation, this integral takes the form (Page 1995and generalizations by Pavlov & Zavlin 2000, Heyl et al. 2003:\nF j (E ∞ , α) = 2π c h 3 R 2 ∞ D 2 E 2 ∞ 1 0 2xdx × 2π 0 dφ 2π I j θ, φ, E ∞ e −Λs ,(29)\nin units of photons cm −2 s −1 keV −1 . In the equation above, the index j represents any of the Stokes parameter (U, V, Q, I), and E ∞ is the energy as seen by the distant observer. The energy emitted at the stellar surface is given by E ∞ e −Λs , with\ne Λs ≡ 1 − R s R ,(30)\nThe local specific intensity I j is equal to zero outside of the boundaries for θ and ϕ defined by equations (24)- (26). The observed flux depends on the geometric angles α R and α M through the angle α in Eq. (23). In the following, we consider an orthogonal rotator, for which α R = α M = 90 0 . This implies that α(t) = γ(t). Given that the emission region is centered around the dipole axis, the angle α will be identified with the angle θ p between the photon propagation and magnetic field directions (see §3).\n\nIf photon-axion conversion occurs with probability P(E ∞ , α), the phase-resolved spectrum is given by\nF(E ∞ , α) = F X (E ∞ , α) + [1 − P(E ∞ , α)] F O (E ∞ , α). (31)\nFor a fixed energy (or integrated over an energy band), α]) yields the light curve as a function of the viewing angle α. The phase averaged spectrum is readily obtained from\nF(E ∞ , α) (or E2 E1 dE ∞ F[E ∞ ,F ave (E ∞ ) = 1 2π 2π 0 dα F(E ∞ , α) .(32)\nFinally, the linear polarization is computed as\nΠ(E ∞ , α) = [1 − P(E ∞ , α)]F O (E ∞ , α) − F X (E ∞ , α) [1 − P(E ∞ , α)]F O (E ∞ , α) + F X (E ∞ , α) .(33)\nTo simplify our notation, we will omit the subscript '∞ ′ from the observed energy. In the discussion that follows, all quoted energies are redshifted energies for a NS of mass M = 1.4M ⊙ and radius R = 10 km. Before concluding this section, we need to remark that Eq. (31) and Eq.(33) assume that the O and X modes remain decoupled once they have emerged from the atmosphere of the NS. The evolution of the modes in the changing dipole magnetic field of the star was computed in papers by Heyl & Shaviv (2000, Heyl, Shaviv & Lloyd (2003), Lai & Ho (2003) and van Adelsberg & Lai (2006). All of the above studies found that, in the birefringent, magnetized vacuum near the star surface, the photon modes are decoupled. In this region, the mode eigenvectors evolve adiabatically along the changing direction of the magnetic field. Further from the star, the mode evolution continues to be adiabatic, up to the \"polarization limiting radius\", R pl , which is defined as the distance at which the polarization modes mix (and are frozen thereafter). Heyl, Shaviv & Lloyd (2003) derive the following expression for R pl\n5 R pl ≈ 3.6 × 10 7 B 2/5 13 E 1/5 1 (sin α) 2/5 cm ,(34)\nwhere E 1 ≡ E/(1 keV) and B 13 ≡ B/(10 13 G). For radii larger than R pl , the mode eigenvectors are fixed, and the polarization no longer evolves (i.e., it is the observed polarization [Ω/(2π)] −1 sin 2 α 1/6 10 7 cm. The scaling of R pl in this equation differs slightly from that in Eq. (34) due to different adiabatic conditions applied by the two sets of authors. Heyl et al. (2003) constrain the change in the relative difference in the mode indices of refraction compared to the magnitude of that difference [see their Eq. (9)], while van Adelsberg & consider the distance at which the mode evolution equations have equal diagonal and off-diagonal terms [see their Eq. (62)]. This results in slightly different magnitudes and scaling relations; for slowly rotating NSs, the precise numerical value of R pl does not change the value of the observables, and both approaches result in the same physics. except for very rapidly rotating stars (i.e., with frequencies of tens of milliseconds) the coupling results in only a very small quantitative change. For example, for a star with period ∼ 0.1 sec (B ∼ 10 12 G, E ∼ 1 keV), the mode intensities I O and I X (∝ |A O | 2 , |A X | 2 ) couple at the level of 4%, and the amount of mixing further decreases at longer periods. Therefore, for the purpose of our study and the observational tests we propose (requiring NSs with relatively long periods in order to perform time-resolved spectroscopy), the formalism that we use provides a very good description of the observed flux and the linear polarization fraction computed here (see also Lai & Heyl 2006). We further note that, if the radius of the emission region is much smaller than the polarization limiting radius, then summing over the Stokes parameters at the detector is well approximated by summing over a mode (X or O) at the NS surface (Heyl et al. 2003). Since our emission region (a hot spot) is always much smaller than R pl for the B field strengths and energies under consideration, we use this approximation in our calculations.\n\n\nRESULTS: THEORETICAL PREDICTIONS AND OBSERVATIONAL TESTS\n\n\nTheoretical predictions\n\nWe present the results of our calculations of NS spectra, light curves, and polarization signals, with and without photon-axion conversion. We explore the dependencies on the stellar magnetic field strength and effective temperature, with particular emphasis on the magnetic field since it has a stronger influence than the temperature on the functional form of the modal energy dependence (see Figs. 1 and 2). We consider emission from a circular region, 10 deg in angular size, that is centered on the magnetic pole of the star. Such hot spot emission is common in NSs, and exhibits a strong dependence on phase. Given the significant angle dependence of the O and X mode intensities (see Figs. 1 and 2), the study of emission from a finite region allows us to extract more information on the effects that photon-axion conversion has on the observable properties of NSs.\n\nAs shown in Fig.3, the conversion probability is a strong function of the axion mass, m a , and the coupling constant, g a . For the thermal spectra of NSs to be affected by photonaxion conversion, the probability of this process must be non-negligible in the soft X-ray band, where the NS thermal spectrum peaks. Therefore, we perform a detailed analysis of the consequences of photon-axion conversion on NS spectral properties for a particular set of axion parameters; these parameters are chosen to yield a substantial conversion probability in the soft X-ray band. We choose the values m a = 2 × 10 −6 eV, and g a = 10 −8 GeV −1 ; the study of this specific case will uncover the main signatures of axion conversion. Once these are identified, we explore a wider area of m a − g a parameter space to identify the main region that can be probed by means of NS observations in the X-ray band. Fig. 4 shows phase-resolved spectra for three magnetic field strengths, B = 4 × 10 13 , 10 13 , and 4 × 10 12 G. For each value of B, we plot the spectra at two phases, coinciding with the angles between the line of sight and the magnetic field axis for the chosen geometry. We show angles α = 2 deg and a typical large viewing angle α = 60 deg. The former angle is essentially a 'phase-on' spectrum (i.e. a spectrum observed in correspondence to the maximum of the pulsation); we do not plot the results for α = 0 • since the conversion probability in that case is zero. For each combination of viewing angle and magnetic field strength, the figure shows the fluxes from the O and X-modes separately, as well as the total composite spectra. Results are plotted with and without photon-axion conversion.\n\nThe relative fraction of O and X mode photons, as a function of energy and angle, plays a fundamental role in determining the effects of axions on phase-dependent spectra and light curves. Before discussing the spectral effects of axions in further detail, we comment on the energy and angular dependence of the two photon modes. Although only the Omode photons are affected by axion conversion, the properties of the X-mode photons also matter for our purposes because of vacuum polarization-induced mode conversion. The X-mode has a complex beam pattern, consisting of the sum of a narrow \"pencil\" beam and a broad \"fan\" beam. The width of the pencil beam scales as ∼ (E/E Be ) 1/2 , decreasing with magnetic field strength (Pavlov et al. 1994). Therefore, the modal intensities vary from being approximately equal to significantly different between widely separated emission angles.\n\nFor a phase-on spectrum, the intensities of the O and X modes peak at roughly similar energies. Thus, the presence of photon-axion conversion results in a dimming of the spectrum without significant spectral distortion. The situation is different at larger viewing angles, as evident in the α = 60 deg panels of the Fig. 4. In these cases, the X mode dominates the spectrum at (redshifted) energies 1.4 keV, while the O mode dominates at higher energies; thus, an inversion of the dominant spectral mode occurs. Suppression of the O mode by photon-axion conversion therefore produces not only a dimming of the spectrum but also a spectral distortion, suppressing the high-energy tail.\n\nSince axion conversion occurs at larger distances from the star than mode conversion due to the vacuum resonance, only O-mode photons are affected as they emerge from the star atmosphere. This is shown in Fig. 4, where the X-mode photons are untouched by photon conversion, while the O-mode intensities are noticeably suppressed. The overall effect on the spectrum is that of flux suppression. However, since the O/X flux ratio, as well as the conversion probability, has an energy dependence which varies with viewing angle, the shape of the spectrum is also affected, causing a shift of the energy peak. While the shift is very small at phase-on viewing angles (since the O-modes and X-modes have a similar energy-dependent intensity for α ∼ 0 deg), it is very pronounced at wider viewing angles, where the O-mode largely dominates the intensity at high energies. In this case, enhanced suppression of high energy photons produces an effective shift of the peak of the spectrum towards lower energies. Figures 5 and 6 quantify the phase-dependent spectral effects discussed above. For the same values of magnetic fields and axion parameters considered in Fig. 4, Fig. 5 shows the ratio of the X-ray flux in the 0.8-10 keV band photon-axion conversion to that without conversion. As expected, at all phases, the ratio F Ax X /F noAx X is smaller than 1. This is because the contribution to the flux from the O-mode photons is non-negligible at all viewing angles with respect to the magnetic field axis. The precise value of the flux ratio, however, shows a significant dependence on the viewing angle, with a mimimum around α ∼ 40 − 50 deg. This is due to the O-mode flux, which is a large fraction of the total flux at E 1 keV, for viewing angles in the 30 − 70 deg range.\n\nThe left panels of Fig. 6 show the (redshifted) energy where the spectrum peaks, E peak , as a function of viewing angle. We -Comparison between phase-resolved spectra of NSs with (thick/red lines) and without (thin/blue lines) photon-axion conversion. In each case, the dotted lines show the flux contribution from the O mode, while the dashed lines display the contribution from the X mode, which is unaffected by conversion. The solid lines show the total spectra. In all panels, the effective temperature of the star is Ts = 5 × 10 6 K and the axion parameters are ma = 2 × 10 −6 eV and ga = 10 −8 GeV −1 , while the B-field strength has three values from top to bottom. The spectra are shown for two angles α between the observer and the dipole axis.\n\nconsider only angles α ≤ 100 deg since the flux from larger angles, which is still visible due to relativistic light deflection, is too low for phase-resolved spectroscopy. We begin by examining the behavior of E peak without photon-axion conversion. At the magnetic field strengths for the models in this paper, the spectral peak of the total (X and O mode) specific intensity follows the beam pattern of the X mode: a narrow pencil beam at small angles and a broad fan beam at large angles. This is because conversion between the X and O modes occurs above the photosphere of both modes and thus does not change the total intensity. The observed variation of E peak with viewing angle deviates somewhat from the emission beam pattern due to the narrowness of the pencil beam (a few deg in width around δ = 0 • ), the gravitational deflection of photon trajectories, and the finite size of the emitting region, which smears out features from nearby emitting points 6 .\n\nWhen axion conversion is taken into account for B = 4 × 10 12 − 4 × 10 13 G, a common feature is a shift of the peak FIG. 5.-Ratio of X-ray flux (in the [0.08-8] keV band) with axion conversion to that without axion conversion. Axion parameters have the same values as in Fig. 4, while the NS temperature and magnetic field are indicated for each curve, with the B 12 ≡ 10 12 G and T 6 = 10 6 K. energy to lower values at all viewing angles except for those close to phase-on and edge-on. For α ≈ 0 deg, the intensity of the O mode photons peaks at similar or slightly lower energies than that of the X-mode photons (see Fig. 4); hence, suppression of O mode photons primarily affects the amplitude of the spectral intensity, as opposed to the location of the spectral peak. In fact, because the O-mode intensity peaks at lower energies, especially at lower fields, at small angles there is actually a shift of the peak towards higher energies after photon-axion conversion.\n\nAnother notable signature in the variation of E peak (α) is a strong decrease (or flattening in the higher B-field case) for viewing angles α ∼ 30−50 deg. Around these angles, the contribution from the O mode photons is dominant at high energies, and the reduction of these photons through photon-axion conversion results in a significantly cooler spectrum. As the viewing angle increases further, the X-mode photons become increasingly dominant (see Fig. 1), and the reduction of high energy photon lessens. As a result, the effective temperature of the spectrum -as measured through E peak -approaches the value it would have without photon-axion conversion.\n\nTo examine the dependence of our results on the effective temperature T s of the star, we consider the values T s = 10 6 , 2.5 × 10 6 , and 5 × 10 6 K (where T s is the unredshifted temperature at the stellar surface) for the magnetic field B = 4 × 10 12 G. We find that the shift in E peak due to axion conversion (compared to the case with equal temperature but no conversion) is more pronounced at higher temperatures. This is not surprising given that the relative fraction of O to Xmode photons is larger at higher temperatures.\n\nThe right panels of Fig. 6 show the phase-resolved effective area [i.e., A eff (α) ∝ F(α)/E 4 peak (α)] that an observer measures when fitting spectra with a blackbody model. To eliminate the dependence of this diagnostic on the often poorly determined distance to the NS, we normalize A eff (α) to its value near phase-on. Since the photon-axion conversion probability is zero for precisely α = 0, we normalize to a small angle α = 2 deg which represents a \"phase-on\" spectrum as an average of small angles around α = 0 deg. When photon-axion conversion is included, the variation of the effective area with phase results from the interplay of two effects: a decrease in A eff due to flux dimming and an increase in A eff due to decreases in the peak energy and inferred temperature. Because of the E 4 peak -dependence, the latter effect dominates the former. This is particularly the case for lower fields, which has the strongest E peak shift with phase.\n\nThe variation of inferred effective area with phase is a particularly interesting and peculiar feature of photon-axion conversion. The specific details of the variation, however, depend on NS and axion parameters. In order to explore a wider region of m a − g a parameter space, we compute the effective area for the phase-averaged spectrum (A ave eff ) with photon axion conversion, normalized to that without axion conversion. This represents a measure of the \"correction\" that photonaxion conversion makes to the inferred effective area (as determined through measurements of phase-averaged flux and effective temperature). For our purposes, the phase-averaged spectrum of a hot spot provides an adequate approximation for emission from the entire NS surface. The results of this calculation are displayed in Fig. 7 for a combination of magnetic fields and effective temperatures: B = 4 × 10 12 G, T s = 5×10 6 K; B = 4×10 12 G, T s = 2.5×10 6 K; and B = 4×10 13 G, T s = 5 × 10 6 K. It is interesting to note that, in all cases, there are regions of parameter space in which the inferred effective areas are smaller/larger than the uncorrected areas. This is expected given the two counteracting effects described above. The m a − g a region shown in Fig. 7 is one for which, at typical NS magnetic fields, there is a larger probability of photonaxion conversion in the soft X-ray band (where NSs are routinely observed). The correction to the effective area becomes negligible for the largest values of m a and g a −1 displayed in Fig. 7. This is expected since the photon-axion conversion probability tends to zero for these values at the keV energies of interest here. The results summarized by this figure have a strong potential to constrain axion signatures for NSs with good distance measurements (see the discussion in §5.2).\n\nIn Fig. 8, we examine the effects of photon-axion conversion on NS light curves at different energies, for the same set of axion parameters as in Fig. 4. The lightcurves are all normalized to the maximum value of the flux among the set of energies considered (0.5, 1.5, 2.5, and 4 keV). The main effect of photon-axion conversion is the relative suppression of the flux at higher energies with respect to that at lower energies.\n\nFinally, we study the effect of photon-axion conversion on the magnitude of linear polarization [see Eq. (33)]. For consistency with Fig. 4, we consider the same set of parameters. The polarization signal with and without conversion is shown in Fig. 9. Since the main effect of axion conversion is suppression of O-mode photons, the net result is a reduction of the magnitude of linear polarization. In addition, it is noteworthy that the presence of photon-axion conversion causes the plane of polarization to rotate from parallel (with respect to the plane formed by the propagation direction and magnetic field) to perpendicular for a phase-on spectrum (small α, left panels of Fig. 9). This can be understood with reference to Fig. 4: for a NS observed phase-on with respect to the emission region, the intensity of the O mode photons slightly exceeds the X-mode photon intensity. When photonaxion conversion is considered, the O mode photon intensity is suppressed and becomes smaller than the X-mode intensity, which results in an inversion of the plane of polarization with respect to the case without conversion. Clearly, the extent to which this occurs depends on the specific axion parameter values, since the magnitude of the conversion probability is a strong function of m a and g a (see Fig. 3). For example, at B = 10 13 G and T s = 5 × 10 6 K, the X-mode intensity of a phase-on spectrum is about 90% of the O-mode intensity at E ∼ 1 keV. Therefore, if photon-axion conversion occurs with probability 10%, it results in an inversion of the plane of polarization. For the combination of axion parameters adopted in Fig. 3, a probability of 10% is achieved when g a 10 −9 GeV −1 (for m a = 2 × 10 −6 eV). When g a = 10 −8 GeV −1 , the probability is 10% for the entire range of axion masses considered (7 × 10 −7 − 1.7 × 10 −5 eV).\n\nAt larger viewing angles on the other hand (right panels of Fig. 9), it becomes evident one of the unique signatures of vacuum polarization on the polarization signal of NSs with magnetic fields B 7 × 10 13 G, that is the rotation at high energies of the plane of polarization from perpendicular to parallel (Lai & Ho 2003b). Photon-axion conversion does not erase this feature; however, as a result of the suppression of the O-mode photons which dominate at higher energies, the shift of the plane occurs at higher energies than it would without conversion. For the axion parameters adopted in Fig. 9, the energy at which the shift occurs with the conversion differs by a factor of ∼ 15 − 40% with respect to the case with no conversion, with the precise value depending on the magnetic field strength. Our theoretical results provide several methods for constraining axion parameters from actual measurements of NS spectra. The foremost method is spectral analysis, often performed on phase-averaged spectra when the resolution low. We showed in §5.1 that the relative suppression of the Omode photons with respect to the X-mode photons noticeably distorts the spectral shape with respect to the case without conversion. This distortion is strongest in the high-energy spectral tail, where the intensity of the O-mode photons is larger for most viewing angles (see Fig. 4). Observed enhancements of high energy spectral tails are generally modeled with a power law (often of unknown origin; e.g. Manzali et al. 2007, Durant et al. 2011 for young and middle-aged pulsars; Perna et al. 2001, Juett al. (2002, Rea et al. 2007 for magnetars). However, a deficit of high energy photons in a thermal fit could be a signature of photon-axion conversion, and the axion parameters could then be determined by means of detailed spectral fits that account for the effect. For this type of analysis, the distance to the NS is not important since the main discriminant is the spectral shape, in particular the deficit of high-energy photons.\n\n\nObservational tests\n\nFor sources with good distance estimates 7 but too few Xray counts to perform detailed phase-resolved spectroscopy, Fig. 7 demonstrates that the determination of the effective emission area (through measurements of flux and peak energy) can rule out large regions of m a − g a parameter space if the inferred NS radius is close to the maximum or minimum value allowed by NS nuclear equations of state (EOS). The minimum radius constraint requires knowing that the observed radiation is being emitted by the entire NS surface and not from a small hot spot. This may be the case when pulsations are not detected or when the angle between the line of sight and magnetic dipole axis is non-negligible and the measurement is made near pulse minimum. The maximum conthe distances to NSs in globular clusters are known to much higher precision. straint on the other hand does not require other independent information.\n\nTo illustrate how to use Fig. 7 with NS observations, suppose R max is the maximum radius allowed by a particular EOS (see, e.g., Lattimer & Prakash 2007, for a review), and A max is the corresponding surface area. Then a NS whose inferred emission area is A ∼ A max would rule out the m a − g a region of the parameter space for which the effective area 'correction' due to photon-axion conversion is larger than 1. For B = 4 × 10 12 G and T s = 5 × 10 6 K (corresponding to a redshifted value ∼ 3.8 × 10 6 K), the parameter space with m a 2 × 10 −5 eV and g a −1 4 × 10 9 GeV (or g a 2.5 × 10 −8 GeV −1 ) would be ruled out. For NSs whose radius determination relies on methods other than flux and peak energy measurements 8 , constraints on axion parameters can be obtained using the method described above without any need for the radius to be near the maximum or minimum allowed by the EOS.\n\nAs an aside, we note that the the effects of axions on the inferred area discussed above suggest that the use of NS effective radii to rule out nuclear EOSs may not be entirely reliable. The existence of axions could severely bias such conclusions.\n\nFor sources with phase-resolved X-ray spectra, observations can be fit with either blackbody or model atmosphere spectra to obtain peak energies E peak and effective areas A eff as a function of rotation phase or viewing angle α. Figure 6 shows a clear signature of photon-axion conversion, i.e., a significant rise and fall of effective area at intermediate viewing angles, especially for B 10 13 G. The important quantity for this diagnostic is the relative change of the effective area with phase; thus, this method does not require knowledge of the distance to the NS. It is also rather insensitive to the NS radius/mass (i.e., surface gravity), since variations in this quantity produce energy shifts that are the same at all phases, leaving the relative phase-dependent signatures of the conversion unaltered. An exact estimate of the range of axion parameters that can be constrained by means of this method cannot be made in absolute terms, since it clearly depends on the quality of the data. However, from a theoretical point of view, photonaxion signatures are imprinted in the NS spectra as long as the conversion probability is non-negligible in the typical X-ray band of observations. Fig. 4 shows that this is the case for an axion coupling strength g a 10 −9 GeV −1 and for a large range of axion masses, up to m a ∼ 10 −4 eV if g a 10 −8 GeV −1 .\n\nThe theoretical framework and methods for extracting axion parameters from observations of magnetic NSs is presented above. Detailed analyses of specific sources require a model appropriate for the specific viewing and emission geometry, magnetic field strength, and surface temperature for each source; a comparison with the data can then be performed by folding the model through the response function of the detector. Such an analysis will be the subject of future work. Below, we briefly review NSs that serve as potential candidates for obtaining constraints on axion physics. These sources possess bright X-ray thermal emission from the NS surface. Among the observed NS population, such sources can be grouped into several classes (which may not be mutually exclusive groups; see, e.g., Kaspi 2010;. One class is the magnetars, which generally possess superstrong magnetic fields B 10 14 G. 9 For these objects, vacuum resonance effects lead to emission that is predominantly in the X-mode. Therefore the possible conversion of O-mode photons to axions will not produce an observable effect on the total radiation spectrum. Another class of NSs are the rotation-powered pulsars; these are the classic radio pulsars that are powered by the loss of 8 Examples of such methods of NS radius measurement (or more generally of the radius-to-mass ratio) include spectral lines (e.g. Cackett et al. 2009), thermal X-ray pulse profiles (e.g. Bogdanov et al. 2008; note that at low energies E 1 keV, these pulse profiles are not very sensitive to the presence of photon axion conversion, as shown in Fig. 8), gravitational wave emission (Lenzi et al. 2009), quasi-periodic oscillations in accreting NSs (Miller et al. 1998), and neutrino emission from proto-stars (Lattimer & Prakash 2007). Masses can be independently inferred from radial velocity studies for NSs in binary systems (e.g. van and pulsar timing (a comprehensive review of all these methods can be found in Lattimer & Prakash 2007). 9 Note however that the bursting behaviour used to characterize magnetars has also been seen in a lower magnetic field object (see Rea et al 2010). rotational energy due to emission of electromagnetic radiation. These NSs can have magnetic fields up to the magnetar regime B 10 14 G, and a small fraction of sources are sufficiently bright in the X-rays (see, e.g., Pavlov et al. 2002;Mereghetti 2007;Zavlin 2009, for reviews). These NSs are good candidates if their X-ray emission is not dominated by magnetospheric emission (as in, e.g., PSR B0656+14).\n\nThe isolated NSs are another interesting class. The seven confirmed isolated NSs have B ≈ (1 − 4) × 10 13 G inferred from timing measurements , while features in their X-ray spectrum suggest B ≈ (5 − 10) × 10 13 G if the features are due to the proton cyclotron resonance (Haberl 2007). We note that the observed X-ray spectra of isolated NSs are generally well-fit (except around spectral lines) by blackbodies, including Wien-like behavior at high energies (van Kerkwijk & Kaplan 2007). Various explanations have been proposed to explain the soft high energy tails as compared to the hard tails predicted by atmosphere models (e.g., Romani 1987), such as emission from a partially optically thin atmosphere (e.g., Motch et al. 2003) or the effect of vacuum polarization (see §2 and Ho & Lai 2003). Here we have shown that photon-axion conversion can suppress high-energy emission and produce softer tails (see Fig. 4). However, this cannot explain the spectra of isolated NSs. These NSs have surface temperatures 10 6 K, and, as we discussed in Sec. 2, the conversion is less effective at low temperatures due to the low fraction of O-mode photons. However, if soft tails (softer than the blackbody) were detected in NSs with relatively high surface temperature, then photon-axion conversion could indeed be a competing explanation.\n\nA final class is the central compact object neutron stars (CCOs), which have B ∼ 10 10 − 10 11 G, obtained from timing and spectral analyses (see, e.g., Halpern & Gotthelf 2010). At these low fields, the vacuum resonance is outside of the atmosphere, and the emerging intensity has a high fraction of O-mode photons; their surface temperatures are constrained to be (2 − 3) × 10 6 K, while hot spot temperatures are quite high [∼ (4 − 6) × 10 6 K]. These characteristics, combined with the fact that the photon-axion conversion probability at these lower fields is still high for a wide range of axion parameters (Jimenez et al. 2011), make CCOs potentially good candidates. Current data on many of the NSs belonging to the classes described above are already of sufficient quality to allow axion constraints to be derived.\n\nFinally, we note that X-ray polarization measurements, like those that will be performed by the forthcoming Gravity and Extreme Magnetism Small Explorer (GEMS) satellite (Jahoda 2010), will increase the robustness of axion constraints, as well as provide an independent measure of the stellar magnetic field strength and geometry (van Adelsberg & Perna 2009).\n\n\nSUMMARY\n\nConstraints on the axion mass and coupling strength by means of observations of magnetized objects have been discussed in the recent literature (e.g., Lai & Heyl 2006;Chelouche et al. 2009;Gill & Heyl 2011;Jimenez et al. 2011) as a means to complement and independently test constraints obtained from other methods. The studies above emphasized different observational aspects and tests. The modification to the X-ray spectra of highly magnetized NSs induced by photon-axion conversion was discussed by Lai & Heyl (2006) using a blackbody spectrum to model the photon spectrum from NSs. Chelouche et al. (2009) discussed spectral features produced in the sub-mm/IR wavelength regime for magnetized NSs, while Gill & Heyl (2011) considered limits that can be derived from polarization measurements of white dwarfs. Jimenez et al. (2011) focused their analysis on constraints that can be obtained through observations of eclipsing white dwarfs in binaries, but they also discussed qualitatively the potential of NSs. Since most of the photon-axion conversion happens at a distance of many stellar radii from the NS, they pointed out that there are two configurations that should be considered: light from a background object passing through the magnetic field of the NS (i.e., occultation) and a binary containing a NS where the companion transits close enough to the line of sight for its light to be influenced by the NS magnetic field. In these two cases the treatment is much simpler as the physics of photon propagation in the NS atmosphere is irrelevant. However, they determined that the occultation probability is too low to be astrophysically relevant and that there are no known binary systems involving a NS that are detached enough to yield a clean constraint.\n\nSince NSs are routinely observed in the X-ray domain, the goal of our paper has been an extensive exploration of the actual constraints on axion physics that can be obtained from observations of NS thermal spectra. We have also explored what can be learned from future X-ray polarization measurements. For these purposes, we used detailed, magnetized atmosphere models, which properly account for the energy and angledependent emerging intensities of the two polarization modes, including mode conversion due to vacuum polarization effects. The emergent O mode intensity was coupled with the energy and angle-dependent photon-axion conversion probability, and this allowed us to make theoretical predictions for phase-resolved spectra with photon axion-conversion in the relevant regime of magnetic fields and temperatures for NSs. Since the relative intensities of the O and X mode photons depend strongly on the angle at which they emerge with respect to the magnetic field direction, we considered emission from a region consisting of a hot spot with an axis coincident with the NS magnetic field axis. Thermal emission from most Xray bright NSs does indeed indicate that we are observing hot spots (this from both non-negligible pulse fractions and small inferred emission areas). In order to extend axion studies by means of comparison with observations of any NS (especially those emitting from the entire surface, with non-dipolar, complicated B topologies, and fast rotators), one would have to include, locally, magnetic fields non-normal to the surface in the atmospheres, investigate how non-dipolar fields influence both the emission spectra as well as the photon-axion conversion, and, for the polarization, explicitly integrate the Stokes parameters to R pl . However, since the magnetic field structure over the entire surface of a NS is not fully known apriori, determination of the axion parameters would be more degenerate under these more general conditions. Our analysis has identified some features in NS spectra that bear the telltale signs of photon-axion conversion:\n\n• The NS spectral shape is noticeably distorted compared to the case without photon-axion conversion, with suppression of the high-energy tail for viewing angles ∼ 20 − 70 deg (with respect to the center of the emission region). Detailed spectral fits can yield axion parameters. This analysis does not require knowledge of the distance to the NS, as the axion signatures are imprinted in the spectral shape.\n\n• The spectral suppression of the O-mode photons by photon-axion conversion dominates the high energy tail of the spectrum for a range of viewing angles. In addition, it results in a shift of the peak energy, E peak , towards lower values for a range of rotation phases (i.e., viewing angles). As a result, the effective area of the spot A eff (as inferred from measurements of the flux and peak spectral energy) is larger than its value without axion conversion. We demonstrated that a clear signature of photon-axion conversion is a significant rise and fall of effective area (normalized to the phase-on value) at intermediate viewing angles, especially for B 10 13 G. Such a measurement requires high signalto-noise phase-resolved spectra, but does not require knowledge of the distance to the NS, since the axion signature appears in the relative change of the effective area with phase.\n\n• For a star emitting from the entire surface, photonaxion conversion can substantially affect the inferred NS emission area by 50%, and the inferred radius by 20%. These values are measured using the flux and peak energy of the thermal spectrum. If the distance is well constrained, and the NS radius is known through different methods, then the inferred emission area can be directly translated into a constraint in the m a − g a plane. If the NS radius is not independently known, then an inferred emission area that exceeds the maximum value allowed by NS equations of state could indicate the presence of photon-axion conversion.\n\n• In the absence of photon-axion conversion, radiation from a hot spot observed phase-on at energies ∼ 2 − 3 keV is linearly polarized in the plane formed by B and the direction of photon propagation. Conversion rotates the plane of polarization and leads to radiation polarized perpendicular to the plane of B and the direction of photon propagation, for a range of axion parameters.\n\nWe concluded §5.2 with a discussion of appropriate sources to study within the theoretical framework developed here for probing axions. While this paper has outlined the main elements for connecting theoretical ideas of photon-axion conversion to actual observations that are made of NSs, studying each suitable source will require a specific suite of models, tailored to the particular object (e.g., its magnetic field, surface temperature, emission and viewing geometry). The theoretical models are then convolved with the detector response, and a process of minimization identifies the most likely values of the parameters. Detailed analysis of the most promising NS candidates will be the subject of future work.\n\nThis work was partially supported by grants NSF AST-1009396, AR1-12003X, and DD1-12053X (RP); WCGH is supported by STFC in the UK; LV and RJ are supported by MICINN grab AYA 2008-03531. RP, RJ and LV thank the Centro de Ciencias de Benasque Pedro Pascual where this work was initiated. We further acknowledge Carlos Peña-Garay, Dong Lai, Jeremy Heyl and Enrique Fernadez-Martinez for insightful discussions, and an anonymous referee for helpful comments on our manuscript. WCGH appreciates the use of the computer facilities at the Kavli Institute for Particle Astrophysics and Cosmology.\n\nFIG. 1 .\n1-Fractional contributions to the flux from the emergent O ( thick/green lines) and X (thin/red lines) modes for four angles δ with respect to the surface normal and three magnetic field strengths. The effective temperature is Ts = 5 × 10 6 K in all cases.\n\nFIG. 2 .\n2-Fractional contributions to the flux from the emergent O (thick/green lines) and X (thin/red lines) modes, for four angles δ with respect to the surface normal and three effective temperatures of the star. The magnetic field strength is B = 4 × 10 12 G in all cases.\n\n\n4. THERMAL SPECTRA, LIGHT CURVES, AND POLARIZATION OF NEUTRON STARS WITH PHOTON-AXION CONVERSION\n\n\nFIG. 4.-Comparison between phase-resolved spectra of NSs with (thick/red lines) and without (thin/blue lines) photon-axion conversion. In each case, the dotted lines show the flux contribution from the O mode, while the dashed lines display the contribution from the X mode, which is unaffected by conversion. The solid lines show the total spectra. In all panels, the effective temperature of the star is Ts = 5 × 10 6 K and the axion parameters are ma = 2 × 10 −6 eV and ga = 10 −8 GeV −1 , while the B-field strength has three values from top to bottom. The spectra are shown for two angles α between the observer and the dipole axis.\n\nFIG. 6 .FIG. 7 .\n67-Left: Peak energy of the observed spectrum as a function of viewing angle α (i.e., phase of the NS). The presence of photon-axion conversion changes and enhances the variation in peak energy as the star rotates. Right: Corresponding phase-resolved effective area that an observer would infer from spectral fitting. Axion parameters have the same values as inFig. 4. -Ratio between the observed effective area with photon axion conversion and without photon-axion conversion for a range of axion parameters ma and ga. The NS magnetic field and temperature are B = 4 × 10 12 G and T = 5 × 10 6 K (top left panel), B = 4 × 10 13 G and T = 5 × 10 6 K (bottom left panel), and B = 4 × 10 12 G and T = 2.5 × 10 6 K (top right panel), respectively.\n\nFIG. 8 .\n8-Comparison of light curves at four energies with (right panels) and without (left panels) photon-axion conversion. The values of the energies are 0.5 keV (solid curves), 1.5 keV (dotted curves), 2.5 keV (dashed curves), 4 keV (long-dashed curves). In each panel, light curves are normalized to the maximum value among the energies considered.\n\nFIG. 9 .\n9-Linear polarization, defined as (F O − F X )/(F O + F X ), for the cases with no photon-axion conversion (solid lines) and with photon-axion conversion (dotted lines). NS and axion parameters are the same as inFig. 2.\nTo be able to clearly observe the very narrow pencil beam, both the size of the emission region and the viewing angle would need to be smaller than (or comparable to) the width of the beam.\nDistances to isolated NSs are usually accurate to, at best, ∼ 10%, while\n\n. S L Adler, Ann. Phys. 67599Adler, S. L. 1971, Ann. Phys., 67, 599\n\n. C Alcock, A Illarionov, ApJ. 235534Alcock, C. & Illarionov, A. 1980, ApJ, 235, 534\n\n. S Andriamonje, J. Cosm. Astropart. Phys. 410Andriamonje, S. et al. 2007, J. Cosm. Astropart. Phys., 4, 10\n\n. A Arvanitaki, S Dimopoulos, S Dubovsky, N Kaloper, J March-Russell, Phys. Rev., D. 123530Arvanitaki, A., Dimopoulos, S., Dubovsky, S., Kaloper, N., March-Russell, J. 2010, Phys. Rev., D, 81, 123530.\n\n. S J Asztalos, Phys. Rev. D. 6911101Asztalos, S. J. et al., 2004, Phys. Rev. D, 69, 011101\n\n. S J Asztalos, Phys. Rev. Lett. 10441301Asztalos, S. J. et al., 2010, Phys. Rev. Lett. 104, 041301\n\n. A Avgoustidis, C Burrage, J Redondo, L Verde, R Jimenez, arXiv:1004.2053JCAP. 24preprintAvgoustidis, A. Burrage, C. Redondo, J., Verde, L., Jimenez, R. 2010, JCAP, 1010, 24, preprint arXiv:1004.2053\n\n. J Bahcall, Neutrino Astrophysics. Cambridge Univ. PressBahcall, J. 1989, Neutrino Astrophysics, (Cambridge: Cambridge Univ. Press)\n\n. F Bernardini, R Perna, E Gotthelf, G Israel, N Rea, L Stella, MNRAS. 418638Bernardini, F., Perna, R., Gotthelf, E., Israel, G. Rea, N., Stella, L. 2011, MNRAS, 418, 638\n\n. S Bogdanov, J E Grindlay, G B Rybicki, ApJ. 689407Bogdanov, S., Grindlay, J. E., Rybicki, G. B. 2008, ApJ, 689, 407\n\n. E F Brown, L Bildsten, P Chang, ApJ. 574920Brown, E. F., Bildsten, L., & Chang, P. 2002, ApJ, 574, 920\n\n. E M Cackett, J M Miller, S Bhattacharyya, J E Grindlay, J Homan, M Van Der Klis, M C Miller, T E Strohmayer, R Wijnands, ApJ. 674415Cackett, E. M., Miller, J. M., Bhattacharyya, S., Grindlay, J. E., Homan, J., van der Klis, M., Miller, M. C., Strohmayer, T. E., Wijnands, R. 2008, ApJ, 674, 415\n\n. P Chang, L Bildsten, P Arras, ApJ. 723719Chang, P., Bildsten, L., & Arras, P. 2010, ApJ, 723, 719\n\n. D Chelouche, R Rabadan, S S Pavlov, F Castejon, ApJ. 1801Chelouche, D., Rabadan, R., Pavlov, S. S., Castejon, F. 2009, ApJ, 180, 1\n\n. S De Panlis, Phys. Rev. Lett. 59839De Panlis, S. et al. 1987, Phys. Rev. Lett. 59, 839\n\n. M Durant, O Kargaltsev, G G Pavlov, arXiv:1109.1984arXiv e-printDurant, M., Kargaltsev, O., Pavlov, G. G., 2011, arXiv e-print (arXiv:1109.1984)\n\n. S Eidelman, Phys. Lett. 5921Eidelman, S. et al. 2004, Phys. Lett. B592, 1\n\n. R Gill, J S Heyl, arXiv:1105.2083Gill, R., & Heyl, J. S. 2011, arXiv:1105.2083\n\n. F Haberl, Astrophys. and Space Science. 308181Haberl, F. 2007, Astrophys. and Space Science, 308, 181\n\n. J P Halpern, E V Gotthelf, ApJ. 709436Halpern, J. P., Gotthelf, E. V. 2010, ApJ, 709, 436\n\n. W C Haxton, ARA&A. 33459Haxton, W. C. 1995, ARA&A, 33, 459\n\n. J S Heyl, N J Shaviv, MNRAS. 311555Heyl, J. S., Shaviv, N. J. 2000, MNRAS, 311, 555\n\n. J S Heyl, N J Shaviv, Phys. Rev.D. 6623002Heyl, J. S., Shaviv, N. J. 2002, Phys. Rev.D, 66, 023002\n\n. J S Heyl, N J Shaviv, D Lloyd, MNRAS. 342134Heyl, J. S., Shaviv, N. J., Lloyd, D. 2003, MNRAS, 342, 134\n\n. W C G Ho, D Lai, MNRAS. 3271081Ho, W. C. G. & Lai, D. 2001, MNRAS, 327, 1081\n\n. W C G Ho, D Lai, MNRAS. 588962Ho, W. C. G. & Lai, D. 2003, MNRAS, 588, 962\n\n. W C G Ho, A Y Potekhin, G Chabrier, ApJS. 178102Ho, W. C. G., Potekhin, A. Y., Chabrier, G. 2008, ApJS, 178, 102\n\nK Jahoda, Proc. SPIE. SPIE773277320Jahoda, K. 2010, Proc. SPIE, 7732, 77320W\n\n. A M Juett, H L Marshall, D Chakrabarty, N Schulz, ApJ. 56831Juett, A. M., Marshall, H. L., Chakrabarty, D., Schulz, N. S 2002, ApJ, 568L, 31\n\n. R Jimenez, C Peña-Garay, L Verde, Physics Letters B. 703232Jimenez, R., Peña-Garay, C., & Verde, L. 2011, Physics Letters B, 703, 232\n\n. D L Kaplan, M H Van Kerkwijk, ApJ. 705798Kaplan, D. L., van Kerkwijk, M. H. 2009, ApJ, 705, 798\n\n. D L Kaplan, M H Van Kerkwijk, ApJ. 74030Kaplan, D. L., van Kerkwijk, M. H. 2011, ApJ, 740, 30\n\n. V M Kaspi, Publ. of the Nat. Acad. of Science. 1077147Kaspi, V. M. 2010, Publ. of the Nat. Acad. of Science, 107, 7147\n\n. D Lai, J Heyl, Phys. Rev. D. 7413003Lai, D. & Heyl, J. 2006, Phys. Rev. D, 2006, 74, 13003\n\n. D Lai, W C G Ho, ApJ. 566373Lai, D. & Ho, W. C. G. 2002, ApJ, 566, 373\n\n. D Lai, W C Ho, ApJ. 588962Lai, D. & Ho, W. C. G. 2003a, ApJ, 588, 962\n\n. D Lai, W C G Ho, Phys. Rev. Lett. 9171101Lai, D. & Ho, W. C. G. 2003b, Phys. Rev. Lett., 91, 071101\n\n. J M Lattimer, M Prakash, Phys. Rep. 4421Lattimer, J. M., & Prakash, M. 2007, Phys. Rep., 442, 1\n\n. C H Lenzi, M Malheiro, R M Marinho, G F Marranghello, C Providencia, Journ. of Phys. 1541Lenzi, C. H., Malheiro, M., Marinho, R. M., Marranghello, G. F., Providencia, C. 2009, Journ. of Phys. 154, 1\n\n. A Manzali, A De Luca, P A Caraveo, ApJ. 669570Manzali, A., De Luca, A., Caraveo, P. A. 2007, ApJ, 669, 570\n\n. S Mereghetti, MmSAI. 78644Mereghetti, S. 2007, MmSAI, 78, 644\n\n. M C Miller, F K Lamb, D Psaltis, ApJ. 508791Miller, M. C., Lamb, F.K., Psaltis, D. 1998, ApJ, 508, 791\n\n. K Mori, W C G Ho, MNRAS. 377905Mori, K. & Ho, W. C. G. 2007, MNRAS, 377, 905\n\n. C Motch, V E Zavlin, F Haberl, A&A. 408323Motch, C., Zavlin, V. E., Haberl, F. 2003, A&A, 408, 323\n\n. D Page, ApJ. 442273Page, D. 1995, ApJ, 442, 273\n\nNeutron Stars in Supernova Remnants. G G Pavlov, D Sanwal, G P Garmire, V E Zavlin, ASP Conference Series. Ed. Patrick O. Slane and Bryan M. Gaensler. San Francisco: ASP271247Pavlov, G. G., Sanwal, D., Garmire, G. P., Zavlin, V. E. 2002, Neutron Stars in Supernova Remnants, ASP Conference Series, Vol. 271, held in Boston, MA, USA, 14-17 August 2001. Ed. Patrick O. Slane and Bryan M. Gaensler. San Francisco: ASP, p.247\n\n. G G Pavlov, Yu A Shibanov, J Ventura, V E Zavlin, A&A. 289837Pavlov, G. G., Shibanov, Yu. A., Ventura, J., Zavlin, V. E. 1994, A&A, 289, 837\n\n. G G Pavlov, V E Zavlin, ApJ. 5291011Pavlov, G. G. & Zavlin, V. E. 2000, ApJ, 529, 1011\n\n. R D Peccei, H R Quinn, Phys. Rev. Lett. 381440Peccei, R. D., Quinn, H. R. 1977, Phys. Rev. Lett., 38, 1440\n\n. K R Pechenick, C Ftaclas, J M Cohen, ApJ. 274846Pechenick, K. R., Ftaclas, C., Cohen, J. M. 1983, ApJ, 274, 846\n\n. R Perna, J S Heyl, L E Hernquist, A M Juett, D Chakrabarty, ApJ. 55718Perna, R., Heyl, J. S., Hernquist, L. E., Juett, A. M., Chakrabarty, D. 2001, ApJ, 557, 18\n\n. R Perna, E V Gotthelf, ApJ. 681522Perna, R. & Gotthelf, E. V. 2008, ApJ, 681, 522\n\n. R Perna, J A Pons, ApJ. 72751Perna, R. & Pons, J. A. 2011, ApJ, 727, L51\n\n. J A Pons, R Perna, ApJ. 741123Pons, J. A. & Perna, R. 2011, ApJ, 741, 123\n\n. A Y Potekhin, D Lai, G Chabrier, Ho, C G Wynn, ApJ. 6121034Potekhin, A. Y., Lai, D., Chabrier, G., Ho, Wynn C. G. 2004, ApJ, 612, 1034\n\n. J Preskill, M B Wise, F Wilczek, Phys. Lett. B. 120127Preskill, J. Wise, M. B., Wilczek, F. 1983, Phys. Lett. B, 120, 127\n\n. G G Raffelt, Lect. Notes Phys. 74151Raffelt, G. G. 2008, Lect. Notes Phys., 741, 51\n\n. G Raffelt, L Stodolski, Phys Rev. D. 371237Raffelt, G. & Stodolski, L. 1988, Phys Rev. D, 37, 1237\n\n. N Rea, MNRAS. 381293Rea, N. et al. 2007, MNRAS, 381, 293\n\n. N Rea, Science. 330944Rea, N. et al. 2010, Science, 330, 944\n\n. R W Romani, ApJ. 313718Romani, R. W. 1987, ApJ, 313, 718\n\n. V Suleimanov, A Y Potekhin, K Werner, A&A. 500891Suleimanov, V., Potekhin, A. Y., & Werner, K. 2009, A&A, 500, 891\n\n. W Y Tsai, T Erber, M Van Adelsberg, D Lai, M Van Adelsberg, R Perna, M H Van Kerkwijk, D L Kaplan, Phys. Rev. D. 12191Ap&SSTsai, W. Y. & Erber, T. 1975, Phys. Rev. D, 12, 1132 van Adelsberg, M. & Lai, D. 2006, MNRAS, 373, 1495 van Adelsberg, M. & Perna, R. 2009, MNRAS, 399, 1523 van Kerkwijk, M. H. & Kaplan, D. L. 2007, Ap&SS, 308, 191\n\n. M H Van Kerkwijk, R P Breton, S R Kulkarni, ApJ. 72895van Kerkwijk, M. H., Breton, R. P., Kulkarni, S. R. 2011, ApJ, 728, 95\n\n. W U Wuensch, Phys. Rev., D. 403153Wuensch, W. U. et al., 1989, Phys. Rev., D, 40, 3153\n\nV E Zavlin, Neutron Stars and Pulsars. Berlin HeidelbergSpringer357181Zavlin, V. E. 2009, in Neutron Stars and Pulsars, Astrophysics and Space Science Library, Vol. 357. Springer Berlin Heidelberg, p. 181\n"},"corpus_id":{"kind":"number","value":119242279,"string":"119,242,279"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1201.5390\",\n \"mag\": 2097726156,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1088/0004-637x/748/2/116\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1201.5390v1.pdf\"\n ],\n \"pdf_sha\": \"bb78eb5c48e8cdaf07f3a4e5f55767da50778065\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1201.5390\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"SIGNATURES OF PHOTON-AXION CONVERSION IN THE THERMAL SPECTRA AND POLARIZATION OF NEUTRON STARS"},"authors":{"kind":"list like","value":["Rosalba Perna","Wynn C G Ho","Licia Verde","Matthew Van Adelsberg","Raul Jimenez"],"string":"[\n \"Rosalba Perna\",\n \"Wynn C G Ho\",\n \"Licia Verde\",\n \"Matthew Van Adelsberg\",\n \"Raul Jimenez\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Conversion of photons into axions under the presence of a strong magnetic field can dim the radiation from magnetized astrophysical objects. Here we perform a detailed calculation aimed at quantifying the signatures of photon-axion conversion in the spectra, light curves, and polarization of neutron stars (NSs). We take into account the energy and angle-dependence of the conversion probability and the surface thermal emission from NSs. The latter is computed from magnetized atmosphere models that include the effect of photon polarization mode conversion due to vacuum polarization. The resulting spectral models, inclusive of the general-relativistic effects of gravitational redshift and light deflection, allow us to make realistic predictions for the effects of photon to axion conversion on observed NS spectra, light curves, and polarization signals. We identify unique signatures of the conversion, such as an increase of the effective area of a hot spot as it rotates away from the observer line of sight. For a star emitting from the entire surface, the conversion produces apparent radii that are either larger or smaller (depending on axion mass and coupling strength) than the limits set by NS equations of state. For an emission region that is observed phase-on, photon-axion conversion results in an inversion of the plane of polarization with respect to the no-conversion case. While the quantitative details of the features that we identify depend on NS properties (magnetic field strength, temperature) and axion parameters, the spectral and polarization signatures induced by photon-axion conversion are distinctive enough to make NSs very interesting and promising probes of axion physics."},"sections":{"kind":"list like","value":[{"title":"INTRODUCTION","content":"Axions with low mass (below 1 meV) are a direct prediction of the solution for the strong CP violation problem. Peccei & Quinn (1977) proposed axions as Pseudo-Goldstone bosons arising from the spontaneous breakdown of the U(1) symmetry. The Peccei-Quinn axion occupies a narrow region in the axion parameter space of mass and coupling strength. However, axions (or pseudo-scalar particles) can exist with more generic values for their mass and coupling strength, and are one type of dark matter candidate (e.g. Arvanitaki et al. 2009). This motivates efforts to constrain axion properties even if the parameters do not reach into the Peccei-Quinn regime.\nSeveral experimental avenues have been used to detect axions and constrain their properties. These fall into two categories: ground-based experiments and cosmologicalastrophysical efforts. Current limits on the axion mass (10 −6 < m a /eV < 10 −3 ) come from the latter approach. The cooling rate observed in supernova 1987A implies a direct axion mass upper limit: if axions are more massive, then cooling would be predominantly due to axions rather than neutrinos (Eidelman et al. 2004). The lower limit comes from cosmological considerations: if axions constitute dark matter, then they must be cold and not overclose the Universe (Preskill et al. 1983). Constraints on the coupling constant can be obtained with both astrophysical observations and ground-based experiments. The existence of horizontal branch stars with extended morphology, that is, stars with a range of surface tem-peratures, constrains the coupling constant of axion-photon interaction to be g a < 10 −10 GeV −1 (Raffelt 2008). Groundbased experiments also place similar constraints, and future experiments are anticipated that will obtain stronger limits (Andriamonje et al. 1983;De Panlis et al 1987;Wuensch et al. 1989;Asztalos et al. 2004Asztalos et al. , 2010.\nRecent developments in understanding radiation dimming (Lai & Heyl 2006;Jimenez et al. 2011) and modification of the polarization pattern (Gill & Heyl 2011) have put further constraints on the m a − g a plane. Limits on cosmological radiation dimming have also been used to place constraints on much lighter axion masses (Avgoustidis et al. 2010).\nThe potential use of neutron stars (NSs) as probes of axion parameters has been noted by a number of authors (e.g., Lai & Heyl 2006;Chelouche et al. 2009;Jimenez et al. 2011). In particular, occultation and eclipsing or transiting of a companion and dimming of the spectrum can potentially produce observable signatures. However, the phenomenon of occultation (when a background object passes through the influence of a NS magnetic field) was deemed to have too low a probability to be astrophysically relevant. In addition, there are no known binary systems involving a NS that are detached enough to yield a clean constraint (Jimenez et al. 2011). On the other hand, Lai & Heyl (2006) showed how the thermal X-ray spectrum of a NS (modeled as simple blackbody emission) can be modified in a significant way by the presence of axions. Spectral features due to axion-photon oscillations have been discussed by Chelouche et al. (2009), and shown to be relevant for highly magnetized NSs in the sub-mm wavelength range for an observationally interesting range of axion parameters.\nIn this paper, we focus on the soft X-ray band in which NSs are routinely observed. The goal of our work is to perform a detailed calculation of the observable effects that the presence of axions would have on NS spectra, light curves, and polar-ization. Our analysis goes beyond previous computations in the literature for the subject by coupling the calculation of photon-axion conversion probabilities with detailed models of magnetized atmospheres and general relativistic models for emission from the NS surface. For the local surface brightness of the NS, we use accurate magnetized atmosphere models, which yield the angular and energy-dependent photon intensities in the two photon polarization modes. These models include the effect of mode conversion due to the vacuum polarization. The local intensity is then imported into a code for computing spectra and light curves from finite emission regions on the NS surface. The analysis incorporates the effects of gravitational redshift and gravitational light deflection and allows for arbitrary viewing geometries. The resulting spectra are then modified by the photon-axion conversion process, accounting for both the energy and angular dependencies of the conversion probability. Note that we solve the conversion between photon modes and the conversion between axions and photons independently, which is shown by Lai & Heyl (2006) to be a very good approximation for the regimes we explore in this paper. With our calculations, we are able to produce realistic models for the spectra, light curves, and polarization signal of NSs, and use these to identify specific features that carry the telltale signs of photon-axion conversion. We discuss these signs in the context of NS observations in §5.2.\nOur paper is organized as follows: In §2, we discuss the formation of the NS thermal spectrum within a magnetized atmosphere, placing particular emphasis on the emergence of two photon polarization modes. In §3, we summarize the calculation of the photon-axion conversion probability for a magnetized object. The computation of the 'observed' thermal spectrum, light curves, and polarization with and without photon-axion conversion is described in §4, and the results are presented in §5. We summarize our findings in §6."},{"title":"PHOTON PROPAGATION IN THE ATMOSPHERE OF","content":""},{"title":"MAGNETIZED NEUTRON STARS","content":"In this section, we describe the main properties of magnetized NS atmospheres and the processes that influence photon propagation in them. Axions do not directly couple to the processes described in this section. However, the details presented below are important for calculating the polarized radiation emerging from the NS atmosphere. As discussed in §3, axions couple only to photons that are linearly polarized in the plane formed by the directions of propagation and the magnetic field; the physics of photon propagation in the NS atmosphere determines the relative fraction of such photons.\nRadiation emerging from the hot surfaces of magnetized NSs is the result of radiative transfer through a layer of ionized plasma. Due to the strong gravitational field of the NS, heavy elements settle quickly (Alcock & Illarionov 1980;Brown et al. 2002), and this atmospheric layer is composed of light elements, unless nuclear burning on the NS surface consumes the light elements (Chang et al. 2010, and references therein). Only recently have self-consistent magnetic atmosphere models for partially ionized hydrogen and mid-Z elements been constructed (see Mori & Ho 2007;Ho et al. 2008, and references therein for details; see also Suleimanov et al. 2009). For simplicity, we focus our calculations on fully ionized hydrogen atmospheres (see below). In this case, the scale height is\nwhere T s is the effective temperature of the star, g * is the gravitational acceleration, and δ is the angle that the ray makes with the surface normal.\nIn a NS atmosphere, radiation propagates in two distinct polarization modes: the extraordinary (X) mode and ordinary (O) mode, which are very nearly linearly polarized perpendicular and parallel, respectively, to the plane formed by the directions of propagation and the magnetic field. The X mode absorption opacity is reduced by a factor of (E Be /E) 2 relative to the O mode opacity, where the electron cyclotron energy is E Be = 115.8B 13 keV and B 13 = B/10 13 G. Thus the X mode radiation decouples from deeper, hotter layers in the NS atmosphere than the O mode radiation. For rays propagating at intermediate angles θ p relative to the magnetic field, the net emission is significantly polarized and dominated by the X mode.\nTo produce surface emissivities for the X and O photon modes, we use the model of van Adelsberg & Lai (2006), which quantitatively incorporates the effects of vacuum polarization on the radiative transfer for fully ionized atmospheres with an external magnetic field parallel to the axis of symmetry. This model is reasonably accurate for temperatures T s 10 6 K. At lower temperatures, partial ionization of atomic species becomes important, and partially ionized models are needed (see Ho et al. 2008, and references therein). Therefore, we use the fully ionized model only in the high temperature regime. It should be noted, however, that this is the effective temperature of the star, and that the measured value at the observer is lower due to gravitational redshift. Colder stars are less interesting for the purpose of this study: First, being much dimmer, their measured spectra are generally of lower statistical significance for detailed spectral studies. Second, and more importantly, since the fraction of Omode photons increases with temperature, spectra of colder stars are less sensitive to axion effects (see the discussion below).\nFor magnetic field strengths B ∼ 4 × 10 13 G, there is a significant contribution from vacuum polarization to the dielectric properties of the medium (Adler 1971;Tsai & Erber 1975). A ray traversing the density gradient of a magnetized NS atmosphere will eventually encounter a layer of the medium in which the plasma and vacuum contributions to the dielectric tensor are of the same magnitude, leading to a resonance. For the models studied in this paper, the resonance occurs at lower density than the X and O mode decoupling depths (where the optical depth τ ν ≈ 1) for most photon energies and propagation angles. In addition, the integrated opacity across the vacuum resonance is negligible at these field strengths. A discussion of vacuum resonance effects on the mode opacities is given by Ho & Lai (2003). As shown by Lai & Ho (2002) and Lai & Ho (2003a), there is coherent mixing of the modes at the resonance, analogous to the Mikheyev-Smirnov-Wolfenstein (MSW) effect for neutrino oscillations (see, e.g., Bahcall 1989;Haxton 1995). Using the geometric optics approximation, and neglecting damping terms in the dielectric tensor (which only affect the width of the resonance), the amplitudes of the two modes, A O and A X , evolve according to\nwhere (e.g. Lai & Heyl 2006)\nIn equations (3)-(5), θ p is the angle between the magnetic field and the photon direction (defined to be the z direction), ω is the photon frequency, v e = (ω pe /ω) 2 and u e = (ω ce /ω) 2 , where ω pe and ω ce are the electron plasma and cyclotron frequencies, respectively. The B-field dependent functions ξ(b) and χ(b) are (Potekhin et al. 2004)\nand\nwhere α f is the fine structure constant and b is the magnetic field strength in units of the critical field B c = m 2 e c 3 /(eh) = 4.414 × 10 13 G. The equations above assume that v e ≪ 1 and u e ≫ 1. Since v e ∼ 10 −3 (ρ/1 g cm −3 ) and u e ∼ 10 4 (B/10 13 G) at 1 keV, these constraints are easily satisfied throughout the soft X-ray band in which NS thermal spectra typically peak.\nEquation (2) determines the evolution of the photon mode amplitudes through the atmosphere (see Ho & Lai 2003;Lai & Ho 2003, for details). The amplitudes are related to the specific intensities of each mode by I X ,O ∝ |A X ,O | 2 . As the photons encounter the resonance, the polarization mode ellipticities, K X ,O (∝ E x /E y ), experience a discontinuity at the resonance. As discussed in Ho & Lai (2003), an alternative description of the polarization state uses ellipticities K ± , which vary continuously through the vacuum resonance. If the variation of the atmosphere density profile is gradual enough, an adiabatic condition is satisfied (see below), and the mode properties are fixed along the curves described by K ± . Before the resonance, a K − photon corresponds to an X mode polarized photon, while a K + photon corresponds to an O mode photon. After the resonance, the correspondence between K ± and K X ,O switches, so that under adiabatic conditions the character of each modes changes. This is analogous to the correspondence between flavor and mass eigenstates in the MSW mechanism for neutrino mixing. The adiabatic condition is set by evaluating the coefficient matrix in Eq. (2) at the resonance and setting the magnitude of the diagonal term equal to the off-diagonal term (Lai & Ho, 2002;2003a). In the non-adiabatic regime, the diagonal terms have a much larger magnitude than the off-diagonal terms, leading to a set of decoupled equations for A X ,O . In the adiabatic regime, the offdiagonal terms dominate over the diagonal ones, leading to a coupled set of equations for A X ,O and mixing of the polarization states.\nThe conversion probability P c is derived from Eq. (2) by setting the radiation completely in one mode (e.g., A O = 1, A X = 0) and evolving the equations to z → ∞. If the atmosphere density profile is linear in z, then v e (∝ ρ) is a linear function of z, and the conversion fraction is given by the well known Landau-Zener formula (see Lai & Ho, 2002;2003a, and the references therein):\nThe adiabatic energy E ad is defined by\nwhere E Bi is the ion cyclotron energy, and f −2 B is a slowly varying function of the magnetic field (see van Adelsberg & Lai, 2006, and the references therein). Because the vacuum resonance is very narrow in energy (and hence density, for a given photon energy and propagation angle), the linear approximation for the density profile is always satisfied for the atmosphere models considered in van Adelsberg & Lai (2006). In addition, the conversion occurs on a distance scale much less than the scale height of the atmosphere; thus the conversion process can be regarded as occurring exactly at the resonance density, and the asymptotic solution to Eq. (2), in the form of the conversion probability Eq. (8), can be used to treat mode conversion effects. This was shown to be an accurate approximation in van Adelsberg & Lai (2006), and we adopt it here in our calculations of mode conversion.\nA photon with energy E encounters the vacuum resonance at a density\nFor a given energy, ρ V is the density at which the vacuum and plasma contributions are of equal magnitude (i.e., when σ 11 = σ 22 ). If E > E ad , then conversion between the modes at this depth is effective. If I X and I O are the values of the mode specific intensities at the resonance, then after conversion they become:\nThis process is automatically incorporated into the models with B = 10 13 G and B = 4 × 10 13 G (T s = 5 × 10 6 K) using the code of van Adelsberg & Lai (2006). For models with B = 4 × 10 12 G, we use the code of Ho & Lai (2001). This code does not include partial mode conversion, but we have included the effect according to the following prescription: The code of Ho & Lai (2001) produces atmosphere profiles ρ d and T d at a set of discrete points in Thomson optical depth, τ d , where d = 1, . . . , D. Since the conversion process occurs at lower densities than both the X and O mode decoupling depths, it does not affect the atmosphere structure and can be treated as a modification operating on the emergent specific intensities. For a given photon energy and propagation angle, we calculate the vacuum resonance density using eq. (10), and then locate a grid point i, such that ρ i ≤ ρ V ≤ ρ i+1 . We then compute a linear approximation to the resonance Thomson depth τ V , using the formula:\nwhere ∆τ = τ i+1 − τ , and ∆ρ = ρ i+1 − ρ i . We relate the resonance temperature T V to the Thomson depth and density in a hydrostatic atmosphere using\nand calculate the atmosphere scale height at the vacuum resonance according to Eq. (1). Finally, the adiabatic energy and mode conversion probabilities are calculated according to Eqs. (8) and (9). The emergent intensities are then mixed using Eq. (11). When the surface magnetic field exceeds the quantum critical field B c ∼ 4 × 10 13 G, there are significant vacuum contributions to the dielectric properties of the medium. At the resonance, if the radiation energy is much greater than the adiabatic energy [Eq. (9)], there is a high probability of conversion between the X and O modes. For magnetic field strengths B 7 × 10 13 G, the vacuum resonance lies between the photospheres of the X and O modes (where the mode opacities τ ν ≈ 1). The modes mix at the resonance with a fraction of Xmode photons converting into O-mode photons and vice-versa according to the probability in Eq. (8). The O-mode photons, after converting into X-mode photons at the resonance, decouple from the atmosphere. The X-mode photons that con-vert to the O-mode, subsequently interact strongly with the atmosphere, as they now experience the large opacity for O mode polarization. Any X-mode photons that do not convert encounter a large integrated opacity at the resonance (under typical conditions; see Ho & Lai, 2003). Thus, one of the net effects of vacuum polarization is to shift the location of the X-mode photosphere to the resonance density, which is at lower density and temperature than the original X-mode photosphere. As a result, the high energy spectrum is softer, and the relative fraction of X to O mode photons decreases. At magnetic field strengths B ≪ 7 × 10 13 G, radiation in both modes decouples from the NS atmosphere before encountering the vacuum resonance. In this case, mode conversion es- sentially switches the photospheres of the two modes, with the O-mode photons now being produced in the deeper (hotter, denser) layers of the star. Therefore, if the emission is mostly polarized in the X mode, for E > E ad , the plane of polarization switches to the O mode after resonant conversion (Lai & Ho 2003a, 2003b. The above considerations are particularly relevant for the subsequent mixing between O-mode photons and axions, as emission which would otherwise have a small fraction of O mode radiation now predominantly consists of O mode photons, depending on the magnetic field strength and NS geometry (see van Adelsberg & Perna 2009, for more discussion). Figure 1 shows the fractional contribution to the emer-gent flux of the X and O modes for three values of the field strength, B = 4 × 10 13 G, 10 13 G, and 4 × 10 12 G. The effective temperature is T s = 5 × 10 6 K in all cases. Each panel shows the results for a separate emission angle δ relative to the magnetic field. In general, the X mode dominates the emergent flux at low energies, except around the ion cyclotron feature E Bi . According to the formula for E ad [eq. (9)], there is an angle-dependent point at which mode conversion becomes efficient and the radiation is composed mostly of O mode photons. Since E ad ∝ tan 2/3 θ p = tan 2/3 δ for a magnetic field perpendicular to the surface, at larger values of δ, mode conversion is efficient only at high energies. Note that as θ p → 0, the properties of the two modes become similar, and the dif-ference in the opacities decreases. Figure 2 shows the fractional contributions to the emergent flux of the X mode and O modes, for a fixed magnetic field strength, B = 4 × 10 12 G, at three effective temperatures T s = 5 × 10 6 K, 2.5 × 10 6 K, and 10 6 K. The general trend is an increase in the O/X flux ratio as the temperature increases. This is straightforward to understand: since a higher temperature atmosphere produces more photons with higher E, the average ratio E Be /E decreases. Thus, the X-mode opacity becomes more similar to the O-mode, leading to an increase in the O/X ratio."},{"title":"PHOTON-AXION CONVERSION IN THE MAGNETIC FIELD OF NEUTRON STARS","content":"In the following we discuss the effect of photon-axion mixing as a photon beam propagates through the dipolar field of a magnetized NS. Firstly, we note that an important assumption of our analysis is that the conversions between the O and X modes and between the O mode and axion can be treated separately (each as a two-state mixing) rather than as a coupled three-state mixing. This assumption is valid as long as the vacuum resonance and the photon-axion resonance are well separated. This was shown to be the case by Lai & Heyl (2006) for highly magnetized NSs at soft X-ray energies and axion masses m a 10 −3 eV. Thus, the assumption of decoupled mixing is satisfactory for our regime of interest, and it will be adopted in our analysis.\nOur formalism for photon-axion conversion in a magnetized medium follows that of Raffelt & Stodolski (1988). To be consistent with the notation adopted in the previous section for the two photon polarization states, we denote the amplitudes of photons polarized parallel and perpendicular to the magnetic field as A O and A X , respectively. Under the assumption that the length scale over which the magnetic field varies is much larger than the photon and axion wavelengths, the evolution of the photon and axion fields in the z direction, with frequency ω, is given by\nin units withh = c = 1, where a is the axion field. The diagonal matrix elements are given by\nwhere m a is the axion mass and ξ(b) is defined by Eq. (6). The off-diagonal component is given by\nwhere g a describes the coupling strength between the O-mode photons and the axions, and r is the distance from the stellar surface. Since mixing occurs only between the axion field and the O-mode photon field, we neglect the evolution of the X-mode field in the discussion that follows. The mixing strength can be measured in terms of the ratio between the off-diagonal term and the difference between the diagonal terms in Eq. (14); a useful parametrization can be made in terms of the mixing angle θ m defined by\nWe begin our analysis by considering a region of space over which the magnetic field can be approximated as homogeneous. Following Raffelt & Stodolsky (1988), the discussion is greatly simplified if we define phases relative to the unmixed component, and neglect a common phase. The solution can be found by first performing a matrix rotation to an eigenstate basis where the propagation matrix is diagonal, propagating the two eigenstates independently, and then rotating back to the photon-axion basis. This yields an evolution equation for the mixing components:\nwhere\nand\nand the plus and minus sign indicates ∆ ′ and ∆ ′ a , respectively.\nThe solution provided by Eq. (19) assumes that the field is homogeneous. The generalization for photon propagation within the dipolar field around the NS, with B ∝ r −3 , can be made by assuming that the total evolution operator is the product of evolution operators within spatial slices of width ∆z j wherein the B field can be approximated as constant.\nWithin each shell, we use Eq. (19) to evolve the photon and axion amplitudes from A O (z j ) and a(z j ) in shell j to A O (z i ) and a(z i ), where z i = (z j + ∆z j ) is the subsequent shell. We tested the code for convergence and found that, for ∆z j ≤ 0.07 km, the amplitudes at infinity converge for the range of B, m a , and g a of interest. Fig. 3 shows the conversion probability for combinations of the axion parameters m a , g a in the soft X-ray band of interest for NS observations. The probability is plotted as a function of photon energy and propagation angle θ p . There is a range in the parameter space of axion mass, coupling strength, and NS properties (such as surface magnetic field strength) where the photon-axion conversion probability is non-negligible. This implies that photon-axion conversion can leave distinctive signatures in NS spectra, making them interesting sources for constraining axion physics. In the next section, we perform a detailed analysis of such signatures. The observed spectrum of a NS is obtained by summing the local emissivities over the emission region. For simplicity, we consider an emitting region that has a constant temperature and magnetic field strength, with the field direction along the surface normal (e.g., a hot magnetic polar cap). Such a region allows us to directly examine the spectral dependencies of photon-axion conversion on these physical quantities. It is often the case that the emission is dominated by a region smaller than the entire star surface (i.e., a hot spot). The spectrum and lightcurves will then be functions of the viewing angle α between the observer and spot axes. As we will show FIG. 3.-Photon-axion conversion probabilities for the magnetic field strength B = 10 13 G and various combinations of the coupling strength, ga, and axion mass, ma. The probability is shown as a function of photon energy and angle θp between B and the direction of photon propagation. In the left panels, the dependence on the coupling strength is explored for fixed axion mass ma = 2 × 10 −6 eV (from top to bottom: ga = 10 −8 , 10 −9 , 10 −10 GeV −1 ). The right panels show the dependence of the conversion probability on the axion mass for fixed coupling strength ga = 10 −8 GeV −1 (from top to bottom: ma = 7 × 10 −7 , 6 × 10 −6 , 1.7 × 10 −5 eV). The dependence of the probability on B is weak for the range considered here (4 × 10 12 − 4 × 10 13 G), so we only display results at the intermediate field strength.\nin § 5, phase dependent methods add considerable diagnostic power for detecting the presence of axions.\nThe calculation of phase-dependent emission from an extended region on the NS surface follows the formalism developed by Pechenick et al (1983), with generalizations by Perna & Gotthelf (2008; see also Bernardini et al. 2011, for a similar geometry). We define the time-dependent rotational phase γ(t) as the azimuthal angle subtended by the magnetic dipole vector µ around the axis of rotation. It is related to the modulus of the NS angular velocity, Ω(t), by γ(t) = Ω(t)t. We choose the coordinate system so that the observer is located along the z axis; the inclination angle of the rotation axis,Ω, with respect to the line of sight is denoted by α R , while the angle between the magnetic dipole vector and the rotation an-gle is denoted by α M . Thus, the angle between µ and the line of sight is given by\nThe emission region is assumed to be circular, with opening angle β, and is centered around the magnetic dipole axis. Hence, α(t) also indicates the time-dependent angular separation between the spot axis and the line of sight as the star rotates. We describe points on the NS surface by means of the polar angle θ, and the azimuthal angle, ϕ, in spherical polar coordinates (see Fig. 1 in Perna & Gotthelf 2008 for a visual representation of this geometry).\nThe emission region is restricted to θ ≤ β for α = 0, while for α = 0 it is identified by the conditions\nand by the condition\nIn the latter case, the outer boundary of the spot, θ ⋆ (α, ϕ, β), must be determined numerically from the expression cos β = sin α sin θ ⋆ cos ϕ + cosα cos θ ⋆ .\nDue to the intense gravitational field of the NS, photons emitted at the NS surface are substantially deflected as they travel to the observer. A photon emitted from colatitude θ will reach the observer (at infinity) if emitted at an angle δ with respect to the surface normal; the relation between the two angles is given by the ray-tracing function\nwhere x ≡ sin δ, R and M are the NS radius and mass, respectively, and R s ≡ 2GM/c 2 is the Schwarzchild radius.\nThe spectrum at the observer is obtained, as a function of the viewing angle, α(γ), by integrating the local emission over the observable surface; this procedure yields the flux seen by an observer at a distance D ≫ R. Accounting for the gravitational redshift of the radiation, this integral takes the form (Page 1995and generalizations by Pavlov & Zavlin 2000, Heyl et al. 2003:\nin units of photons cm −2 s −1 keV −1 . In the equation above, the index j represents any of the Stokes parameter (U, V, Q, I), and E ∞ is the energy as seen by the distant observer. The energy emitted at the stellar surface is given by E ∞ e −Λs , with\nThe local specific intensity I j is equal to zero outside of the boundaries for θ and ϕ defined by equations (24)- (26). The observed flux depends on the geometric angles α R and α M through the angle α in Eq. (23). In the following, we consider an orthogonal rotator, for which α R = α M = 90 0 . This implies that α(t) = γ(t). Given that the emission region is centered around the dipole axis, the angle α will be identified with the angle θ p between the photon propagation and magnetic field directions (see §3).\nIf photon-axion conversion occurs with probability P(E ∞ , α), the phase-resolved spectrum is given by\nFor a fixed energy (or integrated over an energy band), α]) yields the light curve as a function of the viewing angle α. The phase averaged spectrum is readily obtained from\nFinally, the linear polarization is computed as\nTo simplify our notation, we will omit the subscript '∞ ′ from the observed energy. In the discussion that follows, all quoted energies are redshifted energies for a NS of mass M = 1.4M ⊙ and radius R = 10 km. Before concluding this section, we need to remark that Eq. (31) and Eq.(33) assume that the O and X modes remain decoupled once they have emerged from the atmosphere of the NS. The evolution of the modes in the changing dipole magnetic field of the star was computed in papers by Heyl & Shaviv (2000, Heyl, Shaviv & Lloyd (2003), Lai & Ho (2003) and van Adelsberg & Lai (2006). All of the above studies found that, in the birefringent, magnetized vacuum near the star surface, the photon modes are decoupled. In this region, the mode eigenvectors evolve adiabatically along the changing direction of the magnetic field. Further from the star, the mode evolution continues to be adiabatic, up to the \"polarization limiting radius\", R pl , which is defined as the distance at which the polarization modes mix (and are frozen thereafter). Heyl, Shaviv & Lloyd (2003) derive the following expression for R pl\nwhere E 1 ≡ E/(1 keV) and B 13 ≡ B/(10 13 G). For radii larger than R pl , the mode eigenvectors are fixed, and the polarization no longer evolves (i.e., it is the observed polarization [Ω/(2π)] −1 sin 2 α 1/6 10 7 cm. The scaling of R pl in this equation differs slightly from that in Eq. (34) due to different adiabatic conditions applied by the two sets of authors. Heyl et al. (2003) constrain the change in the relative difference in the mode indices of refraction compared to the magnitude of that difference [see their Eq. (9)], while van Adelsberg & consider the distance at which the mode evolution equations have equal diagonal and off-diagonal terms [see their Eq. (62)]. This results in slightly different magnitudes and scaling relations; for slowly rotating NSs, the precise numerical value of R pl does not change the value of the observables, and both approaches result in the same physics. except for very rapidly rotating stars (i.e., with frequencies of tens of milliseconds) the coupling results in only a very small quantitative change. For example, for a star with period ∼ 0.1 sec (B ∼ 10 12 G, E ∼ 1 keV), the mode intensities I O and I X (∝ |A O | 2 , |A X | 2 ) couple at the level of 4%, and the amount of mixing further decreases at longer periods. Therefore, for the purpose of our study and the observational tests we propose (requiring NSs with relatively long periods in order to perform time-resolved spectroscopy), the formalism that we use provides a very good description of the observed flux and the linear polarization fraction computed here (see also Lai & Heyl 2006). We further note that, if the radius of the emission region is much smaller than the polarization limiting radius, then summing over the Stokes parameters at the detector is well approximated by summing over a mode (X or O) at the NS surface (Heyl et al. 2003). Since our emission region (a hot spot) is always much smaller than R pl for the B field strengths and energies under consideration, we use this approximation in our calculations."},{"title":"RESULTS: THEORETICAL PREDICTIONS AND OBSERVATIONAL TESTS","content":""},{"title":"Theoretical predictions","content":"We present the results of our calculations of NS spectra, light curves, and polarization signals, with and without photon-axion conversion. We explore the dependencies on the stellar magnetic field strength and effective temperature, with particular emphasis on the magnetic field since it has a stronger influence than the temperature on the functional form of the modal energy dependence (see Figs. 1 and 2). We consider emission from a circular region, 10 deg in angular size, that is centered on the magnetic pole of the star. Such hot spot emission is common in NSs, and exhibits a strong dependence on phase. Given the significant angle dependence of the O and X mode intensities (see Figs. 1 and 2), the study of emission from a finite region allows us to extract more information on the effects that photon-axion conversion has on the observable properties of NSs.\nAs shown in Fig.3, the conversion probability is a strong function of the axion mass, m a , and the coupling constant, g a . For the thermal spectra of NSs to be affected by photonaxion conversion, the probability of this process must be non-negligible in the soft X-ray band, where the NS thermal spectrum peaks. Therefore, we perform a detailed analysis of the consequences of photon-axion conversion on NS spectral properties for a particular set of axion parameters; these parameters are chosen to yield a substantial conversion probability in the soft X-ray band. We choose the values m a = 2 × 10 −6 eV, and g a = 10 −8 GeV −1 ; the study of this specific case will uncover the main signatures of axion conversion. Once these are identified, we explore a wider area of m a − g a parameter space to identify the main region that can be probed by means of NS observations in the X-ray band. Fig. 4 shows phase-resolved spectra for three magnetic field strengths, B = 4 × 10 13 , 10 13 , and 4 × 10 12 G. For each value of B, we plot the spectra at two phases, coinciding with the angles between the line of sight and the magnetic field axis for the chosen geometry. We show angles α = 2 deg and a typical large viewing angle α = 60 deg. The former angle is essentially a 'phase-on' spectrum (i.e. a spectrum observed in correspondence to the maximum of the pulsation); we do not plot the results for α = 0 • since the conversion probability in that case is zero. For each combination of viewing angle and magnetic field strength, the figure shows the fluxes from the O and X-modes separately, as well as the total composite spectra. Results are plotted with and without photon-axion conversion.\nThe relative fraction of O and X mode photons, as a function of energy and angle, plays a fundamental role in determining the effects of axions on phase-dependent spectra and light curves. Before discussing the spectral effects of axions in further detail, we comment on the energy and angular dependence of the two photon modes. Although only the Omode photons are affected by axion conversion, the properties of the X-mode photons also matter for our purposes because of vacuum polarization-induced mode conversion. The X-mode has a complex beam pattern, consisting of the sum of a narrow \"pencil\" beam and a broad \"fan\" beam. The width of the pencil beam scales as ∼ (E/E Be ) 1/2 , decreasing with magnetic field strength (Pavlov et al. 1994). Therefore, the modal intensities vary from being approximately equal to significantly different between widely separated emission angles.\nFor a phase-on spectrum, the intensities of the O and X modes peak at roughly similar energies. Thus, the presence of photon-axion conversion results in a dimming of the spectrum without significant spectral distortion. The situation is different at larger viewing angles, as evident in the α = 60 deg panels of the Fig. 4. In these cases, the X mode dominates the spectrum at (redshifted) energies 1.4 keV, while the O mode dominates at higher energies; thus, an inversion of the dominant spectral mode occurs. Suppression of the O mode by photon-axion conversion therefore produces not only a dimming of the spectrum but also a spectral distortion, suppressing the high-energy tail.\nSince axion conversion occurs at larger distances from the star than mode conversion due to the vacuum resonance, only O-mode photons are affected as they emerge from the star atmosphere. This is shown in Fig. 4, where the X-mode photons are untouched by photon conversion, while the O-mode intensities are noticeably suppressed. The overall effect on the spectrum is that of flux suppression. However, since the O/X flux ratio, as well as the conversion probability, has an energy dependence which varies with viewing angle, the shape of the spectrum is also affected, causing a shift of the energy peak. While the shift is very small at phase-on viewing angles (since the O-modes and X-modes have a similar energy-dependent intensity for α ∼ 0 deg), it is very pronounced at wider viewing angles, where the O-mode largely dominates the intensity at high energies. In this case, enhanced suppression of high energy photons produces an effective shift of the peak of the spectrum towards lower energies. Figures 5 and 6 quantify the phase-dependent spectral effects discussed above. For the same values of magnetic fields and axion parameters considered in Fig. 4, Fig. 5 shows the ratio of the X-ray flux in the 0.8-10 keV band photon-axion conversion to that without conversion. As expected, at all phases, the ratio F Ax X /F noAx X is smaller than 1. This is because the contribution to the flux from the O-mode photons is non-negligible at all viewing angles with respect to the magnetic field axis. The precise value of the flux ratio, however, shows a significant dependence on the viewing angle, with a mimimum around α ∼ 40 − 50 deg. This is due to the O-mode flux, which is a large fraction of the total flux at E 1 keV, for viewing angles in the 30 − 70 deg range.\nThe left panels of Fig. 6 show the (redshifted) energy where the spectrum peaks, E peak , as a function of viewing angle. We -Comparison between phase-resolved spectra of NSs with (thick/red lines) and without (thin/blue lines) photon-axion conversion. In each case, the dotted lines show the flux contribution from the O mode, while the dashed lines display the contribution from the X mode, which is unaffected by conversion. The solid lines show the total spectra. In all panels, the effective temperature of the star is Ts = 5 × 10 6 K and the axion parameters are ma = 2 × 10 −6 eV and ga = 10 −8 GeV −1 , while the B-field strength has three values from top to bottom. The spectra are shown for two angles α between the observer and the dipole axis.\nconsider only angles α ≤ 100 deg since the flux from larger angles, which is still visible due to relativistic light deflection, is too low for phase-resolved spectroscopy. We begin by examining the behavior of E peak without photon-axion conversion. At the magnetic field strengths for the models in this paper, the spectral peak of the total (X and O mode) specific intensity follows the beam pattern of the X mode: a narrow pencil beam at small angles and a broad fan beam at large angles. This is because conversion between the X and O modes occurs above the photosphere of both modes and thus does not change the total intensity. The observed variation of E peak with viewing angle deviates somewhat from the emission beam pattern due to the narrowness of the pencil beam (a few deg in width around δ = 0 • ), the gravitational deflection of photon trajectories, and the finite size of the emitting region, which smears out features from nearby emitting points 6 .\nWhen axion conversion is taken into account for B = 4 × 10 12 − 4 × 10 13 G, a common feature is a shift of the peak FIG. 5.-Ratio of X-ray flux (in the [0.08-8] keV band) with axion conversion to that without axion conversion. Axion parameters have the same values as in Fig. 4, while the NS temperature and magnetic field are indicated for each curve, with the B 12 ≡ 10 12 G and T 6 = 10 6 K. energy to lower values at all viewing angles except for those close to phase-on and edge-on. For α ≈ 0 deg, the intensity of the O mode photons peaks at similar or slightly lower energies than that of the X-mode photons (see Fig. 4); hence, suppression of O mode photons primarily affects the amplitude of the spectral intensity, as opposed to the location of the spectral peak. In fact, because the O-mode intensity peaks at lower energies, especially at lower fields, at small angles there is actually a shift of the peak towards higher energies after photon-axion conversion.\nAnother notable signature in the variation of E peak (α) is a strong decrease (or flattening in the higher B-field case) for viewing angles α ∼ 30−50 deg. Around these angles, the contribution from the O mode photons is dominant at high energies, and the reduction of these photons through photon-axion conversion results in a significantly cooler spectrum. As the viewing angle increases further, the X-mode photons become increasingly dominant (see Fig. 1), and the reduction of high energy photon lessens. As a result, the effective temperature of the spectrum -as measured through E peak -approaches the value it would have without photon-axion conversion.\nTo examine the dependence of our results on the effective temperature T s of the star, we consider the values T s = 10 6 , 2.5 × 10 6 , and 5 × 10 6 K (where T s is the unredshifted temperature at the stellar surface) for the magnetic field B = 4 × 10 12 G. We find that the shift in E peak due to axion conversion (compared to the case with equal temperature but no conversion) is more pronounced at higher temperatures. This is not surprising given that the relative fraction of O to Xmode photons is larger at higher temperatures.\nThe right panels of Fig. 6 show the phase-resolved effective area [i.e., A eff (α) ∝ F(α)/E 4 peak (α)] that an observer measures when fitting spectra with a blackbody model. To eliminate the dependence of this diagnostic on the often poorly determined distance to the NS, we normalize A eff (α) to its value near phase-on. Since the photon-axion conversion probability is zero for precisely α = 0, we normalize to a small angle α = 2 deg which represents a \"phase-on\" spectrum as an average of small angles around α = 0 deg. When photon-axion conversion is included, the variation of the effective area with phase results from the interplay of two effects: a decrease in A eff due to flux dimming and an increase in A eff due to decreases in the peak energy and inferred temperature. Because of the E 4 peak -dependence, the latter effect dominates the former. This is particularly the case for lower fields, which has the strongest E peak shift with phase.\nThe variation of inferred effective area with phase is a particularly interesting and peculiar feature of photon-axion conversion. The specific details of the variation, however, depend on NS and axion parameters. In order to explore a wider region of m a − g a parameter space, we compute the effective area for the phase-averaged spectrum (A ave eff ) with photon axion conversion, normalized to that without axion conversion. This represents a measure of the \"correction\" that photonaxion conversion makes to the inferred effective area (as determined through measurements of phase-averaged flux and effective temperature). For our purposes, the phase-averaged spectrum of a hot spot provides an adequate approximation for emission from the entire NS surface. The results of this calculation are displayed in Fig. 7 for a combination of magnetic fields and effective temperatures: B = 4 × 10 12 G, T s = 5×10 6 K; B = 4×10 12 G, T s = 2.5×10 6 K; and B = 4×10 13 G, T s = 5 × 10 6 K. It is interesting to note that, in all cases, there are regions of parameter space in which the inferred effective areas are smaller/larger than the uncorrected areas. This is expected given the two counteracting effects described above. The m a − g a region shown in Fig. 7 is one for which, at typical NS magnetic fields, there is a larger probability of photonaxion conversion in the soft X-ray band (where NSs are routinely observed). The correction to the effective area becomes negligible for the largest values of m a and g a −1 displayed in Fig. 7. This is expected since the photon-axion conversion probability tends to zero for these values at the keV energies of interest here. The results summarized by this figure have a strong potential to constrain axion signatures for NSs with good distance measurements (see the discussion in §5.2).\nIn Fig. 8, we examine the effects of photon-axion conversion on NS light curves at different energies, for the same set of axion parameters as in Fig. 4. The lightcurves are all normalized to the maximum value of the flux among the set of energies considered (0.5, 1.5, 2.5, and 4 keV). The main effect of photon-axion conversion is the relative suppression of the flux at higher energies with respect to that at lower energies.\nFinally, we study the effect of photon-axion conversion on the magnitude of linear polarization [see Eq. (33)]. For consistency with Fig. 4, we consider the same set of parameters. The polarization signal with and without conversion is shown in Fig. 9. Since the main effect of axion conversion is suppression of O-mode photons, the net result is a reduction of the magnitude of linear polarization. In addition, it is noteworthy that the presence of photon-axion conversion causes the plane of polarization to rotate from parallel (with respect to the plane formed by the propagation direction and magnetic field) to perpendicular for a phase-on spectrum (small α, left panels of Fig. 9). This can be understood with reference to Fig. 4: for a NS observed phase-on with respect to the emission region, the intensity of the O mode photons slightly exceeds the X-mode photon intensity. When photonaxion conversion is considered, the O mode photon intensity is suppressed and becomes smaller than the X-mode intensity, which results in an inversion of the plane of polarization with respect to the case without conversion. Clearly, the extent to which this occurs depends on the specific axion parameter values, since the magnitude of the conversion probability is a strong function of m a and g a (see Fig. 3). For example, at B = 10 13 G and T s = 5 × 10 6 K, the X-mode intensity of a phase-on spectrum is about 90% of the O-mode intensity at E ∼ 1 keV. Therefore, if photon-axion conversion occurs with probability 10%, it results in an inversion of the plane of polarization. For the combination of axion parameters adopted in Fig. 3, a probability of 10% is achieved when g a 10 −9 GeV −1 (for m a = 2 × 10 −6 eV). When g a = 10 −8 GeV −1 , the probability is 10% for the entire range of axion masses considered (7 × 10 −7 − 1.7 × 10 −5 eV).\nAt larger viewing angles on the other hand (right panels of Fig. 9), it becomes evident one of the unique signatures of vacuum polarization on the polarization signal of NSs with magnetic fields B 7 × 10 13 G, that is the rotation at high energies of the plane of polarization from perpendicular to parallel (Lai & Ho 2003b). Photon-axion conversion does not erase this feature; however, as a result of the suppression of the O-mode photons which dominate at higher energies, the shift of the plane occurs at higher energies than it would without conversion. For the axion parameters adopted in Fig. 9, the energy at which the shift occurs with the conversion differs by a factor of ∼ 15 − 40% with respect to the case with no conversion, with the precise value depending on the magnetic field strength. Our theoretical results provide several methods for constraining axion parameters from actual measurements of NS spectra. The foremost method is spectral analysis, often performed on phase-averaged spectra when the resolution low. We showed in §5.1 that the relative suppression of the Omode photons with respect to the X-mode photons noticeably distorts the spectral shape with respect to the case without conversion. This distortion is strongest in the high-energy spectral tail, where the intensity of the O-mode photons is larger for most viewing angles (see Fig. 4). Observed enhancements of high energy spectral tails are generally modeled with a power law (often of unknown origin; e.g. Manzali et al. 2007, Durant et al. 2011 for young and middle-aged pulsars; Perna et al. 2001, Juett al. (2002, Rea et al. 2007 for magnetars). However, a deficit of high energy photons in a thermal fit could be a signature of photon-axion conversion, and the axion parameters could then be determined by means of detailed spectral fits that account for the effect. For this type of analysis, the distance to the NS is not important since the main discriminant is the spectral shape, in particular the deficit of high-energy photons."},{"title":"Observational tests","content":"For sources with good distance estimates 7 but too few Xray counts to perform detailed phase-resolved spectroscopy, Fig. 7 demonstrates that the determination of the effective emission area (through measurements of flux and peak energy) can rule out large regions of m a − g a parameter space if the inferred NS radius is close to the maximum or minimum value allowed by NS nuclear equations of state (EOS). The minimum radius constraint requires knowing that the observed radiation is being emitted by the entire NS surface and not from a small hot spot. This may be the case when pulsations are not detected or when the angle between the line of sight and magnetic dipole axis is non-negligible and the measurement is made near pulse minimum. The maximum conthe distances to NSs in globular clusters are known to much higher precision. straint on the other hand does not require other independent information.\nTo illustrate how to use Fig. 7 with NS observations, suppose R max is the maximum radius allowed by a particular EOS (see, e.g., Lattimer & Prakash 2007, for a review), and A max is the corresponding surface area. Then a NS whose inferred emission area is A ∼ A max would rule out the m a − g a region of the parameter space for which the effective area 'correction' due to photon-axion conversion is larger than 1. For B = 4 × 10 12 G and T s = 5 × 10 6 K (corresponding to a redshifted value ∼ 3.8 × 10 6 K), the parameter space with m a 2 × 10 −5 eV and g a −1 4 × 10 9 GeV (or g a 2.5 × 10 −8 GeV −1 ) would be ruled out. For NSs whose radius determination relies on methods other than flux and peak energy measurements 8 , constraints on axion parameters can be obtained using the method described above without any need for the radius to be near the maximum or minimum allowed by the EOS.\nAs an aside, we note that the the effects of axions on the inferred area discussed above suggest that the use of NS effective radii to rule out nuclear EOSs may not be entirely reliable. The existence of axions could severely bias such conclusions.\nFor sources with phase-resolved X-ray spectra, observations can be fit with either blackbody or model atmosphere spectra to obtain peak energies E peak and effective areas A eff as a function of rotation phase or viewing angle α. Figure 6 shows a clear signature of photon-axion conversion, i.e., a significant rise and fall of effective area at intermediate viewing angles, especially for B 10 13 G. The important quantity for this diagnostic is the relative change of the effective area with phase; thus, this method does not require knowledge of the distance to the NS. It is also rather insensitive to the NS radius/mass (i.e., surface gravity), since variations in this quantity produce energy shifts that are the same at all phases, leaving the relative phase-dependent signatures of the conversion unaltered. An exact estimate of the range of axion parameters that can be constrained by means of this method cannot be made in absolute terms, since it clearly depends on the quality of the data. However, from a theoretical point of view, photonaxion signatures are imprinted in the NS spectra as long as the conversion probability is non-negligible in the typical X-ray band of observations. Fig. 4 shows that this is the case for an axion coupling strength g a 10 −9 GeV −1 and for a large range of axion masses, up to m a ∼ 10 −4 eV if g a 10 −8 GeV −1 .\nThe theoretical framework and methods for extracting axion parameters from observations of magnetic NSs is presented above. Detailed analyses of specific sources require a model appropriate for the specific viewing and emission geometry, magnetic field strength, and surface temperature for each source; a comparison with the data can then be performed by folding the model through the response function of the detector. Such an analysis will be the subject of future work. Below, we briefly review NSs that serve as potential candidates for obtaining constraints on axion physics. These sources possess bright X-ray thermal emission from the NS surface. Among the observed NS population, such sources can be grouped into several classes (which may not be mutually exclusive groups; see, e.g., Kaspi 2010;. One class is the magnetars, which generally possess superstrong magnetic fields B 10 14 G. 9 For these objects, vacuum resonance effects lead to emission that is predominantly in the X-mode. Therefore the possible conversion of O-mode photons to axions will not produce an observable effect on the total radiation spectrum. Another class of NSs are the rotation-powered pulsars; these are the classic radio pulsars that are powered by the loss of 8 Examples of such methods of NS radius measurement (or more generally of the radius-to-mass ratio) include spectral lines (e.g. Cackett et al. 2009), thermal X-ray pulse profiles (e.g. Bogdanov et al. 2008; note that at low energies E 1 keV, these pulse profiles are not very sensitive to the presence of photon axion conversion, as shown in Fig. 8), gravitational wave emission (Lenzi et al. 2009), quasi-periodic oscillations in accreting NSs (Miller et al. 1998), and neutrino emission from proto-stars (Lattimer & Prakash 2007). Masses can be independently inferred from radial velocity studies for NSs in binary systems (e.g. van and pulsar timing (a comprehensive review of all these methods can be found in Lattimer & Prakash 2007). 9 Note however that the bursting behaviour used to characterize magnetars has also been seen in a lower magnetic field object (see Rea et al 2010). rotational energy due to emission of electromagnetic radiation. These NSs can have magnetic fields up to the magnetar regime B 10 14 G, and a small fraction of sources are sufficiently bright in the X-rays (see, e.g., Pavlov et al. 2002;Mereghetti 2007;Zavlin 2009, for reviews). These NSs are good candidates if their X-ray emission is not dominated by magnetospheric emission (as in, e.g., PSR B0656+14).\nThe isolated NSs are another interesting class. The seven confirmed isolated NSs have B ≈ (1 − 4) × 10 13 G inferred from timing measurements , while features in their X-ray spectrum suggest B ≈ (5 − 10) × 10 13 G if the features are due to the proton cyclotron resonance (Haberl 2007). We note that the observed X-ray spectra of isolated NSs are generally well-fit (except around spectral lines) by blackbodies, including Wien-like behavior at high energies (van Kerkwijk & Kaplan 2007). Various explanations have been proposed to explain the soft high energy tails as compared to the hard tails predicted by atmosphere models (e.g., Romani 1987), such as emission from a partially optically thin atmosphere (e.g., Motch et al. 2003) or the effect of vacuum polarization (see §2 and Ho & Lai 2003). Here we have shown that photon-axion conversion can suppress high-energy emission and produce softer tails (see Fig. 4). However, this cannot explain the spectra of isolated NSs. These NSs have surface temperatures 10 6 K, and, as we discussed in Sec. 2, the conversion is less effective at low temperatures due to the low fraction of O-mode photons. However, if soft tails (softer than the blackbody) were detected in NSs with relatively high surface temperature, then photon-axion conversion could indeed be a competing explanation.\nA final class is the central compact object neutron stars (CCOs), which have B ∼ 10 10 − 10 11 G, obtained from timing and spectral analyses (see, e.g., Halpern & Gotthelf 2010). At these low fields, the vacuum resonance is outside of the atmosphere, and the emerging intensity has a high fraction of O-mode photons; their surface temperatures are constrained to be (2 − 3) × 10 6 K, while hot spot temperatures are quite high [∼ (4 − 6) × 10 6 K]. These characteristics, combined with the fact that the photon-axion conversion probability at these lower fields is still high for a wide range of axion parameters (Jimenez et al. 2011), make CCOs potentially good candidates. Current data on many of the NSs belonging to the classes described above are already of sufficient quality to allow axion constraints to be derived.\nFinally, we note that X-ray polarization measurements, like those that will be performed by the forthcoming Gravity and Extreme Magnetism Small Explorer (GEMS) satellite (Jahoda 2010), will increase the robustness of axion constraints, as well as provide an independent measure of the stellar magnetic field strength and geometry (van Adelsberg & Perna 2009)."},{"title":"SUMMARY","content":"Constraints on the axion mass and coupling strength by means of observations of magnetized objects have been discussed in the recent literature (e.g., Lai & Heyl 2006;Chelouche et al. 2009;Gill & Heyl 2011;Jimenez et al. 2011) as a means to complement and independently test constraints obtained from other methods. The studies above emphasized different observational aspects and tests. The modification to the X-ray spectra of highly magnetized NSs induced by photon-axion conversion was discussed by Lai & Heyl (2006) using a blackbody spectrum to model the photon spectrum from NSs. Chelouche et al. (2009) discussed spectral features produced in the sub-mm/IR wavelength regime for magnetized NSs, while Gill & Heyl (2011) considered limits that can be derived from polarization measurements of white dwarfs. Jimenez et al. (2011) focused their analysis on constraints that can be obtained through observations of eclipsing white dwarfs in binaries, but they also discussed qualitatively the potential of NSs. Since most of the photon-axion conversion happens at a distance of many stellar radii from the NS, they pointed out that there are two configurations that should be considered: light from a background object passing through the magnetic field of the NS (i.e., occultation) and a binary containing a NS where the companion transits close enough to the line of sight for its light to be influenced by the NS magnetic field. In these two cases the treatment is much simpler as the physics of photon propagation in the NS atmosphere is irrelevant. However, they determined that the occultation probability is too low to be astrophysically relevant and that there are no known binary systems involving a NS that are detached enough to yield a clean constraint.\nSince NSs are routinely observed in the X-ray domain, the goal of our paper has been an extensive exploration of the actual constraints on axion physics that can be obtained from observations of NS thermal spectra. We have also explored what can be learned from future X-ray polarization measurements. For these purposes, we used detailed, magnetized atmosphere models, which properly account for the energy and angledependent emerging intensities of the two polarization modes, including mode conversion due to vacuum polarization effects. The emergent O mode intensity was coupled with the energy and angle-dependent photon-axion conversion probability, and this allowed us to make theoretical predictions for phase-resolved spectra with photon axion-conversion in the relevant regime of magnetic fields and temperatures for NSs. Since the relative intensities of the O and X mode photons depend strongly on the angle at which they emerge with respect to the magnetic field direction, we considered emission from a region consisting of a hot spot with an axis coincident with the NS magnetic field axis. Thermal emission from most Xray bright NSs does indeed indicate that we are observing hot spots (this from both non-negligible pulse fractions and small inferred emission areas). In order to extend axion studies by means of comparison with observations of any NS (especially those emitting from the entire surface, with non-dipolar, complicated B topologies, and fast rotators), one would have to include, locally, magnetic fields non-normal to the surface in the atmospheres, investigate how non-dipolar fields influence both the emission spectra as well as the photon-axion conversion, and, for the polarization, explicitly integrate the Stokes parameters to R pl . However, since the magnetic field structure over the entire surface of a NS is not fully known apriori, determination of the axion parameters would be more degenerate under these more general conditions. Our analysis has identified some features in NS spectra that bear the telltale signs of photon-axion conversion:\n• The NS spectral shape is noticeably distorted compared to the case without photon-axion conversion, with suppression of the high-energy tail for viewing angles ∼ 20 − 70 deg (with respect to the center of the emission region). Detailed spectral fits can yield axion parameters. This analysis does not require knowledge of the distance to the NS, as the axion signatures are imprinted in the spectral shape.\n• The spectral suppression of the O-mode photons by photon-axion conversion dominates the high energy tail of the spectrum for a range of viewing angles. In addition, it results in a shift of the peak energy, E peak , towards lower values for a range of rotation phases (i.e., viewing angles). As a result, the effective area of the spot A eff (as inferred from measurements of the flux and peak spectral energy) is larger than its value without axion conversion. We demonstrated that a clear signature of photon-axion conversion is a significant rise and fall of effective area (normalized to the phase-on value) at intermediate viewing angles, especially for B 10 13 G. Such a measurement requires high signalto-noise phase-resolved spectra, but does not require knowledge of the distance to the NS, since the axion signature appears in the relative change of the effective area with phase.\n• For a star emitting from the entire surface, photonaxion conversion can substantially affect the inferred NS emission area by 50%, and the inferred radius by 20%. These values are measured using the flux and peak energy of the thermal spectrum. If the distance is well constrained, and the NS radius is known through different methods, then the inferred emission area can be directly translated into a constraint in the m a − g a plane. If the NS radius is not independently known, then an inferred emission area that exceeds the maximum value allowed by NS equations of state could indicate the presence of photon-axion conversion.\n• In the absence of photon-axion conversion, radiation from a hot spot observed phase-on at energies ∼ 2 − 3 keV is linearly polarized in the plane formed by B and the direction of photon propagation. Conversion rotates the plane of polarization and leads to radiation polarized perpendicular to the plane of B and the direction of photon propagation, for a range of axion parameters.\nWe concluded §5.2 with a discussion of appropriate sources to study within the theoretical framework developed here for probing axions. While this paper has outlined the main elements for connecting theoretical ideas of photon-axion conversion to actual observations that are made of NSs, studying each suitable source will require a specific suite of models, tailored to the particular object (e.g., its magnetic field, surface temperature, emission and viewing geometry). The theoretical models are then convolved with the detector response, and a process of minimization identifies the most likely values of the parameters. Detailed analysis of the most promising NS candidates will be the subject of future work.\nThis work was partially supported by grants NSF AST-1009396, AR1-12003X, and DD1-12053X (RP); WCGH is supported by STFC in the UK; LV and RJ are supported by MICINN grab AYA 2008-03531. RP, RJ and LV thank the Centro de Ciencias de Benasque Pedro Pascual where this work was initiated. We further acknowledge Carlos Peña-Garay, Dong Lai, Jeremy Heyl and Enrique Fernadez-Martinez for insightful discussions, and an anonymous referee for helpful comments on our manuscript. WCGH appreciates the use of the computer facilities at the Kavli Institute for Particle Astrophysics and Cosmology."},{"title":"FIG. 1 .","content":""},{"title":"FIG. 2 .","content":""},{"title":"FIG. 6 .FIG. 7 .","content":""},{"title":"FIG. 8 .","content":""},{"title":"FIG. 9 .","content":""}],"string":"[\n {\n \"title\": \"INTRODUCTION\",\n \"content\": \"Axions with low mass (below 1 meV) are a direct prediction of the solution for the strong CP violation problem. Peccei & Quinn (1977) proposed axions as Pseudo-Goldstone bosons arising from the spontaneous breakdown of the U(1) symmetry. The Peccei-Quinn axion occupies a narrow region in the axion parameter space of mass and coupling strength. However, axions (or pseudo-scalar particles) can exist with more generic values for their mass and coupling strength, and are one type of dark matter candidate (e.g. Arvanitaki et al. 2009). This motivates efforts to constrain axion properties even if the parameters do not reach into the Peccei-Quinn regime.\\nSeveral experimental avenues have been used to detect axions and constrain their properties. These fall into two categories: ground-based experiments and cosmologicalastrophysical efforts. Current limits on the axion mass (10 −6 < m a /eV < 10 −3 ) come from the latter approach. The cooling rate observed in supernova 1987A implies a direct axion mass upper limit: if axions are more massive, then cooling would be predominantly due to axions rather than neutrinos (Eidelman et al. 2004). The lower limit comes from cosmological considerations: if axions constitute dark matter, then they must be cold and not overclose the Universe (Preskill et al. 1983). Constraints on the coupling constant can be obtained with both astrophysical observations and ground-based experiments. The existence of horizontal branch stars with extended morphology, that is, stars with a range of surface tem-peratures, constrains the coupling constant of axion-photon interaction to be g a < 10 −10 GeV −1 (Raffelt 2008). Groundbased experiments also place similar constraints, and future experiments are anticipated that will obtain stronger limits (Andriamonje et al. 1983;De Panlis et al 1987;Wuensch et al. 1989;Asztalos et al. 2004Asztalos et al. , 2010.\\nRecent developments in understanding radiation dimming (Lai & Heyl 2006;Jimenez et al. 2011) and modification of the polarization pattern (Gill & Heyl 2011) have put further constraints on the m a − g a plane. Limits on cosmological radiation dimming have also been used to place constraints on much lighter axion masses (Avgoustidis et al. 2010).\\nThe potential use of neutron stars (NSs) as probes of axion parameters has been noted by a number of authors (e.g., Lai & Heyl 2006;Chelouche et al. 2009;Jimenez et al. 2011). In particular, occultation and eclipsing or transiting of a companion and dimming of the spectrum can potentially produce observable signatures. However, the phenomenon of occultation (when a background object passes through the influence of a NS magnetic field) was deemed to have too low a probability to be astrophysically relevant. In addition, there are no known binary systems involving a NS that are detached enough to yield a clean constraint (Jimenez et al. 2011). On the other hand, Lai & Heyl (2006) showed how the thermal X-ray spectrum of a NS (modeled as simple blackbody emission) can be modified in a significant way by the presence of axions. Spectral features due to axion-photon oscillations have been discussed by Chelouche et al. (2009), and shown to be relevant for highly magnetized NSs in the sub-mm wavelength range for an observationally interesting range of axion parameters.\\nIn this paper, we focus on the soft X-ray band in which NSs are routinely observed. The goal of our work is to perform a detailed calculation of the observable effects that the presence of axions would have on NS spectra, light curves, and polar-ization. Our analysis goes beyond previous computations in the literature for the subject by coupling the calculation of photon-axion conversion probabilities with detailed models of magnetized atmospheres and general relativistic models for emission from the NS surface. For the local surface brightness of the NS, we use accurate magnetized atmosphere models, which yield the angular and energy-dependent photon intensities in the two photon polarization modes. These models include the effect of mode conversion due to the vacuum polarization. The local intensity is then imported into a code for computing spectra and light curves from finite emission regions on the NS surface. The analysis incorporates the effects of gravitational redshift and gravitational light deflection and allows for arbitrary viewing geometries. The resulting spectra are then modified by the photon-axion conversion process, accounting for both the energy and angular dependencies of the conversion probability. Note that we solve the conversion between photon modes and the conversion between axions and photons independently, which is shown by Lai & Heyl (2006) to be a very good approximation for the regimes we explore in this paper. With our calculations, we are able to produce realistic models for the spectra, light curves, and polarization signal of NSs, and use these to identify specific features that carry the telltale signs of photon-axion conversion. We discuss these signs in the context of NS observations in §5.2.\\nOur paper is organized as follows: In §2, we discuss the formation of the NS thermal spectrum within a magnetized atmosphere, placing particular emphasis on the emergence of two photon polarization modes. In §3, we summarize the calculation of the photon-axion conversion probability for a magnetized object. The computation of the 'observed' thermal spectrum, light curves, and polarization with and without photon-axion conversion is described in §4, and the results are presented in §5. We summarize our findings in §6.\"\n },\n {\n \"title\": \"PHOTON PROPAGATION IN THE ATMOSPHERE OF\",\n \"content\": \"\"\n },\n {\n \"title\": \"MAGNETIZED NEUTRON STARS\",\n \"content\": \"In this section, we describe the main properties of magnetized NS atmospheres and the processes that influence photon propagation in them. Axions do not directly couple to the processes described in this section. However, the details presented below are important for calculating the polarized radiation emerging from the NS atmosphere. As discussed in §3, axions couple only to photons that are linearly polarized in the plane formed by the directions of propagation and the magnetic field; the physics of photon propagation in the NS atmosphere determines the relative fraction of such photons.\\nRadiation emerging from the hot surfaces of magnetized NSs is the result of radiative transfer through a layer of ionized plasma. Due to the strong gravitational field of the NS, heavy elements settle quickly (Alcock & Illarionov 1980;Brown et al. 2002), and this atmospheric layer is composed of light elements, unless nuclear burning on the NS surface consumes the light elements (Chang et al. 2010, and references therein). Only recently have self-consistent magnetic atmosphere models for partially ionized hydrogen and mid-Z elements been constructed (see Mori & Ho 2007;Ho et al. 2008, and references therein for details; see also Suleimanov et al. 2009). For simplicity, we focus our calculations on fully ionized hydrogen atmospheres (see below). In this case, the scale height is\\nwhere T s is the effective temperature of the star, g * is the gravitational acceleration, and δ is the angle that the ray makes with the surface normal.\\nIn a NS atmosphere, radiation propagates in two distinct polarization modes: the extraordinary (X) mode and ordinary (O) mode, which are very nearly linearly polarized perpendicular and parallel, respectively, to the plane formed by the directions of propagation and the magnetic field. The X mode absorption opacity is reduced by a factor of (E Be /E) 2 relative to the O mode opacity, where the electron cyclotron energy is E Be = 115.8B 13 keV and B 13 = B/10 13 G. Thus the X mode radiation decouples from deeper, hotter layers in the NS atmosphere than the O mode radiation. For rays propagating at intermediate angles θ p relative to the magnetic field, the net emission is significantly polarized and dominated by the X mode.\\nTo produce surface emissivities for the X and O photon modes, we use the model of van Adelsberg & Lai (2006), which quantitatively incorporates the effects of vacuum polarization on the radiative transfer for fully ionized atmospheres with an external magnetic field parallel to the axis of symmetry. This model is reasonably accurate for temperatures T s 10 6 K. At lower temperatures, partial ionization of atomic species becomes important, and partially ionized models are needed (see Ho et al. 2008, and references therein). Therefore, we use the fully ionized model only in the high temperature regime. It should be noted, however, that this is the effective temperature of the star, and that the measured value at the observer is lower due to gravitational redshift. Colder stars are less interesting for the purpose of this study: First, being much dimmer, their measured spectra are generally of lower statistical significance for detailed spectral studies. Second, and more importantly, since the fraction of Omode photons increases with temperature, spectra of colder stars are less sensitive to axion effects (see the discussion below).\\nFor magnetic field strengths B ∼ 4 × 10 13 G, there is a significant contribution from vacuum polarization to the dielectric properties of the medium (Adler 1971;Tsai & Erber 1975). A ray traversing the density gradient of a magnetized NS atmosphere will eventually encounter a layer of the medium in which the plasma and vacuum contributions to the dielectric tensor are of the same magnitude, leading to a resonance. For the models studied in this paper, the resonance occurs at lower density than the X and O mode decoupling depths (where the optical depth τ ν ≈ 1) for most photon energies and propagation angles. In addition, the integrated opacity across the vacuum resonance is negligible at these field strengths. A discussion of vacuum resonance effects on the mode opacities is given by Ho & Lai (2003). As shown by Lai & Ho (2002) and Lai & Ho (2003a), there is coherent mixing of the modes at the resonance, analogous to the Mikheyev-Smirnov-Wolfenstein (MSW) effect for neutrino oscillations (see, e.g., Bahcall 1989;Haxton 1995). Using the geometric optics approximation, and neglecting damping terms in the dielectric tensor (which only affect the width of the resonance), the amplitudes of the two modes, A O and A X , evolve according to\\nwhere (e.g. Lai & Heyl 2006)\\nIn equations (3)-(5), θ p is the angle between the magnetic field and the photon direction (defined to be the z direction), ω is the photon frequency, v e = (ω pe /ω) 2 and u e = (ω ce /ω) 2 , where ω pe and ω ce are the electron plasma and cyclotron frequencies, respectively. The B-field dependent functions ξ(b) and χ(b) are (Potekhin et al. 2004)\\nand\\nwhere α f is the fine structure constant and b is the magnetic field strength in units of the critical field B c = m 2 e c 3 /(eh) = 4.414 × 10 13 G. The equations above assume that v e ≪ 1 and u e ≫ 1. Since v e ∼ 10 −3 (ρ/1 g cm −3 ) and u e ∼ 10 4 (B/10 13 G) at 1 keV, these constraints are easily satisfied throughout the soft X-ray band in which NS thermal spectra typically peak.\\nEquation (2) determines the evolution of the photon mode amplitudes through the atmosphere (see Ho & Lai 2003;Lai & Ho 2003, for details). The amplitudes are related to the specific intensities of each mode by I X ,O ∝ |A X ,O | 2 . As the photons encounter the resonance, the polarization mode ellipticities, K X ,O (∝ E x /E y ), experience a discontinuity at the resonance. As discussed in Ho & Lai (2003), an alternative description of the polarization state uses ellipticities K ± , which vary continuously through the vacuum resonance. If the variation of the atmosphere density profile is gradual enough, an adiabatic condition is satisfied (see below), and the mode properties are fixed along the curves described by K ± . Before the resonance, a K − photon corresponds to an X mode polarized photon, while a K + photon corresponds to an O mode photon. After the resonance, the correspondence between K ± and K X ,O switches, so that under adiabatic conditions the character of each modes changes. This is analogous to the correspondence between flavor and mass eigenstates in the MSW mechanism for neutrino mixing. The adiabatic condition is set by evaluating the coefficient matrix in Eq. (2) at the resonance and setting the magnitude of the diagonal term equal to the off-diagonal term (Lai & Ho, 2002;2003a). In the non-adiabatic regime, the diagonal terms have a much larger magnitude than the off-diagonal terms, leading to a set of decoupled equations for A X ,O . In the adiabatic regime, the offdiagonal terms dominate over the diagonal ones, leading to a coupled set of equations for A X ,O and mixing of the polarization states.\\nThe conversion probability P c is derived from Eq. (2) by setting the radiation completely in one mode (e.g., A O = 1, A X = 0) and evolving the equations to z → ∞. If the atmosphere density profile is linear in z, then v e (∝ ρ) is a linear function of z, and the conversion fraction is given by the well known Landau-Zener formula (see Lai & Ho, 2002;2003a, and the references therein):\\nThe adiabatic energy E ad is defined by\\nwhere E Bi is the ion cyclotron energy, and f −2 B is a slowly varying function of the magnetic field (see van Adelsberg & Lai, 2006, and the references therein). Because the vacuum resonance is very narrow in energy (and hence density, for a given photon energy and propagation angle), the linear approximation for the density profile is always satisfied for the atmosphere models considered in van Adelsberg & Lai (2006). In addition, the conversion occurs on a distance scale much less than the scale height of the atmosphere; thus the conversion process can be regarded as occurring exactly at the resonance density, and the asymptotic solution to Eq. (2), in the form of the conversion probability Eq. (8), can be used to treat mode conversion effects. This was shown to be an accurate approximation in van Adelsberg & Lai (2006), and we adopt it here in our calculations of mode conversion.\\nA photon with energy E encounters the vacuum resonance at a density\\nFor a given energy, ρ V is the density at which the vacuum and plasma contributions are of equal magnitude (i.e., when σ 11 = σ 22 ). If E > E ad , then conversion between the modes at this depth is effective. If I X and I O are the values of the mode specific intensities at the resonance, then after conversion they become:\\nThis process is automatically incorporated into the models with B = 10 13 G and B = 4 × 10 13 G (T s = 5 × 10 6 K) using the code of van Adelsberg & Lai (2006). For models with B = 4 × 10 12 G, we use the code of Ho & Lai (2001). This code does not include partial mode conversion, but we have included the effect according to the following prescription: The code of Ho & Lai (2001) produces atmosphere profiles ρ d and T d at a set of discrete points in Thomson optical depth, τ d , where d = 1, . . . , D. Since the conversion process occurs at lower densities than both the X and O mode decoupling depths, it does not affect the atmosphere structure and can be treated as a modification operating on the emergent specific intensities. For a given photon energy and propagation angle, we calculate the vacuum resonance density using eq. (10), and then locate a grid point i, such that ρ i ≤ ρ V ≤ ρ i+1 . We then compute a linear approximation to the resonance Thomson depth τ V , using the formula:\\nwhere ∆τ = τ i+1 − τ , and ∆ρ = ρ i+1 − ρ i . We relate the resonance temperature T V to the Thomson depth and density in a hydrostatic atmosphere using\\nand calculate the atmosphere scale height at the vacuum resonance according to Eq. (1). Finally, the adiabatic energy and mode conversion probabilities are calculated according to Eqs. (8) and (9). The emergent intensities are then mixed using Eq. (11). When the surface magnetic field exceeds the quantum critical field B c ∼ 4 × 10 13 G, there are significant vacuum contributions to the dielectric properties of the medium. At the resonance, if the radiation energy is much greater than the adiabatic energy [Eq. (9)], there is a high probability of conversion between the X and O modes. For magnetic field strengths B 7 × 10 13 G, the vacuum resonance lies between the photospheres of the X and O modes (where the mode opacities τ ν ≈ 1). The modes mix at the resonance with a fraction of Xmode photons converting into O-mode photons and vice-versa according to the probability in Eq. (8). The O-mode photons, after converting into X-mode photons at the resonance, decouple from the atmosphere. The X-mode photons that con-vert to the O-mode, subsequently interact strongly with the atmosphere, as they now experience the large opacity for O mode polarization. Any X-mode photons that do not convert encounter a large integrated opacity at the resonance (under typical conditions; see Ho & Lai, 2003). Thus, one of the net effects of vacuum polarization is to shift the location of the X-mode photosphere to the resonance density, which is at lower density and temperature than the original X-mode photosphere. As a result, the high energy spectrum is softer, and the relative fraction of X to O mode photons decreases. At magnetic field strengths B ≪ 7 × 10 13 G, radiation in both modes decouples from the NS atmosphere before encountering the vacuum resonance. In this case, mode conversion es- sentially switches the photospheres of the two modes, with the O-mode photons now being produced in the deeper (hotter, denser) layers of the star. Therefore, if the emission is mostly polarized in the X mode, for E > E ad , the plane of polarization switches to the O mode after resonant conversion (Lai & Ho 2003a, 2003b. The above considerations are particularly relevant for the subsequent mixing between O-mode photons and axions, as emission which would otherwise have a small fraction of O mode radiation now predominantly consists of O mode photons, depending on the magnetic field strength and NS geometry (see van Adelsberg & Perna 2009, for more discussion). Figure 1 shows the fractional contribution to the emer-gent flux of the X and O modes for three values of the field strength, B = 4 × 10 13 G, 10 13 G, and 4 × 10 12 G. The effective temperature is T s = 5 × 10 6 K in all cases. Each panel shows the results for a separate emission angle δ relative to the magnetic field. In general, the X mode dominates the emergent flux at low energies, except around the ion cyclotron feature E Bi . According to the formula for E ad [eq. (9)], there is an angle-dependent point at which mode conversion becomes efficient and the radiation is composed mostly of O mode photons. Since E ad ∝ tan 2/3 θ p = tan 2/3 δ for a magnetic field perpendicular to the surface, at larger values of δ, mode conversion is efficient only at high energies. Note that as θ p → 0, the properties of the two modes become similar, and the dif-ference in the opacities decreases. Figure 2 shows the fractional contributions to the emergent flux of the X mode and O modes, for a fixed magnetic field strength, B = 4 × 10 12 G, at three effective temperatures T s = 5 × 10 6 K, 2.5 × 10 6 K, and 10 6 K. The general trend is an increase in the O/X flux ratio as the temperature increases. This is straightforward to understand: since a higher temperature atmosphere produces more photons with higher E, the average ratio E Be /E decreases. Thus, the X-mode opacity becomes more similar to the O-mode, leading to an increase in the O/X ratio.\"\n },\n {\n \"title\": \"PHOTON-AXION CONVERSION IN THE MAGNETIC FIELD OF NEUTRON STARS\",\n \"content\": \"In the following we discuss the effect of photon-axion mixing as a photon beam propagates through the dipolar field of a magnetized NS. Firstly, we note that an important assumption of our analysis is that the conversions between the O and X modes and between the O mode and axion can be treated separately (each as a two-state mixing) rather than as a coupled three-state mixing. This assumption is valid as long as the vacuum resonance and the photon-axion resonance are well separated. This was shown to be the case by Lai & Heyl (2006) for highly magnetized NSs at soft X-ray energies and axion masses m a 10 −3 eV. Thus, the assumption of decoupled mixing is satisfactory for our regime of interest, and it will be adopted in our analysis.\\nOur formalism for photon-axion conversion in a magnetized medium follows that of Raffelt & Stodolski (1988). To be consistent with the notation adopted in the previous section for the two photon polarization states, we denote the amplitudes of photons polarized parallel and perpendicular to the magnetic field as A O and A X , respectively. Under the assumption that the length scale over which the magnetic field varies is much larger than the photon and axion wavelengths, the evolution of the photon and axion fields in the z direction, with frequency ω, is given by\\nin units withh = c = 1, where a is the axion field. The diagonal matrix elements are given by\\nwhere m a is the axion mass and ξ(b) is defined by Eq. (6). The off-diagonal component is given by\\nwhere g a describes the coupling strength between the O-mode photons and the axions, and r is the distance from the stellar surface. Since mixing occurs only between the axion field and the O-mode photon field, we neglect the evolution of the X-mode field in the discussion that follows. The mixing strength can be measured in terms of the ratio between the off-diagonal term and the difference between the diagonal terms in Eq. (14); a useful parametrization can be made in terms of the mixing angle θ m defined by\\nWe begin our analysis by considering a region of space over which the magnetic field can be approximated as homogeneous. Following Raffelt & Stodolsky (1988), the discussion is greatly simplified if we define phases relative to the unmixed component, and neglect a common phase. The solution can be found by first performing a matrix rotation to an eigenstate basis where the propagation matrix is diagonal, propagating the two eigenstates independently, and then rotating back to the photon-axion basis. This yields an evolution equation for the mixing components:\\nwhere\\nand\\nand the plus and minus sign indicates ∆ ′ and ∆ ′ a , respectively.\\nThe solution provided by Eq. (19) assumes that the field is homogeneous. The generalization for photon propagation within the dipolar field around the NS, with B ∝ r −3 , can be made by assuming that the total evolution operator is the product of evolution operators within spatial slices of width ∆z j wherein the B field can be approximated as constant.\\nWithin each shell, we use Eq. (19) to evolve the photon and axion amplitudes from A O (z j ) and a(z j ) in shell j to A O (z i ) and a(z i ), where z i = (z j + ∆z j ) is the subsequent shell. We tested the code for convergence and found that, for ∆z j ≤ 0.07 km, the amplitudes at infinity converge for the range of B, m a , and g a of interest. Fig. 3 shows the conversion probability for combinations of the axion parameters m a , g a in the soft X-ray band of interest for NS observations. The probability is plotted as a function of photon energy and propagation angle θ p . There is a range in the parameter space of axion mass, coupling strength, and NS properties (such as surface magnetic field strength) where the photon-axion conversion probability is non-negligible. This implies that photon-axion conversion can leave distinctive signatures in NS spectra, making them interesting sources for constraining axion physics. In the next section, we perform a detailed analysis of such signatures. The observed spectrum of a NS is obtained by summing the local emissivities over the emission region. For simplicity, we consider an emitting region that has a constant temperature and magnetic field strength, with the field direction along the surface normal (e.g., a hot magnetic polar cap). Such a region allows us to directly examine the spectral dependencies of photon-axion conversion on these physical quantities. It is often the case that the emission is dominated by a region smaller than the entire star surface (i.e., a hot spot). The spectrum and lightcurves will then be functions of the viewing angle α between the observer and spot axes. As we will show FIG. 3.-Photon-axion conversion probabilities for the magnetic field strength B = 10 13 G and various combinations of the coupling strength, ga, and axion mass, ma. The probability is shown as a function of photon energy and angle θp between B and the direction of photon propagation. In the left panels, the dependence on the coupling strength is explored for fixed axion mass ma = 2 × 10 −6 eV (from top to bottom: ga = 10 −8 , 10 −9 , 10 −10 GeV −1 ). The right panels show the dependence of the conversion probability on the axion mass for fixed coupling strength ga = 10 −8 GeV −1 (from top to bottom: ma = 7 × 10 −7 , 6 × 10 −6 , 1.7 × 10 −5 eV). The dependence of the probability on B is weak for the range considered here (4 × 10 12 − 4 × 10 13 G), so we only display results at the intermediate field strength.\\nin § 5, phase dependent methods add considerable diagnostic power for detecting the presence of axions.\\nThe calculation of phase-dependent emission from an extended region on the NS surface follows the formalism developed by Pechenick et al (1983), with generalizations by Perna & Gotthelf (2008; see also Bernardini et al. 2011, for a similar geometry). We define the time-dependent rotational phase γ(t) as the azimuthal angle subtended by the magnetic dipole vector µ around the axis of rotation. It is related to the modulus of the NS angular velocity, Ω(t), by γ(t) = Ω(t)t. We choose the coordinate system so that the observer is located along the z axis; the inclination angle of the rotation axis,Ω, with respect to the line of sight is denoted by α R , while the angle between the magnetic dipole vector and the rotation an-gle is denoted by α M . Thus, the angle between µ and the line of sight is given by\\nThe emission region is assumed to be circular, with opening angle β, and is centered around the magnetic dipole axis. Hence, α(t) also indicates the time-dependent angular separation between the spot axis and the line of sight as the star rotates. We describe points on the NS surface by means of the polar angle θ, and the azimuthal angle, ϕ, in spherical polar coordinates (see Fig. 1 in Perna & Gotthelf 2008 for a visual representation of this geometry).\\nThe emission region is restricted to θ ≤ β for α = 0, while for α = 0 it is identified by the conditions\\nand by the condition\\nIn the latter case, the outer boundary of the spot, θ ⋆ (α, ϕ, β), must be determined numerically from the expression cos β = sin α sin θ ⋆ cos ϕ + cosα cos θ ⋆ .\\nDue to the intense gravitational field of the NS, photons emitted at the NS surface are substantially deflected as they travel to the observer. A photon emitted from colatitude θ will reach the observer (at infinity) if emitted at an angle δ with respect to the surface normal; the relation between the two angles is given by the ray-tracing function\\nwhere x ≡ sin δ, R and M are the NS radius and mass, respectively, and R s ≡ 2GM/c 2 is the Schwarzchild radius.\\nThe spectrum at the observer is obtained, as a function of the viewing angle, α(γ), by integrating the local emission over the observable surface; this procedure yields the flux seen by an observer at a distance D ≫ R. Accounting for the gravitational redshift of the radiation, this integral takes the form (Page 1995and generalizations by Pavlov & Zavlin 2000, Heyl et al. 2003:\\nin units of photons cm −2 s −1 keV −1 . In the equation above, the index j represents any of the Stokes parameter (U, V, Q, I), and E ∞ is the energy as seen by the distant observer. The energy emitted at the stellar surface is given by E ∞ e −Λs , with\\nThe local specific intensity I j is equal to zero outside of the boundaries for θ and ϕ defined by equations (24)- (26). The observed flux depends on the geometric angles α R and α M through the angle α in Eq. (23). In the following, we consider an orthogonal rotator, for which α R = α M = 90 0 . This implies that α(t) = γ(t). Given that the emission region is centered around the dipole axis, the angle α will be identified with the angle θ p between the photon propagation and magnetic field directions (see §3).\\nIf photon-axion conversion occurs with probability P(E ∞ , α), the phase-resolved spectrum is given by\\nFor a fixed energy (or integrated over an energy band), α]) yields the light curve as a function of the viewing angle α. The phase averaged spectrum is readily obtained from\\nFinally, the linear polarization is computed as\\nTo simplify our notation, we will omit the subscript '∞ ′ from the observed energy. In the discussion that follows, all quoted energies are redshifted energies for a NS of mass M = 1.4M ⊙ and radius R = 10 km. Before concluding this section, we need to remark that Eq. (31) and Eq.(33) assume that the O and X modes remain decoupled once they have emerged from the atmosphere of the NS. The evolution of the modes in the changing dipole magnetic field of the star was computed in papers by Heyl & Shaviv (2000, Heyl, Shaviv & Lloyd (2003), Lai & Ho (2003) and van Adelsberg & Lai (2006). All of the above studies found that, in the birefringent, magnetized vacuum near the star surface, the photon modes are decoupled. In this region, the mode eigenvectors evolve adiabatically along the changing direction of the magnetic field. Further from the star, the mode evolution continues to be adiabatic, up to the \\\"polarization limiting radius\\\", R pl , which is defined as the distance at which the polarization modes mix (and are frozen thereafter). Heyl, Shaviv & Lloyd (2003) derive the following expression for R pl\\nwhere E 1 ≡ E/(1 keV) and B 13 ≡ B/(10 13 G). For radii larger than R pl , the mode eigenvectors are fixed, and the polarization no longer evolves (i.e., it is the observed polarization [Ω/(2π)] −1 sin 2 α 1/6 10 7 cm. The scaling of R pl in this equation differs slightly from that in Eq. (34) due to different adiabatic conditions applied by the two sets of authors. Heyl et al. (2003) constrain the change in the relative difference in the mode indices of refraction compared to the magnitude of that difference [see their Eq. (9)], while van Adelsberg & consider the distance at which the mode evolution equations have equal diagonal and off-diagonal terms [see their Eq. (62)]. This results in slightly different magnitudes and scaling relations; for slowly rotating NSs, the precise numerical value of R pl does not change the value of the observables, and both approaches result in the same physics. except for very rapidly rotating stars (i.e., with frequencies of tens of milliseconds) the coupling results in only a very small quantitative change. For example, for a star with period ∼ 0.1 sec (B ∼ 10 12 G, E ∼ 1 keV), the mode intensities I O and I X (∝ |A O | 2 , |A X | 2 ) couple at the level of 4%, and the amount of mixing further decreases at longer periods. Therefore, for the purpose of our study and the observational tests we propose (requiring NSs with relatively long periods in order to perform time-resolved spectroscopy), the formalism that we use provides a very good description of the observed flux and the linear polarization fraction computed here (see also Lai & Heyl 2006). We further note that, if the radius of the emission region is much smaller than the polarization limiting radius, then summing over the Stokes parameters at the detector is well approximated by summing over a mode (X or O) at the NS surface (Heyl et al. 2003). Since our emission region (a hot spot) is always much smaller than R pl for the B field strengths and energies under consideration, we use this approximation in our calculations.\"\n },\n {\n \"title\": \"RESULTS: THEORETICAL PREDICTIONS AND OBSERVATIONAL TESTS\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theoretical predictions\",\n \"content\": \"We present the results of our calculations of NS spectra, light curves, and polarization signals, with and without photon-axion conversion. We explore the dependencies on the stellar magnetic field strength and effective temperature, with particular emphasis on the magnetic field since it has a stronger influence than the temperature on the functional form of the modal energy dependence (see Figs. 1 and 2). We consider emission from a circular region, 10 deg in angular size, that is centered on the magnetic pole of the star. Such hot spot emission is common in NSs, and exhibits a strong dependence on phase. Given the significant angle dependence of the O and X mode intensities (see Figs. 1 and 2), the study of emission from a finite region allows us to extract more information on the effects that photon-axion conversion has on the observable properties of NSs.\\nAs shown in Fig.3, the conversion probability is a strong function of the axion mass, m a , and the coupling constant, g a . For the thermal spectra of NSs to be affected by photonaxion conversion, the probability of this process must be non-negligible in the soft X-ray band, where the NS thermal spectrum peaks. Therefore, we perform a detailed analysis of the consequences of photon-axion conversion on NS spectral properties for a particular set of axion parameters; these parameters are chosen to yield a substantial conversion probability in the soft X-ray band. We choose the values m a = 2 × 10 −6 eV, and g a = 10 −8 GeV −1 ; the study of this specific case will uncover the main signatures of axion conversion. Once these are identified, we explore a wider area of m a − g a parameter space to identify the main region that can be probed by means of NS observations in the X-ray band. Fig. 4 shows phase-resolved spectra for three magnetic field strengths, B = 4 × 10 13 , 10 13 , and 4 × 10 12 G. For each value of B, we plot the spectra at two phases, coinciding with the angles between the line of sight and the magnetic field axis for the chosen geometry. We show angles α = 2 deg and a typical large viewing angle α = 60 deg. The former angle is essentially a 'phase-on' spectrum (i.e. a spectrum observed in correspondence to the maximum of the pulsation); we do not plot the results for α = 0 • since the conversion probability in that case is zero. For each combination of viewing angle and magnetic field strength, the figure shows the fluxes from the O and X-modes separately, as well as the total composite spectra. Results are plotted with and without photon-axion conversion.\\nThe relative fraction of O and X mode photons, as a function of energy and angle, plays a fundamental role in determining the effects of axions on phase-dependent spectra and light curves. Before discussing the spectral effects of axions in further detail, we comment on the energy and angular dependence of the two photon modes. Although only the Omode photons are affected by axion conversion, the properties of the X-mode photons also matter for our purposes because of vacuum polarization-induced mode conversion. The X-mode has a complex beam pattern, consisting of the sum of a narrow \\\"pencil\\\" beam and a broad \\\"fan\\\" beam. The width of the pencil beam scales as ∼ (E/E Be ) 1/2 , decreasing with magnetic field strength (Pavlov et al. 1994). Therefore, the modal intensities vary from being approximately equal to significantly different between widely separated emission angles.\\nFor a phase-on spectrum, the intensities of the O and X modes peak at roughly similar energies. Thus, the presence of photon-axion conversion results in a dimming of the spectrum without significant spectral distortion. The situation is different at larger viewing angles, as evident in the α = 60 deg panels of the Fig. 4. In these cases, the X mode dominates the spectrum at (redshifted) energies 1.4 keV, while the O mode dominates at higher energies; thus, an inversion of the dominant spectral mode occurs. Suppression of the O mode by photon-axion conversion therefore produces not only a dimming of the spectrum but also a spectral distortion, suppressing the high-energy tail.\\nSince axion conversion occurs at larger distances from the star than mode conversion due to the vacuum resonance, only O-mode photons are affected as they emerge from the star atmosphere. This is shown in Fig. 4, where the X-mode photons are untouched by photon conversion, while the O-mode intensities are noticeably suppressed. The overall effect on the spectrum is that of flux suppression. However, since the O/X flux ratio, as well as the conversion probability, has an energy dependence which varies with viewing angle, the shape of the spectrum is also affected, causing a shift of the energy peak. While the shift is very small at phase-on viewing angles (since the O-modes and X-modes have a similar energy-dependent intensity for α ∼ 0 deg), it is very pronounced at wider viewing angles, where the O-mode largely dominates the intensity at high energies. In this case, enhanced suppression of high energy photons produces an effective shift of the peak of the spectrum towards lower energies. Figures 5 and 6 quantify the phase-dependent spectral effects discussed above. For the same values of magnetic fields and axion parameters considered in Fig. 4, Fig. 5 shows the ratio of the X-ray flux in the 0.8-10 keV band photon-axion conversion to that without conversion. As expected, at all phases, the ratio F Ax X /F noAx X is smaller than 1. This is because the contribution to the flux from the O-mode photons is non-negligible at all viewing angles with respect to the magnetic field axis. The precise value of the flux ratio, however, shows a significant dependence on the viewing angle, with a mimimum around α ∼ 40 − 50 deg. This is due to the O-mode flux, which is a large fraction of the total flux at E 1 keV, for viewing angles in the 30 − 70 deg range.\\nThe left panels of Fig. 6 show the (redshifted) energy where the spectrum peaks, E peak , as a function of viewing angle. We -Comparison between phase-resolved spectra of NSs with (thick/red lines) and without (thin/blue lines) photon-axion conversion. In each case, the dotted lines show the flux contribution from the O mode, while the dashed lines display the contribution from the X mode, which is unaffected by conversion. The solid lines show the total spectra. In all panels, the effective temperature of the star is Ts = 5 × 10 6 K and the axion parameters are ma = 2 × 10 −6 eV and ga = 10 −8 GeV −1 , while the B-field strength has three values from top to bottom. The spectra are shown for two angles α between the observer and the dipole axis.\\nconsider only angles α ≤ 100 deg since the flux from larger angles, which is still visible due to relativistic light deflection, is too low for phase-resolved spectroscopy. We begin by examining the behavior of E peak without photon-axion conversion. At the magnetic field strengths for the models in this paper, the spectral peak of the total (X and O mode) specific intensity follows the beam pattern of the X mode: a narrow pencil beam at small angles and a broad fan beam at large angles. This is because conversion between the X and O modes occurs above the photosphere of both modes and thus does not change the total intensity. The observed variation of E peak with viewing angle deviates somewhat from the emission beam pattern due to the narrowness of the pencil beam (a few deg in width around δ = 0 • ), the gravitational deflection of photon trajectories, and the finite size of the emitting region, which smears out features from nearby emitting points 6 .\\nWhen axion conversion is taken into account for B = 4 × 10 12 − 4 × 10 13 G, a common feature is a shift of the peak FIG. 5.-Ratio of X-ray flux (in the [0.08-8] keV band) with axion conversion to that without axion conversion. Axion parameters have the same values as in Fig. 4, while the NS temperature and magnetic field are indicated for each curve, with the B 12 ≡ 10 12 G and T 6 = 10 6 K. energy to lower values at all viewing angles except for those close to phase-on and edge-on. For α ≈ 0 deg, the intensity of the O mode photons peaks at similar or slightly lower energies than that of the X-mode photons (see Fig. 4); hence, suppression of O mode photons primarily affects the amplitude of the spectral intensity, as opposed to the location of the spectral peak. In fact, because the O-mode intensity peaks at lower energies, especially at lower fields, at small angles there is actually a shift of the peak towards higher energies after photon-axion conversion.\\nAnother notable signature in the variation of E peak (α) is a strong decrease (or flattening in the higher B-field case) for viewing angles α ∼ 30−50 deg. Around these angles, the contribution from the O mode photons is dominant at high energies, and the reduction of these photons through photon-axion conversion results in a significantly cooler spectrum. As the viewing angle increases further, the X-mode photons become increasingly dominant (see Fig. 1), and the reduction of high energy photon lessens. As a result, the effective temperature of the spectrum -as measured through E peak -approaches the value it would have without photon-axion conversion.\\nTo examine the dependence of our results on the effective temperature T s of the star, we consider the values T s = 10 6 , 2.5 × 10 6 , and 5 × 10 6 K (where T s is the unredshifted temperature at the stellar surface) for the magnetic field B = 4 × 10 12 G. We find that the shift in E peak due to axion conversion (compared to the case with equal temperature but no conversion) is more pronounced at higher temperatures. This is not surprising given that the relative fraction of O to Xmode photons is larger at higher temperatures.\\nThe right panels of Fig. 6 show the phase-resolved effective area [i.e., A eff (α) ∝ F(α)/E 4 peak (α)] that an observer measures when fitting spectra with a blackbody model. To eliminate the dependence of this diagnostic on the often poorly determined distance to the NS, we normalize A eff (α) to its value near phase-on. Since the photon-axion conversion probability is zero for precisely α = 0, we normalize to a small angle α = 2 deg which represents a \\\"phase-on\\\" spectrum as an average of small angles around α = 0 deg. When photon-axion conversion is included, the variation of the effective area with phase results from the interplay of two effects: a decrease in A eff due to flux dimming and an increase in A eff due to decreases in the peak energy and inferred temperature. Because of the E 4 peak -dependence, the latter effect dominates the former. This is particularly the case for lower fields, which has the strongest E peak shift with phase.\\nThe variation of inferred effective area with phase is a particularly interesting and peculiar feature of photon-axion conversion. The specific details of the variation, however, depend on NS and axion parameters. In order to explore a wider region of m a − g a parameter space, we compute the effective area for the phase-averaged spectrum (A ave eff ) with photon axion conversion, normalized to that without axion conversion. This represents a measure of the \\\"correction\\\" that photonaxion conversion makes to the inferred effective area (as determined through measurements of phase-averaged flux and effective temperature). For our purposes, the phase-averaged spectrum of a hot spot provides an adequate approximation for emission from the entire NS surface. The results of this calculation are displayed in Fig. 7 for a combination of magnetic fields and effective temperatures: B = 4 × 10 12 G, T s = 5×10 6 K; B = 4×10 12 G, T s = 2.5×10 6 K; and B = 4×10 13 G, T s = 5 × 10 6 K. It is interesting to note that, in all cases, there are regions of parameter space in which the inferred effective areas are smaller/larger than the uncorrected areas. This is expected given the two counteracting effects described above. The m a − g a region shown in Fig. 7 is one for which, at typical NS magnetic fields, there is a larger probability of photonaxion conversion in the soft X-ray band (where NSs are routinely observed). The correction to the effective area becomes negligible for the largest values of m a and g a −1 displayed in Fig. 7. This is expected since the photon-axion conversion probability tends to zero for these values at the keV energies of interest here. The results summarized by this figure have a strong potential to constrain axion signatures for NSs with good distance measurements (see the discussion in §5.2).\\nIn Fig. 8, we examine the effects of photon-axion conversion on NS light curves at different energies, for the same set of axion parameters as in Fig. 4. The lightcurves are all normalized to the maximum value of the flux among the set of energies considered (0.5, 1.5, 2.5, and 4 keV). The main effect of photon-axion conversion is the relative suppression of the flux at higher energies with respect to that at lower energies.\\nFinally, we study the effect of photon-axion conversion on the magnitude of linear polarization [see Eq. (33)]. For consistency with Fig. 4, we consider the same set of parameters. The polarization signal with and without conversion is shown in Fig. 9. Since the main effect of axion conversion is suppression of O-mode photons, the net result is a reduction of the magnitude of linear polarization. In addition, it is noteworthy that the presence of photon-axion conversion causes the plane of polarization to rotate from parallel (with respect to the plane formed by the propagation direction and magnetic field) to perpendicular for a phase-on spectrum (small α, left panels of Fig. 9). This can be understood with reference to Fig. 4: for a NS observed phase-on with respect to the emission region, the intensity of the O mode photons slightly exceeds the X-mode photon intensity. When photonaxion conversion is considered, the O mode photon intensity is suppressed and becomes smaller than the X-mode intensity, which results in an inversion of the plane of polarization with respect to the case without conversion. Clearly, the extent to which this occurs depends on the specific axion parameter values, since the magnitude of the conversion probability is a strong function of m a and g a (see Fig. 3). For example, at B = 10 13 G and T s = 5 × 10 6 K, the X-mode intensity of a phase-on spectrum is about 90% of the O-mode intensity at E ∼ 1 keV. Therefore, if photon-axion conversion occurs with probability 10%, it results in an inversion of the plane of polarization. For the combination of axion parameters adopted in Fig. 3, a probability of 10% is achieved when g a 10 −9 GeV −1 (for m a = 2 × 10 −6 eV). When g a = 10 −8 GeV −1 , the probability is 10% for the entire range of axion masses considered (7 × 10 −7 − 1.7 × 10 −5 eV).\\nAt larger viewing angles on the other hand (right panels of Fig. 9), it becomes evident one of the unique signatures of vacuum polarization on the polarization signal of NSs with magnetic fields B 7 × 10 13 G, that is the rotation at high energies of the plane of polarization from perpendicular to parallel (Lai & Ho 2003b). Photon-axion conversion does not erase this feature; however, as a result of the suppression of the O-mode photons which dominate at higher energies, the shift of the plane occurs at higher energies than it would without conversion. For the axion parameters adopted in Fig. 9, the energy at which the shift occurs with the conversion differs by a factor of ∼ 15 − 40% with respect to the case with no conversion, with the precise value depending on the magnetic field strength. Our theoretical results provide several methods for constraining axion parameters from actual measurements of NS spectra. The foremost method is spectral analysis, often performed on phase-averaged spectra when the resolution low. We showed in §5.1 that the relative suppression of the Omode photons with respect to the X-mode photons noticeably distorts the spectral shape with respect to the case without conversion. This distortion is strongest in the high-energy spectral tail, where the intensity of the O-mode photons is larger for most viewing angles (see Fig. 4). Observed enhancements of high energy spectral tails are generally modeled with a power law (often of unknown origin; e.g. Manzali et al. 2007, Durant et al. 2011 for young and middle-aged pulsars; Perna et al. 2001, Juett al. (2002, Rea et al. 2007 for magnetars). However, a deficit of high energy photons in a thermal fit could be a signature of photon-axion conversion, and the axion parameters could then be determined by means of detailed spectral fits that account for the effect. For this type of analysis, the distance to the NS is not important since the main discriminant is the spectral shape, in particular the deficit of high-energy photons.\"\n },\n {\n \"title\": \"Observational tests\",\n \"content\": \"For sources with good distance estimates 7 but too few Xray counts to perform detailed phase-resolved spectroscopy, Fig. 7 demonstrates that the determination of the effective emission area (through measurements of flux and peak energy) can rule out large regions of m a − g a parameter space if the inferred NS radius is close to the maximum or minimum value allowed by NS nuclear equations of state (EOS). The minimum radius constraint requires knowing that the observed radiation is being emitted by the entire NS surface and not from a small hot spot. This may be the case when pulsations are not detected or when the angle between the line of sight and magnetic dipole axis is non-negligible and the measurement is made near pulse minimum. The maximum conthe distances to NSs in globular clusters are known to much higher precision. straint on the other hand does not require other independent information.\\nTo illustrate how to use Fig. 7 with NS observations, suppose R max is the maximum radius allowed by a particular EOS (see, e.g., Lattimer & Prakash 2007, for a review), and A max is the corresponding surface area. Then a NS whose inferred emission area is A ∼ A max would rule out the m a − g a region of the parameter space for which the effective area 'correction' due to photon-axion conversion is larger than 1. For B = 4 × 10 12 G and T s = 5 × 10 6 K (corresponding to a redshifted value ∼ 3.8 × 10 6 K), the parameter space with m a 2 × 10 −5 eV and g a −1 4 × 10 9 GeV (or g a 2.5 × 10 −8 GeV −1 ) would be ruled out. For NSs whose radius determination relies on methods other than flux and peak energy measurements 8 , constraints on axion parameters can be obtained using the method described above without any need for the radius to be near the maximum or minimum allowed by the EOS.\\nAs an aside, we note that the the effects of axions on the inferred area discussed above suggest that the use of NS effective radii to rule out nuclear EOSs may not be entirely reliable. The existence of axions could severely bias such conclusions.\\nFor sources with phase-resolved X-ray spectra, observations can be fit with either blackbody or model atmosphere spectra to obtain peak energies E peak and effective areas A eff as a function of rotation phase or viewing angle α. Figure 6 shows a clear signature of photon-axion conversion, i.e., a significant rise and fall of effective area at intermediate viewing angles, especially for B 10 13 G. The important quantity for this diagnostic is the relative change of the effective area with phase; thus, this method does not require knowledge of the distance to the NS. It is also rather insensitive to the NS radius/mass (i.e., surface gravity), since variations in this quantity produce energy shifts that are the same at all phases, leaving the relative phase-dependent signatures of the conversion unaltered. An exact estimate of the range of axion parameters that can be constrained by means of this method cannot be made in absolute terms, since it clearly depends on the quality of the data. However, from a theoretical point of view, photonaxion signatures are imprinted in the NS spectra as long as the conversion probability is non-negligible in the typical X-ray band of observations. Fig. 4 shows that this is the case for an axion coupling strength g a 10 −9 GeV −1 and for a large range of axion masses, up to m a ∼ 10 −4 eV if g a 10 −8 GeV −1 .\\nThe theoretical framework and methods for extracting axion parameters from observations of magnetic NSs is presented above. Detailed analyses of specific sources require a model appropriate for the specific viewing and emission geometry, magnetic field strength, and surface temperature for each source; a comparison with the data can then be performed by folding the model through the response function of the detector. Such an analysis will be the subject of future work. Below, we briefly review NSs that serve as potential candidates for obtaining constraints on axion physics. These sources possess bright X-ray thermal emission from the NS surface. Among the observed NS population, such sources can be grouped into several classes (which may not be mutually exclusive groups; see, e.g., Kaspi 2010;. One class is the magnetars, which generally possess superstrong magnetic fields B 10 14 G. 9 For these objects, vacuum resonance effects lead to emission that is predominantly in the X-mode. Therefore the possible conversion of O-mode photons to axions will not produce an observable effect on the total radiation spectrum. Another class of NSs are the rotation-powered pulsars; these are the classic radio pulsars that are powered by the loss of 8 Examples of such methods of NS radius measurement (or more generally of the radius-to-mass ratio) include spectral lines (e.g. Cackett et al. 2009), thermal X-ray pulse profiles (e.g. Bogdanov et al. 2008; note that at low energies E 1 keV, these pulse profiles are not very sensitive to the presence of photon axion conversion, as shown in Fig. 8), gravitational wave emission (Lenzi et al. 2009), quasi-periodic oscillations in accreting NSs (Miller et al. 1998), and neutrino emission from proto-stars (Lattimer & Prakash 2007). Masses can be independently inferred from radial velocity studies for NSs in binary systems (e.g. van and pulsar timing (a comprehensive review of all these methods can be found in Lattimer & Prakash 2007). 9 Note however that the bursting behaviour used to characterize magnetars has also been seen in a lower magnetic field object (see Rea et al 2010). rotational energy due to emission of electromagnetic radiation. These NSs can have magnetic fields up to the magnetar regime B 10 14 G, and a small fraction of sources are sufficiently bright in the X-rays (see, e.g., Pavlov et al. 2002;Mereghetti 2007;Zavlin 2009, for reviews). These NSs are good candidates if their X-ray emission is not dominated by magnetospheric emission (as in, e.g., PSR B0656+14).\\nThe isolated NSs are another interesting class. The seven confirmed isolated NSs have B ≈ (1 − 4) × 10 13 G inferred from timing measurements , while features in their X-ray spectrum suggest B ≈ (5 − 10) × 10 13 G if the features are due to the proton cyclotron resonance (Haberl 2007). We note that the observed X-ray spectra of isolated NSs are generally well-fit (except around spectral lines) by blackbodies, including Wien-like behavior at high energies (van Kerkwijk & Kaplan 2007). Various explanations have been proposed to explain the soft high energy tails as compared to the hard tails predicted by atmosphere models (e.g., Romani 1987), such as emission from a partially optically thin atmosphere (e.g., Motch et al. 2003) or the effect of vacuum polarization (see §2 and Ho & Lai 2003). Here we have shown that photon-axion conversion can suppress high-energy emission and produce softer tails (see Fig. 4). However, this cannot explain the spectra of isolated NSs. These NSs have surface temperatures 10 6 K, and, as we discussed in Sec. 2, the conversion is less effective at low temperatures due to the low fraction of O-mode photons. However, if soft tails (softer than the blackbody) were detected in NSs with relatively high surface temperature, then photon-axion conversion could indeed be a competing explanation.\\nA final class is the central compact object neutron stars (CCOs), which have B ∼ 10 10 − 10 11 G, obtained from timing and spectral analyses (see, e.g., Halpern & Gotthelf 2010). At these low fields, the vacuum resonance is outside of the atmosphere, and the emerging intensity has a high fraction of O-mode photons; their surface temperatures are constrained to be (2 − 3) × 10 6 K, while hot spot temperatures are quite high [∼ (4 − 6) × 10 6 K]. These characteristics, combined with the fact that the photon-axion conversion probability at these lower fields is still high for a wide range of axion parameters (Jimenez et al. 2011), make CCOs potentially good candidates. Current data on many of the NSs belonging to the classes described above are already of sufficient quality to allow axion constraints to be derived.\\nFinally, we note that X-ray polarization measurements, like those that will be performed by the forthcoming Gravity and Extreme Magnetism Small Explorer (GEMS) satellite (Jahoda 2010), will increase the robustness of axion constraints, as well as provide an independent measure of the stellar magnetic field strength and geometry (van Adelsberg & Perna 2009).\"\n },\n {\n \"title\": \"SUMMARY\",\n \"content\": \"Constraints on the axion mass and coupling strength by means of observations of magnetized objects have been discussed in the recent literature (e.g., Lai & Heyl 2006;Chelouche et al. 2009;Gill & Heyl 2011;Jimenez et al. 2011) as a means to complement and independently test constraints obtained from other methods. The studies above emphasized different observational aspects and tests. The modification to the X-ray spectra of highly magnetized NSs induced by photon-axion conversion was discussed by Lai & Heyl (2006) using a blackbody spectrum to model the photon spectrum from NSs. Chelouche et al. (2009) discussed spectral features produced in the sub-mm/IR wavelength regime for magnetized NSs, while Gill & Heyl (2011) considered limits that can be derived from polarization measurements of white dwarfs. Jimenez et al. (2011) focused their analysis on constraints that can be obtained through observations of eclipsing white dwarfs in binaries, but they also discussed qualitatively the potential of NSs. Since most of the photon-axion conversion happens at a distance of many stellar radii from the NS, they pointed out that there are two configurations that should be considered: light from a background object passing through the magnetic field of the NS (i.e., occultation) and a binary containing a NS where the companion transits close enough to the line of sight for its light to be influenced by the NS magnetic field. In these two cases the treatment is much simpler as the physics of photon propagation in the NS atmosphere is irrelevant. However, they determined that the occultation probability is too low to be astrophysically relevant and that there are no known binary systems involving a NS that are detached enough to yield a clean constraint.\\nSince NSs are routinely observed in the X-ray domain, the goal of our paper has been an extensive exploration of the actual constraints on axion physics that can be obtained from observations of NS thermal spectra. We have also explored what can be learned from future X-ray polarization measurements. For these purposes, we used detailed, magnetized atmosphere models, which properly account for the energy and angledependent emerging intensities of the two polarization modes, including mode conversion due to vacuum polarization effects. The emergent O mode intensity was coupled with the energy and angle-dependent photon-axion conversion probability, and this allowed us to make theoretical predictions for phase-resolved spectra with photon axion-conversion in the relevant regime of magnetic fields and temperatures for NSs. Since the relative intensities of the O and X mode photons depend strongly on the angle at which they emerge with respect to the magnetic field direction, we considered emission from a region consisting of a hot spot with an axis coincident with the NS magnetic field axis. Thermal emission from most Xray bright NSs does indeed indicate that we are observing hot spots (this from both non-negligible pulse fractions and small inferred emission areas). In order to extend axion studies by means of comparison with observations of any NS (especially those emitting from the entire surface, with non-dipolar, complicated B topologies, and fast rotators), one would have to include, locally, magnetic fields non-normal to the surface in the atmospheres, investigate how non-dipolar fields influence both the emission spectra as well as the photon-axion conversion, and, for the polarization, explicitly integrate the Stokes parameters to R pl . However, since the magnetic field structure over the entire surface of a NS is not fully known apriori, determination of the axion parameters would be more degenerate under these more general conditions. Our analysis has identified some features in NS spectra that bear the telltale signs of photon-axion conversion:\\n• The NS spectral shape is noticeably distorted compared to the case without photon-axion conversion, with suppression of the high-energy tail for viewing angles ∼ 20 − 70 deg (with respect to the center of the emission region). Detailed spectral fits can yield axion parameters. This analysis does not require knowledge of the distance to the NS, as the axion signatures are imprinted in the spectral shape.\\n• The spectral suppression of the O-mode photons by photon-axion conversion dominates the high energy tail of the spectrum for a range of viewing angles. In addition, it results in a shift of the peak energy, E peak , towards lower values for a range of rotation phases (i.e., viewing angles). As a result, the effective area of the spot A eff (as inferred from measurements of the flux and peak spectral energy) is larger than its value without axion conversion. We demonstrated that a clear signature of photon-axion conversion is a significant rise and fall of effective area (normalized to the phase-on value) at intermediate viewing angles, especially for B 10 13 G. Such a measurement requires high signalto-noise phase-resolved spectra, but does not require knowledge of the distance to the NS, since the axion signature appears in the relative change of the effective area with phase.\\n• For a star emitting from the entire surface, photonaxion conversion can substantially affect the inferred NS emission area by 50%, and the inferred radius by 20%. These values are measured using the flux and peak energy of the thermal spectrum. If the distance is well constrained, and the NS radius is known through different methods, then the inferred emission area can be directly translated into a constraint in the m a − g a plane. If the NS radius is not independently known, then an inferred emission area that exceeds the maximum value allowed by NS equations of state could indicate the presence of photon-axion conversion.\\n• In the absence of photon-axion conversion, radiation from a hot spot observed phase-on at energies ∼ 2 − 3 keV is linearly polarized in the plane formed by B and the direction of photon propagation. Conversion rotates the plane of polarization and leads to radiation polarized perpendicular to the plane of B and the direction of photon propagation, for a range of axion parameters.\\nWe concluded §5.2 with a discussion of appropriate sources to study within the theoretical framework developed here for probing axions. While this paper has outlined the main elements for connecting theoretical ideas of photon-axion conversion to actual observations that are made of NSs, studying each suitable source will require a specific suite of models, tailored to the particular object (e.g., its magnetic field, surface temperature, emission and viewing geometry). The theoretical models are then convolved with the detector response, and a process of minimization identifies the most likely values of the parameters. Detailed analysis of the most promising NS candidates will be the subject of future work.\\nThis work was partially supported by grants NSF AST-1009396, AR1-12003X, and DD1-12053X (RP); WCGH is supported by STFC in the UK; LV and RJ are supported by MICINN grab AYA 2008-03531. RP, RJ and LV thank the Centro de Ciencias de Benasque Pedro Pascual where this work was initiated. We further acknowledge Carlos Peña-Garay, Dong Lai, Jeremy Heyl and Enrique Fernadez-Martinez for insightful discussions, and an anonymous referee for helpful comments on our manuscript. WCGH appreciates the use of the computer facilities at the Kavli Institute for Particle Astrophysics and Cosmology.\"\n },\n {\n \"title\": \"FIG. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 6 .FIG. 7 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 8 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 9 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"-Fractional contributions to the flux from the emergent O ( thick/green lines) and X (thin/red lines) modes for four angles δ with respect to the surface normal and three magnetic field strengths. The effective temperature is Ts = 5 × 10 6 K in all cases.","content":"FIG. 1 .\n1-Fractional contributions to the flux from the emergent O ( thick/green lines) and X (thin/red lines) modes for four angles δ with respect to the surface normal and three magnetic field strengths. The effective temperature is Ts = 5 × 10 6 K in all cases."},{"id":"","caption":"-Fractional contributions to the flux from the emergent O (thick/green lines) and X (thin/red lines) modes, for four angles δ with respect to the surface normal and three effective temperatures of the star. The magnetic field strength is B = 4 × 10 12 G in all cases.","content":"FIG. 2 .\n2-Fractional contributions to the flux from the emergent O (thick/green lines) and X (thin/red lines) modes, for four angles δ with respect to the surface normal and three effective temperatures of the star. The magnetic field strength is B = 4 × 10 12 G in all cases."},{"id":"","caption":"4. THERMAL SPECTRA, LIGHT CURVES, AND POLARIZATION OF NEUTRON STARS WITH PHOTON-AXION CONVERSION","content":"4. THERMAL SPECTRA, LIGHT CURVES, AND POLARIZATION OF NEUTRON STARS WITH PHOTON-AXION CONVERSION"},{"id":"","caption":"FIG. 4.-Comparison between phase-resolved spectra of NSs with (thick/red lines) and without (thin/blue lines) photon-axion conversion. In each case, the dotted lines show the flux contribution from the O mode, while the dashed lines display the contribution from the X mode, which is unaffected by conversion. The solid lines show the total spectra. In all panels, the effective temperature of the star is Ts = 5 × 10 6 K and the axion parameters are ma = 2 × 10 −6 eV and ga = 10 −8 GeV −1 , while the B-field strength has three values from top to bottom. The spectra are shown for two angles α between the observer and the dipole axis.","content":"FIG. 4.-Comparison between phase-resolved spectra of NSs with (thick/red lines) and without (thin/blue lines) photon-axion conversion. In each case, the dotted lines show the flux contribution from the O mode, while the dashed lines display the contribution from the X mode, which is unaffected by conversion. The solid lines show the total spectra. In all panels, the effective temperature of the star is Ts = 5 × 10 6 K and the axion parameters are ma = 2 × 10 −6 eV and ga = 10 −8 GeV −1 , while the B-field strength has three values from top to bottom. The spectra are shown for two angles α between the observer and the dipole axis."},{"id":"","caption":"-Left: Peak energy of the observed spectrum as a function of viewing angle α (i.e., phase of the NS). The presence of photon-axion conversion changes and enhances the variation in peak energy as the star rotates. Right: Corresponding phase-resolved effective area that an observer would infer from spectral fitting. Axion parameters have the same values as inFig. 4. -Ratio between the observed effective area with photon axion conversion and without photon-axion conversion for a range of axion parameters ma and ga. The NS magnetic field and temperature are B = 4 × 10 12 G and T = 5 × 10 6 K (top left panel), B = 4 × 10 13 G and T = 5 × 10 6 K (bottom left panel), and B = 4 × 10 12 G and T = 2.5 × 10 6 K (top right panel), respectively.","content":"FIG. 6 .FIG. 7 .\n67-Left: Peak energy of the observed spectrum as a function of viewing angle α (i.e., phase of the NS). The presence of photon-axion conversion changes and enhances the variation in peak energy as the star rotates. Right: Corresponding phase-resolved effective area that an observer would infer from spectral fitting. Axion parameters have the same values as inFig. 4. -Ratio between the observed effective area with photon axion conversion and without photon-axion conversion for a range of axion parameters ma and ga. The NS magnetic field and temperature are B = 4 × 10 12 G and T = 5 × 10 6 K (top left panel), B = 4 × 10 13 G and T = 5 × 10 6 K (bottom left panel), and B = 4 × 10 12 G and T = 2.5 × 10 6 K (top right panel), respectively."},{"id":"","caption":"-Comparison of light curves at four energies with (right panels) and without (left panels) photon-axion conversion. The values of the energies are 0.5 keV (solid curves), 1.5 keV (dotted curves), 2.5 keV (dashed curves), 4 keV (long-dashed curves). In each panel, light curves are normalized to the maximum value among the energies considered.","content":"FIG. 8 .\n8-Comparison of light curves at four energies with (right panels) and without (left panels) photon-axion conversion. The values of the energies are 0.5 keV (solid curves), 1.5 keV (dotted curves), 2.5 keV (dashed curves), 4 keV (long-dashed curves). In each panel, light curves are normalized to the maximum value among the energies considered."},{"id":"","caption":"-Linear polarization, defined as (F O − F X )/(F O + F X ), for the cases with no photon-axion conversion (solid lines) and with photon-axion conversion (dotted lines). NS and axion parameters are the same as inFig. 2.","content":"FIG. 9 .\n9-Linear polarization, defined as (F O − F X )/(F O + F X ), for the cases with no photon-axion conversion (solid lines) and with photon-axion conversion (dotted lines). NS and axion parameters are the same as inFig. 2."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"-Fractional contributions to the flux from the emergent O ( thick/green lines) and X (thin/red lines) modes for four angles δ with respect to the surface normal and three magnetic field strengths. The effective temperature is Ts = 5 × 10 6 K in all cases.\",\n \"content\": \"FIG. 1 .\\n1-Fractional contributions to the flux from the emergent O ( thick/green lines) and X (thin/red lines) modes for four angles δ with respect to the surface normal and three magnetic field strengths. The effective temperature is Ts = 5 × 10 6 K in all cases.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"-Fractional contributions to the flux from the emergent O (thick/green lines) and X (thin/red lines) modes, for four angles δ with respect to the surface normal and three effective temperatures of the star. The magnetic field strength is B = 4 × 10 12 G in all cases.\",\n \"content\": \"FIG. 2 .\\n2-Fractional contributions to the flux from the emergent O (thick/green lines) and X (thin/red lines) modes, for four angles δ with respect to the surface normal and three effective temperatures of the star. The magnetic field strength is B = 4 × 10 12 G in all cases.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"4. THERMAL SPECTRA, LIGHT CURVES, AND POLARIZATION OF NEUTRON STARS WITH PHOTON-AXION CONVERSION\",\n \"content\": \"4. THERMAL SPECTRA, LIGHT CURVES, AND POLARIZATION OF NEUTRON STARS WITH PHOTON-AXION CONVERSION\"\n },\n {\n \"id\": \"\",\n \"caption\": \"FIG. 4.-Comparison between phase-resolved spectra of NSs with (thick/red lines) and without (thin/blue lines) photon-axion conversion. In each case, the dotted lines show the flux contribution from the O mode, while the dashed lines display the contribution from the X mode, which is unaffected by conversion. The solid lines show the total spectra. In all panels, the effective temperature of the star is Ts = 5 × 10 6 K and the axion parameters are ma = 2 × 10 −6 eV and ga = 10 −8 GeV −1 , while the B-field strength has three values from top to bottom. The spectra are shown for two angles α between the observer and the dipole axis.\",\n \"content\": \"FIG. 4.-Comparison between phase-resolved spectra of NSs with (thick/red lines) and without (thin/blue lines) photon-axion conversion. In each case, the dotted lines show the flux contribution from the O mode, while the dashed lines display the contribution from the X mode, which is unaffected by conversion. The solid lines show the total spectra. In all panels, the effective temperature of the star is Ts = 5 × 10 6 K and the axion parameters are ma = 2 × 10 −6 eV and ga = 10 −8 GeV −1 , while the B-field strength has three values from top to bottom. The spectra are shown for two angles α between the observer and the dipole axis.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"-Left: Peak energy of the observed spectrum as a function of viewing angle α (i.e., phase of the NS). The presence of photon-axion conversion changes and enhances the variation in peak energy as the star rotates. Right: Corresponding phase-resolved effective area that an observer would infer from spectral fitting. Axion parameters have the same values as inFig. 4. -Ratio between the observed effective area with photon axion conversion and without photon-axion conversion for a range of axion parameters ma and ga. The NS magnetic field and temperature are B = 4 × 10 12 G and T = 5 × 10 6 K (top left panel), B = 4 × 10 13 G and T = 5 × 10 6 K (bottom left panel), and B = 4 × 10 12 G and T = 2.5 × 10 6 K (top right panel), respectively.\",\n \"content\": \"FIG. 6 .FIG. 7 .\\n67-Left: Peak energy of the observed spectrum as a function of viewing angle α (i.e., phase of the NS). The presence of photon-axion conversion changes and enhances the variation in peak energy as the star rotates. Right: Corresponding phase-resolved effective area that an observer would infer from spectral fitting. Axion parameters have the same values as inFig. 4. -Ratio between the observed effective area with photon axion conversion and without photon-axion conversion for a range of axion parameters ma and ga. The NS magnetic field and temperature are B = 4 × 10 12 G and T = 5 × 10 6 K (top left panel), B = 4 × 10 13 G and T = 5 × 10 6 K (bottom left panel), and B = 4 × 10 12 G and T = 2.5 × 10 6 K (top right panel), respectively.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"-Comparison of light curves at four energies with (right panels) and without (left panels) photon-axion conversion. The values of the energies are 0.5 keV (solid curves), 1.5 keV (dotted curves), 2.5 keV (dashed curves), 4 keV (long-dashed curves). In each panel, light curves are normalized to the maximum value among the energies considered.\",\n \"content\": \"FIG. 8 .\\n8-Comparison of light curves at four energies with (right panels) and without (left panels) photon-axion conversion. The values of the energies are 0.5 keV (solid curves), 1.5 keV (dotted curves), 2.5 keV (dashed curves), 4 keV (long-dashed curves). In each panel, light curves are normalized to the maximum value among the energies considered.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"-Linear polarization, defined as (F O − F X )/(F O + F X ), for the cases with no photon-axion conversion (solid lines) and with photon-axion conversion (dotted lines). NS and axion parameters are the same as inFig. 2.\",\n \"content\": \"FIG. 9 .\\n9-Linear polarization, defined as (F O − F X )/(F O + F X ), for the cases with no photon-axion conversion (solid lines) and with photon-axion conversion (dotted lines). NS and axion parameters are the same as inFig. 2.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". S L Adler, Ann. Phys. 67599Adler, S. L. 1971, Ann. Phys., 67, 599","doi":""},{"text":". C Alcock, A Illarionov, ApJ. 235534Alcock, C. & Illarionov, A. 1980, ApJ, 235, 534","doi":""},{"text":". S Andriamonje, J. Cosm. Astropart. Phys. 410Andriamonje, S. et al. 2007, J. Cosm. Astropart. Phys., 4, 10","doi":""},{"text":". A Arvanitaki, S Dimopoulos, S Dubovsky, N Kaloper, J March-Russell, Phys. Rev., D. 123530Arvanitaki, A., Dimopoulos, S., Dubovsky, S., Kaloper, N., March-Russell, J. 2010, Phys. Rev., D, 81, 123530.","doi":""},{"text":". S J Asztalos, Phys. Rev. D. 6911101Asztalos, S. J. et al., 2004, Phys. Rev. D, 69, 011101","doi":""},{"text":". S J Asztalos, Phys. Rev. Lett. 10441301Asztalos, S. J. et al., 2010, Phys. Rev. Lett. 104, 041301","doi":""},{"text":". A Avgoustidis, C Burrage, J Redondo, L Verde, R Jimenez, arXiv:1004.2053JCAP. 24preprintAvgoustidis, A. Burrage, C. Redondo, J., Verde, L., Jimenez, R. 2010, JCAP, 1010, 24, preprint arXiv:1004.2053","doi":"arXiv:1004.2053"},{"text":". J Bahcall, Neutrino Astrophysics. Cambridge Univ. PressBahcall, J. 1989, Neutrino Astrophysics, (Cambridge: Cambridge Univ. Press)","doi":""},{"text":". F Bernardini, R Perna, E Gotthelf, G Israel, N Rea, L Stella, MNRAS. 418638Bernardini, F., Perna, R., Gotthelf, E., Israel, G. Rea, N., Stella, L. 2011, MNRAS, 418, 638","doi":""},{"text":". S Bogdanov, J E Grindlay, G B Rybicki, ApJ. 689407Bogdanov, S., Grindlay, J. E., Rybicki, G. B. 2008, ApJ, 689, 407","doi":""},{"text":". E F Brown, L Bildsten, P Chang, ApJ. 574920Brown, E. F., Bildsten, L., & Chang, P. 2002, ApJ, 574, 920","doi":""},{"text":". E M Cackett, J M Miller, S Bhattacharyya, J E Grindlay, J Homan, M Van Der Klis, M C Miller, T E Strohmayer, R Wijnands, ApJ. 674415Cackett, E. M., Miller, J. M., Bhattacharyya, S., Grindlay, J. E., Homan, J., van der Klis, M., Miller, M. C., Strohmayer, T. E., Wijnands, R. 2008, ApJ, 674, 415","doi":""},{"text":". P Chang, L Bildsten, P Arras, ApJ. 723719Chang, P., Bildsten, L., & Arras, P. 2010, ApJ, 723, 719","doi":""},{"text":". D Chelouche, R Rabadan, S S Pavlov, F Castejon, ApJ. 1801Chelouche, D., Rabadan, R., Pavlov, S. S., Castejon, F. 2009, ApJ, 180, 1","doi":""},{"text":". S De Panlis, Phys. Rev. Lett. 59839De Panlis, S. et al. 1987, Phys. Rev. Lett. 59, 839","doi":""},{"text":". M Durant, O Kargaltsev, G G Pavlov, arXiv:1109.1984arXiv e-printDurant, M., Kargaltsev, O., Pavlov, G. G., 2011, arXiv e-print (arXiv:1109.1984)","doi":"arXiv:1109.1984"},{"text":". S Eidelman, Phys. Lett. 5921Eidelman, S. et al. 2004, Phys. Lett. B592, 1","doi":""},{"text":". R Gill, J S Heyl, arXiv:1105.2083Gill, R., & Heyl, J. S. 2011, arXiv:1105.2083","doi":"arXiv:1105.2083"},{"text":". F Haberl, Astrophys. and Space Science. 308181Haberl, F. 2007, Astrophys. and Space Science, 308, 181","doi":""},{"text":". J P Halpern, E V Gotthelf, ApJ. 709436Halpern, J. P., Gotthelf, E. V. 2010, ApJ, 709, 436","doi":""},{"text":". W C Haxton, ARA&A. 33459Haxton, W. C. 1995, ARA&A, 33, 459","doi":""},{"text":". J S Heyl, N J Shaviv, MNRAS. 311555Heyl, J. S., Shaviv, N. J. 2000, MNRAS, 311, 555","doi":""},{"text":". J S Heyl, N J Shaviv, Phys. Rev.D. 6623002Heyl, J. S., Shaviv, N. J. 2002, Phys. Rev.D, 66, 023002","doi":""},{"text":". J S Heyl, N J Shaviv, D Lloyd, MNRAS. 342134Heyl, J. S., Shaviv, N. J., Lloyd, D. 2003, MNRAS, 342, 134","doi":""},{"text":". W C G Ho, D Lai, MNRAS. 3271081Ho, W. C. G. & Lai, D. 2001, MNRAS, 327, 1081","doi":""},{"text":". W C G Ho, D Lai, MNRAS. 588962Ho, W. C. G. & Lai, D. 2003, MNRAS, 588, 962","doi":""},{"text":". W C G Ho, A Y Potekhin, G Chabrier, ApJS. 178102Ho, W. C. G., Potekhin, A. Y., Chabrier, G. 2008, ApJS, 178, 102","doi":""},{"text":"K Jahoda, Proc. SPIE. SPIE773277320Jahoda, K. 2010, Proc. SPIE, 7732, 77320W","doi":""},{"text":". A M Juett, H L Marshall, D Chakrabarty, N Schulz, ApJ. 56831Juett, A. M., Marshall, H. L., Chakrabarty, D., Schulz, N. S 2002, ApJ, 568L, 31","doi":""},{"text":". R Jimenez, C Peña-Garay, L Verde, Physics Letters B. 703232Jimenez, R., Peña-Garay, C., & Verde, L. 2011, Physics Letters B, 703, 232","doi":""},{"text":". D L Kaplan, M H Van Kerkwijk, ApJ. 705798Kaplan, D. L., van Kerkwijk, M. H. 2009, ApJ, 705, 798","doi":""},{"text":". D L Kaplan, M H Van Kerkwijk, ApJ. 74030Kaplan, D. L., van Kerkwijk, M. H. 2011, ApJ, 740, 30","doi":""},{"text":". V M Kaspi, Publ. of the Nat. Acad. of Science. 1077147Kaspi, V. M. 2010, Publ. of the Nat. Acad. of Science, 107, 7147","doi":""},{"text":". D Lai, J Heyl, Phys. Rev. D. 7413003Lai, D. & Heyl, J. 2006, Phys. Rev. D, 2006, 74, 13003","doi":""},{"text":". D Lai, W C G Ho, ApJ. 566373Lai, D. & Ho, W. C. G. 2002, ApJ, 566, 373","doi":""},{"text":". D Lai, W C Ho, ApJ. 588962Lai, D. & Ho, W. C. G. 2003a, ApJ, 588, 962","doi":""},{"text":". D Lai, W C G Ho, Phys. Rev. Lett. 9171101Lai, D. & Ho, W. C. G. 2003b, Phys. Rev. Lett., 91, 071101","doi":""},{"text":". J M Lattimer, M Prakash, Phys. Rep. 4421Lattimer, J. M., & Prakash, M. 2007, Phys. Rep., 442, 1","doi":""},{"text":". C H Lenzi, M Malheiro, R M Marinho, G F Marranghello, C Providencia, Journ. of Phys. 1541Lenzi, C. H., Malheiro, M., Marinho, R. M., Marranghello, G. F., Providencia, C. 2009, Journ. of Phys. 154, 1","doi":""},{"text":". A Manzali, A De Luca, P A Caraveo, ApJ. 669570Manzali, A., De Luca, A., Caraveo, P. A. 2007, ApJ, 669, 570","doi":""},{"text":". S Mereghetti, MmSAI. 78644Mereghetti, S. 2007, MmSAI, 78, 644","doi":""},{"text":". M C Miller, F K Lamb, D Psaltis, ApJ. 508791Miller, M. C., Lamb, F.K., Psaltis, D. 1998, ApJ, 508, 791","doi":""},{"text":". K Mori, W C G Ho, MNRAS. 377905Mori, K. & Ho, W. C. G. 2007, MNRAS, 377, 905","doi":""},{"text":". C Motch, V E Zavlin, F Haberl, A&A. 408323Motch, C., Zavlin, V. E., Haberl, F. 2003, A&A, 408, 323","doi":""},{"text":". D Page, ApJ. 442273Page, D. 1995, ApJ, 442, 273","doi":""},{"text":"Neutron Stars in Supernova Remnants. G G Pavlov, D Sanwal, G P Garmire, V E Zavlin, ASP Conference Series. Ed. Patrick O. Slane and Bryan M. Gaensler. San Francisco: ASP271247Pavlov, G. G., Sanwal, D., Garmire, G. P., Zavlin, V. E. 2002, Neutron Stars in Supernova Remnants, ASP Conference Series, Vol. 271, held in Boston, MA, USA, 14-17 August 2001. Ed. Patrick O. Slane and Bryan M. Gaensler. San Francisco: ASP, p.247","doi":""},{"text":". G G Pavlov, Yu A Shibanov, J Ventura, V E Zavlin, A&A. 289837Pavlov, G. G., Shibanov, Yu. A., Ventura, J., Zavlin, V. E. 1994, A&A, 289, 837","doi":""},{"text":". G G Pavlov, V E Zavlin, ApJ. 5291011Pavlov, G. G. & Zavlin, V. E. 2000, ApJ, 529, 1011","doi":""},{"text":". R D Peccei, H R Quinn, Phys. Rev. Lett. 381440Peccei, R. D., Quinn, H. R. 1977, Phys. Rev. Lett., 38, 1440","doi":""},{"text":". K R Pechenick, C Ftaclas, J M Cohen, ApJ. 274846Pechenick, K. R., Ftaclas, C., Cohen, J. M. 1983, ApJ, 274, 846","doi":""},{"text":". R Perna, J S Heyl, L E Hernquist, A M Juett, D Chakrabarty, ApJ. 55718Perna, R., Heyl, J. S., Hernquist, L. E., Juett, A. M., Chakrabarty, D. 2001, ApJ, 557, 18","doi":""},{"text":". R Perna, E V Gotthelf, ApJ. 681522Perna, R. & Gotthelf, E. V. 2008, ApJ, 681, 522","doi":""},{"text":". R Perna, J A Pons, ApJ. 72751Perna, R. & Pons, J. A. 2011, ApJ, 727, L51","doi":""},{"text":". J A Pons, R Perna, ApJ. 741123Pons, J. A. & Perna, R. 2011, ApJ, 741, 123","doi":""},{"text":". A Y Potekhin, D Lai, G Chabrier, Ho, C G Wynn, ApJ. 6121034Potekhin, A. Y., Lai, D., Chabrier, G., Ho, Wynn C. G. 2004, ApJ, 612, 1034","doi":""},{"text":". J Preskill, M B Wise, F Wilczek, Phys. Lett. B. 120127Preskill, J. Wise, M. B., Wilczek, F. 1983, Phys. Lett. B, 120, 127","doi":""},{"text":". G G Raffelt, Lect. Notes Phys. 74151Raffelt, G. G. 2008, Lect. Notes Phys., 741, 51","doi":""},{"text":". G Raffelt, L Stodolski, Phys Rev. D. 371237Raffelt, G. & Stodolski, L. 1988, Phys Rev. D, 37, 1237","doi":""},{"text":". N Rea, MNRAS. 381293Rea, N. et al. 2007, MNRAS, 381, 293","doi":""},{"text":". N Rea, Science. 330944Rea, N. et al. 2010, Science, 330, 944","doi":""},{"text":". R W Romani, ApJ. 313718Romani, R. W. 1987, ApJ, 313, 718","doi":""},{"text":". V Suleimanov, A Y Potekhin, K Werner, A&A. 500891Suleimanov, V., Potekhin, A. Y., & Werner, K. 2009, A&A, 500, 891","doi":""},{"text":". W Y Tsai, T Erber, M Van Adelsberg, D Lai, M Van Adelsberg, R Perna, M H Van Kerkwijk, D L Kaplan, Phys. Rev. D. 12191Ap&SSTsai, W. Y. & Erber, T. 1975, Phys. Rev. D, 12, 1132 van Adelsberg, M. & Lai, D. 2006, MNRAS, 373, 1495 van Adelsberg, M. & Perna, R. 2009, MNRAS, 399, 1523 van Kerkwijk, M. H. & Kaplan, D. L. 2007, Ap&SS, 308, 191","doi":""},{"text":". M H Van Kerkwijk, R P Breton, S R Kulkarni, ApJ. 72895van Kerkwijk, M. H., Breton, R. P., Kulkarni, S. R. 2011, ApJ, 728, 95","doi":""},{"text":". W U Wuensch, Phys. Rev., D. 403153Wuensch, W. U. et al., 1989, Phys. Rev., D, 40, 3153","doi":""},{"text":"V E Zavlin, Neutron Stars and Pulsars. Berlin HeidelbergSpringer357181Zavlin, V. E. 2009, in Neutron Stars and Pulsars, Astrophysics and Space Science Library, Vol. 357. Springer Berlin Heidelberg, p. 181","doi":""}],"string":"[\n {\n \"text\": \". S L Adler, Ann. Phys. 67599Adler, S. L. 1971, Ann. Phys., 67, 599\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Alcock, A Illarionov, ApJ. 235534Alcock, C. & Illarionov, A. 1980, ApJ, 235, 534\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Andriamonje, J. Cosm. Astropart. Phys. 410Andriamonje, S. et al. 2007, J. Cosm. Astropart. Phys., 4, 10\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Arvanitaki, S Dimopoulos, S Dubovsky, N Kaloper, J March-Russell, Phys. Rev., D. 123530Arvanitaki, A., Dimopoulos, S., Dubovsky, S., Kaloper, N., March-Russell, J. 2010, Phys. Rev., D, 81, 123530.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S J Asztalos, Phys. Rev. D. 6911101Asztalos, S. J. et al., 2004, Phys. Rev. D, 69, 011101\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S J Asztalos, Phys. Rev. Lett. 10441301Asztalos, S. J. et al., 2010, Phys. Rev. Lett. 104, 041301\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Avgoustidis, C Burrage, J Redondo, L Verde, R Jimenez, arXiv:1004.2053JCAP. 24preprintAvgoustidis, A. Burrage, C. Redondo, J., Verde, L., Jimenez, R. 2010, JCAP, 1010, 24, preprint arXiv:1004.2053\",\n \"doi\": \"arXiv:1004.2053\"\n },\n {\n \"text\": \". J Bahcall, Neutrino Astrophysics. Cambridge Univ. PressBahcall, J. 1989, Neutrino Astrophysics, (Cambridge: Cambridge Univ. Press)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Bernardini, R Perna, E Gotthelf, G Israel, N Rea, L Stella, MNRAS. 418638Bernardini, F., Perna, R., Gotthelf, E., Israel, G. Rea, N., Stella, L. 2011, MNRAS, 418, 638\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Bogdanov, J E Grindlay, G B Rybicki, ApJ. 689407Bogdanov, S., Grindlay, J. E., Rybicki, G. B. 2008, ApJ, 689, 407\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E F Brown, L Bildsten, P Chang, ApJ. 574920Brown, E. F., Bildsten, L., & Chang, P. 2002, ApJ, 574, 920\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E M Cackett, J M Miller, S Bhattacharyya, J E Grindlay, J Homan, M Van Der Klis, M C Miller, T E Strohmayer, R Wijnands, ApJ. 674415Cackett, E. M., Miller, J. M., Bhattacharyya, S., Grindlay, J. E., Homan, J., van der Klis, M., Miller, M. C., Strohmayer, T. E., Wijnands, R. 2008, ApJ, 674, 415\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Chang, L Bildsten, P Arras, ApJ. 723719Chang, P., Bildsten, L., & Arras, P. 2010, ApJ, 723, 719\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Chelouche, R Rabadan, S S Pavlov, F Castejon, ApJ. 1801Chelouche, D., Rabadan, R., Pavlov, S. S., Castejon, F. 2009, ApJ, 180, 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S De Panlis, Phys. Rev. Lett. 59839De Panlis, S. et al. 1987, Phys. Rev. Lett. 59, 839\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Durant, O Kargaltsev, G G Pavlov, arXiv:1109.1984arXiv e-printDurant, M., Kargaltsev, O., Pavlov, G. G., 2011, arXiv e-print (arXiv:1109.1984)\",\n \"doi\": \"arXiv:1109.1984\"\n },\n {\n \"text\": \". S Eidelman, Phys. Lett. 5921Eidelman, S. et al. 2004, Phys. Lett. B592, 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Gill, J S Heyl, arXiv:1105.2083Gill, R., & Heyl, J. S. 2011, arXiv:1105.2083\",\n \"doi\": \"arXiv:1105.2083\"\n },\n {\n \"text\": \". F Haberl, Astrophys. and Space Science. 308181Haberl, F. 2007, Astrophys. and Space Science, 308, 181\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J P Halpern, E V Gotthelf, ApJ. 709436Halpern, J. P., Gotthelf, E. V. 2010, ApJ, 709, 436\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W C Haxton, ARA&A. 33459Haxton, W. C. 1995, ARA&A, 33, 459\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J S Heyl, N J Shaviv, MNRAS. 311555Heyl, J. S., Shaviv, N. J. 2000, MNRAS, 311, 555\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J S Heyl, N J Shaviv, Phys. Rev.D. 6623002Heyl, J. S., Shaviv, N. J. 2002, Phys. Rev.D, 66, 023002\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J S Heyl, N J Shaviv, D Lloyd, MNRAS. 342134Heyl, J. S., Shaviv, N. J., Lloyd, D. 2003, MNRAS, 342, 134\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W C G Ho, D Lai, MNRAS. 3271081Ho, W. C. G. & Lai, D. 2001, MNRAS, 327, 1081\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W C G Ho, D Lai, MNRAS. 588962Ho, W. C. G. & Lai, D. 2003, MNRAS, 588, 962\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W C G Ho, A Y Potekhin, G Chabrier, ApJS. 178102Ho, W. C. G., Potekhin, A. Y., Chabrier, G. 2008, ApJS, 178, 102\",\n \"doi\": \"\"\n },\n {\n \"text\": \"K Jahoda, Proc. SPIE. SPIE773277320Jahoda, K. 2010, Proc. SPIE, 7732, 77320W\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A M Juett, H L Marshall, D Chakrabarty, N Schulz, ApJ. 56831Juett, A. M., Marshall, H. L., Chakrabarty, D., Schulz, N. S 2002, ApJ, 568L, 31\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Jimenez, C Peña-Garay, L Verde, Physics Letters B. 703232Jimenez, R., Peña-Garay, C., & Verde, L. 2011, Physics Letters B, 703, 232\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D L Kaplan, M H Van Kerkwijk, ApJ. 705798Kaplan, D. L., van Kerkwijk, M. H. 2009, ApJ, 705, 798\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D L Kaplan, M H Van Kerkwijk, ApJ. 74030Kaplan, D. L., van Kerkwijk, M. H. 2011, ApJ, 740, 30\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V M Kaspi, Publ. of the Nat. Acad. of Science. 1077147Kaspi, V. M. 2010, Publ. of the Nat. Acad. of Science, 107, 7147\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Lai, J Heyl, Phys. Rev. D. 7413003Lai, D. & Heyl, J. 2006, Phys. Rev. D, 2006, 74, 13003\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Lai, W C G Ho, ApJ. 566373Lai, D. & Ho, W. C. G. 2002, ApJ, 566, 373\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Lai, W C Ho, ApJ. 588962Lai, D. & Ho, W. C. G. 2003a, ApJ, 588, 962\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Lai, W C G Ho, Phys. Rev. Lett. 9171101Lai, D. & Ho, W. C. G. 2003b, Phys. Rev. Lett., 91, 071101\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J M Lattimer, M Prakash, Phys. Rep. 4421Lattimer, J. M., & Prakash, M. 2007, Phys. Rep., 442, 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C H Lenzi, M Malheiro, R M Marinho, G F Marranghello, C Providencia, Journ. of Phys. 1541Lenzi, C. H., Malheiro, M., Marinho, R. M., Marranghello, G. F., Providencia, C. 2009, Journ. of Phys. 154, 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Manzali, A De Luca, P A Caraveo, ApJ. 669570Manzali, A., De Luca, A., Caraveo, P. A. 2007, ApJ, 669, 570\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Mereghetti, MmSAI. 78644Mereghetti, S. 2007, MmSAI, 78, 644\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M C Miller, F K Lamb, D Psaltis, ApJ. 508791Miller, M. C., Lamb, F.K., Psaltis, D. 1998, ApJ, 508, 791\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Mori, W C G Ho, MNRAS. 377905Mori, K. & Ho, W. C. G. 2007, MNRAS, 377, 905\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Motch, V E Zavlin, F Haberl, A&A. 408323Motch, C., Zavlin, V. E., Haberl, F. 2003, A&A, 408, 323\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Page, ApJ. 442273Page, D. 1995, ApJ, 442, 273\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Neutron Stars in Supernova Remnants. G G Pavlov, D Sanwal, G P Garmire, V E Zavlin, ASP Conference Series. Ed. Patrick O. Slane and Bryan M. Gaensler. San Francisco: ASP271247Pavlov, G. G., Sanwal, D., Garmire, G. P., Zavlin, V. E. 2002, Neutron Stars in Supernova Remnants, ASP Conference Series, Vol. 271, held in Boston, MA, USA, 14-17 August 2001. Ed. Patrick O. Slane and Bryan M. Gaensler. San Francisco: ASP, p.247\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G G Pavlov, Yu A Shibanov, J Ventura, V E Zavlin, A&A. 289837Pavlov, G. G., Shibanov, Yu. A., Ventura, J., Zavlin, V. E. 1994, A&A, 289, 837\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G G Pavlov, V E Zavlin, ApJ. 5291011Pavlov, G. G. & Zavlin, V. E. 2000, ApJ, 529, 1011\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R D Peccei, H R Quinn, Phys. Rev. Lett. 381440Peccei, R. D., Quinn, H. R. 1977, Phys. Rev. Lett., 38, 1440\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K R Pechenick, C Ftaclas, J M Cohen, ApJ. 274846Pechenick, K. R., Ftaclas, C., Cohen, J. M. 1983, ApJ, 274, 846\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Perna, J S Heyl, L E Hernquist, A M Juett, D Chakrabarty, ApJ. 55718Perna, R., Heyl, J. S., Hernquist, L. E., Juett, A. M., Chakrabarty, D. 2001, ApJ, 557, 18\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Perna, E V Gotthelf, ApJ. 681522Perna, R. & Gotthelf, E. V. 2008, ApJ, 681, 522\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Perna, J A Pons, ApJ. 72751Perna, R. & Pons, J. A. 2011, ApJ, 727, L51\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J A Pons, R Perna, ApJ. 741123Pons, J. A. & Perna, R. 2011, ApJ, 741, 123\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Y Potekhin, D Lai, G Chabrier, Ho, C G Wynn, ApJ. 6121034Potekhin, A. Y., Lai, D., Chabrier, G., Ho, Wynn C. G. 2004, ApJ, 612, 1034\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Preskill, M B Wise, F Wilczek, Phys. Lett. B. 120127Preskill, J. Wise, M. B., Wilczek, F. 1983, Phys. Lett. B, 120, 127\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G G Raffelt, Lect. Notes Phys. 74151Raffelt, G. G. 2008, Lect. Notes Phys., 741, 51\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Raffelt, L Stodolski, Phys Rev. D. 371237Raffelt, G. & Stodolski, L. 1988, Phys Rev. D, 37, 1237\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Rea, MNRAS. 381293Rea, N. et al. 2007, MNRAS, 381, 293\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Rea, Science. 330944Rea, N. et al. 2010, Science, 330, 944\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R W Romani, ApJ. 313718Romani, R. W. 1987, ApJ, 313, 718\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Suleimanov, A Y Potekhin, K Werner, A&A. 500891Suleimanov, V., Potekhin, A. Y., & Werner, K. 2009, A&A, 500, 891\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W Y Tsai, T Erber, M Van Adelsberg, D Lai, M Van Adelsberg, R Perna, M H Van Kerkwijk, D L Kaplan, Phys. Rev. D. 12191Ap&SSTsai, W. Y. & Erber, T. 1975, Phys. Rev. D, 12, 1132 van Adelsberg, M. & Lai, D. 2006, MNRAS, 373, 1495 van Adelsberg, M. & Perna, R. 2009, MNRAS, 399, 1523 van Kerkwijk, M. H. & Kaplan, D. L. 2007, Ap&SS, 308, 191\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M H Van Kerkwijk, R P Breton, S R Kulkarni, ApJ. 72895van Kerkwijk, M. H., Breton, R. P., Kulkarni, S. R. 2011, ApJ, 728, 95\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W U Wuensch, Phys. Rev., D. 403153Wuensch, W. U. et al., 1989, Phys. Rev., D, 40, 3153\",\n \"doi\": \"\"\n },\n {\n \"text\": \"V E Zavlin, Neutron Stars and Pulsars. Berlin HeidelbergSpringer357181Zavlin, V. E. 2009, in Neutron Stars and Pulsars, Astrophysics and Space Science Library, Vol. 357. Springer Berlin Heidelberg, p. 181\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":479,"cells":{"text":{"kind":"string","value":"\n\n21 Oct 2017\n21 Oct 2017EQUIVALENCE OF VISCOSITY AND WEAK SOLUTIONS FOR THE NORMALIZED p(x)-LAPLACIAN JARKKO SILTAKOSKI\nWe show that viscosity solutions to the normalized p(x)-Laplace equation coincide with distributional weak solutions to the strong p(x)-Laplace equation when p is Lipschitz and inf p > 1. This yields C 1,α regularity for the viscosity solutions of the normalized p(x)-Laplace equation. As an additional application, we prove a Radó-type removability theorem.\n\nIntroduction\n\nIn this paper, we study viscosity solutions to the normalized p(x)-Laplace equation which is defined by There has been recent interest in normalized equations, see for example [JS17,IJS,BG15]. We are partly motivated by the connection to stochastic tug-of-war games [PSSW09] as the case of space dependent probabilities leads to (1.1) [AHP17]. The objective of this work is to show that viscosity solutions to (1.1) coincide with solutions of its counterpart in the theory of distributional weak solutions. One approach to this kind of equivalence results [JLM01,Ish95] is based on the uniqueness of solutions. However, it seems difficult to use uniqueness in our case because the uniqueness of solutions is an open problem for the equation (1.1) as pointed out in [JLP10]. The equation (1.1) is in the non-divergence form. In order to find the weak counterpart, we note that for u ∈ C 2 (Ω) with non-vanishing gradient it holds that − |Du| p(x)−2 ∆ N p(x) u =−div |Du| p(x)−2 Du + |Du| p(x)−2 log (|Du|) Du · Dp. Thus the weak counterpart of (1.1) should be the strong p(x)-Laplace equation − ∆ S p(x) u := −div(|Du| p(x)−2 Du) + |Du| p(x)−2 log |Du| Du · Dp = 0.\n\n(1.2)\n\nOur main result, Theorem 5.9, is that viscosity solutions to (1.1) coincide with weak solutions to (1.2) when the function p is Lipschitz with inf p > 1. With these assumptions weak solutions to (1.2) in a domain are locally C 1,α continuous [ZZ12]. Thus our equivalence result yields local C 1,α regularity also for viscosity solutions to (1.1). As an application, we prove a Radó-type removability theorem for the strong p(x)-Laplacian. The theorem follows from the equivalence result since in the definition of a viscosity solution we may ignore the test functions whose gradient vanishes.\n\nThat viscosity solutions to (1.1) are weak solutions to (1.2) is proven by applying the method of [JJ12]. The idea is to approximate a viscosity solution through a sequence of inf-convolutions, show that the inf-convolutions are essentially weak supersolutions, and then pass to the limit.\n\nFirst, in Lemma 5.3 we show that the inf-convolution u ε of a viscosity supersolution u to (1.1) is still, in essence, a viscosity supersolution up to some error. This fact is a key part of our proof. If there was no x-dependence in (1.1), it would be straightforward to see that the inf-convolution of a viscosity supersolution is still a viscosity supersolution. This is because a test function that touches the inf-convolution from below also touches the original function from below at a nearby point once we add some constant to it. From this it would follow that the inf-convolution is a supersolution to the original equation. However, the equation (1.1) has x-dependence caused by p(x). Thus the inf-convolution no longer satisfies the original equation.\n\nIn Lemma 5.5 we use the standard mollification on u ε and p to deduce from Lemma 5.3 that u ε is \"almost\" a weak solution to −∆ S p(x) u ε ≥ 0. Applying Caccioppoli type estimates and vector inequalities we are then able to deduce that the sequence of inf-convolutions converges to the viscosity supersolution in W 1,p(·) loc (Ω) as ε → 0. This allows us to pass to the limit and conclude that u satisfies −∆ S p(x) u ≥ 0 in the weak sense. Due to the variable exponent, the operator ∆ S p(x) can be singular in some subsets and degenerate in others. Therefore we apply different arguments in the cases p(x) < 2 and p(x) ≥ 2, and finally need to be able to combine them.\n\nThe equivalence of weak and viscosity solutions to the usual p-Laplace equation was first proven by Juutinen, Lindqvist and Manfredi [JLM01]. Later Julin and Juutinen [JJ12] presented a more direct way to show that viscosity solutions to −∆ p u = f are also weak solutions. This proof was adapted in [APR17] to show that viscosity solutions to −∆ N p u = f coincide with weak solutions to −∆ p u = |Du| p−2 f when p ≥ 2. Similar arguments were also used in [MO] to study the equivalence of solutions to −∆ p u = f (x, u, Du). The variable exponent case was explored in [JLP10] where the equivalence of weak and viscosity solutions was proven for the p(x)-Laplace equation using techniques of [JLM01].\n\nThe equation (1.2) was introduced by Adamowicz and Hästö [AH10,AH11] in connection with mappings of finite distortion. It has been further studied for example in [ZZ12,PL13].\n\nThe paper is organized as follows: in Section 2 we recall the variable exponent Lebesgue and Sobolev spaces. Section 3 contains the rigorous definitions of solutions to equations (1.1) and (1.2). In Section 4 we show that weak solutions of (1.1) are viscosity solutions to (1.2) and the converse statement is proven in Section 5. Finally, in Section 6 we formulate and prove a Radó-type removability theorem for weak solutions of (1.2).\n\n\nVariable exponent lebesgue and sobolev spaces\n\nWe briefly recall basic facts about these spaces. For general reference see e.g. [DHHR11]. Let Ω ⊂ R N be an open and bounded set and let p : Ω → (1, ∞) be a measurable function. We denote p + := ess sup x∈Ω p(x) and p − := ess inf p(x).\n\n\nx∈Ω\n\nThe variable exponent Lebesgue space L p(·) (Ω) is defined as the set of measurable functions u : Ω → R for which the p(·)-modular ̺ p(·) (u) := Ω |u| p(x) dx is finite. It is a Banach space equipped with the Luxemburg norm\nu L p(·) (Ω) := inf λ > 0 : Ω u λ p(x) dx ≤ 1 .\nGiven that p + < ∞ or ̺ p(·) (u) > 0, the norm and the modular satisfy the inequality (see [DHHR11,p75])\nmin ̺ p(·) (u) 1 p − , ̺ p(·) (u) 1 p + ≤ u L p(·) (Ω) ≤ max ̺ p(·) (u) 1 p − , ̺ p(·) (u) 1 p + . (2.1)\nA version of Hölder's inequality holds [DHHR11,p81] \n: if u ∈ L p(·) (Ω) and v ∈ L p ′ (·) (Ω), where 1 p(x) + 1 p ′ (x) = 1 for a.e. x ∈ Ω, then Ω |u| |v| dx ≤ 2 u L p(·) (Ω) v L p ′ (·) (Ω) .\nAs a consequence of the Hölder's inequality we have that\nu L q(·) (Ω) ≤ 2 (1 + |Ω|) u L p(·) (Ω) for all u ∈ L p(·) (Ω) if q(x) ≤ p(x) for a.e. x ∈ Ω. If 1 < p − ≤ p + < ∞, then L p(·) (Ω) is reflexive and the dual of L p(·) (Ω) is L p ′ (·) (Ω).\nThe variable exponent Sobolev space W 1,p(·) (Ω) is the set of functions in u ∈ L p(·) (Ω) for which the weak gradient Du belongs in L p(·) (Ω). It is a Banach space equipped with the norm\nu W 1,p(·) (Ω) := u L p(·) (Ω) + Du L p(·) (Ω) .\nThe space W 1,p 0 (Ω) is the closure of compactly supported Sobolev functions in the space W 1,p(·) (Ω). A function belongs to the the local Lebesgue space L From now on we assume that p is Lipschitz continuous and p − > 1.\np(·) loc (Ω) if it belongs to L p(·) (Ω ′ ) for all Ω ′ ⋐ Ω. The local Sobolev space W 1,p(·) loc (Ω) isDefinition 3.1. A function u ∈ W 1,p(·) loc (Ω) is a weak supersolution to −∆ S p(x) u ≥ 0 in Ω if Ω |Du| p(x)−2 Du · Dϕ + |Du| p(x)−2 log (|Du|) Du · Dp ϕ dx ≥ 0\nfor all non-negative ϕ ∈ W 1,p(·) (Ω) with compact support. We say that u is a weak subsolution to −∆ S p(x) u ≤ 0 if −u is a supersolution and that u is a weak solution to −∆ S p(x) u = 0 if u is both supersolution and subsolution.\n\nLemma 3.2. It is enough to consider C ∞ 0 (Ω) test functions in the previous definition.\n\nProof. Assume that ϕ ∈ W 1,p(·) (Ω) has a compact support in an open set Ω ′ ⋐ Ω. Since p is log-Hölder continuous and bounded as a Lipschitz function, there is a sequence of functions ϕ j ∈ C ∞ 0 (Ω ′ ) such that ϕ j → ϕ in W 1,p(·) (Ω ′ ) (see [DHHR11,p347]). We set ψ j := ϕ − ϕ j . Then it is enough to show that\nΩ ′ |Du| p(x)−2 Du · Dψ j dx + Ω ′ |Du| p(x)−2 log (|Du|) Du · Dp ψ j dx → 0\nas j → ∞. The first integral convergences to zero by Hölder's inequality so we focus on the second integral. We may assume that N > 1. We set q(x) := p(x) p(x)−1+ 1 N . Using the inequality a s log a ≤ N a s+ 1 N + 1 s for a, s > 0 we get\nΩ ′ |Du| p(x)−1 |log |Du|| |Dp| |ψ j | dx ≤ Dp L ∞ (Ω ′ ) Ω ′ |ψ j | p(x) − 1 dx + N Ω ′ |Du| p(x)−1+ 1 N |ψ j | dx ≤ C(p, Ω) ψ j L p(·) (Ω ′ ) + |Du| p(x)−1+ 1 N L q(·) (Ω ′ ) ψ j L q ′ (·) (Ω ′ ) .\nWe take r ∈ (1, N ) such that q ′+ ≤ r * :\n= N r N −r . Then we have q ′ (x) = N p(x) N −1 ≤ min(p * (x), r * ), where p * (x) := N p(x) N −p(x) . Therefore ψ j L q ′ (·) (Ω ′ ) ≤ 2 (1 + |Ω|) ψ j L min(p * (·),r * ) (Ω ′ ) .\nSince ψ j ∈ W 1,min(p(·),r) 0\n\n(Ω ′ ) , we have by a variable exponent version of the Sobolev inequality (see e.g. [DHHR11,p265])\nψ j L min(p * (·),r * ) (Ω ′ ) ≤ C Dψ j L min(p(·),r) (Ω ′ ) ≤ 2C(1 + |Ω|) Dψ j L p(·) (Ω ′ ) .\nThese estimates imply the claim since ψ j W 1,p (Ω ′ ) → 0 as j → ∞.\n\nIn order to define viscosity solutions to −∆ N p(x) u = 0, we set\nF (x, η, X) := − trX + p(x) − 2 |η| 2 Xη, η for all (x, η, X) ∈ Ω × R N \\ {0} × S N where S N is the set of symmetric N × N matrices.\nWe also recall the concept of semi-jets. The subjet of a function u :\nΩ → R at x is defined by setting (η, X) ∈ J 2,− u(x) if u(y) ≥ u(x) + η · (y − x) + 1 2 X(y − x), (y − x) + o(|y − x| 2 ) as y → x. (3.1)\nThe closure of a subjet is defined by setting (η, X) ∈ J 2,− u(x) if there is a sequence\n(η i , X i ) ∈ J 2,− u(x i ) such that (x i , η i , X i ) → (x, η, X).\nThe superjet J 2,+ u(x) and its closure J 2,+ u(x) are defined in the same manner except that the inequality (3.1) is reversed.\n\nDefinition 3.3. A lower semicontinuous function u : Ω → R is a viscosity su- Remark. Observe that in the previous definition we require nothing in the case (0, X) ∈ J 2,− u(x).\npersolution to −∆ N p(x) u ≥ 0 in Ω if, whenever (η, X) ∈ J 2,− u(x) with x ∈ Ω and η = 0, then F (x, η, X) ≥ 0. A function u is a viscosity subsolution to −∆ N p(x) u ≤ 0 if −u is\nViscosity solutions may be equivalently defined using the jet-closures or test functions. For the next proposition, see e.g. [Koi12, Prop 2.6].\n\nProposition 3.4. Let u : Ω → R be lower semicontinuous. Then the following conditions are equivalent.\n\n(i) The function u is a viscosity supersolution to\n−∆ N p(x) u ≥ 0 in Ω. (ii) Whenever (η, X) ∈ J 2,− u(x) with x ∈ Ω, η = 0, we have F (x, η, X) ≥ 0. (iii) Whenever ϕ ∈ C 2 (Ω) is such that ϕ(x) = u(x), Dϕ(x) = 0 and ϕ(y) < u(y) for all y = x, it holds F (x, Dϕ(x), D 2 ϕ(x)) ≥ 0.\nWhen ϕ is as in the third condition above, we say that ϕ touches u from below at x.\n\n\nWeak solutions are Viscosity solutions\n\nWe show that if u is a weak solution to −∆ S p(x) u = 0, then it is a viscosity solution to −∆ N p(x) u = 0. Juutinen, Lukkari and Parviainen [JLP10] showed that weak solutions to the standard p(x)-Laplace equation are also viscosity solutions. This was accomplished with the help of the comparison principle. For if u is a weak supersolution to −∆ p(x) u ≥ 0 that is not a viscosity supersolution, then there is a test function ϕ ∈ C 2 touching u from below at x so that −∆ p(x) ϕ < 0 in some ball B(x). Lifting ϕ slightly produces a new functionφ still satisfying\n−∆ p(x)φ < 0 in B(x) andφ ≤ u in ∂B(x). Comparison principle now implies thatφ ≤ u in B(x) which is a contradiction sinceφ(x) > ϕ(x) = u(x).\nOur difficulty is that, to the best of our knowledge, the comparison principle is an open problem for the strong p(x)-Laplacian. Our strategy is therefore to consider a ball so small that the gradient of the test function does not vanish. Then the comparison principle holds and we arrive at a contradiction.\nTheorem 4.1. If u ∈ W 1,p(·) loc (Ω) is a weak solution to −∆ S p(x) u = 0, then it is a viscosity solution to −∆ N p(x) u = 0 in Ω.\nProof. Zhang and Zhou [ZZ12] showed that weak solutions of −∆ S p(x) u = 0 are in C 1 (Ω). Therefore it suffices to show that if u ∈ C 1 (Ω) is a weak supersolution to −∆ S p(x) u ≥ 0, then it is also a viscosity supersolution to −∆ N p(x) u ≥ 0. Assume on the contrary that there is ϕ ∈ C 2 (Ω) touching u from below at x 0 ∈ Ω, Dϕ(x 0 ) = 0 and\n0 > −h > F (x 0 , Dϕ(x 0 ), D 2 ϕ(x 0 )).\n\nThen by continuity there is\nr > 0 such that in B r (x 0 ) it holds −h |Dϕ| p(x)−2 ≥ − |Dϕ| p(x)−2 ∆ϕ + p(x) − 2 |Dϕ| 2 ∆ ∞ ϕ . (4.1) Since Du(x 0 ) = Dϕ(x 0 ) = 0, we may also assume that there is m > 0 such that inf x∈Br(x 0 ) |Dϕ| p(x)−2 ≥ m (4.2)\nand ess sup\nx∈Br(x 0 ) |Dp| |Dϕ| p(x)−2 log (|Dϕ|) Dϕ − |Du| p(x)−2 log (|Du|) Du ≤ hm 2 . (4.3) Let l := min x∈∂Br(x 0 ) (u − ϕ) > 0 and set ψ(x) := max (ϕ(x) + l − u(x), 0) . Then ψ ∈ W 1,2 0 (B r (x 0 )) so there are ψ j ∈ C ∞ 0 (B r (x 0 )) such that ψ j → ψ in W 1,2 (B r (x 0 )). Let p j be the standard mollification of p. Multiplying (4.1) by ψ and integrating over B r (x 0 ) yields −h Br(x 0 ) |Dϕ| p(x)−2 ψ dx ≥ Br(x 0 ) − |Dϕ| p(x)−2 ∆ϕ + p(x) − 2 |Dϕ| 2 ∆ ∞ ϕ ψ dx = lim j→∞ Br(x 0 ) − |Dϕ| p j (x)−2 ∆ϕ + p j (x) − 2 |Dϕ| 2 ∆ ∞ ϕ ψ j dx, (4.4)\nwhere the last equality holds because ψ j → ψ in W 1,2 (B r (x 0 )) and p j → p uniformly in B r (x 0 ). Calculating the divergence of |Dϕ| p j (x)−2 Dϕ and integrating by parts we get\nBr(x 0 ) − |Dϕ| p j (x)−2 ∆ϕ + p j (x) − 2 |Dϕ| 2 ∆ ∞ ϕ ψ j dx = Br(x 0 ) −div |Dϕ| p j (x)−2 Dϕ ψ j + |Dϕ| p j (x)−2 log (|Dϕ|) Dϕ · Dp j ψ j dx = Br(x 0 ) |Dϕ| p j (x)−2 Dϕ · (Dψ j + log (|Dϕ|) Dp j ψ j ) dx. (4.5)\nBy the convergence of ψ j and p j , it follows from (4.4) and (4.5) that\n−h Br(x 0 ) |Dϕ| p(x)−2 ψ dx ≥ Br(x 0 ) |Dϕ| p(x)−2 Dϕ · (Dψ + log (|Dϕ|) Dp ψ) dx.\n(4.6)\n\nSince u is a weak supersolution to ∆ S p(x) u = 0 and ψ ∈ W 1,p(·) (Ω) has a compact support in Ω, we have\nBr(x 0 ) |Du| p(x)−2 Du · (Dψ + log |Du| Dp ψ) dx ≥ 0.\n(4.7)\n\nDenoting A := {x ∈ B r (x 0 ) : ψ(x) > 0} and combining (4.6) and (4.7) we arrive at\nA |Dϕ| p(x)−2 Dϕ − |Du| p(x)−2 Du · (Dϕ − Du) dx ≤ A |Du| p(x)−2 log (|Du|) Du − |Dϕ| p(x)−2 log (|Dϕ|) Dϕ |Dp| ψ dx − h A |Dϕ| p(x)−2 ψ dx ≤ − hm 2 A ψ dx, (4.8)\nwhere the last inequality follows from (4.2) and (4.3). Since\n|a| p(x)−2 a − |b| p(x)−2 b · (a − b) ≥ 0\nfor any two vectors a, b ∈ R N when p(x) > 1, it follows from (4.8) that |A| = 0. But this is impossible since ϕ(x 0 ) = u(x 0 ) and l > 0.\n\n\nViscosity solutions are Weak solutions\n\nWe show that if u is a viscosity supersolution to −∆ N p(x) u ≥ 0, then it is a weak supersolution to −∆ S p(x) u ≥ 0. The same statement for subsolutions then follows by analogy.\n\nWe recall the usual partial ordering for symmetric N × N matrices by setting X ≤ Y if Xξ, ξ ≤ Y ξ, ξ for all ξ ∈ R N . For a matrix X we also set X := max {|λ| : λ is an eigenvalue of X} and for vectors ξ, η ∈ R N we use the notation ξ ⊗ η := ξη ′ , i.e. ξ ⊗ η is an N × N matrix whose (i, j) entry is ξ i η j .\n\nDefinition 5.1 (Inf-convolution). Let q ≥ 2 and ε > 0. The inf-convolution of a bounded function u ∈ C(Ω) is defined by\nu ε (x) := inf y∈Ω u(y) + 1 qε q−1 |x − y| q . (5.1)\nThe inf-convolution is well known to provide good approximations of viscosity supersolutions and often one only needs to consider it for q = 2 (see e.g. [CIL92]). However, as the authors in [JJ12] observed, considering large enough q essentially cancels the singularity in the usual p-Laplace operator when 1 < p < 2. In similar fashion it also cancels the singularity of the operator ∆ S p(x) . This is due to the property (v) in the next lemma. We also list some other basic properties of the inf-convolution.\n\nLemma 5.2. Let u ∈ C(Ω) be a bounded function. Then the inf-convolution u ε as defined in (5.1) has the following properties.\n(i) We have u ε ≤ u in Ω and u ε → u locally uniformly in Ω as ε → 0. (ii) There exists r(ε) > 0 such that u ε (x) = inf y∈B r(ε) (x)∩Ω u(y) + 1 qε q−1 |x − y| q and r(ε) → 0 as ε → 0. In fact we can choose r(ε) = qε q−1 osc Ω u 1 q . (iii) The function u ε is semi-concave in Ω r(ε) , that is, the function x → u ε (x)− q−1 2ε q−1 r(ε) q−2 |x| 2 is concave. (iv) If x ∈ Ω r(ε) := {x ∈ Ω : dist(x, ∂Ω) < r(ε)}, then there exists a point x ε ∈ B r(ε) (x) such that u ε (x) = u(x ε ) + 1 qε q−1 |x − x ε | q . (v) If (η, X) ∈ J 2,− u ε (x) with x ∈ Ω r(ε) , then η = (x−xε) ε q−1 |x ε − x| q−2 and X ≤ q−1 ε |η| q−2 q−1 I, where x ε is as in (iv).\nThese properties are well known, see appendix of [JJ12] and also [Kat15b] where more general \"flat inf-convolution\" is considered. Regardless, we give a proof of (v) based on [Kat15a,p53] due to its critical role in the proof of Lemma 5.5.\n\nProof of property (v) in Lemma 5.2. Let (η, X) ∈ J 2,− u ε (x). Then there is a function ϕ ∈ C 2 (R N ) such that it touches u ε from below at x and Dϕ(x) = η, D 2 ϕ(x) = X. Therefore for all y, z ∈ Ω we have\nu(y) + |y − z| q qε q−1 − ϕ(z) ≥u ε (z) − ϕ(z) ≥ 0.\nChoosing y = x ε , we obtain\nϕ(z) − |x ε − z| q qε q−1 ≤ u(x ε ) for all z ∈ Ω.\nSince ϕ(x) = u ε (x) = u(x ε )+ |xε−x| q qε q−1 , the above inequality means that the function\nz → ϕ(z) − |x ε − z| q qε q−1 =: ϕ(z) − ψ(z)\nhas a maximum at x. Thus η = Dψ(x) = (x−xε) ε q−1 |x ε − x| q−2 and\nX ≤ D 2 ψ(x) = 1 ε q−1 |x ε − x| q−4 (q − 2) (x ε − x) ⊗ (x ε − x) + |x ε − x| 2 I ≤ 1 ε q−1 |x ε − x| q−4 (q − 2) (x ε − x) ⊗ (x ε − x) I + |x ε − x| 2 I = q − 1 ε q−1 |x ε − x| q−2 I = q − 1 ε q−1 ε |η| 1 q−1 q−2 I = q − 1 ε |η| q−2 q−1 I.\nWe will show that the inf-convolution provides approximations of viscosity supersolutions to −∆ N p(x) u ≥ 0. If there was no x-dependence in the equation, it would be straightforward to show that the inf-convolution of a supersolution is still a supersolution. However, the equation −∆ N p(x) u ≥ 0 has x-dependence caused by p(x). Regardless, in [Ish95,Thm 3] it is shown that with some assumptions on G, the inf-convolution u ε of a viscosity supersolution to G(x, u, Du, D 2 u) ≥ 0 is still a viscosity supersolution to G(\nx, u ε , Du ε , D 2 u ε ) ≥ E(ε), where E(ε) → 0 as ε → 0.\nWe prove a modified version of this theorem for the solutions of −∆ N p(x) u ≥ 0. The important modification is the term |η| min(p(x)−2,0) in (5.2) as it cancels a singular gradient term that appears due to the error term in the proof of Lemma 5.5, see (5.14). Another difference is that we consider inf-convolution with the exponent q ≥ 2. where E(ε) → 0 as ε → 0. The error function E depends only on p, q and the modulus of continuity of u.\n\nProof. Fix x ∈ Ω r(ε) and (η, X) ∈ J 2,− u ε (x), η = 0. Then by Lemma 5.2 there is\nx ε ∈ B r(ε) (x) such that u ε (x) = u(x ε ) + |x ε − x| q qε q−1 (5.3) and η = (x−xε) ε q−1 |x ε − x| q−2 .\nThere exists a function ϕ ∈ C 2 (R N ) such that it touches u ε from below at x and Dϕ(x) = η, D 2 ϕ(x) = X. By the definition of inf-convolution\nu(y) − ϕ(z)+ |y − z| q qε q−1 ≥ u ε (z) − ϕ(z) ≥ 0 for all y, z ∈ Ω r(ε) . (5.4)\nSince by (5.3) we have u(x ε ) = ϕ(x) − |xε−x| q qε q−1 , it follows from (5.4) that the expression u(y) − ϕ(z) + |y−z| q qε q−1 reaches its minimum at (y, z) = (x ε , x). Thus\nmax (y,z)∈Ω r(ε) ×Ω r(ε) −u(y) + ϕ(z) − |y − z| q qε q−1 = −u(x ε ) + ϕ(x) − |x ε − x| q qε q−1 .\nWe denote Φ(y, z) := 1 qε q−1 |y − z| q and invoke the Theorem of sums (see [CIL92]). There exist Y, Z ∈ S N such that\n(η, −Y ) ∈ J 2,− u(x ε ), (η, −Z) ∈ J 2,+ ϕ(x) and Y 0 0 −Z ≤ D 2 Φ(x ε , x) + ε q−1 D 2 Φ(x ε , x) 2 (5.5) where D 2 Φ(x ε , x) = M −M −M M with M = 1 ε q−1 |x ε − x| q−4 (q − 2) (x ε − x) ⊗ (x ε − x) + |x ε − x| 2 I and D 2 Φ(x ε , x) 2 = 2 M 2 −M 2 −M 2 M 2 . The above implies Y ≤ Z ≤ −D 2 ϕ(x) = −X. Multiplying (5.5) by the R 2N vector ( η |η| p(x ε ) − 1, η |η| p(x) − 1) from both sides yields (p(x ε ) − 1) |η| 2 Y η, η − (p(x) − 1) |η| 2 Zη, η ≤ Λ 2 M + 2ε q−1 M 2 η |η| , η |η| , (5.6) where Λ = p(x) − 1 − p(x ε ) − 1. We have 0 ≤F (x ε , η, −Y ) =F (x, η, Z) − F (x ε , η, Y ) − F (x, η, Z) = (p(x ε ) − 1) Y η |η| , η |η| − (p(x) − 1) Z η |η| , η |η| + tr(Y ) − Y η |η| , η |η| − tr(Z) + Z η |η| , η |η| + F (x, η, −Z) ≤Λ 2 M + 2ε q−1 M 2 η |η| , η |η| + F (x, η, X), (5.7)\nwhere we used (5.6) and the fact that Y ≤ Z implies\ntr (Y − Z) − (Y − Z) η |η| , η |η| ≤ 0.\nWe have the estimate\nM ≤ 1 ε q−1 |x ε − x| q−4 (q − 2) (x ε − x) ⊗ (x ε − x) + |x ε − x| 2 I = q − 1 ε q−1 |x ε − x| q−2 .\nSince p is Lipschtiz continuous and p − > 1, we have also\nΛ 2 = |p(x) − p(x ε )| 2 p(x) − 1 + p(x ε ) − 1 2 ≤ C(p) |x − x ε | 2 .\nCombining these with (5.7) we get (we may assume that r(ε) < 1)\n−F (x, η, X) ≤Λ 2 M + 2ε q−1 M 2 ≤Λ 2 q − 1 ε q−1 |x ε − x| q−2 + 2ε q−1 q − 1 ε q−1 2 |x ε − x| 2(q−2) ≤ 3 (q − 1) 2 ε q−1 Λ 2 |x ε − x| q−2 ≤C(p, q) 1 ε q−1 |x ε − x| q .\n(5.8)\n\nMoreover, by uniform continuity of u there is a modulus of continuity ω such that ω(t) → 0 as t → 0 and |u(y) − u(z)| ≤ ω(|y − z|) for all y, z ∈ Ω. Hence by (5.3)\n|x ε − x| ≤ qε q−1 (u(x) − u(x ε )) 1 q ≤ q 1 q ε q−1 q ω(r(ε)) 1 q . (5.9)\nWe now consider the situations p(x) ≤ 2 and p(x) > 2 separately. If p(x) ≤ 2, we multiply (5.8) by |η| p(x)−2 and estimate using (5.9). We get\n− |η| p(x)−2 F (x, η, X) ≤C(p, q) 1 ε q−1 |x ε − x| q |η| p(x)−2 =C(p, q) 1 ε q−1 |x ε − x| q 1 ε q−1 (x − x ε ) |x ε − x| q−2 p(x)−2 =C(p, q) 1 ε (q−1)(p(x)−1) |x ε − x| q+(q−1)(p(x)−2) ≤C(p, q) 1 ε (q−1)(p(x)−1) q 1 q ε q−1 q ω(r(ε)) 1 q q+(q−1)(p(x)−2) =C(p, q) 1 ε q−1 q (p(x)−2) ω(r(ε)) q+(q−1)(p(x)−2) q ≤C(p, q)ω(r(ε)) q+(q−1)(p − −2) q ,\nwhere the last inequality is true when ε < 1 is so small that ω(r(ε)) < 1. This proves (5.2) when p(x) ≤ 2.\n\nIf p(x) > 2, we estimate (5.8) directly using (5.9). We get −F (x, η, X) ≤C(p, q) 1 ε q−1 q 1 q ε q−1 q ω(r(ε)) 1 q q = C(p, q)ω(r(ε))), which proves (5.2) when p(x) > 2.\n\nNext we will use the previous lemma to show that inf-convolution of a viscosity supersolution to −∆ N p(x) u ≥ 0 in Ω is a weak supersolution to −∆ S p(x) u ≥ 0 in Ω r(ε) up to some error term. Before proceeding we make some remarks about the point-wise differentiability of inf-convolution.\n\nRemark 5.4. It follows from semi-concavity that the inf-convolution u ε is locally Lipschitz in Ω r(ε) (see [EG15,p267]). Therefore it belongs in W 1,∞ loc (Ω r(ε) ), is differentiable almost everywhere in Ω r(ε) , and its derivative agrees with its Sobolev derivative almost everywhere in Ω r(ε) (see [EG15,p155 and p265]).\n\nBy Lemma 5.2 the function φ(x) := u ε (x) − C(q, ε, u) |x| 2 is concave in Ω r(ε) . Thus Alexandrov's theorem implies that u ε is twice differentiable almost everywhere in Ω r(ε) . Furthermore, the proof of Alexandrov's theorem in [EG15,p273] establishes that if φ j is the standard mollification of φ, then D 2 φ j → D 2 φ almost everywhere in Ω r(ε) .\n\nLemma 5.5. Assume that u is a uniformly continuous viscosity supersolution to −∆ N p(x) u ≥ 0 in Ω. Let q > 2 be so large that p − − 2 + q−2 q−1 ≥ 0 and let u ε be the inf-convolution of u as defined in (5.1). Then\nΩ r(ε) |Du ε | p(x)−2 Du ε · (Dϕ + log |Du ε | Dp ϕ) dx ≥ E(ε) Ω r(ε) |Du ε | s(x) ϕ dx\nfor all non-negative ϕ ∈ W 1,p(·) (Ω r(ε) ) with compact support, where E(ε) → 0 as ε → 0 and s(x) = max(p(x) − 2, 0).\n\nProof. It is enough to consider ϕ ∈ C ∞ 0 (Ω r(ε) ). This can be seen in the same way as Lemma 3.2, but since u ε ∈ W 1,∞ loc (Ω r(ε) ), the proof is even simpler. (Step 1) We show that u ε satisfies the auxiliary inequality (5.11) for all 0 < δ < 1. As mentioned in Remark 5.4, the function φ(x) := u ε (x) − C(q, ε, u) |x| 2 is concave in Ω r(ε) and we can approximate it by smooth concave functions φ j so that φ j , Dφ j , D 2 φ j → φ, Dφ, D 2 φ almost everywhere in Ω r(ε) . We define u ε,j (x) := φ j (x) + C(q, ε, u) |x| 2 and denote by p j the standard mollification of p. Since u ε,j and p j are smooth, we calculate\nΩ r(ε) − δ + |Du ε,j | 2 p j (x)−2 2 ∆u ε,j + p j (x) − 2 δ + |Du ε,j | 2 ∆ ∞ u ε,j ϕ dx = Ω r(ε) −div δ + |Du ε,j | 2 p j (x)−2 2\nDu ε,j ϕ\n+ 1 2 δ + |Du ε,j | 2 p j (x)−2 2 log δ + |Du ε,j | 2 Du ε,j · Dp j ϕ dx = Ω r(ε) δ + |Du ε,j | 2 p j (x)−2 2\nDu ε,j · Dϕ + 1 2 log δ + |Du ε,j | 2 Dp j ϕ dx.\n\n(5.10)\n\nWe let j → ∞ in (5.10) and intend to use Fatou's lemma at the LHS and the Dominated convergence theorem at the RHS. This results in the auxiliary inequality\nΩ r(ε) − δ + |Du ε | 2 p(x)−2 2 ∆u ε + p(x) − 2 δ + |Du ε | 2 ∆ ∞ u ε ϕ dx ≤ Ω r(ε) δ + |Du ε | 2 p(x)−2 2\nDu ε · Dϕ + 1 2 log δ + |Du ε | 2 Dp ϕ dx,\n(5.11)\nwhere D 2 u ε is the Hessian of u ε in the Alexandrov's sense. We still need to check that the assumptions of the Dominated convergence theorem and Fatou's lemma hold. By Lipschitz continuity of u ε and p there is M ≥ 1 such that\nsup j Du ε,j L ∞ (supp ϕ) , sup j Dp j L ∞ (supp ϕ) ≤ M.\nThis justifies our use of the Dominated convergence theorem. In order to justify our use of Fatou's lemma, we notice first that by concavity of φ j we have D 2 u ε,j ≤ C(q, ε, u)I. Thus the integrand at the LHS of (5.10) is clearly bounded from below by a constant independent of j if Du ε,j = 0. If Du ε,j = 0, we have\nδ + |Du ε,j | 2 p j (x)−2 2 ∆u ε,j + p j (x) − 2 δ + |Du ε,j | 2 ∆ ∞ u ε,j = δ + |Du ε,j | 2 p j (x)−2 2 δ + |Du ε,j | 2 |Du ε,j | 2 ∆u ε,j + p j (x) − 2 |Du ε,j | 2 ∆ ∞ u ε,j + δ∆u ε,j ≤ δ p j (x)−2 2 + δ + M 2 p j (x)−2 2 δ + |Du ε,j | 2 C(q, ε, u) |Du ε,j | 2 (N + p j (x) − 2) + δN ≤ C(q, ε, u) δ p − −2 2 + δ + M 2 p + −2 2 2N + p + − 2 ,\nwhere the first inequality follows like estimate (5.7) since p j ≥ p − > 1. (Step 2) We let δ → 0 in the auxiliary inequality (5.11). The RHS becomes\nΩ r(ε) \\{Duε=0} |Du ε | p(x)−2 Du ε · (Dϕ + log |Du ε | Dp ϕ) dx\nby the Lebesgue's dominated convergence theorem. We intend to apply Fatou's lemma on the LHS. We have Du ε (x), D 2 u ε (x) ∈ J 2,− u ε (x) for almost every x ∈ Ω r(ε) . Therefore by Lemma 5.3 it holds that\n|Du ε | min(p(x)−2,0) F (x, Du ε , D 2 u ε ) ≥ E(ε) in x ∈ Ω r(ε) : Du ε = 0 (5.12)\nand by the property (v) in Lemma 5.2 we have\nD 2 u ε ≤ q − 1 ε |Du ε | q−2 q−1 I. (5.13)\nObserve that since q > 2, the condition (5.13) implies that the Hessian D 2 u ε is negative semi-definite in the set where the gradient Du ε vanishes. Using this fact, Fatou's lemma and (5.12) we get\nlim inf δ→0 Ω r(ε) − |Du ε | 2 + δ p(x)−2 2 ∆u ε + p(x) − 2 |Du ε | 2 + δ ∆ ∞ u ε ϕ dx ≥ lim inf δ→0 {Duε =0} − |Du ε | 2 + δ p(x)−2 2 ∆u ε + p(x) − 2 |Du ε | 2 + δ ∆ ∞ u ε ϕ dx + lim inf δ→0 {Duε=0} −δ p(x)−2 2 ∆u ε ϕ dx ≥ {Duε =0} − |Du ε | p(x)−2 ∆u ε + p(x) − 2 |Du ε | 2 ∆ ∞ u ε ϕ dx ≥E(ε) {Duε =0}\n|Du ε | max(p(x)−2,0) ϕ dx, (5.14) and thus we arrive at the desired inequality. Our use of Fatou's lemma is justified since if Du ε = 0 and p(x) ≤ 2, we have by (5.13)\n|Du ε | 2 + δ p(x)−2 2 ∆u ε + p(x) − 2 |Du ε | 2 + δ ∆ ∞ u ε = |Du ε | 2 + δ |Du ε | 2 + δ p(x)−2 2 |Du ε | 2 ∆u ε + p(x) − 2 |Du ε | 2 ∆ ∞ u ε + δ∆u ε ≤ |Du ε | 2 + δ |Du ε | 2 + δ p(x)−2 2 q − 1 ε |Du ε | q−2 q−1 +2 (N + p(x) − 2) + |Du ε | q−2 q−1 δN ≤ |Du ε | p(x)−2+ q−2 q−1 q − 1 ε (2N + p(x) − 2) ≤ Du ε L ∞ (supp ϕ) + 1 p + −2+ q−2 q−1 q − 1 ε 2N + p + − 2 ,\nwhere the last inequality follows from p − − 2 + q−2 q−1 ≥ 0. If Du ε = 0 and p(x) > 2, we have simply\n|Du ε | 2 + δ p(x)−2 2 ∆u ε + p(x) − 2 |Du ε | 2 + δ ∆ ∞ u ε ≤ Du ε 2 L ∞ (supp ϕ) + 1 p + −2 2 + q−2 q−1 ( q − 1 ε ) N + p + − 2 .\nIn the next two lemmas we use Caccioppoli type estimates and algebraic inequalities to show that the sequence of inf-convolutions converges to the viscosity supersolution in W 1,p(·) loc (Ω).\n\nLemma 5.6. Under the assumptions of Lemma 5.5, the function u belongs in W 1,p(·) loc (Ω) and for any Ω ′ ⋐ Ω we have Du ε → Du weakly in L p(·) (Ω ′ ) for some subsequence.\n\nProof. Take a cut-off function ξ ∈ C ∞ 0 (Ω ′ ) such that 0 ≤ ξ ≤ 1 in Ω and ξ ≡ 1 in Ω ′ . Then assume that ε is so small that supp ξ =: K ⊂ Ω r(ε) . We define a test function ϕ := (L − u ε )ξ p + where L := sup ε,x∈Ω ′ |u ε (x)| is finite since u ε → u locally uniformly. We have Dϕ = −Du ε ξ p + + (L − u ε )p + ξ p + −1 Dξ and therefore by Lemma 5.5\nΩ r(ε) |Du ε | p(x) ξ p + dx ≤ Ω r(ε) |Du ε | p(x)−1 ξ p + −1 (L − u ε ) p + |Dξ| dx + Ω r(ε) |Du ε | p(x)−1 |log |Du ε || |Dp| (L − u ε ) ξ p + dx + |E(ε)| Ω r(ε) |Du ε | max(p(x)−2,0) (L − u ε ) ξ p + dx =:I 1 + I 2 + I 3 .\nWe estimate these integrals using Young's inequality. The first integral is estimated by the facts p(x)(p + −1)\n\np(x)−1 ≥ p + and ξ ≤ 1 as follows\nI 1 ≤ Ω r(ε) δ |Du ε | p(x) ξ p(x)(p + −1) p(x)−1 + 2 δ Lp + |Dξ| p(x) dx ≤δ Ω r(ε) |Du ε | p(x) ξ p + dx + C(δ, p, L, Dξ).\nTo estimate I 2 , we also use the inequality a s |log a| ≤ a s+ 1 2 + 1 s for a > 0 and s > 0,\nI 2 ≤ Ω r(ε) |Du ε | p(x)− 1 2 + 1 p(x) − 1 ξ p + |Dp| 2L dx ≤ Ω r(ε) δ |Du ε | p(x) ξ p + p(x) p(x)− 1 2 + 2 δ |Dp| L 2p(x) + 2L |Dp| ξ p + p − − 1 dx ≤δ Ω r(ε) |Du ε | p(x) ξ p + dx + C(δ, p, Dp, L).\nThe last integral is estimated by the two alternatives in max(p(x) − 2, 0) as follows (we may assume that |E(ε)| ≤ 1)\nI 3 ≤ Ω r(ε) ∩{p(x)>2} |Du ε | p(x)−2 ξ p + 2L dx + Ω r(ε) ∩{p(x)≤2} 2Lξ p + dx ≤ Ω r(ε) ∩{p(x)>2} δ |Du ε | p(x) ξ p + p(x) p(x)−2 + 2 δ L p(x) 2 dx + C(p, L) ≤δ Ω r(ε) |Du ε | p(x) ξ p + dx + C(δ, p, L).\nTaking small δ we conclude that Du ε is bounded in L p(·) (Ω ′ ) with respect to ε. Since L p(·) (Ω ′ ) is a reflexive Banach space [DHHR11, p76 and p89], it follows that there is a function Du ∈ L p(·) (Ω ′ ) such that Du ε → Du weakly in L p(·) (Ω ′ ) for some subsequence. Consequently u ∈ W 1,p(·) (Ω ′ ) with Du as its weak derivative.\n\nLemma 5.7. Under the assumptions of Lemma 5.5, for any Ω ′ ⋐ Ω we have Du ε → Du in L p(·) (Ω ′ ) for some subsequence.\n\nProof. Take a cut-off function ξ ∈ C ∞ 0 (Ω) such that ξ ≡ 1 in Ω ′ and define a test function ϕ := (u − u ε )ξ. Then assume that ε is so small that supp ξ =: K ⊂ Ω r(ε) .\n\nSince ϕ ∈ W 1,p(·) (Ω r(ε) ) with compact support it follows from Lemma 5.5 that\nΩ r(ε) |Du| p(x)−2 Du − |Du ε | p(x)−2 Du ε · (Du − Du ε ) ξ dx ≤ Ω r(ε) |Du ε | p(x)−2 Du ε · Dξ (u − u ε ) dx + Ω r(ε) |Du ε | p(x)−2 log (|Du ε |) Du ε · Dp (u − u ε )ξ dx + |E(ε)| Ω r(ε) |Du ε | max(p(x)−2,0) (u − u ε )ξ dx + Ω r(ε) |Du| p(x)−2 Du · (Du − Du ε ) ξ dx ≤ u − u ε L ∞ (K) K C(p − ) + |Du ε | p(x) (Dξ + |Dp| + |E(ε)|) dx + K |Du| p(x)−2 Du · (Du − Du ε ) ξ dx.\n(5.15) According to Lemma 5.6 we have u ε → u locally uniformly and Du ε → Du weakly in L p(·) (K) for a subsequence. Thus by passing to a subsequence we may assume that the right hand side of (5.15) converges to zero. The claim now follows from the inequalities (see e.g. [Lin17, Chapter 12])\n|a| p(x)−2 a − |b| p(x)−2 b · (a − b) ≥ (p(x) − 1) |a − b| 2 1 + |a| 2 + |b| 2 p(x)−2 2 p(x) < 2 2 2−p(x) |a − b| p(x) p(x) ≥ 2\nfor a, b ∈ R N . Indeed, we immediately get that Ω ′ ∩{p(x)≥2} |Du − Du ε | p(x) dx → 0. To deal with the set {p(x) < 2}, we first apply the above algebraic inequality and then estimate using Hölder's inequality, the modular inequality (2.1) and the definition of the · L p(·) -norm. We get\nΩ ′ ∩{p(x)<2} |Du − Du ε | p(x) dx ≤ Ω ′ ∩{p(x)<2} |Du| p(x)−2 Du − |Du ε | p(x)−2 Du ε · (Du − Du ε ) p(x) 2 · 1 p(x) − 1 p(x) 2 1 + |Du| 2 + |Du ε | 2 p(x)(2−p(x)) 4 dx ≤ |Du| p(x)−2 Du − |Du ε | p(x)−2 Du ε · (Du − Du ε ) p(x) 2 L 2 p(·) (Ω ′ ∩{p(x)<2}) · 2 p − − 1 1 + |Du| 2 + |Du ε | 2 p(x)(2−p(x)) 4 L 2 2−p(·) (Ω ′ ∩{p(x)<2}) ≤ Ω r(ε) |Du| p(x)−2 Du − |Du ε | p(x)−2 Du ε · (Du − Du ε ) ξ dx s · 2 p − − 1 1 + Ω ′ ∩{p(x)<2} 1 + |Du| 2 + |Du ε | 2 p(x) 2 dx ,\nwhere s ∈ p + 2 , p − 2 . The last integral is bounded since the sequence Du ε is bounded in L p(·) (Ω ′ ) by its weak convergence. The RHS therefore converges to zero by (5.15).\n\nNext, we use the previous convergence result to pass to the limit in the inequality of Lemma 5.5 and conclude that viscosity supersolutions to −∆ N p(x) u ≥ 0 are weak supersolutions to −∆ S p(x) u ≥ 0.\nTheorem 5.8. If u ∈ C(Ω) is a viscosity supersolution to −∆ N p(x) u ≥ 0 in Ω, then u is a weak supersolution to −∆ S p(x) u ≥ 0 in Ω.\nProof. It is clear from the definition of weak supersolutions to −∆ S p(x) u ≥ 0 that we can without loss of generality assume that u is uniformly continuous in Ω by restricting to a smaller domain. Fix a non-negative test function ϕ ∈ C ∞ 0 (Ω) and take open Ω ′ ⋐ Ω such that supp ϕ ⊂ Ω ′ . Let q and u ε be as in Lemma 5.5 and assume that ε is so small that Ω ′ ⊂ Ω r(ε) . Then the claim follows from Lemma 5.5 if we show that \nlim ε→0 Ω ′ |Du ε | p(x)−2 Du ε · Dϕ dx = Ω ′ |Du| p(x)−2 Du · Dϕ dx (5.16) and lim ε→0 Ω ′ |Du ε | p(x)−2 log (|Du ε |) Du ε · Dp ϕ dx = Ω ′ |Du| p(x)−2 log (|Du|) Du ·|a| p(x)−2 a − |b| p(x)−2 b ≤ 2 2−p(x) |a − b| p(x)−1 p(x) < 2 2 −1 |a| p(x)−2 + |b| p(x)−2 |a − b| p(x) ≥ 2 (5.19)\nfor a, b ∈ R N . Indeed, when ε is so small that Ω ′ |Du ε − Du| p(x) dx < 1 we have by Hölder's inequality and the modular inequality\nΩ ′ |Du ε | p(x)−2 Du ε − |Du| p(x)−2 Du dx ≤2 Ω ′ ∩{p(x)<2} |Du ε − Du| p(x)−1 dx + 2 −1 Ω ′ ∩{p(x)≥2} |Du ε | p(x)−2 + |Du| p(x)−2 |Du ε − Du| dx ≤C(p, Ω) Ω ′ |Du ε − Du| p(x) dx 1 p + + C(p, Ω) 1 + Ω ′ |Du ε | p(x) + |Du| p(x) dx Du ε − Du L p(·) (Ω ′ ) .\nClaim (5.18) holds since Ω ′ |Du ε | p(x) dx is bounded and E(ε) → 0. To this end, fix 0 < ǫ < 1. The mapping (a, x) → |a| p(x)−2 log (|a|) a is uniformly continuous in bounded sets of R N × Ω ′ . Hence there exists δ = δ(ǫ) < ǫ such that whenever x ∈ Ω ′ and a, b ∈ B(0, 3) satisfy |a − b| < δ, it holds\n|a| p(x)−2 log (|a|) a − |b| p(x)−2 log (|b|) b ≤ ǫ. (5.21)\nIf |a| , |b| ≥ 1 and |a − b| < δ, then we use (5.19) to get the estimate\n|a| p(x)−2 log (|a|) a − |b| p(x)−2 log (|b|) b ≤ |b| p(x)−1 |log |a| − log |b|| + |log |a|| |a| p(x)−2 a − |b| p(x)−2 b ≤ |b| p(x) |a − b| + |a| · 2 2−p(x) |a − b| p(x)−1 , p(x) < 2 2 −1 |a| p(x)−2 + |b| p(x)−2 |a − b| , p(x) ≥ 2 ≤(1 + 2 −1 ) |a| p(x) + |b| p(x) |a − b| + 2 |a| |a − b| p(x)−1 ≤C |a| p(x) + |b| p(x) ǫ min(p − −1,1) . (5.22)\nWe denote\nF ε = x ∈ Ω ′ : |Du ε (x) − Du(x)| ≥ δ .\nThe strong convergence of Du ε to Du in L p(·) (Ω ′ ) implies that Du ε → Du in measure in Ω ′ (see [DHHR11, Lemma 3.2.10]). Thus there is ε 0 = ε 0 (δ) such that for all ε < ε 0 it holds |F ε | ≤ δ. Using the inequality a s |log a| ≤ a s+ 1 2 + 1 s for a > 0 and s > 0 we get for all ε < ε 0\nFε |Du ε | p(x)−2 log (|Du ε |) Du ε − |Du| p(x)−2 log (|Du|) Du dx ≤ Fε 2 p(x) − 1 + |Du ε | p(x)− 1 2 + |Du| p(x)− 1 2 dx ≤C(p − ) |F ε | + 1 L 2p(·) (Fε) Du ε L p(·) p(·)− 1 2 (Fε) + Du L p(·) p(·)− 1 2 (Fε) ≤C(p − ) |F ε | + |F ε | 1 2p + 2 + Fε |Du ε | p(x) + |Du| p(x) dx ≤C(p − ) 1 + Ω ′ |Du ε | p(x) + |Du| p(x) dx ǫ 1 2p + .\n(5.23) If x ∈ Ω ′ \\ F ε , then either |Du ε | , |Du| ≤ 3 or |Du ε | , |Du| ≥ 1. Hence by (5.21) and (5.22) we have\nΩ ′ \\Fε |Du ε | p(x)−2 log (|Du ε |) Du ε − |Du| p(x)−2 log (|Du|) Du dx ≤C Ω ′ |Du ε | p(x) + |Du| p(x) + 1 dx ǫ min(p − −1,1) . (5.24)\nCombining (5.24) and (5.23) proves (5.20) since ǫ was arbitrary.\n\nMerging Theorems 4.1 and 5.8 yields the following equivalence result.\n\nTheorem 5.9. A function u is a viscosity solution to −∆ N p(x) u = 0 in Ω if and only if it is a weak solution to −∆ S p(x) u = 0 in Ω. Since the weak solutions to the strong p(x)-Laplace equation are locally C 1,α continuous [ZZ12], our equivalence result yields local C 1,α regularity also for viscosity solutions of the normalized p(x)-Laplace equation.\n\nCorollary 5.10. If u is a viscosity solution to −∆ N p(x) u = 0 in a bounded domain Ω, then u ∈ C 1,α (Ω) with α ∈ (0, 1).\n\n\nAn Application: A Radó-type removability theorem\n\nThe classical theorem of Radó says that if a continuous complex-valued function f defined on a domain Ω ⊂ C is holomorphic in Ω \\{f = 0}, then it is holomorphic in the whole Ω. Similar results have been proven for solutions of partial differential equations. We prove a Radó-type removability theorem for the strong p(x)-Laplace equation. It is worth pointing out that it could be difficult to show this kind of result without appealing to viscosity solutions whereas it is straightforward to do so with the help of the equivalence result. The theorem follows by observing that weak solutions to ∆ S p(x) u = 0 coincide with viscosity solutions of an equation that satisfies the assumptions of a Radó-type removability theorem in [JL05].\n\nRecall that we ignore the test functions whose gradient vanishes at the point of touching in the Definition 3.3 of viscosity solutions to −∆ N p(x) u = 0. Sometimes this kind of solutions are called feeble viscosity solutions (e.g. [JL05,Kat15b]). We will observe that these feeble viscosity solutions to −∆ N p(x) u = 0 are exactly the usual viscosity solutions to − tr(A(x, Du)D 2 u) = 0, (6.1)\n\nwhere A(x, Du) := |Du| 2 I + (p(x) − 2) Du ⊗ Du. To be precise, we define the viscosity solutions to (6.1).\n\nDefinition 6.1. A lower semicontinuous function u is a viscosity supersolution to (6.1) in Ω if, whenever (η, X) ∈ J 2,− u(x) with x ∈ Ω, then −tr(A(x, η)X) ≥ 0.\n\nA function u is a viscosity subsolution to (6.1) if −u is a supersolution, and a viscosity solution if it is both viscosity super-and subsolution.\n\nLemma 6.2. A function u is a viscosity solution to −∆ N p(x) u = 0 if and only if it is a viscosity solution to (6.1).\n\nProof. It is enough to consider supersolutions. Take (η, X) ∈ J 2,− u(x) with x ∈ Ω. If η = 0, then the conditions for both definitions are satisfied, so we may assume that η = 0. Then we have F (x, η, X) ≥ 0 if and only if − |η| 2 tr(X) + (p(x) − 2) Xη, η ≥ 0, where |η| 2 tr(X) + (p(x) − 2) Xη, η = |η| 2 tr(X) + (p(x) − 2) tr(η ⊗ ηX) =tr |η| 2 I + (p(x) − 2) η ⊗ η X .\n\nHence the definitions are equivalent. Theorem 6.3 (A Radó-type removability theorem). Let u ∈ C 1 (Ω) be a weak solution to −∆ S p(x) u = 0 in Ω \\ {u = 0}. Then u is a weak solution to −∆ S p(x) u = 0 in the whole Ω.\n\nProof. By Lemma 6.2 and our equivalence result weak solutions to −∆ S p(x) u = 0 coincide with viscosity solutions to (6.1). Therefore it suffices to show that if u is a viscosity solution to (6.1) in Ω \\ {u = 0}, it is a viscosity solution to (6.1) in the whole Ω. This on the other hand follows from [JL05, Theorem 2.2]. The matrix A satisfies the assumptions of the theorem as it is symmetric, has continuous entries and A(x, 0, 0) = 0 for all x ∈ Ω. It is also positive semi-definite since for all ξ ∈ R N we have ξ ′ |η| 2 I + (p(x) − 2) η ⊗ η ξ ≥ξ ′ |η| 2 I − η ⊗ η ξ ≥ |ξ| 2 |η| 2 − η ⊗ η = 0. \n\n−\n∆ N p(x) u := −∆u − p(x) − 2 |Du| 2 ∆ ∞ u := −∆u − p(x) − 2 |Du| 2 D 2 uDu, Du = 0. (1.1)\n\n\nThe strong and normalized p(x)-Laplace equations In this section, we define weak solutions to the strong p(x)-Laplace equation and viscosity solutions to the normalized p(x)-Laplace equation.\n\n\na viscosity supersolution, and a viscosity solution to −∆ N p(x) u = 0 if it is both viscosity super-and subsolution.\n\nLemma 5. 3 .\n3Assume that u is a uniformly continuous viscosity supersolution to −∆ N p(x) u ≥ 0 in Ω. Then, whenever (η, X) ∈ J 2,− u ε (x), η = 0 and x ∈ Ω r(ε) , it holds |η| min(p(x)−2,0) F (x, η, X) ≥ E(ε), (5.2)\n\n\nClaim (5.17) follows if we show that lim ε→0 Ω ′ |Du ε | p(x)−2 log (|Du ε |) Du ε − |Du| p(x)−2 log (|Du|) Du dx = 0. (5.20)\n\n\nBy Lemma 5.7 we have that u ε → u in W 1,p(·) (Ω ′ ). Claim (5.16) follows from the inequalities (see e.g. [Lin17, Chapter 12])Dp ϕ dx \n(5.17) \n\nas well as \nlim \n\nε→0 \n\nE(ε) \n\nΩ ′ \n\n|Du ε | max(p(x)−2,0) ϕ dx = 0. \n(5.18) \n\n\n\n\nJarkko Siltakoski, Department of Mathematics and Statistics, P.O.Box 35, FIN-40014, University of Jyväskylä, Finland E-mail address: jarkko.j.m.siltakoski@student.jyu.fi\n\nMappings of finite distortion and PDE with nonstandard growth. T Adamowicz, P Hästö, Int. Math. Res. Not. IMRN. T. Adamowicz and P. Hästö. Mappings of finite distortion and PDE with nonstandard growth. Int. Math. Res. Not. IMRN, pages 1940-1965, 2010.\n\nHarnack's inequality and the strong p(·)-Laplacian. T Adamowicz, P Hästö, J. Differential Equations. 250T. Adamowicz and P. Hästö. Harnack's inequality and the strong p(·)-Laplacian. J. Differential Equations, 250:1631-1649, 2011.\n\nTug-of-war games with varying probabilities and the normalized p(x)-Laplacian. Á Arroyo, J Heino, M Parviainen, Commun. Pure Appl. Anal. 163Á. Arroyo, J. Heino, and M. Parviainen. Tug-of-war games with varying probabilities and the normalized p(x)-Laplacian. Commun. Pure Appl. Anal., 16(3):915-944, 2017.\n\nC 1,α regularity for the normalized p-Poisson problem. A Attouchi, M Parviainen, E Ruosteenoja, J. Math. Pures Appl. 1084A. Attouchi, M. Parviainen, and E. Ruosteenoja. C 1,α regularity for the normalized p-Poisson problem. J. Math. Pures Appl., 108(4):553-591, 2017.\n\nModica type gradient estimates for an inhomogeneus variant of the normalized p-Laplacian evolution. A Banerjee, N Garofalo, Nonlinear Anal. 121A. Banerjee and N. Garofalo. Modica type gradient estimates for an inhomogeneus variant of the normalized p-Laplacian evolution. Nonlinear Anal., 121:458-468, 2015.\n\nUser's guide to viscosity solutions of second order partial differential equations. M G Crandall, H Ishii, P.-L Lions, Bull. Amer. Math. Soc. 271M. G. Crandall, H. Ishii, and P.-L. Lions. User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc., 27(1):1-67, 1992.\n\nLebesgue and Sobolev Spaces with Variable Exponents. L Diening, P Harjulehto, P Hästö, M Růžička, Lecture Notes in Mathematics. 2017Springer-VerlagL. Diening, P. Harjulehto, P. Hästö, and M. Růžička. Lebesgue and Sobolev Spaces with Variable Exponents, volume 2017 of Lecture Notes in Mathematics. Springer- Verlag, 2011.\n\nMeasure Theory and Fine Properties of Functions. L C Evans, R F Gariepy, CRC Pressrevised editionL. C. Evans and R. F. Gariepy. Measure Theory and Fine Properties of Functions. CRC Press, revised edition, 2015.\n\nHölder gradient estimates for a class of singular or degenerate parabolic equations. C Imbert, T Jin, L Silvestre, To appear in Adv. Nonlinear AnalC. Imbert, T. Jin, and L. Silvestre. Hölder gradient estimates for a class of singular or degenerate parabolic equations. To appear in Adv. Nonlinear Anal.\n\nOn the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions. H Ishii, Funkcialaj Ekvacioj. 38H. Ishii. On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions. Funkcialaj Ekvacioj, 38:101-120, 1995.\n\nA new proof for the equivalence of weak and viscosity solutions for the p-Laplace equation. V Julin, P Juutinen, Comm. Partial Differential Equations. 375V. Julin and P. Juutinen. A new proof for the equivalence of weak and viscosity so- lutions for the p-Laplace equation. Comm. Partial Differential Equations, 37(5):934- 946, 2012.\n\nRemovability of a level set for solutions of quasilinear equations. P Juutinen, P Lindqvist, Comm. Partial Differential Equations. 30P. Juutinen and P. Lindqvist. Removability of a level set for solutions of quasilinear equations. Comm. Partial Differential Equations, 30:305-321, 2005.\n\nOn the equivalence of viscosity solutions and weak solutions for a quasi-linear equation. P Juutinen, P Lindqvist, J J Manfredi, SIAM J. Math. Anal. 333P. Juutinen, P. Lindqvist, and J.J. Manfredi. On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation. SIAM J. Math. Anal., 33(3):699-717, 2001.\n\nEquivalence of viscosity and weak solutions for the p(x)-Laplacian. P Juutinen, T Lukkari, M Parviainen, Ann. Inst. H. Poincaré Anal. Non Linéaire. 276P. Juutinen, T. Lukkari, and M. Parviainen. Equivalence of viscosity and weak solu- tions for the p(x)-Laplacian. Ann. Inst. H. Poincaré Anal. Non Linéaire, 27(6):1471- 1487, 2010.\n\nHölder gradient estimates for parabolic homogeneous p-Laplacian equations. T Jin, L Silvestre, J. Math. Pures Appl. 1083T. Jin and L. Silvestre. Hölder gradient estimates for parabolic homogeneous p- Laplacian equations. J. Math. Pures Appl., 108(3):63-87, 2017.\n\nAn Introduction To Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L ∞. N Katzourakis, SpringerN. Katzourakis. An Introduction To Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L ∞ . Springer, 2015.\n\nNonsmooth convex functionals and feeble viscosity solutions of singular Euler-Lagrange equations. N Katzourakis, Calc. Var. 541N. Katzourakis. Nonsmooth convex functionals and feeble viscosity solutions of sin- gular Euler-Lagrange equations. Calc. Var., 54(1):275-298, 2015.\n\nA Beginner's Guide to the Theory of Viscosity Solutions. Shigeaki Koike, 2nd editionShigeaki Koike. A Beginner's Guide to the Theory of Viscosity Solutions. 2nd edition, 2012.\n\nP Lindqvist, Notes on the p-Laplace equation. 161Report / University of Jyväskylä, Department of Mathematics and Statistics. University of Jyväskyläsecond editionP. Lindqvist. Notes on the p-Laplace equation (second edition), volume 161 of Report / University of Jyväskylä, Department of Mathematics and Statistics. University of Jyväskylä, 2017.\n\nOn viscosity and weak solutions for non-homogeneous p-Laplace equations. M Medina, P Ochoa, To appear in Adv. Nonlinear AnalM. Medina and P. Ochoa. On viscosity and weak solutions for non-homogeneous p-Laplace equations. To appear in Adv. Nonlinear Anal.\n\nA homogenization process for the strong p(x)-Laplacian. M Pérez-Llanos, Nonlinear Anal. 76M. Pérez-Llanos. A homogenization process for the strong p(x)-Laplacian. Nonlinear Anal., 76:105-114, 2013.\n\nTug-of-war and the infinity Laplacian. Y Peres, O Schramm, S Sheffield, D B Wilson, J. Amer. Math. Soc. 221Y. Peres, O. Schramm, S. Sheffield, and D.B. Wilson. Tug-of-war and the infinity Laplacian. J. Amer. Math. Soc., 22(1):167-210, 2009.\n\nHölder regularity for the gradients of solutions of the strong p(x)-Laplacian. C Zhang, S Zhou, J. Math. Anal. Appl. 3892C. Zhang and S. Zhou. Hölder regularity for the gradients of solutions of the strong p(x)-Laplacian. J. Math. Anal. Appl., 389(2):1066-1077, 2012.\n"},"corpus_id":{"kind":"number","value":119566555,"string":"119,566,555"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1710.07760\",\n \"mag\": 2963997759,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1007/s00526-018-1375-1\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1710.07760v1.pdf\"\n ],\n \"pdf_sha\": \"30a77bdf240d7376b838c3d10ea98da9698126e1\",\n \"open_access\": {\n \"license\": \"CC0\",\n \"url\": \"https://jyx.jyu.fi/bitstream/123456789/62921/1/siltakoskiequivalence%20revised.pdf\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":""},"authors":{"kind":"list like","value":[],"string":"[]","isConversational":false},"abstract":{"kind":"string","value":"We show that viscosity solutions to the normalized p(x)-Laplace equation coincide with distributional weak solutions to the strong p(x)-Laplace equation when p is Lipschitz and inf p > 1. This yields C 1,α regularity for the viscosity solutions of the normalized p(x)-Laplace equation. As an additional application, we prove a Radó-type removability theorem."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"In this paper, we study viscosity solutions to the normalized p(x)-Laplace equation which is defined by There has been recent interest in normalized equations, see for example [JS17,IJS,BG15]. We are partly motivated by the connection to stochastic tug-of-war games [PSSW09] as the case of space dependent probabilities leads to (1.1) [AHP17]. The objective of this work is to show that viscosity solutions to (1.1) coincide with solutions of its counterpart in the theory of distributional weak solutions. One approach to this kind of equivalence results [JLM01,Ish95] is based on the uniqueness of solutions. However, it seems difficult to use uniqueness in our case because the uniqueness of solutions is an open problem for the equation (1.1) as pointed out in [JLP10]. The equation (1.1) is in the non-divergence form. In order to find the weak counterpart, we note that for u ∈ C 2 (Ω) with non-vanishing gradient it holds that − |Du| p(x)−2 ∆ N p(x) u =−div |Du| p(x)−2 Du + |Du| p(x)−2 log (|Du|) Du · Dp. Thus the weak counterpart of (1.1) should be the strong p(x)-Laplace equation − ∆ S p(x) u := −div(|Du| p(x)−2 Du) + |Du| p(x)−2 log |Du| Du · Dp = 0.\n(1.2)\nOur main result, Theorem 5.9, is that viscosity solutions to (1.1) coincide with weak solutions to (1.2) when the function p is Lipschitz with inf p > 1. With these assumptions weak solutions to (1.2) in a domain are locally C 1,α continuous [ZZ12]. Thus our equivalence result yields local C 1,α regularity also for viscosity solutions to (1.1). As an application, we prove a Radó-type removability theorem for the strong p(x)-Laplacian. The theorem follows from the equivalence result since in the definition of a viscosity solution we may ignore the test functions whose gradient vanishes.\nThat viscosity solutions to (1.1) are weak solutions to (1.2) is proven by applying the method of [JJ12]. The idea is to approximate a viscosity solution through a sequence of inf-convolutions, show that the inf-convolutions are essentially weak supersolutions, and then pass to the limit.\nFirst, in Lemma 5.3 we show that the inf-convolution u ε of a viscosity supersolution u to (1.1) is still, in essence, a viscosity supersolution up to some error. This fact is a key part of our proof. If there was no x-dependence in (1.1), it would be straightforward to see that the inf-convolution of a viscosity supersolution is still a viscosity supersolution. This is because a test function that touches the inf-convolution from below also touches the original function from below at a nearby point once we add some constant to it. From this it would follow that the inf-convolution is a supersolution to the original equation. However, the equation (1.1) has x-dependence caused by p(x). Thus the inf-convolution no longer satisfies the original equation.\nIn Lemma 5.5 we use the standard mollification on u ε and p to deduce from Lemma 5.3 that u ε is \"almost\" a weak solution to −∆ S p(x) u ε ≥ 0. Applying Caccioppoli type estimates and vector inequalities we are then able to deduce that the sequence of inf-convolutions converges to the viscosity supersolution in W 1,p(·) loc (Ω) as ε → 0. This allows us to pass to the limit and conclude that u satisfies −∆ S p(x) u ≥ 0 in the weak sense. Due to the variable exponent, the operator ∆ S p(x) can be singular in some subsets and degenerate in others. Therefore we apply different arguments in the cases p(x) < 2 and p(x) ≥ 2, and finally need to be able to combine them.\nThe equivalence of weak and viscosity solutions to the usual p-Laplace equation was first proven by Juutinen, Lindqvist and Manfredi [JLM01]. Later Julin and Juutinen [JJ12] presented a more direct way to show that viscosity solutions to −∆ p u = f are also weak solutions. This proof was adapted in [APR17] to show that viscosity solutions to −∆ N p u = f coincide with weak solutions to −∆ p u = |Du| p−2 f when p ≥ 2. Similar arguments were also used in [MO] to study the equivalence of solutions to −∆ p u = f (x, u, Du). The variable exponent case was explored in [JLP10] where the equivalence of weak and viscosity solutions was proven for the p(x)-Laplace equation using techniques of [JLM01].\nThe equation (1.2) was introduced by Adamowicz and Hästö [AH10,AH11] in connection with mappings of finite distortion. It has been further studied for example in [ZZ12,PL13].\nThe paper is organized as follows: in Section 2 we recall the variable exponent Lebesgue and Sobolev spaces. Section 3 contains the rigorous definitions of solutions to equations (1.1) and (1.2). In Section 4 we show that weak solutions of (1.1) are viscosity solutions to (1.2) and the converse statement is proven in Section 5. Finally, in Section 6 we formulate and prove a Radó-type removability theorem for weak solutions of (1.2)."},{"title":"Variable exponent lebesgue and sobolev spaces","content":"We briefly recall basic facts about these spaces. For general reference see e.g. [DHHR11]. Let Ω ⊂ R N be an open and bounded set and let p : Ω → (1, ∞) be a measurable function. We denote p + := ess sup x∈Ω p(x) and p − := ess inf p(x)."},{"title":"x∈Ω","content":"The variable exponent Lebesgue space L p(·) (Ω) is defined as the set of measurable functions u : Ω → R for which the p(·)-modular ̺ p(·) (u) := Ω |u| p(x) dx is finite. It is a Banach space equipped with the Luxemburg norm\nGiven that p + < ∞ or ̺ p(·) (u) > 0, the norm and the modular satisfy the inequality (see [DHHR11,p75])\nA version of Hölder's inequality holds [DHHR11,p81]\nAs a consequence of the Hölder's inequality we have that\nThe variable exponent Sobolev space W 1,p(·) (Ω) is the set of functions in u ∈ L p(·) (Ω) for which the weak gradient Du belongs in L p(·) (Ω). It is a Banach space equipped with the norm\nThe space W 1,p 0 (Ω) is the closure of compactly supported Sobolev functions in the space W 1,p(·) (Ω). A function belongs to the the local Lebesgue space L From now on we assume that p is Lipschitz continuous and p − > 1.\nfor all non-negative ϕ ∈ W 1,p(·) (Ω) with compact support. We say that u is a weak subsolution to −∆ S p(x) u ≤ 0 if −u is a supersolution and that u is a weak solution to −∆ S p(x) u = 0 if u is both supersolution and subsolution.\nLemma 3.2. It is enough to consider C ∞ 0 (Ω) test functions in the previous definition.\nProof. Assume that ϕ ∈ W 1,p(·) (Ω) has a compact support in an open set Ω ′ ⋐ Ω. Since p is log-Hölder continuous and bounded as a Lipschitz function, there is a sequence of functions ϕ j ∈ C ∞ 0 (Ω ′ ) such that ϕ j → ϕ in W 1,p(·) (Ω ′ ) (see [DHHR11,p347]). We set ψ j := ϕ − ϕ j . Then it is enough to show that\nas j → ∞. The first integral convergences to zero by Hölder's inequality so we focus on the second integral. We may assume that N > 1. We set q(x) := p(x) p(x)−1+ 1 N . Using the inequality a s log a ≤ N a s+ 1 N + 1 s for a, s > 0 we get\nWe take r ∈ (1, N ) such that q ′+ ≤ r * :\nSince ψ j ∈ W 1,min(p(·),r) 0\n(Ω ′ ) , we have by a variable exponent version of the Sobolev inequality (see e.g. [DHHR11,p265])\nThese estimates imply the claim since ψ j W 1,p (Ω ′ ) → 0 as j → ∞.\nIn order to define viscosity solutions to −∆ N p(x) u = 0, we set\nWe also recall the concept of semi-jets. The subjet of a function u :\nThe closure of a subjet is defined by setting (η, X) ∈ J 2,− u(x) if there is a sequence\nThe superjet J 2,+ u(x) and its closure J 2,+ u(x) are defined in the same manner except that the inequality (3.1) is reversed.\nDefinition 3.3. A lower semicontinuous function u : Ω → R is a viscosity su- Remark. Observe that in the previous definition we require nothing in the case (0, X) ∈ J 2,− u(x).\nViscosity solutions may be equivalently defined using the jet-closures or test functions. For the next proposition, see e.g. [Koi12, Prop 2.6].\nProposition 3.4. Let u : Ω → R be lower semicontinuous. Then the following conditions are equivalent.\n(i) The function u is a viscosity supersolution to\nWhen ϕ is as in the third condition above, we say that ϕ touches u from below at x."},{"title":"Weak solutions are Viscosity solutions","content":"We show that if u is a weak solution to −∆ S p(x) u = 0, then it is a viscosity solution to −∆ N p(x) u = 0. Juutinen, Lukkari and Parviainen [JLP10] showed that weak solutions to the standard p(x)-Laplace equation are also viscosity solutions. This was accomplished with the help of the comparison principle. For if u is a weak supersolution to −∆ p(x) u ≥ 0 that is not a viscosity supersolution, then there is a test function ϕ ∈ C 2 touching u from below at x so that −∆ p(x) ϕ < 0 in some ball B(x). Lifting ϕ slightly produces a new functionφ still satisfying\nOur difficulty is that, to the best of our knowledge, the comparison principle is an open problem for the strong p(x)-Laplacian. Our strategy is therefore to consider a ball so small that the gradient of the test function does not vanish. Then the comparison principle holds and we arrive at a contradiction.\nProof. Zhang and Zhou [ZZ12] showed that weak solutions of −∆ S p(x) u = 0 are in C 1 (Ω). Therefore it suffices to show that if u ∈ C 1 (Ω) is a weak supersolution to −∆ S p(x) u ≥ 0, then it is also a viscosity supersolution to −∆ N p(x) u ≥ 0. Assume on the contrary that there is ϕ ∈ C 2 (Ω) touching u from below at x 0 ∈ Ω, Dϕ(x 0 ) = 0 and"},{"title":"Then by continuity there is","content":"and ess sup\nwhere the last equality holds because ψ j → ψ in W 1,2 (B r (x 0 )) and p j → p uniformly in B r (x 0 ). Calculating the divergence of |Dϕ| p j (x)−2 Dϕ and integrating by parts we get\nBy the convergence of ψ j and p j , it follows from (4.4) and (4.5) that\n(4.6)\nSince u is a weak supersolution to ∆ S p(x) u = 0 and ψ ∈ W 1,p(·) (Ω) has a compact support in Ω, we have\n(4.7)\nDenoting A := {x ∈ B r (x 0 ) : ψ(x) > 0} and combining (4.6) and (4.7) we arrive at\nwhere the last inequality follows from (4.2) and (4.3). Since\nfor any two vectors a, b ∈ R N when p(x) > 1, it follows from (4.8) that |A| = 0. But this is impossible since ϕ(x 0 ) = u(x 0 ) and l > 0."},{"title":"Viscosity solutions are Weak solutions","content":"We show that if u is a viscosity supersolution to −∆ N p(x) u ≥ 0, then it is a weak supersolution to −∆ S p(x) u ≥ 0. The same statement for subsolutions then follows by analogy.\nWe recall the usual partial ordering for symmetric N × N matrices by setting X ≤ Y if Xξ, ξ ≤ Y ξ, ξ for all ξ ∈ R N . For a matrix X we also set X := max {|λ| : λ is an eigenvalue of X} and for vectors ξ, η ∈ R N we use the notation ξ ⊗ η := ξη ′ , i.e. ξ ⊗ η is an N × N matrix whose (i, j) entry is ξ i η j .\nDefinition 5.1 (Inf-convolution). Let q ≥ 2 and ε > 0. The inf-convolution of a bounded function u ∈ C(Ω) is defined by\nThe inf-convolution is well known to provide good approximations of viscosity supersolutions and often one only needs to consider it for q = 2 (see e.g. [CIL92]). However, as the authors in [JJ12] observed, considering large enough q essentially cancels the singularity in the usual p-Laplace operator when 1 < p < 2. In similar fashion it also cancels the singularity of the operator ∆ S p(x) . This is due to the property (v) in the next lemma. We also list some other basic properties of the inf-convolution.\nLemma 5.2. Let u ∈ C(Ω) be a bounded function. Then the inf-convolution u ε as defined in (5.1) has the following properties.\nThese properties are well known, see appendix of [JJ12] and also [Kat15b] where more general \"flat inf-convolution\" is considered. Regardless, we give a proof of (v) based on [Kat15a,p53] due to its critical role in the proof of Lemma 5.5.\nProof of property (v) in Lemma 5.2. Let (η, X) ∈ J 2,− u ε (x). Then there is a function ϕ ∈ C 2 (R N ) such that it touches u ε from below at x and Dϕ(x) = η, D 2 ϕ(x) = X. Therefore for all y, z ∈ Ω we have\nChoosing y = x ε , we obtain\nSince ϕ(x) = u ε (x) = u(x ε )+ |xε−x| q qε q−1 , the above inequality means that the function\nhas a maximum at x. Thus η = Dψ(x) = (x−xε) ε q−1 |x ε − x| q−2 and\nWe will show that the inf-convolution provides approximations of viscosity supersolutions to −∆ N p(x) u ≥ 0. If there was no x-dependence in the equation, it would be straightforward to show that the inf-convolution of a supersolution is still a supersolution. However, the equation −∆ N p(x) u ≥ 0 has x-dependence caused by p(x). Regardless, in [Ish95,Thm 3] it is shown that with some assumptions on G, the inf-convolution u ε of a viscosity supersolution to G(x, u, Du, D 2 u) ≥ 0 is still a viscosity supersolution to G(\nWe prove a modified version of this theorem for the solutions of −∆ N p(x) u ≥ 0. The important modification is the term |η| min(p(x)−2,0) in (5.2) as it cancels a singular gradient term that appears due to the error term in the proof of Lemma 5.5, see (5.14). Another difference is that we consider inf-convolution with the exponent q ≥ 2. where E(ε) → 0 as ε → 0. The error function E depends only on p, q and the modulus of continuity of u.\nProof. Fix x ∈ Ω r(ε) and (η, X) ∈ J 2,− u ε (x), η = 0. Then by Lemma 5.2 there is\nThere exists a function ϕ ∈ C 2 (R N ) such that it touches u ε from below at x and Dϕ(x) = η, D 2 ϕ(x) = X. By the definition of inf-convolution\nSince by (5.3) we have u(x ε ) = ϕ(x) − |xε−x| q qε q−1 , it follows from (5.4) that the expression u(y) − ϕ(z) + |y−z| q qε q−1 reaches its minimum at (y, z) = (x ε , x). Thus\nWe denote Φ(y, z) := 1 qε q−1 |y − z| q and invoke the Theorem of sums (see [CIL92]). There exist Y, Z ∈ S N such that\nwhere we used (5.6) and the fact that Y ≤ Z implies\nWe have the estimate\nSince p is Lipschtiz continuous and p − > 1, we have also\nCombining these with (5.7) we get (we may assume that r(ε) < 1)\n(5.8)\nMoreover, by uniform continuity of u there is a modulus of continuity ω such that ω(t) → 0 as t → 0 and |u(y) − u(z)| ≤ ω(|y − z|) for all y, z ∈ Ω. Hence by (5.3)\nWe now consider the situations p(x) ≤ 2 and p(x) > 2 separately. If p(x) ≤ 2, we multiply (5.8) by |η| p(x)−2 and estimate using (5.9). We get\nwhere the last inequality is true when ε < 1 is so small that ω(r(ε)) < 1. This proves (5.2) when p(x) ≤ 2.\nIf p(x) > 2, we estimate (5.8) directly using (5.9). We get −F (x, η, X) ≤C(p, q) 1 ε q−1 q 1 q ε q−1 q ω(r(ε)) 1 q q = C(p, q)ω(r(ε))), which proves (5.2) when p(x) > 2.\nNext we will use the previous lemma to show that inf-convolution of a viscosity supersolution to −∆ N p(x) u ≥ 0 in Ω is a weak supersolution to −∆ S p(x) u ≥ 0 in Ω r(ε) up to some error term. Before proceeding we make some remarks about the point-wise differentiability of inf-convolution.\nRemark 5.4. It follows from semi-concavity that the inf-convolution u ε is locally Lipschitz in Ω r(ε) (see [EG15,p267]). Therefore it belongs in W 1,∞ loc (Ω r(ε) ), is differentiable almost everywhere in Ω r(ε) , and its derivative agrees with its Sobolev derivative almost everywhere in Ω r(ε) (see [EG15,p155 and p265]).\nBy Lemma 5.2 the function φ(x) := u ε (x) − C(q, ε, u) |x| 2 is concave in Ω r(ε) . Thus Alexandrov's theorem implies that u ε is twice differentiable almost everywhere in Ω r(ε) . Furthermore, the proof of Alexandrov's theorem in [EG15,p273] establishes that if φ j is the standard mollification of φ, then D 2 φ j → D 2 φ almost everywhere in Ω r(ε) .\nLemma 5.5. Assume that u is a uniformly continuous viscosity supersolution to −∆ N p(x) u ≥ 0 in Ω. Let q > 2 be so large that p − − 2 + q−2 q−1 ≥ 0 and let u ε be the inf-convolution of u as defined in (5.1). Then\nfor all non-negative ϕ ∈ W 1,p(·) (Ω r(ε) ) with compact support, where E(ε) → 0 as ε → 0 and s(x) = max(p(x) − 2, 0).\nProof. It is enough to consider ϕ ∈ C ∞ 0 (Ω r(ε) ). This can be seen in the same way as Lemma 3.2, but since u ε ∈ W 1,∞ loc (Ω r(ε) ), the proof is even simpler. (Step 1) We show that u ε satisfies the auxiliary inequality (5.11) for all 0 < δ < 1. As mentioned in Remark 5.4, the function φ(x) := u ε (x) − C(q, ε, u) |x| 2 is concave in Ω r(ε) and we can approximate it by smooth concave functions φ j so that φ j , Dφ j , D 2 φ j → φ, Dφ, D 2 φ almost everywhere in Ω r(ε) . We define u ε,j (x) := φ j (x) + C(q, ε, u) |x| 2 and denote by p j the standard mollification of p. Since u ε,j and p j are smooth, we calculate\nDu ε,j ϕ\nDu ε,j · Dϕ + 1 2 log δ + |Du ε,j | 2 Dp j ϕ dx.\n(5.10)\nWe let j → ∞ in (5.10) and intend to use Fatou's lemma at the LHS and the Dominated convergence theorem at the RHS. This results in the auxiliary inequality\nDu ε · Dϕ + 1 2 log δ + |Du ε | 2 Dp ϕ dx,\nwhere D 2 u ε is the Hessian of u ε in the Alexandrov's sense. We still need to check that the assumptions of the Dominated convergence theorem and Fatou's lemma hold. By Lipschitz continuity of u ε and p there is M ≥ 1 such that\nThis justifies our use of the Dominated convergence theorem. In order to justify our use of Fatou's lemma, we notice first that by concavity of φ j we have D 2 u ε,j ≤ C(q, ε, u)I. Thus the integrand at the LHS of (5.10) is clearly bounded from below by a constant independent of j if Du ε,j = 0. If Du ε,j = 0, we have\nwhere the first inequality follows like estimate (5.7) since p j ≥ p − > 1. (Step 2) We let δ → 0 in the auxiliary inequality (5.11). The RHS becomes\nby the Lebesgue's dominated convergence theorem. We intend to apply Fatou's lemma on the LHS. We have Du ε (x), D 2 u ε (x) ∈ J 2,− u ε (x) for almost every x ∈ Ω r(ε) . Therefore by Lemma 5.3 it holds that\nand by the property (v) in Lemma 5.2 we have\nObserve that since q > 2, the condition (5.13) implies that the Hessian D 2 u ε is negative semi-definite in the set where the gradient Du ε vanishes. Using this fact, Fatou's lemma and (5.12) we get\n|Du ε | max(p(x)−2,0) ϕ dx, (5.14) and thus we arrive at the desired inequality. Our use of Fatou's lemma is justified since if Du ε = 0 and p(x) ≤ 2, we have by (5.13)\nwhere the last inequality follows from p − − 2 + q−2 q−1 ≥ 0. If Du ε = 0 and p(x) > 2, we have simply\nIn the next two lemmas we use Caccioppoli type estimates and algebraic inequalities to show that the sequence of inf-convolutions converges to the viscosity supersolution in W 1,p(·) loc (Ω).\nLemma 5.6. Under the assumptions of Lemma 5.5, the function u belongs in W 1,p(·) loc (Ω) and for any Ω ′ ⋐ Ω we have Du ε → Du weakly in L p(·) (Ω ′ ) for some subsequence.\nProof. Take a cut-off function ξ ∈ C ∞ 0 (Ω ′ ) such that 0 ≤ ξ ≤ 1 in Ω and ξ ≡ 1 in Ω ′ . Then assume that ε is so small that supp ξ =: K ⊂ Ω r(ε) . We define a test function ϕ := (L − u ε )ξ p + where L := sup ε,x∈Ω ′ |u ε (x)| is finite since u ε → u locally uniformly. We have Dϕ = −Du ε ξ p + + (L − u ε )p + ξ p + −1 Dξ and therefore by Lemma 5.5\nWe estimate these integrals using Young's inequality. The first integral is estimated by the facts p(x)(p + −1)\np(x)−1 ≥ p + and ξ ≤ 1 as follows\nTo estimate I 2 , we also use the inequality a s |log a| ≤ a s+ 1 2 + 1 s for a > 0 and s > 0,\nThe last integral is estimated by the two alternatives in max(p(x) − 2, 0) as follows (we may assume that |E(ε)| ≤ 1)\nTaking small δ we conclude that Du ε is bounded in L p(·) (Ω ′ ) with respect to ε. Since L p(·) (Ω ′ ) is a reflexive Banach space [DHHR11, p76 and p89], it follows that there is a function Du ∈ L p(·) (Ω ′ ) such that Du ε → Du weakly in L p(·) (Ω ′ ) for some subsequence. Consequently u ∈ W 1,p(·) (Ω ′ ) with Du as its weak derivative.\nLemma 5.7. Under the assumptions of Lemma 5.5, for any Ω ′ ⋐ Ω we have Du ε → Du in L p(·) (Ω ′ ) for some subsequence.\nProof. Take a cut-off function ξ ∈ C ∞ 0 (Ω) such that ξ ≡ 1 in Ω ′ and define a test function ϕ := (u − u ε )ξ. Then assume that ε is so small that supp ξ =: K ⊂ Ω r(ε) .\nSince ϕ ∈ W 1,p(·) (Ω r(ε) ) with compact support it follows from Lemma 5.5 that\n(5.15) According to Lemma 5.6 we have u ε → u locally uniformly and Du ε → Du weakly in L p(·) (K) for a subsequence. Thus by passing to a subsequence we may assume that the right hand side of (5.15) converges to zero. The claim now follows from the inequalities (see e.g. [Lin17, Chapter 12])\nfor a, b ∈ R N . Indeed, we immediately get that Ω ′ ∩{p(x)≥2} |Du − Du ε | p(x) dx → 0. To deal with the set {p(x) < 2}, we first apply the above algebraic inequality and then estimate using Hölder's inequality, the modular inequality (2.1) and the definition of the · L p(·) -norm. We get\nwhere s ∈ p + 2 , p − 2 . The last integral is bounded since the sequence Du ε is bounded in L p(·) (Ω ′ ) by its weak convergence. The RHS therefore converges to zero by (5.15).\nNext, we use the previous convergence result to pass to the limit in the inequality of Lemma 5.5 and conclude that viscosity supersolutions to −∆ N p(x) u ≥ 0 are weak supersolutions to −∆ S p(x) u ≥ 0.\nProof. It is clear from the definition of weak supersolutions to −∆ S p(x) u ≥ 0 that we can without loss of generality assume that u is uniformly continuous in Ω by restricting to a smaller domain. Fix a non-negative test function ϕ ∈ C ∞ 0 (Ω) and take open Ω ′ ⋐ Ω such that supp ϕ ⊂ Ω ′ . Let q and u ε be as in Lemma 5.5 and assume that ε is so small that Ω ′ ⊂ Ω r(ε) . Then the claim follows from Lemma 5.5 if we show that\nfor a, b ∈ R N . Indeed, when ε is so small that Ω ′ |Du ε − Du| p(x) dx < 1 we have by Hölder's inequality and the modular inequality\nClaim (5.18) holds since Ω ′ |Du ε | p(x) dx is bounded and E(ε) → 0. To this end, fix 0 < ǫ < 1. The mapping (a, x) → |a| p(x)−2 log (|a|) a is uniformly continuous in bounded sets of R N × Ω ′ . Hence there exists δ = δ(ǫ) < ǫ such that whenever x ∈ Ω ′ and a, b ∈ B(0, 3) satisfy |a − b| < δ, it holds\nIf |a| , |b| ≥ 1 and |a − b| < δ, then we use (5.19) to get the estimate\nWe denote\nThe strong convergence of Du ε to Du in L p(·) (Ω ′ ) implies that Du ε → Du in measure in Ω ′ (see [DHHR11, Lemma 3.2.10]). Thus there is ε 0 = ε 0 (δ) such that for all ε < ε 0 it holds |F ε | ≤ δ. Using the inequality a s |log a| ≤ a s+ 1 2 + 1 s for a > 0 and s > 0 we get for all ε < ε 0\n(5.23) If x ∈ Ω ′ \\ F ε , then either |Du ε | , |Du| ≤ 3 or |Du ε | , |Du| ≥ 1. Hence by (5.21) and (5.22) we have\nCombining (5.24) and (5.23) proves (5.20) since ǫ was arbitrary.\nMerging Theorems 4.1 and 5.8 yields the following equivalence result.\nTheorem 5.9. A function u is a viscosity solution to −∆ N p(x) u = 0 in Ω if and only if it is a weak solution to −∆ S p(x) u = 0 in Ω. Since the weak solutions to the strong p(x)-Laplace equation are locally C 1,α continuous [ZZ12], our equivalence result yields local C 1,α regularity also for viscosity solutions of the normalized p(x)-Laplace equation.\nCorollary 5.10. If u is a viscosity solution to −∆ N p(x) u = 0 in a bounded domain Ω, then u ∈ C 1,α (Ω) with α ∈ (0, 1)."},{"title":"An Application: A Radó-type removability theorem","content":"The classical theorem of Radó says that if a continuous complex-valued function f defined on a domain Ω ⊂ C is holomorphic in Ω \\{f = 0}, then it is holomorphic in the whole Ω. Similar results have been proven for solutions of partial differential equations. We prove a Radó-type removability theorem for the strong p(x)-Laplace equation. It is worth pointing out that it could be difficult to show this kind of result without appealing to viscosity solutions whereas it is straightforward to do so with the help of the equivalence result. The theorem follows by observing that weak solutions to ∆ S p(x) u = 0 coincide with viscosity solutions of an equation that satisfies the assumptions of a Radó-type removability theorem in [JL05].\nRecall that we ignore the test functions whose gradient vanishes at the point of touching in the Definition 3.3 of viscosity solutions to −∆ N p(x) u = 0. Sometimes this kind of solutions are called feeble viscosity solutions (e.g. [JL05,Kat15b]). We will observe that these feeble viscosity solutions to −∆ N p(x) u = 0 are exactly the usual viscosity solutions to − tr(A(x, Du)D 2 u) = 0, (6.1)\nwhere A(x, Du) := |Du| 2 I + (p(x) − 2) Du ⊗ Du. To be precise, we define the viscosity solutions to (6.1).\nDefinition 6.1. A lower semicontinuous function u is a viscosity supersolution to (6.1) in Ω if, whenever (η, X) ∈ J 2,− u(x) with x ∈ Ω, then −tr(A(x, η)X) ≥ 0.\nA function u is a viscosity subsolution to (6.1) if −u is a supersolution, and a viscosity solution if it is both viscosity super-and subsolution.\nLemma 6.2. A function u is a viscosity solution to −∆ N p(x) u = 0 if and only if it is a viscosity solution to (6.1).\nProof. It is enough to consider supersolutions. Take (η, X) ∈ J 2,− u(x) with x ∈ Ω. If η = 0, then the conditions for both definitions are satisfied, so we may assume that η = 0. Then we have F (x, η, X) ≥ 0 if and only if − |η| 2 tr(X) + (p(x) − 2) Xη, η ≥ 0, where |η| 2 tr(X) + (p(x) − 2) Xη, η = |η| 2 tr(X) + (p(x) − 2) tr(η ⊗ ηX) =tr |η| 2 I + (p(x) − 2) η ⊗ η X .\nHence the definitions are equivalent. Theorem 6.3 (A Radó-type removability theorem). Let u ∈ C 1 (Ω) be a weak solution to −∆ S p(x) u = 0 in Ω \\ {u = 0}. Then u is a weak solution to −∆ S p(x) u = 0 in the whole Ω.\nProof. By Lemma 6.2 and our equivalence result weak solutions to −∆ S p(x) u = 0 coincide with viscosity solutions to (6.1). Therefore it suffices to show that if u is a viscosity solution to (6.1) in Ω \\ {u = 0}, it is a viscosity solution to (6.1) in the whole Ω. This on the other hand follows from [JL05, Theorem 2.2]. The matrix A satisfies the assumptions of the theorem as it is symmetric, has continuous entries and A(x, 0, 0) = 0 for all x ∈ Ω. It is also positive semi-definite since for all ξ ∈ R N we have ξ ′ |η| 2 I + (p(x) − 2) η ⊗ η ξ ≥ξ ′ |η| 2 I − η ⊗ η ξ ≥ |ξ| 2 |η| 2 − η ⊗ η = 0."},{"title":"−","content":""},{"title":"Lemma 5. 3 .","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"In this paper, we study viscosity solutions to the normalized p(x)-Laplace equation which is defined by There has been recent interest in normalized equations, see for example [JS17,IJS,BG15]. We are partly motivated by the connection to stochastic tug-of-war games [PSSW09] as the case of space dependent probabilities leads to (1.1) [AHP17]. The objective of this work is to show that viscosity solutions to (1.1) coincide with solutions of its counterpart in the theory of distributional weak solutions. One approach to this kind of equivalence results [JLM01,Ish95] is based on the uniqueness of solutions. However, it seems difficult to use uniqueness in our case because the uniqueness of solutions is an open problem for the equation (1.1) as pointed out in [JLP10]. The equation (1.1) is in the non-divergence form. In order to find the weak counterpart, we note that for u ∈ C 2 (Ω) with non-vanishing gradient it holds that − |Du| p(x)−2 ∆ N p(x) u =−div |Du| p(x)−2 Du + |Du| p(x)−2 log (|Du|) Du · Dp. Thus the weak counterpart of (1.1) should be the strong p(x)-Laplace equation − ∆ S p(x) u := −div(|Du| p(x)−2 Du) + |Du| p(x)−2 log |Du| Du · Dp = 0.\\n(1.2)\\nOur main result, Theorem 5.9, is that viscosity solutions to (1.1) coincide with weak solutions to (1.2) when the function p is Lipschitz with inf p > 1. With these assumptions weak solutions to (1.2) in a domain are locally C 1,α continuous [ZZ12]. Thus our equivalence result yields local C 1,α regularity also for viscosity solutions to (1.1). As an application, we prove a Radó-type removability theorem for the strong p(x)-Laplacian. The theorem follows from the equivalence result since in the definition of a viscosity solution we may ignore the test functions whose gradient vanishes.\\nThat viscosity solutions to (1.1) are weak solutions to (1.2) is proven by applying the method of [JJ12]. The idea is to approximate a viscosity solution through a sequence of inf-convolutions, show that the inf-convolutions are essentially weak supersolutions, and then pass to the limit.\\nFirst, in Lemma 5.3 we show that the inf-convolution u ε of a viscosity supersolution u to (1.1) is still, in essence, a viscosity supersolution up to some error. This fact is a key part of our proof. If there was no x-dependence in (1.1), it would be straightforward to see that the inf-convolution of a viscosity supersolution is still a viscosity supersolution. This is because a test function that touches the inf-convolution from below also touches the original function from below at a nearby point once we add some constant to it. From this it would follow that the inf-convolution is a supersolution to the original equation. However, the equation (1.1) has x-dependence caused by p(x). Thus the inf-convolution no longer satisfies the original equation.\\nIn Lemma 5.5 we use the standard mollification on u ε and p to deduce from Lemma 5.3 that u ε is \\\"almost\\\" a weak solution to −∆ S p(x) u ε ≥ 0. Applying Caccioppoli type estimates and vector inequalities we are then able to deduce that the sequence of inf-convolutions converges to the viscosity supersolution in W 1,p(·) loc (Ω) as ε → 0. This allows us to pass to the limit and conclude that u satisfies −∆ S p(x) u ≥ 0 in the weak sense. Due to the variable exponent, the operator ∆ S p(x) can be singular in some subsets and degenerate in others. Therefore we apply different arguments in the cases p(x) < 2 and p(x) ≥ 2, and finally need to be able to combine them.\\nThe equivalence of weak and viscosity solutions to the usual p-Laplace equation was first proven by Juutinen, Lindqvist and Manfredi [JLM01]. Later Julin and Juutinen [JJ12] presented a more direct way to show that viscosity solutions to −∆ p u = f are also weak solutions. This proof was adapted in [APR17] to show that viscosity solutions to −∆ N p u = f coincide with weak solutions to −∆ p u = |Du| p−2 f when p ≥ 2. Similar arguments were also used in [MO] to study the equivalence of solutions to −∆ p u = f (x, u, Du). The variable exponent case was explored in [JLP10] where the equivalence of weak and viscosity solutions was proven for the p(x)-Laplace equation using techniques of [JLM01].\\nThe equation (1.2) was introduced by Adamowicz and Hästö [AH10,AH11] in connection with mappings of finite distortion. It has been further studied for example in [ZZ12,PL13].\\nThe paper is organized as follows: in Section 2 we recall the variable exponent Lebesgue and Sobolev spaces. Section 3 contains the rigorous definitions of solutions to equations (1.1) and (1.2). In Section 4 we show that weak solutions of (1.1) are viscosity solutions to (1.2) and the converse statement is proven in Section 5. Finally, in Section 6 we formulate and prove a Radó-type removability theorem for weak solutions of (1.2).\"\n },\n {\n \"title\": \"Variable exponent lebesgue and sobolev spaces\",\n \"content\": \"We briefly recall basic facts about these spaces. For general reference see e.g. [DHHR11]. Let Ω ⊂ R N be an open and bounded set and let p : Ω → (1, ∞) be a measurable function. We denote p + := ess sup x∈Ω p(x) and p − := ess inf p(x).\"\n },\n {\n \"title\": \"x∈Ω\",\n \"content\": \"The variable exponent Lebesgue space L p(·) (Ω) is defined as the set of measurable functions u : Ω → R for which the p(·)-modular ̺ p(·) (u) := Ω |u| p(x) dx is finite. It is a Banach space equipped with the Luxemburg norm\\nGiven that p + < ∞ or ̺ p(·) (u) > 0, the norm and the modular satisfy the inequality (see [DHHR11,p75])\\nA version of Hölder's inequality holds [DHHR11,p81]\\nAs a consequence of the Hölder's inequality we have that\\nThe variable exponent Sobolev space W 1,p(·) (Ω) is the set of functions in u ∈ L p(·) (Ω) for which the weak gradient Du belongs in L p(·) (Ω). It is a Banach space equipped with the norm\\nThe space W 1,p 0 (Ω) is the closure of compactly supported Sobolev functions in the space W 1,p(·) (Ω). A function belongs to the the local Lebesgue space L From now on we assume that p is Lipschitz continuous and p − > 1.\\nfor all non-negative ϕ ∈ W 1,p(·) (Ω) with compact support. We say that u is a weak subsolution to −∆ S p(x) u ≤ 0 if −u is a supersolution and that u is a weak solution to −∆ S p(x) u = 0 if u is both supersolution and subsolution.\\nLemma 3.2. It is enough to consider C ∞ 0 (Ω) test functions in the previous definition.\\nProof. Assume that ϕ ∈ W 1,p(·) (Ω) has a compact support in an open set Ω ′ ⋐ Ω. Since p is log-Hölder continuous and bounded as a Lipschitz function, there is a sequence of functions ϕ j ∈ C ∞ 0 (Ω ′ ) such that ϕ j → ϕ in W 1,p(·) (Ω ′ ) (see [DHHR11,p347]). We set ψ j := ϕ − ϕ j . Then it is enough to show that\\nas j → ∞. The first integral convergences to zero by Hölder's inequality so we focus on the second integral. We may assume that N > 1. We set q(x) := p(x) p(x)−1+ 1 N . Using the inequality a s log a ≤ N a s+ 1 N + 1 s for a, s > 0 we get\\nWe take r ∈ (1, N ) such that q ′+ ≤ r * :\\nSince ψ j ∈ W 1,min(p(·),r) 0\\n(Ω ′ ) , we have by a variable exponent version of the Sobolev inequality (see e.g. [DHHR11,p265])\\nThese estimates imply the claim since ψ j W 1,p (Ω ′ ) → 0 as j → ∞.\\nIn order to define viscosity solutions to −∆ N p(x) u = 0, we set\\nWe also recall the concept of semi-jets. The subjet of a function u :\\nThe closure of a subjet is defined by setting (η, X) ∈ J 2,− u(x) if there is a sequence\\nThe superjet J 2,+ u(x) and its closure J 2,+ u(x) are defined in the same manner except that the inequality (3.1) is reversed.\\nDefinition 3.3. A lower semicontinuous function u : Ω → R is a viscosity su- Remark. Observe that in the previous definition we require nothing in the case (0, X) ∈ J 2,− u(x).\\nViscosity solutions may be equivalently defined using the jet-closures or test functions. For the next proposition, see e.g. [Koi12, Prop 2.6].\\nProposition 3.4. Let u : Ω → R be lower semicontinuous. Then the following conditions are equivalent.\\n(i) The function u is a viscosity supersolution to\\nWhen ϕ is as in the third condition above, we say that ϕ touches u from below at x.\"\n },\n {\n \"title\": \"Weak solutions are Viscosity solutions\",\n \"content\": \"We show that if u is a weak solution to −∆ S p(x) u = 0, then it is a viscosity solution to −∆ N p(x) u = 0. Juutinen, Lukkari and Parviainen [JLP10] showed that weak solutions to the standard p(x)-Laplace equation are also viscosity solutions. This was accomplished with the help of the comparison principle. For if u is a weak supersolution to −∆ p(x) u ≥ 0 that is not a viscosity supersolution, then there is a test function ϕ ∈ C 2 touching u from below at x so that −∆ p(x) ϕ < 0 in some ball B(x). Lifting ϕ slightly produces a new functionφ still satisfying\\nOur difficulty is that, to the best of our knowledge, the comparison principle is an open problem for the strong p(x)-Laplacian. Our strategy is therefore to consider a ball so small that the gradient of the test function does not vanish. Then the comparison principle holds and we arrive at a contradiction.\\nProof. Zhang and Zhou [ZZ12] showed that weak solutions of −∆ S p(x) u = 0 are in C 1 (Ω). Therefore it suffices to show that if u ∈ C 1 (Ω) is a weak supersolution to −∆ S p(x) u ≥ 0, then it is also a viscosity supersolution to −∆ N p(x) u ≥ 0. Assume on the contrary that there is ϕ ∈ C 2 (Ω) touching u from below at x 0 ∈ Ω, Dϕ(x 0 ) = 0 and\"\n },\n {\n \"title\": \"Then by continuity there is\",\n \"content\": \"and ess sup\\nwhere the last equality holds because ψ j → ψ in W 1,2 (B r (x 0 )) and p j → p uniformly in B r (x 0 ). Calculating the divergence of |Dϕ| p j (x)−2 Dϕ and integrating by parts we get\\nBy the convergence of ψ j and p j , it follows from (4.4) and (4.5) that\\n(4.6)\\nSince u is a weak supersolution to ∆ S p(x) u = 0 and ψ ∈ W 1,p(·) (Ω) has a compact support in Ω, we have\\n(4.7)\\nDenoting A := {x ∈ B r (x 0 ) : ψ(x) > 0} and combining (4.6) and (4.7) we arrive at\\nwhere the last inequality follows from (4.2) and (4.3). Since\\nfor any two vectors a, b ∈ R N when p(x) > 1, it follows from (4.8) that |A| = 0. But this is impossible since ϕ(x 0 ) = u(x 0 ) and l > 0.\"\n },\n {\n \"title\": \"Viscosity solutions are Weak solutions\",\n \"content\": \"We show that if u is a viscosity supersolution to −∆ N p(x) u ≥ 0, then it is a weak supersolution to −∆ S p(x) u ≥ 0. The same statement for subsolutions then follows by analogy.\\nWe recall the usual partial ordering for symmetric N × N matrices by setting X ≤ Y if Xξ, ξ ≤ Y ξ, ξ for all ξ ∈ R N . For a matrix X we also set X := max {|λ| : λ is an eigenvalue of X} and for vectors ξ, η ∈ R N we use the notation ξ ⊗ η := ξη ′ , i.e. ξ ⊗ η is an N × N matrix whose (i, j) entry is ξ i η j .\\nDefinition 5.1 (Inf-convolution). Let q ≥ 2 and ε > 0. The inf-convolution of a bounded function u ∈ C(Ω) is defined by\\nThe inf-convolution is well known to provide good approximations of viscosity supersolutions and often one only needs to consider it for q = 2 (see e.g. [CIL92]). However, as the authors in [JJ12] observed, considering large enough q essentially cancels the singularity in the usual p-Laplace operator when 1 < p < 2. In similar fashion it also cancels the singularity of the operator ∆ S p(x) . This is due to the property (v) in the next lemma. We also list some other basic properties of the inf-convolution.\\nLemma 5.2. Let u ∈ C(Ω) be a bounded function. Then the inf-convolution u ε as defined in (5.1) has the following properties.\\nThese properties are well known, see appendix of [JJ12] and also [Kat15b] where more general \\\"flat inf-convolution\\\" is considered. Regardless, we give a proof of (v) based on [Kat15a,p53] due to its critical role in the proof of Lemma 5.5.\\nProof of property (v) in Lemma 5.2. Let (η, X) ∈ J 2,− u ε (x). Then there is a function ϕ ∈ C 2 (R N ) such that it touches u ε from below at x and Dϕ(x) = η, D 2 ϕ(x) = X. Therefore for all y, z ∈ Ω we have\\nChoosing y = x ε , we obtain\\nSince ϕ(x) = u ε (x) = u(x ε )+ |xε−x| q qε q−1 , the above inequality means that the function\\nhas a maximum at x. Thus η = Dψ(x) = (x−xε) ε q−1 |x ε − x| q−2 and\\nWe will show that the inf-convolution provides approximations of viscosity supersolutions to −∆ N p(x) u ≥ 0. If there was no x-dependence in the equation, it would be straightforward to show that the inf-convolution of a supersolution is still a supersolution. However, the equation −∆ N p(x) u ≥ 0 has x-dependence caused by p(x). Regardless, in [Ish95,Thm 3] it is shown that with some assumptions on G, the inf-convolution u ε of a viscosity supersolution to G(x, u, Du, D 2 u) ≥ 0 is still a viscosity supersolution to G(\\nWe prove a modified version of this theorem for the solutions of −∆ N p(x) u ≥ 0. The important modification is the term |η| min(p(x)−2,0) in (5.2) as it cancels a singular gradient term that appears due to the error term in the proof of Lemma 5.5, see (5.14). Another difference is that we consider inf-convolution with the exponent q ≥ 2. where E(ε) → 0 as ε → 0. The error function E depends only on p, q and the modulus of continuity of u.\\nProof. Fix x ∈ Ω r(ε) and (η, X) ∈ J 2,− u ε (x), η = 0. Then by Lemma 5.2 there is\\nThere exists a function ϕ ∈ C 2 (R N ) such that it touches u ε from below at x and Dϕ(x) = η, D 2 ϕ(x) = X. By the definition of inf-convolution\\nSince by (5.3) we have u(x ε ) = ϕ(x) − |xε−x| q qε q−1 , it follows from (5.4) that the expression u(y) − ϕ(z) + |y−z| q qε q−1 reaches its minimum at (y, z) = (x ε , x). Thus\\nWe denote Φ(y, z) := 1 qε q−1 |y − z| q and invoke the Theorem of sums (see [CIL92]). There exist Y, Z ∈ S N such that\\nwhere we used (5.6) and the fact that Y ≤ Z implies\\nWe have the estimate\\nSince p is Lipschtiz continuous and p − > 1, we have also\\nCombining these with (5.7) we get (we may assume that r(ε) < 1)\\n(5.8)\\nMoreover, by uniform continuity of u there is a modulus of continuity ω such that ω(t) → 0 as t → 0 and |u(y) − u(z)| ≤ ω(|y − z|) for all y, z ∈ Ω. Hence by (5.3)\\nWe now consider the situations p(x) ≤ 2 and p(x) > 2 separately. If p(x) ≤ 2, we multiply (5.8) by |η| p(x)−2 and estimate using (5.9). We get\\nwhere the last inequality is true when ε < 1 is so small that ω(r(ε)) < 1. This proves (5.2) when p(x) ≤ 2.\\nIf p(x) > 2, we estimate (5.8) directly using (5.9). We get −F (x, η, X) ≤C(p, q) 1 ε q−1 q 1 q ε q−1 q ω(r(ε)) 1 q q = C(p, q)ω(r(ε))), which proves (5.2) when p(x) > 2.\\nNext we will use the previous lemma to show that inf-convolution of a viscosity supersolution to −∆ N p(x) u ≥ 0 in Ω is a weak supersolution to −∆ S p(x) u ≥ 0 in Ω r(ε) up to some error term. Before proceeding we make some remarks about the point-wise differentiability of inf-convolution.\\nRemark 5.4. It follows from semi-concavity that the inf-convolution u ε is locally Lipschitz in Ω r(ε) (see [EG15,p267]). Therefore it belongs in W 1,∞ loc (Ω r(ε) ), is differentiable almost everywhere in Ω r(ε) , and its derivative agrees with its Sobolev derivative almost everywhere in Ω r(ε) (see [EG15,p155 and p265]).\\nBy Lemma 5.2 the function φ(x) := u ε (x) − C(q, ε, u) |x| 2 is concave in Ω r(ε) . Thus Alexandrov's theorem implies that u ε is twice differentiable almost everywhere in Ω r(ε) . Furthermore, the proof of Alexandrov's theorem in [EG15,p273] establishes that if φ j is the standard mollification of φ, then D 2 φ j → D 2 φ almost everywhere in Ω r(ε) .\\nLemma 5.5. Assume that u is a uniformly continuous viscosity supersolution to −∆ N p(x) u ≥ 0 in Ω. Let q > 2 be so large that p − − 2 + q−2 q−1 ≥ 0 and let u ε be the inf-convolution of u as defined in (5.1). Then\\nfor all non-negative ϕ ∈ W 1,p(·) (Ω r(ε) ) with compact support, where E(ε) → 0 as ε → 0 and s(x) = max(p(x) − 2, 0).\\nProof. It is enough to consider ϕ ∈ C ∞ 0 (Ω r(ε) ). This can be seen in the same way as Lemma 3.2, but since u ε ∈ W 1,∞ loc (Ω r(ε) ), the proof is even simpler. (Step 1) We show that u ε satisfies the auxiliary inequality (5.11) for all 0 < δ < 1. As mentioned in Remark 5.4, the function φ(x) := u ε (x) − C(q, ε, u) |x| 2 is concave in Ω r(ε) and we can approximate it by smooth concave functions φ j so that φ j , Dφ j , D 2 φ j → φ, Dφ, D 2 φ almost everywhere in Ω r(ε) . We define u ε,j (x) := φ j (x) + C(q, ε, u) |x| 2 and denote by p j the standard mollification of p. Since u ε,j and p j are smooth, we calculate\\nDu ε,j ϕ\\nDu ε,j · Dϕ + 1 2 log δ + |Du ε,j | 2 Dp j ϕ dx.\\n(5.10)\\nWe let j → ∞ in (5.10) and intend to use Fatou's lemma at the LHS and the Dominated convergence theorem at the RHS. This results in the auxiliary inequality\\nDu ε · Dϕ + 1 2 log δ + |Du ε | 2 Dp ϕ dx,\\nwhere D 2 u ε is the Hessian of u ε in the Alexandrov's sense. We still need to check that the assumptions of the Dominated convergence theorem and Fatou's lemma hold. By Lipschitz continuity of u ε and p there is M ≥ 1 such that\\nThis justifies our use of the Dominated convergence theorem. In order to justify our use of Fatou's lemma, we notice first that by concavity of φ j we have D 2 u ε,j ≤ C(q, ε, u)I. Thus the integrand at the LHS of (5.10) is clearly bounded from below by a constant independent of j if Du ε,j = 0. If Du ε,j = 0, we have\\nwhere the first inequality follows like estimate (5.7) since p j ≥ p − > 1. (Step 2) We let δ → 0 in the auxiliary inequality (5.11). The RHS becomes\\nby the Lebesgue's dominated convergence theorem. We intend to apply Fatou's lemma on the LHS. We have Du ε (x), D 2 u ε (x) ∈ J 2,− u ε (x) for almost every x ∈ Ω r(ε) . Therefore by Lemma 5.3 it holds that\\nand by the property (v) in Lemma 5.2 we have\\nObserve that since q > 2, the condition (5.13) implies that the Hessian D 2 u ε is negative semi-definite in the set where the gradient Du ε vanishes. Using this fact, Fatou's lemma and (5.12) we get\\n|Du ε | max(p(x)−2,0) ϕ dx, (5.14) and thus we arrive at the desired inequality. Our use of Fatou's lemma is justified since if Du ε = 0 and p(x) ≤ 2, we have by (5.13)\\nwhere the last inequality follows from p − − 2 + q−2 q−1 ≥ 0. If Du ε = 0 and p(x) > 2, we have simply\\nIn the next two lemmas we use Caccioppoli type estimates and algebraic inequalities to show that the sequence of inf-convolutions converges to the viscosity supersolution in W 1,p(·) loc (Ω).\\nLemma 5.6. Under the assumptions of Lemma 5.5, the function u belongs in W 1,p(·) loc (Ω) and for any Ω ′ ⋐ Ω we have Du ε → Du weakly in L p(·) (Ω ′ ) for some subsequence.\\nProof. Take a cut-off function ξ ∈ C ∞ 0 (Ω ′ ) such that 0 ≤ ξ ≤ 1 in Ω and ξ ≡ 1 in Ω ′ . Then assume that ε is so small that supp ξ =: K ⊂ Ω r(ε) . We define a test function ϕ := (L − u ε )ξ p + where L := sup ε,x∈Ω ′ |u ε (x)| is finite since u ε → u locally uniformly. We have Dϕ = −Du ε ξ p + + (L − u ε )p + ξ p + −1 Dξ and therefore by Lemma 5.5\\nWe estimate these integrals using Young's inequality. The first integral is estimated by the facts p(x)(p + −1)\\np(x)−1 ≥ p + and ξ ≤ 1 as follows\\nTo estimate I 2 , we also use the inequality a s |log a| ≤ a s+ 1 2 + 1 s for a > 0 and s > 0,\\nThe last integral is estimated by the two alternatives in max(p(x) − 2, 0) as follows (we may assume that |E(ε)| ≤ 1)\\nTaking small δ we conclude that Du ε is bounded in L p(·) (Ω ′ ) with respect to ε. Since L p(·) (Ω ′ ) is a reflexive Banach space [DHHR11, p76 and p89], it follows that there is a function Du ∈ L p(·) (Ω ′ ) such that Du ε → Du weakly in L p(·) (Ω ′ ) for some subsequence. Consequently u ∈ W 1,p(·) (Ω ′ ) with Du as its weak derivative.\\nLemma 5.7. Under the assumptions of Lemma 5.5, for any Ω ′ ⋐ Ω we have Du ε → Du in L p(·) (Ω ′ ) for some subsequence.\\nProof. Take a cut-off function ξ ∈ C ∞ 0 (Ω) such that ξ ≡ 1 in Ω ′ and define a test function ϕ := (u − u ε )ξ. Then assume that ε is so small that supp ξ =: K ⊂ Ω r(ε) .\\nSince ϕ ∈ W 1,p(·) (Ω r(ε) ) with compact support it follows from Lemma 5.5 that\\n(5.15) According to Lemma 5.6 we have u ε → u locally uniformly and Du ε → Du weakly in L p(·) (K) for a subsequence. Thus by passing to a subsequence we may assume that the right hand side of (5.15) converges to zero. The claim now follows from the inequalities (see e.g. [Lin17, Chapter 12])\\nfor a, b ∈ R N . Indeed, we immediately get that Ω ′ ∩{p(x)≥2} |Du − Du ε | p(x) dx → 0. To deal with the set {p(x) < 2}, we first apply the above algebraic inequality and then estimate using Hölder's inequality, the modular inequality (2.1) and the definition of the · L p(·) -norm. We get\\nwhere s ∈ p + 2 , p − 2 . The last integral is bounded since the sequence Du ε is bounded in L p(·) (Ω ′ ) by its weak convergence. The RHS therefore converges to zero by (5.15).\\nNext, we use the previous convergence result to pass to the limit in the inequality of Lemma 5.5 and conclude that viscosity supersolutions to −∆ N p(x) u ≥ 0 are weak supersolutions to −∆ S p(x) u ≥ 0.\\nProof. It is clear from the definition of weak supersolutions to −∆ S p(x) u ≥ 0 that we can without loss of generality assume that u is uniformly continuous in Ω by restricting to a smaller domain. Fix a non-negative test function ϕ ∈ C ∞ 0 (Ω) and take open Ω ′ ⋐ Ω such that supp ϕ ⊂ Ω ′ . Let q and u ε be as in Lemma 5.5 and assume that ε is so small that Ω ′ ⊂ Ω r(ε) . Then the claim follows from Lemma 5.5 if we show that\\nfor a, b ∈ R N . Indeed, when ε is so small that Ω ′ |Du ε − Du| p(x) dx < 1 we have by Hölder's inequality and the modular inequality\\nClaim (5.18) holds since Ω ′ |Du ε | p(x) dx is bounded and E(ε) → 0. To this end, fix 0 < ǫ < 1. The mapping (a, x) → |a| p(x)−2 log (|a|) a is uniformly continuous in bounded sets of R N × Ω ′ . Hence there exists δ = δ(ǫ) < ǫ such that whenever x ∈ Ω ′ and a, b ∈ B(0, 3) satisfy |a − b| < δ, it holds\\nIf |a| , |b| ≥ 1 and |a − b| < δ, then we use (5.19) to get the estimate\\nWe denote\\nThe strong convergence of Du ε to Du in L p(·) (Ω ′ ) implies that Du ε → Du in measure in Ω ′ (see [DHHR11, Lemma 3.2.10]). Thus there is ε 0 = ε 0 (δ) such that for all ε < ε 0 it holds |F ε | ≤ δ. Using the inequality a s |log a| ≤ a s+ 1 2 + 1 s for a > 0 and s > 0 we get for all ε < ε 0\\n(5.23) If x ∈ Ω ′ \\\\ F ε , then either |Du ε | , |Du| ≤ 3 or |Du ε | , |Du| ≥ 1. Hence by (5.21) and (5.22) we have\\nCombining (5.24) and (5.23) proves (5.20) since ǫ was arbitrary.\\nMerging Theorems 4.1 and 5.8 yields the following equivalence result.\\nTheorem 5.9. A function u is a viscosity solution to −∆ N p(x) u = 0 in Ω if and only if it is a weak solution to −∆ S p(x) u = 0 in Ω. Since the weak solutions to the strong p(x)-Laplace equation are locally C 1,α continuous [ZZ12], our equivalence result yields local C 1,α regularity also for viscosity solutions of the normalized p(x)-Laplace equation.\\nCorollary 5.10. If u is a viscosity solution to −∆ N p(x) u = 0 in a bounded domain Ω, then u ∈ C 1,α (Ω) with α ∈ (0, 1).\"\n },\n {\n \"title\": \"An Application: A Radó-type removability theorem\",\n \"content\": \"The classical theorem of Radó says that if a continuous complex-valued function f defined on a domain Ω ⊂ C is holomorphic in Ω \\\\{f = 0}, then it is holomorphic in the whole Ω. Similar results have been proven for solutions of partial differential equations. We prove a Radó-type removability theorem for the strong p(x)-Laplace equation. It is worth pointing out that it could be difficult to show this kind of result without appealing to viscosity solutions whereas it is straightforward to do so with the help of the equivalence result. The theorem follows by observing that weak solutions to ∆ S p(x) u = 0 coincide with viscosity solutions of an equation that satisfies the assumptions of a Radó-type removability theorem in [JL05].\\nRecall that we ignore the test functions whose gradient vanishes at the point of touching in the Definition 3.3 of viscosity solutions to −∆ N p(x) u = 0. Sometimes this kind of solutions are called feeble viscosity solutions (e.g. [JL05,Kat15b]). We will observe that these feeble viscosity solutions to −∆ N p(x) u = 0 are exactly the usual viscosity solutions to − tr(A(x, Du)D 2 u) = 0, (6.1)\\nwhere A(x, Du) := |Du| 2 I + (p(x) − 2) Du ⊗ Du. To be precise, we define the viscosity solutions to (6.1).\\nDefinition 6.1. A lower semicontinuous function u is a viscosity supersolution to (6.1) in Ω if, whenever (η, X) ∈ J 2,− u(x) with x ∈ Ω, then −tr(A(x, η)X) ≥ 0.\\nA function u is a viscosity subsolution to (6.1) if −u is a supersolution, and a viscosity solution if it is both viscosity super-and subsolution.\\nLemma 6.2. A function u is a viscosity solution to −∆ N p(x) u = 0 if and only if it is a viscosity solution to (6.1).\\nProof. It is enough to consider supersolutions. Take (η, X) ∈ J 2,− u(x) with x ∈ Ω. If η = 0, then the conditions for both definitions are satisfied, so we may assume that η = 0. Then we have F (x, η, X) ≥ 0 if and only if − |η| 2 tr(X) + (p(x) − 2) Xη, η ≥ 0, where |η| 2 tr(X) + (p(x) − 2) Xη, η = |η| 2 tr(X) + (p(x) − 2) tr(η ⊗ ηX) =tr |η| 2 I + (p(x) − 2) η ⊗ η X .\\nHence the definitions are equivalent. Theorem 6.3 (A Radó-type removability theorem). Let u ∈ C 1 (Ω) be a weak solution to −∆ S p(x) u = 0 in Ω \\\\ {u = 0}. Then u is a weak solution to −∆ S p(x) u = 0 in the whole Ω.\\nProof. By Lemma 6.2 and our equivalence result weak solutions to −∆ S p(x) u = 0 coincide with viscosity solutions to (6.1). Therefore it suffices to show that if u is a viscosity solution to (6.1) in Ω \\\\ {u = 0}, it is a viscosity solution to (6.1) in the whole Ω. This on the other hand follows from [JL05, Theorem 2.2]. The matrix A satisfies the assumptions of the theorem as it is symmetric, has continuous entries and A(x, 0, 0) = 0 for all x ∈ Ω. It is also positive semi-definite since for all ξ ∈ R N we have ξ ′ |η| 2 I + (p(x) − 2) η ⊗ η ξ ≥ξ ′ |η| 2 I − η ⊗ η ξ ≥ |ξ| 2 |η| 2 − η ⊗ η = 0.\"\n },\n {\n \"title\": \"−\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 5. 3 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"∆ N p(x) u := −∆u − p(x) − 2 |Du| 2 ∆ ∞ u := −∆u − p(x) − 2 |Du| 2 D 2 uDu, Du = 0. (1.1)","content":"−\n∆ N p(x) u := −∆u − p(x) − 2 |Du| 2 ∆ ∞ u := −∆u − p(x) − 2 |Du| 2 D 2 uDu, Du = 0. (1.1)"},{"id":"","caption":"The strong and normalized p(x)-Laplace equations In this section, we define weak solutions to the strong p(x)-Laplace equation and viscosity solutions to the normalized p(x)-Laplace equation.","content":"The strong and normalized p(x)-Laplace equations In this section, we define weak solutions to the strong p(x)-Laplace equation and viscosity solutions to the normalized p(x)-Laplace equation."},{"id":"","caption":"a viscosity supersolution, and a viscosity solution to −∆ N p(x) u = 0 if it is both viscosity super-and subsolution.","content":"a viscosity supersolution, and a viscosity solution to −∆ N p(x) u = 0 if it is both viscosity super-and subsolution."},{"id":"","caption":"Assume that u is a uniformly continuous viscosity supersolution to −∆ N p(x) u ≥ 0 in Ω. Then, whenever (η, X) ∈ J 2,− u ε (x), η = 0 and x ∈ Ω r(ε) , it holds |η| min(p(x)−2,0) F (x, η, X) ≥ E(ε), (5.2)","content":"Lemma 5. 3 .\n3Assume that u is a uniformly continuous viscosity supersolution to −∆ N p(x) u ≥ 0 in Ω. Then, whenever (η, X) ∈ J 2,− u ε (x), η = 0 and x ∈ Ω r(ε) , it holds |η| min(p(x)−2,0) F (x, η, X) ≥ E(ε), (5.2)"},{"id":"","caption":"Claim (5.17) follows if we show that lim ε→0 Ω ′ |Du ε | p(x)−2 log (|Du ε |) Du ε − |Du| p(x)−2 log (|Du|) Du dx = 0. (5.20)","content":"Claim (5.17) follows if we show that lim ε→0 Ω ′ |Du ε | p(x)−2 log (|Du ε |) Du ε − |Du| p(x)−2 log (|Du|) Du dx = 0. (5.20)"},{"id":"","caption":"By Lemma 5.7 we have that u ε → u in W 1,p(·) (Ω ′ ). Claim (5.16) follows from the inequalities (see e.g. [Lin17, Chapter 12])","content":"By Lemma 5.7 we have that u ε → u in W 1,p(·) (Ω ′ ). Claim (5.16) follows from the inequalities (see e.g. [Lin17, Chapter 12])Dp ϕ dx \n(5.17) \n\nas well as \nlim \n\nε→0 \n\nE(ε) \n\nΩ ′ \n\n|Du ε | max(p(x)−2,0) ϕ dx = 0. \n(5.18)"},{"id":"","caption":"Jarkko Siltakoski, Department of Mathematics and Statistics, P.O.Box 35, FIN-40014, University of Jyväskylä, Finland E-mail address: jarkko.j.m.siltakoski@student.jyu.fi","content":"Jarkko Siltakoski, Department of Mathematics and Statistics, P.O.Box 35, FIN-40014, University of Jyväskylä, Finland E-mail address: jarkko.j.m.siltakoski@student.jyu.fi"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"∆ N p(x) u := −∆u − p(x) − 2 |Du| 2 ∆ ∞ u := −∆u − p(x) − 2 |Du| 2 D 2 uDu, Du = 0. (1.1)\",\n \"content\": \"−\\n∆ N p(x) u := −∆u − p(x) − 2 |Du| 2 ∆ ∞ u := −∆u − p(x) − 2 |Du| 2 D 2 uDu, Du = 0. (1.1)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The strong and normalized p(x)-Laplace equations In this section, we define weak solutions to the strong p(x)-Laplace equation and viscosity solutions to the normalized p(x)-Laplace equation.\",\n \"content\": \"The strong and normalized p(x)-Laplace equations In this section, we define weak solutions to the strong p(x)-Laplace equation and viscosity solutions to the normalized p(x)-Laplace equation.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"a viscosity supersolution, and a viscosity solution to −∆ N p(x) u = 0 if it is both viscosity super-and subsolution.\",\n \"content\": \"a viscosity supersolution, and a viscosity solution to −∆ N p(x) u = 0 if it is both viscosity super-and subsolution.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Assume that u is a uniformly continuous viscosity supersolution to −∆ N p(x) u ≥ 0 in Ω. Then, whenever (η, X) ∈ J 2,− u ε (x), η = 0 and x ∈ Ω r(ε) , it holds |η| min(p(x)−2,0) F (x, η, X) ≥ E(ε), (5.2)\",\n \"content\": \"Lemma 5. 3 .\\n3Assume that u is a uniformly continuous viscosity supersolution to −∆ N p(x) u ≥ 0 in Ω. Then, whenever (η, X) ∈ J 2,− u ε (x), η = 0 and x ∈ Ω r(ε) , it holds |η| min(p(x)−2,0) F (x, η, X) ≥ E(ε), (5.2)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Claim (5.17) follows if we show that lim ε→0 Ω ′ |Du ε | p(x)−2 log (|Du ε |) Du ε − |Du| p(x)−2 log (|Du|) Du dx = 0. (5.20)\",\n \"content\": \"Claim (5.17) follows if we show that lim ε→0 Ω ′ |Du ε | p(x)−2 log (|Du ε |) Du ε − |Du| p(x)−2 log (|Du|) Du dx = 0. (5.20)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"By Lemma 5.7 we have that u ε → u in W 1,p(·) (Ω ′ ). Claim (5.16) follows from the inequalities (see e.g. [Lin17, Chapter 12])\",\n \"content\": \"By Lemma 5.7 we have that u ε → u in W 1,p(·) (Ω ′ ). Claim (5.16) follows from the inequalities (see e.g. [Lin17, Chapter 12])Dp ϕ dx \\n(5.17) \\n\\nas well as \\nlim \\n\\nε→0 \\n\\nE(ε) \\n\\nΩ ′ \\n\\n|Du ε | max(p(x)−2,0) ϕ dx = 0. \\n(5.18)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Jarkko Siltakoski, Department of Mathematics and Statistics, P.O.Box 35, FIN-40014, University of Jyväskylä, Finland E-mail address: jarkko.j.m.siltakoski@student.jyu.fi\",\n \"content\": \"Jarkko Siltakoski, Department of Mathematics and Statistics, P.O.Box 35, FIN-40014, University of Jyväskylä, Finland E-mail address: jarkko.j.m.siltakoski@student.jyu.fi\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Mappings of finite distortion and PDE with nonstandard growth. T Adamowicz, P Hästö, Int. Math. Res. Not. IMRN. T. Adamowicz and P. Hästö. Mappings of finite distortion and PDE with nonstandard growth. Int. Math. Res. Not. IMRN, pages 1940-1965, 2010.","doi":""},{"text":"Harnack's inequality and the strong p(·)-Laplacian. T Adamowicz, P Hästö, J. Differential Equations. 250T. Adamowicz and P. Hästö. Harnack's inequality and the strong p(·)-Laplacian. J. Differential Equations, 250:1631-1649, 2011.","doi":""},{"text":"Tug-of-war games with varying probabilities and the normalized p(x)-Laplacian. Á Arroyo, J Heino, M Parviainen, Commun. Pure Appl. Anal. 163Á. Arroyo, J. Heino, and M. Parviainen. Tug-of-war games with varying probabilities and the normalized p(x)-Laplacian. Commun. Pure Appl. Anal., 16(3):915-944, 2017.","doi":""},{"text":"C 1,α regularity for the normalized p-Poisson problem. A Attouchi, M Parviainen, E Ruosteenoja, J. Math. Pures Appl. 1084A. Attouchi, M. Parviainen, and E. Ruosteenoja. C 1,α regularity for the normalized p-Poisson problem. J. Math. Pures Appl., 108(4):553-591, 2017.","doi":""},{"text":"Modica type gradient estimates for an inhomogeneus variant of the normalized p-Laplacian evolution. A Banerjee, N Garofalo, Nonlinear Anal. 121A. Banerjee and N. Garofalo. Modica type gradient estimates for an inhomogeneus variant of the normalized p-Laplacian evolution. Nonlinear Anal., 121:458-468, 2015.","doi":""},{"text":"User's guide to viscosity solutions of second order partial differential equations. M G Crandall, H Ishii, P.-L Lions, Bull. Amer. Math. Soc. 271M. G. Crandall, H. Ishii, and P.-L. Lions. User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc., 27(1):1-67, 1992.","doi":""},{"text":"Lebesgue and Sobolev Spaces with Variable Exponents. L Diening, P Harjulehto, P Hästö, M Růžička, Lecture Notes in Mathematics. 2017Springer-VerlagL. Diening, P. Harjulehto, P. Hästö, and M. Růžička. Lebesgue and Sobolev Spaces with Variable Exponents, volume 2017 of Lecture Notes in Mathematics. Springer- Verlag, 2011.","doi":""},{"text":"Measure Theory and Fine Properties of Functions. L C Evans, R F Gariepy, CRC Pressrevised editionL. C. Evans and R. F. Gariepy. Measure Theory and Fine Properties of Functions. CRC Press, revised edition, 2015.","doi":""},{"text":"Hölder gradient estimates for a class of singular or degenerate parabolic equations. C Imbert, T Jin, L Silvestre, To appear in Adv. Nonlinear AnalC. Imbert, T. Jin, and L. Silvestre. Hölder gradient estimates for a class of singular or degenerate parabolic equations. To appear in Adv. Nonlinear Anal.","doi":""},{"text":"On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions. H Ishii, Funkcialaj Ekvacioj. 38H. Ishii. On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions. Funkcialaj Ekvacioj, 38:101-120, 1995.","doi":""},{"text":"A new proof for the equivalence of weak and viscosity solutions for the p-Laplace equation. V Julin, P Juutinen, Comm. Partial Differential Equations. 375V. Julin and P. Juutinen. A new proof for the equivalence of weak and viscosity so- lutions for the p-Laplace equation. Comm. Partial Differential Equations, 37(5):934- 946, 2012.","doi":""},{"text":"Removability of a level set for solutions of quasilinear equations. P Juutinen, P Lindqvist, Comm. Partial Differential Equations. 30P. Juutinen and P. Lindqvist. Removability of a level set for solutions of quasilinear equations. Comm. Partial Differential Equations, 30:305-321, 2005.","doi":""},{"text":"On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation. P Juutinen, P Lindqvist, J J Manfredi, SIAM J. Math. Anal. 333P. Juutinen, P. Lindqvist, and J.J. Manfredi. On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation. SIAM J. Math. Anal., 33(3):699-717, 2001.","doi":""},{"text":"Equivalence of viscosity and weak solutions for the p(x)-Laplacian. P Juutinen, T Lukkari, M Parviainen, Ann. Inst. H. Poincaré Anal. Non Linéaire. 276P. Juutinen, T. Lukkari, and M. Parviainen. Equivalence of viscosity and weak solu- tions for the p(x)-Laplacian. Ann. Inst. H. Poincaré Anal. Non Linéaire, 27(6):1471- 1487, 2010.","doi":""},{"text":"Hölder gradient estimates for parabolic homogeneous p-Laplacian equations. T Jin, L Silvestre, J. Math. Pures Appl. 1083T. Jin and L. Silvestre. Hölder gradient estimates for parabolic homogeneous p- Laplacian equations. J. Math. Pures Appl., 108(3):63-87, 2017.","doi":""},{"text":"An Introduction To Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L ∞. N Katzourakis, SpringerN. Katzourakis. An Introduction To Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L ∞ . Springer, 2015.","doi":""},{"text":"Nonsmooth convex functionals and feeble viscosity solutions of singular Euler-Lagrange equations. N Katzourakis, Calc. Var. 541N. Katzourakis. Nonsmooth convex functionals and feeble viscosity solutions of sin- gular Euler-Lagrange equations. Calc. Var., 54(1):275-298, 2015.","doi":""},{"text":"A Beginner's Guide to the Theory of Viscosity Solutions. Shigeaki Koike, 2nd editionShigeaki Koike. A Beginner's Guide to the Theory of Viscosity Solutions. 2nd edition, 2012.","doi":""},{"text":"P Lindqvist, Notes on the p-Laplace equation. 161Report / University of Jyväskylä, Department of Mathematics and Statistics. University of Jyväskyläsecond editionP. Lindqvist. Notes on the p-Laplace equation (second edition), volume 161 of Report / University of Jyväskylä, Department of Mathematics and Statistics. University of Jyväskylä, 2017.","doi":""},{"text":"On viscosity and weak solutions for non-homogeneous p-Laplace equations. M Medina, P Ochoa, To appear in Adv. Nonlinear AnalM. Medina and P. Ochoa. On viscosity and weak solutions for non-homogeneous p-Laplace equations. To appear in Adv. Nonlinear Anal.","doi":""},{"text":"A homogenization process for the strong p(x)-Laplacian. M Pérez-Llanos, Nonlinear Anal. 76M. Pérez-Llanos. A homogenization process for the strong p(x)-Laplacian. Nonlinear Anal., 76:105-114, 2013.","doi":""},{"text":"Tug-of-war and the infinity Laplacian. Y Peres, O Schramm, S Sheffield, D B Wilson, J. Amer. Math. Soc. 221Y. Peres, O. Schramm, S. Sheffield, and D.B. Wilson. Tug-of-war and the infinity Laplacian. J. Amer. Math. Soc., 22(1):167-210, 2009.","doi":""},{"text":"Hölder regularity for the gradients of solutions of the strong p(x)-Laplacian. C Zhang, S Zhou, J. Math. Anal. Appl. 3892C. Zhang and S. Zhou. Hölder regularity for the gradients of solutions of the strong p(x)-Laplacian. J. Math. Anal. Appl., 389(2):1066-1077, 2012.","doi":""}],"string":"[\n {\n \"text\": \"Mappings of finite distortion and PDE with nonstandard growth. T Adamowicz, P Hästö, Int. Math. Res. Not. IMRN. T. Adamowicz and P. Hästö. Mappings of finite distortion and PDE with nonstandard growth. Int. Math. Res. Not. IMRN, pages 1940-1965, 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Harnack's inequality and the strong p(·)-Laplacian. T Adamowicz, P Hästö, J. Differential Equations. 250T. Adamowicz and P. Hästö. Harnack's inequality and the strong p(·)-Laplacian. J. Differential Equations, 250:1631-1649, 2011.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Tug-of-war games with varying probabilities and the normalized p(x)-Laplacian. Á Arroyo, J Heino, M Parviainen, Commun. Pure Appl. Anal. 163Á. Arroyo, J. Heino, and M. Parviainen. Tug-of-war games with varying probabilities and the normalized p(x)-Laplacian. Commun. Pure Appl. Anal., 16(3):915-944, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"C 1,α regularity for the normalized p-Poisson problem. A Attouchi, M Parviainen, E Ruosteenoja, J. Math. Pures Appl. 1084A. Attouchi, M. Parviainen, and E. Ruosteenoja. C 1,α regularity for the normalized p-Poisson problem. J. Math. Pures Appl., 108(4):553-591, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Modica type gradient estimates for an inhomogeneus variant of the normalized p-Laplacian evolution. A Banerjee, N Garofalo, Nonlinear Anal. 121A. Banerjee and N. Garofalo. Modica type gradient estimates for an inhomogeneus variant of the normalized p-Laplacian evolution. Nonlinear Anal., 121:458-468, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"User's guide to viscosity solutions of second order partial differential equations. M G Crandall, H Ishii, P.-L Lions, Bull. Amer. Math. Soc. 271M. G. Crandall, H. Ishii, and P.-L. Lions. User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc., 27(1):1-67, 1992.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Lebesgue and Sobolev Spaces with Variable Exponents. L Diening, P Harjulehto, P Hästö, M Růžička, Lecture Notes in Mathematics. 2017Springer-VerlagL. Diening, P. Harjulehto, P. Hästö, and M. Růžička. Lebesgue and Sobolev Spaces with Variable Exponents, volume 2017 of Lecture Notes in Mathematics. Springer- Verlag, 2011.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Measure Theory and Fine Properties of Functions. L C Evans, R F Gariepy, CRC Pressrevised editionL. C. Evans and R. F. Gariepy. Measure Theory and Fine Properties of Functions. CRC Press, revised edition, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hölder gradient estimates for a class of singular or degenerate parabolic equations. C Imbert, T Jin, L Silvestre, To appear in Adv. Nonlinear AnalC. Imbert, T. Jin, and L. Silvestre. Hölder gradient estimates for a class of singular or degenerate parabolic equations. To appear in Adv. Nonlinear Anal.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions. H Ishii, Funkcialaj Ekvacioj. 38H. Ishii. On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions. Funkcialaj Ekvacioj, 38:101-120, 1995.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A new proof for the equivalence of weak and viscosity solutions for the p-Laplace equation. V Julin, P Juutinen, Comm. Partial Differential Equations. 375V. Julin and P. Juutinen. A new proof for the equivalence of weak and viscosity so- lutions for the p-Laplace equation. Comm. Partial Differential Equations, 37(5):934- 946, 2012.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Removability of a level set for solutions of quasilinear equations. P Juutinen, P Lindqvist, Comm. Partial Differential Equations. 30P. Juutinen and P. Lindqvist. Removability of a level set for solutions of quasilinear equations. Comm. Partial Differential Equations, 30:305-321, 2005.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation. P Juutinen, P Lindqvist, J J Manfredi, SIAM J. Math. Anal. 333P. Juutinen, P. Lindqvist, and J.J. Manfredi. On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation. SIAM J. Math. Anal., 33(3):699-717, 2001.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Equivalence of viscosity and weak solutions for the p(x)-Laplacian. P Juutinen, T Lukkari, M Parviainen, Ann. Inst. H. Poincaré Anal. Non Linéaire. 276P. Juutinen, T. Lukkari, and M. Parviainen. Equivalence of viscosity and weak solu- tions for the p(x)-Laplacian. Ann. Inst. H. Poincaré Anal. Non Linéaire, 27(6):1471- 1487, 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hölder gradient estimates for parabolic homogeneous p-Laplacian equations. T Jin, L Silvestre, J. Math. Pures Appl. 1083T. Jin and L. Silvestre. Hölder gradient estimates for parabolic homogeneous p- Laplacian equations. J. Math. Pures Appl., 108(3):63-87, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An Introduction To Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L ∞. N Katzourakis, SpringerN. Katzourakis. An Introduction To Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L ∞ . Springer, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nonsmooth convex functionals and feeble viscosity solutions of singular Euler-Lagrange equations. N Katzourakis, Calc. Var. 541N. Katzourakis. Nonsmooth convex functionals and feeble viscosity solutions of sin- gular Euler-Lagrange equations. Calc. Var., 54(1):275-298, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Beginner's Guide to the Theory of Viscosity Solutions. Shigeaki Koike, 2nd editionShigeaki Koike. A Beginner's Guide to the Theory of Viscosity Solutions. 2nd edition, 2012.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"P Lindqvist, Notes on the p-Laplace equation. 161Report / University of Jyväskylä, Department of Mathematics and Statistics. University of Jyväskyläsecond editionP. Lindqvist. Notes on the p-Laplace equation (second edition), volume 161 of Report / University of Jyväskylä, Department of Mathematics and Statistics. University of Jyväskylä, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On viscosity and weak solutions for non-homogeneous p-Laplace equations. M Medina, P Ochoa, To appear in Adv. Nonlinear AnalM. Medina and P. Ochoa. On viscosity and weak solutions for non-homogeneous p-Laplace equations. To appear in Adv. Nonlinear Anal.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A homogenization process for the strong p(x)-Laplacian. M Pérez-Llanos, Nonlinear Anal. 76M. Pérez-Llanos. A homogenization process for the strong p(x)-Laplacian. Nonlinear Anal., 76:105-114, 2013.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Tug-of-war and the infinity Laplacian. Y Peres, O Schramm, S Sheffield, D B Wilson, J. Amer. Math. Soc. 221Y. Peres, O. Schramm, S. Sheffield, and D.B. Wilson. Tug-of-war and the infinity Laplacian. J. Amer. Math. Soc., 22(1):167-210, 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hölder regularity for the gradients of solutions of the strong p(x)-Laplacian. C Zhang, S Zhou, J. Math. Anal. Appl. 3892C. Zhang and S. Zhou. Hölder regularity for the gradients of solutions of the strong p(x)-Laplacian. J. Math. Anal. Appl., 389(2):1066-1077, 2012.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":480,"cells":{"text":{"kind":"string","value":"\nPDD Graph: Bridging Electronic Medical Records and Biomedical Knowledge Graphs via Entity Linking\n\n\nMeng Wang wangmengsd@stu.xjtu.edu.cn \nMOEKLINNS lab\nXi'an Jiaotong University\nXi'anChina\n\nJiaheng Zhang \nMOEKLINNS lab\nXi'an Jiaotong University\nXi'anChina\n\nJun Liu \nMOEKLINNS lab\nXi'an Jiaotong University\nXi'anChina\n\nWei Hu \nState Key Laboratory for Novel Software Technology\nNanjing University\nNanjingChina\n\nSen Wang \nGriffith Universtiy\nGold Coast Campus\nAustralia\n\nXue Li \nThe Universtiy of Queensland\nBrisbaneAustralia\n\nWenqiang Liu \nMOEKLINNS lab\nXi'an Jiaotong University\nXi'anChina\n\nPDD Graph: Bridging Electronic Medical Records and Biomedical Knowledge Graphs via Entity Linking\nLinked Data · MIMIC-III · EMR · Drug · Disease\nElectronic medical records contain multi-format electronic medical data that consist of an abundance of medical knowledge. Facing with patients symptoms, experienced caregivers make right medical decisions based on their professional knowledge that accurately grasps relationships between symptoms, diagnosis and corresponding treatments. In this paper, we aim to capture these relationships by constructing a large and high-quality heterogenous graph linking patients, diseases, and drugs (PDD) in EMRs. Specifically, we propose a novel framework to extract important medical entities from MIMIC-III (Medical Information Mart for Intensive Care III) and automatically link them with the existing biomedical knowledge graphs, including ICD-9 ontology and DrugBank. The PDD graph presented in this paper is accessible on the Web via the SPARQL endpoint, and provides a pathway for medical discovery and applications, such as effective treatment recommendations.\n\nIntroduction\n\nBig data vendors collect and store large number of electronic medical records (EMRs) in hospital, with the goal of instantly accessing to comprehensive medical patient histories for caregivers at a lower cost. Public availability of EMRs collections has attracted much attention for different research purposes, including clinical research [14], mortality risk prediction [7], disease diagnosis [15], etc. An EMR database is normally a rich source of multi-format electronic data but remains limitations in scope and content. For example, MIMIC-III (Medical Information Mart for Intensive Care III) [8] collected bedside monitor trends, electronic medical notes, laboratory test results and waveforms from the ICUs (Intensive Care Units) of Beth Israel Deaconess Medical Center between 2001 and 2012. Abundant medical entities (symptoms, drugs and diseases) can be extracted from EMRs (clinical notes, prescriptions, and disease diagnoses). Most of the existing studies only focus on a specific entity, ignoring the relationship between entities. Given clinical data in MIMIC-III, discovering relationship between extracted entities (e.g. sepsis symptoms, pneumonia diagnosis, glucocorticoid drug and aspirin medicine) in wider scope can empower caregivers to make better decisions. Obviously, only focusing on EMR data is far from adequate to fully unveil entity relationships due to the limited scope of EMRs.\n\nMeanwhile, many biomedical knowledge graphs (KGs) are published as Linked Data [1] on the Web using the Resource Description Framework (RDF) [4], such as DrugBank [9] and ICD-9 ontology [13]. Linked Data is about using the Web to set RDF links between entities in different KGs, thereby forming a large heterogeneous graph 1 , where the nodes are entities (drugs, diseases, protein targets, side effects, pathways, etc.), and the edges (or links) represent various relations between entities such as drug-drug interactions. Unfortunately, such biomedical KGs only cover the basic medical facts, and contain little information about clinical outcomes. For instance, there is a relationship \"adverse interaction\" between glucocorticoid and aspirin in DrugBank, but no further information about how the adverse interaction affect the treatment of the patient who took both of the drugs in the same period. Clinical data can practically offer an opportunity to provide the missing relationship between KGs and clinical outcomes.\n\nAs mentioned above, biomedical KGs focus on the medical facts, whereas MIMIC-III only provides clinical data and physiological waveforms. There exists a gap between clinical data and biomedical KGs prohibiting further exploring medical entity relationship on ether side (see Fig.1). To solve this problem, we proposed a novel framework to construct a patient-drug-disease graph dataset (called PDD) in this paper. We summarize contributions of this paper as follows:\n\n-To our best knowledge, we are the first to bridge EMRs and biomedical KGs together. The result is a big and high-quality PDD graph dataset, which provides a salient opportunity to uncover associations of biomedical interest in wider scope. -We propose a novel framework to construct the PDD graph. The process starts by extracting medical entities from prescriptions, clinical notes and diagnoses respectively. RDF links are then set between the extracted medical entities and the corresponding entities in DrugBank and ICD-9 ontology. -We publish the PDD graph as an open resource 2 , and provide a SPARQL query endpoint using Apache Jena Fuseki 3 . Researchers can retrieve data distributed over biomedical KGs and MIMIC-III, ranging from drug-drug interactions, to the outcomes of drugs in clinical trials.\n\nIt is necessary to mention that MIMIC-III contains clinical information of patients. Although the protected health information was de-identified, researchers who seek to use more clinical data should complete an on-line training course and then apply for the permission to download the complete MIMIC-III dataset 4 .\n\nThe rest of this paper is organized as follows. Section 2 describes the proposed framework and details. The statistics and evaluation is reported in Section 3. Section 4 describes related work and finally, Section 5 concludes the paper and identifies topics for further work.\n\n\nPDD Construction\n\nWe first follow the RDF model [4] and introduce the PDD definition.\n\nPDD Definition: PDD is an RDF graph consisting of PDD facts, where a PDD fact is represented by an RDF triple to indicate that a patient takes a drug or a patient is diagnosed with a disease. For instance, pdd :274671, pdd :diagnosed, sepsis 5 . Fig.2 illustrates the general process of the PDD dataset generation, mainly includes two steps: PDD facts generation (described in Section 2.1), and linking PDD to biomedical KGs (described in Section 2.2).\n\n\nPDD Facts Generation\n\nAccording to the PDD definition, we need to extract three types of entities from MIMIC-III (patients, drugs, and diseases), and generate RDF triples of the prescription/diagnosis facts.\n\nPatients IRI Creation: MIMIC-III contains 46,520 distinct patients, and each patient is attached with an unique ID. We add IRI prefix to each patient ID to form a patient entity in PDD.\n\nPrescription Triple Generation: In MIMIC-III, the prescriptions table contains all the prescribed drugs for the treatments of patients. Each prescription record contains the patient's unique ID, the drug's name, the duration, and the dosage. We extracted all distinct drug names as the drug entities in PDD. Then we added a prescription triple in to PDD. An example is pdd :18740, pdd :prescribed, aspirin , where pdd :18740 is a patient entity, and aspirin is the drug's name. Diagnosis Triple Generation: MIMIC-III provides a diagnosed table that contains ICD-9 diagnosis codes for patients. There is an average of 13.9 ICD-9 codes per patient, but with a highly skewed distribution, as shown in Fig. 3. Beyond that, each patient has a set of clinical notes. These notes contain the diagnosis information. We use the named entity recognition (NER) tool C-TAKES [12] to extract diseases from clinical notes. C-TAKES is the most commonly used NER tool in the clinical domain. Then we use the model [15] (our previous work) to assign ICD-9 codes for extracted diseases. We extracted all ICD-9 diagnosis codes as the disease entities in PDD. Then we added a diagnosis triple into PDD. An example is pdd :18740, pdd :diagnosed, icd99592 , where pdd :18740 is a patient entity, and icd99592 is the ICD-9 code of sepsis.\n\n\nLinking PDD to Biomedical Knowledge Graphs\n\nAfter extracting entities, we need to tackle the task of finding sameAs links [5] between the entities in PDD and other biomedical KGs. For drugs, we focused on linking drugs of PDD to the DrugBank of Bio2RDF [6] version, as the project Bio2RDF provides a gateway to other biomedical KGs. Following the analogous reason, we interlinked diseases of PDD with the ICD-9 ontology in Bio2RDF.\n\nDrug Entity Linking: In MIMIC-III, drug names are various and often contain some insignificant words (10%, 200mg, glass bottle, etc.), which challenges the drug entity linking if the label matching method is directly used. In order to overcome this problem, we proposed an entity name model (ENM) based on [2] to link MIMIC-III drugs to DrugBank. The ENM is a statistical translation model which can capture the variations of an drug's name. Given a drugs name m in MIMIC-III, the ENM model assumes that it is a translation of the drugs name d in DrugBank, and each word of the drug name could be translated through three ways: 1) Retained (translated into itself); 2) Omitted (translated into the word NULL); 3) Converted (translated into its alias). Fig. 4 shows how the drug name Glucose in DrugBank translated into Dextrose 5% in MIMIC-III.\n\nBased on the above three ways of translations, we define the probability of drug name d being translated to m as follows:\nP (m|d) = ε (1 d + 1) lm lm j=1 l d i=0 t(m i |d j )(1)\nwhere ε is a normalization factor, l m is the length of m, l d is the length of d, m i is the i th word of m, d j is the j th word of d, and t(m i |d j ) is the lexical translation probability which indicates the probability of a word d j in DrugBank being written as m i in MIMIC-III. DrugBank contains a large amount of drug aliases information, which can be used as training sets to compute the translation probability t(m i |d j ). After training the ENM from sample data, a drug name in MIMIC-III will be more likely to be translated to itself or aliases in DrugBank, whereas the insignificant words tend to be translated to NULL. Hence, our ENM can reduce the effects of insignificant words for drugs entity linking.\n\nIn addition, we propose two constraint rules when selecting candidate drugs for m, and discard those at odds with the rules.\n\nRule 1: One of the drug indications in DrugBank must be in accordance with one of the diagnoses of the patients who took the corresponding drug in MIMIC-III at least .\n\n\nRule 2:\n\nThe dosage of a drug that patients took in MIMIC-III must be in accordance with one of the standard dosages listed in DrugBank.\n\nFinally, we will choose the drug name d in DrugBank for the given drug m in MIMIC-III with maximal P (m|d), and d satisfies the two constraint rules.\n\nDisease IRI Resolution: In our previous work [15], we have assigned ICD-9 disease codes for extracted disease entities. Since the ICD-9 code is the international standard classification of diseases, and each code is unique. We can directly link the ICD-9 codes of PDD to ICD-9 ontology by string matching.\n\n\nStatistics and Evaluation\n\nIn this section, we report the statistics of PDD and make the evaluation on its accuracy . At present PDD includes 58,030 entities and 2.3 million RDF triples. Table 1 shows the result of entities linked to the DrugBank and ICD-9 ontology. For drugs in PDD, 3,449 drugs are linked to 972 distinct drugs in DrugBank. For diseases in PDD, 6,983 diseases are connected to ICD-9 ontology. The only two failures of matching ICD-9 codes in MIMIC-III are '71970' and 'NULL', which are not included in ICD-9 ontology. Table 2 shows the result of RDF triples in PDD. In particular, 1,259,702 RDF triples contain drugs that have sameAs links to DrugBank, and 650,939 RDF triples have ICD-9 diseases codes. It indicates 83.4% drug-taken records in MIMIC-III can find corresponding entity in DrugBank, and 99.9% diagnosed information can link to ICD-9 ontology. A subgraph of PDD is illustrated in Fig. 5 to better understand the PDD graph. To evaluate the ENM model, 500 samples are randomly selected, manually verified and adjusted. The ratio of positive samples to negative samples is 4:1, where positive means the entity can be linked to DrugBank. The precision is 94% and the recall is 85%. For linked entities in PDD we randomly chose 200 of them and manually evaluated the correctness of them, and the precision of entity links is 93% which is in an accordance with the result of our examples.\n\nThe overall accuracy of entity linking will be affected by the performance of the entity recognition tool. No entity recognition tools so far can achieve 100% accuracy. The average accuracy of C-TAKES (we used in this paper) is 94%. Therefore, the overall precision and recall may be lower.\n\nIn order to find out why those 1,076 drugs have not been linked to DrugBank yet, we extract 100 of them that hold the highest usage frequency. The observation shows that most of them are not just contained in DrugBank. For instance, DrugBank does not consider NS (normal saline) as a drug, but PDD contains several expressions of NS (NS, 1/2 NS, NS (Mini Bag Plus), NS (Glass Bottle), etc.). For drugs wrongly linked to DrugBank, the names of those drugs are too short, e.g. N i.e nitrogen. These short names provide little information and affect the performance of ENM directly. Also, the training data from DrugBank does not include the usage frequency of each drug name. That might lead to some inconsistence with applications in MIMIC-III and cause linking errors.\n\n\nRelated Work\n\nIn order to bring the advantages of Semantic Web to the life science community, a number of biomedical KGs have been constructed over the last years, such as Bio2RDF [6] and Chem2Bio2RDF [3]. These datasets make the interconnection and exploration of different biomedical data sources possible. However, there is little patients clinical information within these biomedical KGs. STRIDE2RDF [10] and MCLSS2RDF [11] apply Linked Data Principles to represent patients electronic health records, but the interlinks from clinical data to existing biomedical KGs are still very limited. Hence, none of the existing linked datasets are bridging the gap between clinical and biomedical data.\n\n\nConclusion and Future Work\n\nThis paper presents the process to construct a high-quality patient-drug-disease (PDD) graph linking entities in MIMIC-III to Linked Data Cloud, which satisfies the demand to provide information of clinical outcomes in biomedical KGs, when previous no relationship exists between the medical entities in MIMIC-III. With abundant clinical data of over forty thousand patients linked to open datasets, our work provides more convenient data access for further researches based on clinical outcomes, such as personalized medication and disease correlation analysis. The PDD dataset is currently accessible on the Web via the SPARQL endpoint. In future work, our plan is to improve the linking accuracy of ENM model by feeding more data into its training system.\n\nFig. 1 .\n1Left part is the Linked Data Cloud 1 , which contains interlinked biomedical knowledge graphs. Right part is the MIMIC-III database.\n\nFig. 2 .\n2Overview of PDD bridging MIMIC-III and biomedical knowledge graphs.\n\nFig. 3 .\n3The distribution of assigned ICD-9 codes per patient.\n\nFig. 4 .\n4The translation from Glucose to Dextrose 5%.\n\nFig. 5 .\n5An annotated subgraph of PDD.\n\nTable 1 .\n1Statistics of Entities#Overall \n#Drug/disease \nlinked to KG \nPatient 46,520 \nDrug \n4,525 \n3,449 \nDisease 6,985 \n6,983 \n\n\n\nTable 2 .\n2Statistics of RDF triples#Overall \n#Drug/disease \nlinked to KG \nDemographics \n165,526 \nPatients-Drugs \n1,517,702 \n1,259,702 \nPatients-Diseases 650,987 \n650,939 \n\n\nLinking Open Data cloud diagram 2017. http://lod-cloud.net/\nhttp://kmap.xjtudlc.com/pdd 3 https://jena.apache.org/documentation/fuseki2/index.html 4 https://mimic.physionet.org/ 5 pdd is the IRI prefix http://kmap.xjtudlc.com/pdd_data/\nAcknowledgment\nLinked data-the story so far. Semantic services, interoperability and web applications: emerging concepts pp. C Bizer, T Heath, T Berners-Lee, Bizer, C., Heath, T., Berners-Lee, T.: Linked data-the story so far. Semantic ser- vices, interoperability and web applications: emerging concepts pp. 205-227 (2009)\n\nThe mathematics of statistical machine translation: Parameter estimation. P F Brown, V J D Pietra, S A D Pietra, R L Mercer, Computational linguistics. 192Brown, P.F., Pietra, V.J.D., Pietra, S.A.D., Mercer, R.L.: The mathematics of statistical machine translation: Parameter estimation. Computational linguistics 19(2), 263-311 (1993)\n\nChem2bio2rdf: a semantic framework for linking and data mining chemogenomic and systems chemical biology data. B Chen, X Dong, D Jiao, H Wang, Q Zhu, Y Ding, D J Wild, BMC bioinformatics. 111255Chen, B., Dong, X., Jiao, D., Wang, H., Zhu, Q., Ding, Y., Wild, D.J.: Chem2bio2rdf: a semantic framework for linking and data mining chemogenomic and systems chemical biology data. BMC bioinformatics 11(1), 255 (2010)\n\nW W W Consortium, Rdf 1.1 concepts and abstract syntax. Consortium, W.W.W., et al.: Rdf 1.1 concepts and abstract syntax (2014)\n\nSameas networks and beyond: analyzing deployment status and implications of owl: sameas in linked data. The Semantic Web-ISWC. L Ding, J Shinavier, Z Shangguan, D Mcguinness, Ding, L., Shinavier, J., Shangguan, Z., McGuinness, D.: Sameas networks and beyond: analyzing deployment status and implications of owl: sameas in linked data. The Semantic Web-ISWC 2010 pp. 145-160 (2010)\n\nBio2rdf release 3: a larger connected network of linked data for the life sciences. M Dumontier, A Callahan, J Cruz-Toledo, P Ansell, V Emonet, F Belleau, A Droit, CEUR-WS. orgProceedings of the 2014 International Conference on Posters & Demonstrations Track. the 2014 International Conference on Posters & Demonstrations Track1272Dumontier, M., Callahan, A., Cruz-Toledo, J., Ansell, P., Emonet, V., Belleau, F., Droit, A.: Bio2rdf release 3: a larger connected network of linked data for the life sciences. In: Proceedings of the 2014 International Conference on Posters & Demonstrations Track-Volume 1272. pp. 401-404. CEUR-WS. org (2014)\n\nUnfolding physiological state: Mortality modelling in intensive care units. M Ghassemi, T Naumann, F Doshi-Velez, N Brimmer, R Joshi, A Rumshisky, P Szolovits, Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining. the 20th ACM SIGKDD international conference on Knowledge discovery and data miningACMGhassemi, M., Naumann, T., Doshi-Velez, F., Brimmer, N., Joshi, R., Rumshisky, A., Szolovits, P.: Unfolding physiological state: Mortality modelling in intensive care units. In: Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining. pp. 75-84. ACM (2014)\n\nMimic-iii, a freely accessible critical care database. A E Johnson, T J Pollard, L Shen, L W H Lehman, M Feng, M Ghassemi, B Moody, P Szolovits, L A Celi, R G Mark, 3Johnson, A.E., Pollard, T.J., Shen, L., Lehman, L.w.H., Feng, M., Ghassemi, M., Moody, B., Szolovits, P., Celi, L.A., Mark, R.G.: Mimic-iii, a freely accessible critical care database. Scientific data 3 (2016)\n\nV Law, C Knox, Y Djoumbou, T Jewison, A C Guo, Y Liu, A Maciejewski, D Arndt, M Wilson, V Neveu, Drugbank 4.0: shedding new light on drug metabolism. 42Law, V., Knox, C., Djoumbou, Y., Jewison, T., Guo, A.C., Liu, Y., Maciejewski, A., Arndt, D., Wilson, M., Neveu, V., et al.: Drugbank 4.0: shedding new light on drug metabolism. Nucleic acids research 42(D1), D1091-D1097 (2014)\n\nMining electronic health records using linked data. D J Odgers, M Dumontier, AMIA Summits on Translational Science Proceedings. 217Odgers, D.J., Dumontier, M.: Mining electronic health records using linked data. AMIA Summits on Translational Science Proceedings 2015, 217 (2015)\n\nApplying linked data principles to represent patient's electronic health records at mayo clinic: a case report. J Pathak, R C Kiefer, C G Chute, Proceedings of the 2nd ACM SIGHIT International Health Informatics Symposium. the 2nd ACM SIGHIT International Health Informatics SymposiumACMPathak, J., Kiefer, R.C., Chute, C.G.: Applying linked data principles to represent patient's electronic health records at mayo clinic: a case report. In: Proceedings of the 2nd ACM SIGHIT International Health Informatics Symposium. pp. 455-464. ACM (2012)\n\nMayo clinical text analysis and knowledge extraction system (ctakes): architecture, component evaluation and applications. G K Savova, J J Masanz, P V Ogren, J Zheng, S Sohn, K C Kipper-Schuler, C G Chute, Journal of the American Medical Informatics Association. 175Savova, G.K., Masanz, J.J., Ogren, P.V., Zheng, J., Sohn, S., Kipper-Schuler, K.C., Chute, C.G.: Mayo clinical text analysis and knowledge extraction system (ctakes): architecture, component evaluation and applications. Journal of the American Medical Informatics Association 17(5), 507-513 (2010)\n\nDisease ontology: a backbone for disease semantic integration. L M Schriml, C Arze, S Nadendla, Y W W Chang, M Mazaitis, V Felix, G Feng, W A Kibbe, Nucleic acids research. 40D1Schriml, L.M., Arze, C., Nadendla, S., Chang, Y.W.W., Mazaitis, M., Felix, V., Feng, G., Kibbe, W.A.: Disease ontology: a backbone for disease semantic integra- tion. Nucleic acids research 40(D1), D940-D946 (2012)\n\nA cost-benefit analysis of electronic medical records in primary care. S J Wang, B Middleton, L A Prosser, C G Bardon, C D Spurr, P J Carchidi, A F Kittler, R C Goldszer, D G Fairchild, A J Sussman, The American journal of medicine. 1145Wang, S.J., Middleton, B., Prosser, L.A., Bardon, C.G., Spurr, C.D., Carchidi, P.J., Kittler, A.F., Goldszer, R.C., Fairchild, D.G., Sussman, A.J., et al.: A cost-benefit analysis of electronic medical records in primary care. The American journal of medicine 114(5), 397-403 (2003)\n\nDiagnosis code assignment using sparsity-based disease correlation embedding. S Wang, X Chang, X Li, G Long, L Yao, Q Z Sheng, IEEE Transactions on Knowledge and Data Engineering. 2812Wang, S., Chang, X., Li, X., Long, G., Yao, L., Sheng, Q.Z.: Diagnosis code as- signment using sparsity-based disease correlation embedding. IEEE Transactions on Knowledge and Data Engineering 28(12), 3191-3202 (2016)\n"},"corpus_id":{"kind":"number","value":237451,"string":"237,451"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1707.05340\",\n \"mag\": 2964109414,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"journals/corr/WangZLHWLL17\",\n \"doi\": \"10.1007/978-3-319-68204-4_23\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1707.05340v2.pdf\"\n ],\n \"pdf_sha\": \"5b8c7905fe427aa7c26a41d11894e849b487e72a\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1707.05340\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"PDD Graph: Bridging Electronic Medical Records and Biomedical Knowledge Graphs via Entity Linking"},"authors":{"kind":"list like","value":["Meng Wang wangmengsd@stu.xjtu.edu.cn \nMOEKLINNS lab\nXi'an Jiaotong University\nXi'anChina","Jiaheng Zhang \nMOEKLINNS lab\nXi'an Jiaotong University\nXi'anChina","Jun Liu \nMOEKLINNS lab\nXi'an Jiaotong University\nXi'anChina","Wei Hu \nState Key Laboratory for Novel Software Technology\nNanjing University\nNanjingChina","Sen Wang \nGriffith Universtiy\nGold Coast Campus\nAustralia","Xue Li \nThe Universtiy of Queensland\nBrisbaneAustralia","Wenqiang Liu \nMOEKLINNS lab\nXi'an Jiaotong University\nXi'anChina"],"string":"[\n \"Meng Wang wangmengsd@stu.xjtu.edu.cn \\nMOEKLINNS lab\\nXi'an Jiaotong University\\nXi'anChina\",\n \"Jiaheng Zhang \\nMOEKLINNS lab\\nXi'an Jiaotong University\\nXi'anChina\",\n \"Jun Liu \\nMOEKLINNS lab\\nXi'an Jiaotong University\\nXi'anChina\",\n \"Wei Hu \\nState Key Laboratory for Novel Software Technology\\nNanjing University\\nNanjingChina\",\n \"Sen Wang \\nGriffith Universtiy\\nGold Coast Campus\\nAustralia\",\n \"Xue Li \\nThe Universtiy of Queensland\\nBrisbaneAustralia\",\n \"Wenqiang Liu \\nMOEKLINNS lab\\nXi'an Jiaotong University\\nXi'anChina\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Electronic medical records contain multi-format electronic medical data that consist of an abundance of medical knowledge. Facing with patients symptoms, experienced caregivers make right medical decisions based on their professional knowledge that accurately grasps relationships between symptoms, diagnosis and corresponding treatments. In this paper, we aim to capture these relationships by constructing a large and high-quality heterogenous graph linking patients, diseases, and drugs (PDD) in EMRs. Specifically, we propose a novel framework to extract important medical entities from MIMIC-III (Medical Information Mart for Intensive Care III) and automatically link them with the existing biomedical knowledge graphs, including ICD-9 ontology and DrugBank. The PDD graph presented in this paper is accessible on the Web via the SPARQL endpoint, and provides a pathway for medical discovery and applications, such as effective treatment recommendations."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Big data vendors collect and store large number of electronic medical records (EMRs) in hospital, with the goal of instantly accessing to comprehensive medical patient histories for caregivers at a lower cost. Public availability of EMRs collections has attracted much attention for different research purposes, including clinical research [14], mortality risk prediction [7], disease diagnosis [15], etc. An EMR database is normally a rich source of multi-format electronic data but remains limitations in scope and content. For example, MIMIC-III (Medical Information Mart for Intensive Care III) [8] collected bedside monitor trends, electronic medical notes, laboratory test results and waveforms from the ICUs (Intensive Care Units) of Beth Israel Deaconess Medical Center between 2001 and 2012. Abundant medical entities (symptoms, drugs and diseases) can be extracted from EMRs (clinical notes, prescriptions, and disease diagnoses). Most of the existing studies only focus on a specific entity, ignoring the relationship between entities. Given clinical data in MIMIC-III, discovering relationship between extracted entities (e.g. sepsis symptoms, pneumonia diagnosis, glucocorticoid drug and aspirin medicine) in wider scope can empower caregivers to make better decisions. Obviously, only focusing on EMR data is far from adequate to fully unveil entity relationships due to the limited scope of EMRs.\nMeanwhile, many biomedical knowledge graphs (KGs) are published as Linked Data [1] on the Web using the Resource Description Framework (RDF) [4], such as DrugBank [9] and ICD-9 ontology [13]. Linked Data is about using the Web to set RDF links between entities in different KGs, thereby forming a large heterogeneous graph 1 , where the nodes are entities (drugs, diseases, protein targets, side effects, pathways, etc.), and the edges (or links) represent various relations between entities such as drug-drug interactions. Unfortunately, such biomedical KGs only cover the basic medical facts, and contain little information about clinical outcomes. For instance, there is a relationship \"adverse interaction\" between glucocorticoid and aspirin in DrugBank, but no further information about how the adverse interaction affect the treatment of the patient who took both of the drugs in the same period. Clinical data can practically offer an opportunity to provide the missing relationship between KGs and clinical outcomes.\nAs mentioned above, biomedical KGs focus on the medical facts, whereas MIMIC-III only provides clinical data and physiological waveforms. There exists a gap between clinical data and biomedical KGs prohibiting further exploring medical entity relationship on ether side (see Fig.1). To solve this problem, we proposed a novel framework to construct a patient-drug-disease graph dataset (called PDD) in this paper. We summarize contributions of this paper as follows:\n-To our best knowledge, we are the first to bridge EMRs and biomedical KGs together. The result is a big and high-quality PDD graph dataset, which provides a salient opportunity to uncover associations of biomedical interest in wider scope. -We propose a novel framework to construct the PDD graph. The process starts by extracting medical entities from prescriptions, clinical notes and diagnoses respectively. RDF links are then set between the extracted medical entities and the corresponding entities in DrugBank and ICD-9 ontology. -We publish the PDD graph as an open resource 2 , and provide a SPARQL query endpoint using Apache Jena Fuseki 3 . Researchers can retrieve data distributed over biomedical KGs and MIMIC-III, ranging from drug-drug interactions, to the outcomes of drugs in clinical trials.\nIt is necessary to mention that MIMIC-III contains clinical information of patients. Although the protected health information was de-identified, researchers who seek to use more clinical data should complete an on-line training course and then apply for the permission to download the complete MIMIC-III dataset 4 .\nThe rest of this paper is organized as follows. Section 2 describes the proposed framework and details. The statistics and evaluation is reported in Section 3. Section 4 describes related work and finally, Section 5 concludes the paper and identifies topics for further work."},{"title":"PDD Construction","content":"We first follow the RDF model [4] and introduce the PDD definition.\nPDD Definition: PDD is an RDF graph consisting of PDD facts, where a PDD fact is represented by an RDF triple to indicate that a patient takes a drug or a patient is diagnosed with a disease. For instance, pdd :274671, pdd :diagnosed, sepsis 5 . Fig.2 illustrates the general process of the PDD dataset generation, mainly includes two steps: PDD facts generation (described in Section 2.1), and linking PDD to biomedical KGs (described in Section 2.2)."},{"title":"PDD Facts Generation","content":"According to the PDD definition, we need to extract three types of entities from MIMIC-III (patients, drugs, and diseases), and generate RDF triples of the prescription/diagnosis facts.\nPatients IRI Creation: MIMIC-III contains 46,520 distinct patients, and each patient is attached with an unique ID. We add IRI prefix to each patient ID to form a patient entity in PDD.\nPrescription Triple Generation: In MIMIC-III, the prescriptions table contains all the prescribed drugs for the treatments of patients. Each prescription record contains the patient's unique ID, the drug's name, the duration, and the dosage. We extracted all distinct drug names as the drug entities in PDD. Then we added a prescription triple in to PDD. An example is pdd :18740, pdd :prescribed, aspirin , where pdd :18740 is a patient entity, and aspirin is the drug's name. Diagnosis Triple Generation: MIMIC-III provides a diagnosed table that contains ICD-9 diagnosis codes for patients. There is an average of 13.9 ICD-9 codes per patient, but with a highly skewed distribution, as shown in Fig. 3. Beyond that, each patient has a set of clinical notes. These notes contain the diagnosis information. We use the named entity recognition (NER) tool C-TAKES [12] to extract diseases from clinical notes. C-TAKES is the most commonly used NER tool in the clinical domain. Then we use the model [15] (our previous work) to assign ICD-9 codes for extracted diseases. We extracted all ICD-9 diagnosis codes as the disease entities in PDD. Then we added a diagnosis triple into PDD. An example is pdd :18740, pdd :diagnosed, icd99592 , where pdd :18740 is a patient entity, and icd99592 is the ICD-9 code of sepsis."},{"title":"Linking PDD to Biomedical Knowledge Graphs","content":"After extracting entities, we need to tackle the task of finding sameAs links [5] between the entities in PDD and other biomedical KGs. For drugs, we focused on linking drugs of PDD to the DrugBank of Bio2RDF [6] version, as the project Bio2RDF provides a gateway to other biomedical KGs. Following the analogous reason, we interlinked diseases of PDD with the ICD-9 ontology in Bio2RDF.\nDrug Entity Linking: In MIMIC-III, drug names are various and often contain some insignificant words (10%, 200mg, glass bottle, etc.), which challenges the drug entity linking if the label matching method is directly used. In order to overcome this problem, we proposed an entity name model (ENM) based on [2] to link MIMIC-III drugs to DrugBank. The ENM is a statistical translation model which can capture the variations of an drug's name. Given a drugs name m in MIMIC-III, the ENM model assumes that it is a translation of the drugs name d in DrugBank, and each word of the drug name could be translated through three ways: 1) Retained (translated into itself); 2) Omitted (translated into the word NULL); 3) Converted (translated into its alias). Fig. 4 shows how the drug name Glucose in DrugBank translated into Dextrose 5% in MIMIC-III.\nBased on the above three ways of translations, we define the probability of drug name d being translated to m as follows:\nwhere ε is a normalization factor, l m is the length of m, l d is the length of d, m i is the i th word of m, d j is the j th word of d, and t(m i |d j ) is the lexical translation probability which indicates the probability of a word d j in DrugBank being written as m i in MIMIC-III. DrugBank contains a large amount of drug aliases information, which can be used as training sets to compute the translation probability t(m i |d j ). After training the ENM from sample data, a drug name in MIMIC-III will be more likely to be translated to itself or aliases in DrugBank, whereas the insignificant words tend to be translated to NULL. Hence, our ENM can reduce the effects of insignificant words for drugs entity linking.\nIn addition, we propose two constraint rules when selecting candidate drugs for m, and discard those at odds with the rules.\nRule 1: One of the drug indications in DrugBank must be in accordance with one of the diagnoses of the patients who took the corresponding drug in MIMIC-III at least ."},{"title":"Rule 2:","content":"The dosage of a drug that patients took in MIMIC-III must be in accordance with one of the standard dosages listed in DrugBank.\nFinally, we will choose the drug name d in DrugBank for the given drug m in MIMIC-III with maximal P (m|d), and d satisfies the two constraint rules.\nDisease IRI Resolution: In our previous work [15], we have assigned ICD-9 disease codes for extracted disease entities. Since the ICD-9 code is the international standard classification of diseases, and each code is unique. We can directly link the ICD-9 codes of PDD to ICD-9 ontology by string matching."},{"title":"Statistics and Evaluation","content":"In this section, we report the statistics of PDD and make the evaluation on its accuracy . At present PDD includes 58,030 entities and 2.3 million RDF triples. Table 1 shows the result of entities linked to the DrugBank and ICD-9 ontology. For drugs in PDD, 3,449 drugs are linked to 972 distinct drugs in DrugBank. For diseases in PDD, 6,983 diseases are connected to ICD-9 ontology. The only two failures of matching ICD-9 codes in MIMIC-III are '71970' and 'NULL', which are not included in ICD-9 ontology. Table 2 shows the result of RDF triples in PDD. In particular, 1,259,702 RDF triples contain drugs that have sameAs links to DrugBank, and 650,939 RDF triples have ICD-9 diseases codes. It indicates 83.4% drug-taken records in MIMIC-III can find corresponding entity in DrugBank, and 99.9% diagnosed information can link to ICD-9 ontology. A subgraph of PDD is illustrated in Fig. 5 to better understand the PDD graph. To evaluate the ENM model, 500 samples are randomly selected, manually verified and adjusted. The ratio of positive samples to negative samples is 4:1, where positive means the entity can be linked to DrugBank. The precision is 94% and the recall is 85%. For linked entities in PDD we randomly chose 200 of them and manually evaluated the correctness of them, and the precision of entity links is 93% which is in an accordance with the result of our examples.\nThe overall accuracy of entity linking will be affected by the performance of the entity recognition tool. No entity recognition tools so far can achieve 100% accuracy. The average accuracy of C-TAKES (we used in this paper) is 94%. Therefore, the overall precision and recall may be lower.\nIn order to find out why those 1,076 drugs have not been linked to DrugBank yet, we extract 100 of them that hold the highest usage frequency. The observation shows that most of them are not just contained in DrugBank. For instance, DrugBank does not consider NS (normal saline) as a drug, but PDD contains several expressions of NS (NS, 1/2 NS, NS (Mini Bag Plus), NS (Glass Bottle), etc.). For drugs wrongly linked to DrugBank, the names of those drugs are too short, e.g. N i.e nitrogen. These short names provide little information and affect the performance of ENM directly. Also, the training data from DrugBank does not include the usage frequency of each drug name. That might lead to some inconsistence with applications in MIMIC-III and cause linking errors."},{"title":"Related Work","content":"In order to bring the advantages of Semantic Web to the life science community, a number of biomedical KGs have been constructed over the last years, such as Bio2RDF [6] and Chem2Bio2RDF [3]. These datasets make the interconnection and exploration of different biomedical data sources possible. However, there is little patients clinical information within these biomedical KGs. STRIDE2RDF [10] and MCLSS2RDF [11] apply Linked Data Principles to represent patients electronic health records, but the interlinks from clinical data to existing biomedical KGs are still very limited. Hence, none of the existing linked datasets are bridging the gap between clinical and biomedical data."},{"title":"Conclusion and Future Work","content":"This paper presents the process to construct a high-quality patient-drug-disease (PDD) graph linking entities in MIMIC-III to Linked Data Cloud, which satisfies the demand to provide information of clinical outcomes in biomedical KGs, when previous no relationship exists between the medical entities in MIMIC-III. With abundant clinical data of over forty thousand patients linked to open datasets, our work provides more convenient data access for further researches based on clinical outcomes, such as personalized medication and disease correlation analysis. The PDD dataset is currently accessible on the Web via the SPARQL endpoint. In future work, our plan is to improve the linking accuracy of ENM model by feeding more data into its training system."},{"title":"Fig. 1 .","content":""},{"title":"Fig. 2 .","content":""},{"title":"Fig. 3 .","content":""},{"title":"Fig. 4 .","content":""},{"title":"Fig. 5 .","content":""},{"title":"Table 1 .","content":""},{"title":"Table 2 .","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Big data vendors collect and store large number of electronic medical records (EMRs) in hospital, with the goal of instantly accessing to comprehensive medical patient histories for caregivers at a lower cost. Public availability of EMRs collections has attracted much attention for different research purposes, including clinical research [14], mortality risk prediction [7], disease diagnosis [15], etc. An EMR database is normally a rich source of multi-format electronic data but remains limitations in scope and content. For example, MIMIC-III (Medical Information Mart for Intensive Care III) [8] collected bedside monitor trends, electronic medical notes, laboratory test results and waveforms from the ICUs (Intensive Care Units) of Beth Israel Deaconess Medical Center between 2001 and 2012. Abundant medical entities (symptoms, drugs and diseases) can be extracted from EMRs (clinical notes, prescriptions, and disease diagnoses). Most of the existing studies only focus on a specific entity, ignoring the relationship between entities. Given clinical data in MIMIC-III, discovering relationship between extracted entities (e.g. sepsis symptoms, pneumonia diagnosis, glucocorticoid drug and aspirin medicine) in wider scope can empower caregivers to make better decisions. Obviously, only focusing on EMR data is far from adequate to fully unveil entity relationships due to the limited scope of EMRs.\\nMeanwhile, many biomedical knowledge graphs (KGs) are published as Linked Data [1] on the Web using the Resource Description Framework (RDF) [4], such as DrugBank [9] and ICD-9 ontology [13]. Linked Data is about using the Web to set RDF links between entities in different KGs, thereby forming a large heterogeneous graph 1 , where the nodes are entities (drugs, diseases, protein targets, side effects, pathways, etc.), and the edges (or links) represent various relations between entities such as drug-drug interactions. Unfortunately, such biomedical KGs only cover the basic medical facts, and contain little information about clinical outcomes. For instance, there is a relationship \\\"adverse interaction\\\" between glucocorticoid and aspirin in DrugBank, but no further information about how the adverse interaction affect the treatment of the patient who took both of the drugs in the same period. Clinical data can practically offer an opportunity to provide the missing relationship between KGs and clinical outcomes.\\nAs mentioned above, biomedical KGs focus on the medical facts, whereas MIMIC-III only provides clinical data and physiological waveforms. There exists a gap between clinical data and biomedical KGs prohibiting further exploring medical entity relationship on ether side (see Fig.1). To solve this problem, we proposed a novel framework to construct a patient-drug-disease graph dataset (called PDD) in this paper. We summarize contributions of this paper as follows:\\n-To our best knowledge, we are the first to bridge EMRs and biomedical KGs together. The result is a big and high-quality PDD graph dataset, which provides a salient opportunity to uncover associations of biomedical interest in wider scope. -We propose a novel framework to construct the PDD graph. The process starts by extracting medical entities from prescriptions, clinical notes and diagnoses respectively. RDF links are then set between the extracted medical entities and the corresponding entities in DrugBank and ICD-9 ontology. -We publish the PDD graph as an open resource 2 , and provide a SPARQL query endpoint using Apache Jena Fuseki 3 . Researchers can retrieve data distributed over biomedical KGs and MIMIC-III, ranging from drug-drug interactions, to the outcomes of drugs in clinical trials.\\nIt is necessary to mention that MIMIC-III contains clinical information of patients. Although the protected health information was de-identified, researchers who seek to use more clinical data should complete an on-line training course and then apply for the permission to download the complete MIMIC-III dataset 4 .\\nThe rest of this paper is organized as follows. Section 2 describes the proposed framework and details. The statistics and evaluation is reported in Section 3. Section 4 describes related work and finally, Section 5 concludes the paper and identifies topics for further work.\"\n },\n {\n \"title\": \"PDD Construction\",\n \"content\": \"We first follow the RDF model [4] and introduce the PDD definition.\\nPDD Definition: PDD is an RDF graph consisting of PDD facts, where a PDD fact is represented by an RDF triple to indicate that a patient takes a drug or a patient is diagnosed with a disease. For instance, pdd :274671, pdd :diagnosed, sepsis 5 . Fig.2 illustrates the general process of the PDD dataset generation, mainly includes two steps: PDD facts generation (described in Section 2.1), and linking PDD to biomedical KGs (described in Section 2.2).\"\n },\n {\n \"title\": \"PDD Facts Generation\",\n \"content\": \"According to the PDD definition, we need to extract three types of entities from MIMIC-III (patients, drugs, and diseases), and generate RDF triples of the prescription/diagnosis facts.\\nPatients IRI Creation: MIMIC-III contains 46,520 distinct patients, and each patient is attached with an unique ID. We add IRI prefix to each patient ID to form a patient entity in PDD.\\nPrescription Triple Generation: In MIMIC-III, the prescriptions table contains all the prescribed drugs for the treatments of patients. Each prescription record contains the patient's unique ID, the drug's name, the duration, and the dosage. We extracted all distinct drug names as the drug entities in PDD. Then we added a prescription triple in to PDD. An example is pdd :18740, pdd :prescribed, aspirin , where pdd :18740 is a patient entity, and aspirin is the drug's name. Diagnosis Triple Generation: MIMIC-III provides a diagnosed table that contains ICD-9 diagnosis codes for patients. There is an average of 13.9 ICD-9 codes per patient, but with a highly skewed distribution, as shown in Fig. 3. Beyond that, each patient has a set of clinical notes. These notes contain the diagnosis information. We use the named entity recognition (NER) tool C-TAKES [12] to extract diseases from clinical notes. C-TAKES is the most commonly used NER tool in the clinical domain. Then we use the model [15] (our previous work) to assign ICD-9 codes for extracted diseases. We extracted all ICD-9 diagnosis codes as the disease entities in PDD. Then we added a diagnosis triple into PDD. An example is pdd :18740, pdd :diagnosed, icd99592 , where pdd :18740 is a patient entity, and icd99592 is the ICD-9 code of sepsis.\"\n },\n {\n \"title\": \"Linking PDD to Biomedical Knowledge Graphs\",\n \"content\": \"After extracting entities, we need to tackle the task of finding sameAs links [5] between the entities in PDD and other biomedical KGs. For drugs, we focused on linking drugs of PDD to the DrugBank of Bio2RDF [6] version, as the project Bio2RDF provides a gateway to other biomedical KGs. Following the analogous reason, we interlinked diseases of PDD with the ICD-9 ontology in Bio2RDF.\\nDrug Entity Linking: In MIMIC-III, drug names are various and often contain some insignificant words (10%, 200mg, glass bottle, etc.), which challenges the drug entity linking if the label matching method is directly used. In order to overcome this problem, we proposed an entity name model (ENM) based on [2] to link MIMIC-III drugs to DrugBank. The ENM is a statistical translation model which can capture the variations of an drug's name. Given a drugs name m in MIMIC-III, the ENM model assumes that it is a translation of the drugs name d in DrugBank, and each word of the drug name could be translated through three ways: 1) Retained (translated into itself); 2) Omitted (translated into the word NULL); 3) Converted (translated into its alias). Fig. 4 shows how the drug name Glucose in DrugBank translated into Dextrose 5% in MIMIC-III.\\nBased on the above three ways of translations, we define the probability of drug name d being translated to m as follows:\\nwhere ε is a normalization factor, l m is the length of m, l d is the length of d, m i is the i th word of m, d j is the j th word of d, and t(m i |d j ) is the lexical translation probability which indicates the probability of a word d j in DrugBank being written as m i in MIMIC-III. DrugBank contains a large amount of drug aliases information, which can be used as training sets to compute the translation probability t(m i |d j ). After training the ENM from sample data, a drug name in MIMIC-III will be more likely to be translated to itself or aliases in DrugBank, whereas the insignificant words tend to be translated to NULL. Hence, our ENM can reduce the effects of insignificant words for drugs entity linking.\\nIn addition, we propose two constraint rules when selecting candidate drugs for m, and discard those at odds with the rules.\\nRule 1: One of the drug indications in DrugBank must be in accordance with one of the diagnoses of the patients who took the corresponding drug in MIMIC-III at least .\"\n },\n {\n \"title\": \"Rule 2:\",\n \"content\": \"The dosage of a drug that patients took in MIMIC-III must be in accordance with one of the standard dosages listed in DrugBank.\\nFinally, we will choose the drug name d in DrugBank for the given drug m in MIMIC-III with maximal P (m|d), and d satisfies the two constraint rules.\\nDisease IRI Resolution: In our previous work [15], we have assigned ICD-9 disease codes for extracted disease entities. Since the ICD-9 code is the international standard classification of diseases, and each code is unique. We can directly link the ICD-9 codes of PDD to ICD-9 ontology by string matching.\"\n },\n {\n \"title\": \"Statistics and Evaluation\",\n \"content\": \"In this section, we report the statistics of PDD and make the evaluation on its accuracy . At present PDD includes 58,030 entities and 2.3 million RDF triples. Table 1 shows the result of entities linked to the DrugBank and ICD-9 ontology. For drugs in PDD, 3,449 drugs are linked to 972 distinct drugs in DrugBank. For diseases in PDD, 6,983 diseases are connected to ICD-9 ontology. The only two failures of matching ICD-9 codes in MIMIC-III are '71970' and 'NULL', which are not included in ICD-9 ontology. Table 2 shows the result of RDF triples in PDD. In particular, 1,259,702 RDF triples contain drugs that have sameAs links to DrugBank, and 650,939 RDF triples have ICD-9 diseases codes. It indicates 83.4% drug-taken records in MIMIC-III can find corresponding entity in DrugBank, and 99.9% diagnosed information can link to ICD-9 ontology. A subgraph of PDD is illustrated in Fig. 5 to better understand the PDD graph. To evaluate the ENM model, 500 samples are randomly selected, manually verified and adjusted. The ratio of positive samples to negative samples is 4:1, where positive means the entity can be linked to DrugBank. The precision is 94% and the recall is 85%. For linked entities in PDD we randomly chose 200 of them and manually evaluated the correctness of them, and the precision of entity links is 93% which is in an accordance with the result of our examples.\\nThe overall accuracy of entity linking will be affected by the performance of the entity recognition tool. No entity recognition tools so far can achieve 100% accuracy. The average accuracy of C-TAKES (we used in this paper) is 94%. Therefore, the overall precision and recall may be lower.\\nIn order to find out why those 1,076 drugs have not been linked to DrugBank yet, we extract 100 of them that hold the highest usage frequency. The observation shows that most of them are not just contained in DrugBank. For instance, DrugBank does not consider NS (normal saline) as a drug, but PDD contains several expressions of NS (NS, 1/2 NS, NS (Mini Bag Plus), NS (Glass Bottle), etc.). For drugs wrongly linked to DrugBank, the names of those drugs are too short, e.g. N i.e nitrogen. These short names provide little information and affect the performance of ENM directly. Also, the training data from DrugBank does not include the usage frequency of each drug name. That might lead to some inconsistence with applications in MIMIC-III and cause linking errors.\"\n },\n {\n \"title\": \"Related Work\",\n \"content\": \"In order to bring the advantages of Semantic Web to the life science community, a number of biomedical KGs have been constructed over the last years, such as Bio2RDF [6] and Chem2Bio2RDF [3]. These datasets make the interconnection and exploration of different biomedical data sources possible. However, there is little patients clinical information within these biomedical KGs. STRIDE2RDF [10] and MCLSS2RDF [11] apply Linked Data Principles to represent patients electronic health records, but the interlinks from clinical data to existing biomedical KGs are still very limited. Hence, none of the existing linked datasets are bridging the gap between clinical and biomedical data.\"\n },\n {\n \"title\": \"Conclusion and Future Work\",\n \"content\": \"This paper presents the process to construct a high-quality patient-drug-disease (PDD) graph linking entities in MIMIC-III to Linked Data Cloud, which satisfies the demand to provide information of clinical outcomes in biomedical KGs, when previous no relationship exists between the medical entities in MIMIC-III. With abundant clinical data of over forty thousand patients linked to open datasets, our work provides more convenient data access for further researches based on clinical outcomes, such as personalized medication and disease correlation analysis. The PDD dataset is currently accessible on the Web via the SPARQL endpoint. In future work, our plan is to improve the linking accuracy of ENM model by feeding more data into its training system.\"\n },\n {\n \"title\": \"Fig. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 2 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Left part is the Linked Data Cloud 1 , which contains interlinked biomedical knowledge graphs. Right part is the MIMIC-III database.","content":"Fig. 1 .\n1Left part is the Linked Data Cloud 1 , which contains interlinked biomedical knowledge graphs. Right part is the MIMIC-III database."},{"id":"","caption":"Overview of PDD bridging MIMIC-III and biomedical knowledge graphs.","content":"Fig. 2 .\n2Overview of PDD bridging MIMIC-III and biomedical knowledge graphs."},{"id":"","caption":"The distribution of assigned ICD-9 codes per patient.","content":"Fig. 3 .\n3The distribution of assigned ICD-9 codes per patient."},{"id":"","caption":"The translation from Glucose to Dextrose 5%.","content":"Fig. 4 .\n4The translation from Glucose to Dextrose 5%."},{"id":"","caption":"An annotated subgraph of PDD.","content":"Fig. 5 .\n5An annotated subgraph of PDD."},{"id":"","caption":"Statistics of Entities","content":"Table 1 .\n1Statistics of Entities#Overall \n#Drug/disease \nlinked to KG \nPatient 46,520 \nDrug \n4,525 \n3,449 \nDisease 6,985 \n6,983"},{"id":"","caption":"Statistics of RDF triples","content":"Table 2 .\n2Statistics of RDF triples#Overall \n#Drug/disease \nlinked to KG \nDemographics \n165,526 \nPatients-Drugs \n1,517,702 \n1,259,702 \nPatients-Diseases 650,987 \n650,939"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Left part is the Linked Data Cloud 1 , which contains interlinked biomedical knowledge graphs. Right part is the MIMIC-III database.\",\n \"content\": \"Fig. 1 .\\n1Left part is the Linked Data Cloud 1 , which contains interlinked biomedical knowledge graphs. Right part is the MIMIC-III database.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Overview of PDD bridging MIMIC-III and biomedical knowledge graphs.\",\n \"content\": \"Fig. 2 .\\n2Overview of PDD bridging MIMIC-III and biomedical knowledge graphs.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The distribution of assigned ICD-9 codes per patient.\",\n \"content\": \"Fig. 3 .\\n3The distribution of assigned ICD-9 codes per patient.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The translation from Glucose to Dextrose 5%.\",\n \"content\": \"Fig. 4 .\\n4The translation from Glucose to Dextrose 5%.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"An annotated subgraph of PDD.\",\n \"content\": \"Fig. 5 .\\n5An annotated subgraph of PDD.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Statistics of Entities\",\n \"content\": \"Table 1 .\\n1Statistics of Entities#Overall \\n#Drug/disease \\nlinked to KG \\nPatient 46,520 \\nDrug \\n4,525 \\n3,449 \\nDisease 6,985 \\n6,983\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Statistics of RDF triples\",\n \"content\": \"Table 2 .\\n2Statistics of RDF triples#Overall \\n#Drug/disease \\nlinked to KG \\nDemographics \\n165,526 \\nPatients-Drugs \\n1,517,702 \\n1,259,702 \\nPatients-Diseases 650,987 \\n650,939\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Linked data-the story so far. Semantic services, interoperability and web applications: emerging concepts pp. C Bizer, T Heath, T Berners-Lee, Bizer, C., Heath, T., Berners-Lee, T.: Linked data-the story so far. Semantic ser- vices, interoperability and web applications: emerging concepts pp. 205-227 (2009)","doi":""},{"text":"The mathematics of statistical machine translation: Parameter estimation. P F Brown, V J D Pietra, S A D Pietra, R L Mercer, Computational linguistics. 192Brown, P.F., Pietra, V.J.D., Pietra, S.A.D., Mercer, R.L.: The mathematics of statistical machine translation: Parameter estimation. Computational linguistics 19(2), 263-311 (1993)","doi":""},{"text":"Chem2bio2rdf: a semantic framework for linking and data mining chemogenomic and systems chemical biology data. B Chen, X Dong, D Jiao, H Wang, Q Zhu, Y Ding, D J Wild, BMC bioinformatics. 111255Chen, B., Dong, X., Jiao, D., Wang, H., Zhu, Q., Ding, Y., Wild, D.J.: Chem2bio2rdf: a semantic framework for linking and data mining chemogenomic and systems chemical biology data. BMC bioinformatics 11(1), 255 (2010)","doi":""},{"text":"W W W Consortium, Rdf 1.1 concepts and abstract syntax. Consortium, W.W.W., et al.: Rdf 1.1 concepts and abstract syntax (2014)","doi":""},{"text":"Sameas networks and beyond: analyzing deployment status and implications of owl: sameas in linked data. The Semantic Web-ISWC. L Ding, J Shinavier, Z Shangguan, D Mcguinness, Ding, L., Shinavier, J., Shangguan, Z., McGuinness, D.: Sameas networks and beyond: analyzing deployment status and implications of owl: sameas in linked data. The Semantic Web-ISWC 2010 pp. 145-160 (2010)","doi":""},{"text":"Bio2rdf release 3: a larger connected network of linked data for the life sciences. M Dumontier, A Callahan, J Cruz-Toledo, P Ansell, V Emonet, F Belleau, A Droit, CEUR-WS. orgProceedings of the 2014 International Conference on Posters & Demonstrations Track. the 2014 International Conference on Posters & Demonstrations Track1272Dumontier, M., Callahan, A., Cruz-Toledo, J., Ansell, P., Emonet, V., Belleau, F., Droit, A.: Bio2rdf release 3: a larger connected network of linked data for the life sciences. In: Proceedings of the 2014 International Conference on Posters & Demonstrations Track-Volume 1272. pp. 401-404. CEUR-WS. org (2014)","doi":"CEUR-WS. org"},{"text":"Unfolding physiological state: Mortality modelling in intensive care units. M Ghassemi, T Naumann, F Doshi-Velez, N Brimmer, R Joshi, A Rumshisky, P Szolovits, Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining. the 20th ACM SIGKDD international conference on Knowledge discovery and data miningACMGhassemi, M., Naumann, T., Doshi-Velez, F., Brimmer, N., Joshi, R., Rumshisky, A., Szolovits, P.: Unfolding physiological state: Mortality modelling in intensive care units. In: Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining. pp. 75-84. ACM (2014)","doi":""},{"text":"Mimic-iii, a freely accessible critical care database. A E Johnson, T J Pollard, L Shen, L W H Lehman, M Feng, M Ghassemi, B Moody, P Szolovits, L A Celi, R G Mark, 3Johnson, A.E., Pollard, T.J., Shen, L., Lehman, L.w.H., Feng, M., Ghassemi, M., Moody, B., Szolovits, P., Celi, L.A., Mark, R.G.: Mimic-iii, a freely accessible critical care database. Scientific data 3 (2016)","doi":""},{"text":"V Law, C Knox, Y Djoumbou, T Jewison, A C Guo, Y Liu, A Maciejewski, D Arndt, M Wilson, V Neveu, Drugbank 4.0: shedding new light on drug metabolism. 42Law, V., Knox, C., Djoumbou, Y., Jewison, T., Guo, A.C., Liu, Y., Maciejewski, A., Arndt, D., Wilson, M., Neveu, V., et al.: Drugbank 4.0: shedding new light on drug metabolism. Nucleic acids research 42(D1), D1091-D1097 (2014)","doi":""},{"text":"Mining electronic health records using linked data. D J Odgers, M Dumontier, AMIA Summits on Translational Science Proceedings. 217Odgers, D.J., Dumontier, M.: Mining electronic health records using linked data. AMIA Summits on Translational Science Proceedings 2015, 217 (2015)","doi":""},{"text":"Applying linked data principles to represent patient's electronic health records at mayo clinic: a case report. J Pathak, R C Kiefer, C G Chute, Proceedings of the 2nd ACM SIGHIT International Health Informatics Symposium. the 2nd ACM SIGHIT International Health Informatics SymposiumACMPathak, J., Kiefer, R.C., Chute, C.G.: Applying linked data principles to represent patient's electronic health records at mayo clinic: a case report. In: Proceedings of the 2nd ACM SIGHIT International Health Informatics Symposium. pp. 455-464. ACM (2012)","doi":""},{"text":"Mayo clinical text analysis and knowledge extraction system (ctakes): architecture, component evaluation and applications. G K Savova, J J Masanz, P V Ogren, J Zheng, S Sohn, K C Kipper-Schuler, C G Chute, Journal of the American Medical Informatics Association. 175Savova, G.K., Masanz, J.J., Ogren, P.V., Zheng, J., Sohn, S., Kipper-Schuler, K.C., Chute, C.G.: Mayo clinical text analysis and knowledge extraction system (ctakes): architecture, component evaluation and applications. Journal of the American Medical Informatics Association 17(5), 507-513 (2010)","doi":""},{"text":"Disease ontology: a backbone for disease semantic integration. L M Schriml, C Arze, S Nadendla, Y W W Chang, M Mazaitis, V Felix, G Feng, W A Kibbe, Nucleic acids research. 40D1Schriml, L.M., Arze, C., Nadendla, S., Chang, Y.W.W., Mazaitis, M., Felix, V., Feng, G., Kibbe, W.A.: Disease ontology: a backbone for disease semantic integra- tion. Nucleic acids research 40(D1), D940-D946 (2012)","doi":""},{"text":"A cost-benefit analysis of electronic medical records in primary care. S J Wang, B Middleton, L A Prosser, C G Bardon, C D Spurr, P J Carchidi, A F Kittler, R C Goldszer, D G Fairchild, A J Sussman, The American journal of medicine. 1145Wang, S.J., Middleton, B., Prosser, L.A., Bardon, C.G., Spurr, C.D., Carchidi, P.J., Kittler, A.F., Goldszer, R.C., Fairchild, D.G., Sussman, A.J., et al.: A cost-benefit analysis of electronic medical records in primary care. The American journal of medicine 114(5), 397-403 (2003)","doi":""},{"text":"Diagnosis code assignment using sparsity-based disease correlation embedding. S Wang, X Chang, X Li, G Long, L Yao, Q Z Sheng, IEEE Transactions on Knowledge and Data Engineering. 2812Wang, S., Chang, X., Li, X., Long, G., Yao, L., Sheng, Q.Z.: Diagnosis code as- signment using sparsity-based disease correlation embedding. IEEE Transactions on Knowledge and Data Engineering 28(12), 3191-3202 (2016)","doi":""}],"string":"[\n {\n \"text\": \"Linked data-the story so far. Semantic services, interoperability and web applications: emerging concepts pp. C Bizer, T Heath, T Berners-Lee, Bizer, C., Heath, T., Berners-Lee, T.: Linked data-the story so far. Semantic ser- vices, interoperability and web applications: emerging concepts pp. 205-227 (2009)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The mathematics of statistical machine translation: Parameter estimation. P F Brown, V J D Pietra, S A D Pietra, R L Mercer, Computational linguistics. 192Brown, P.F., Pietra, V.J.D., Pietra, S.A.D., Mercer, R.L.: The mathematics of statistical machine translation: Parameter estimation. Computational linguistics 19(2), 263-311 (1993)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Chem2bio2rdf: a semantic framework for linking and data mining chemogenomic and systems chemical biology data. B Chen, X Dong, D Jiao, H Wang, Q Zhu, Y Ding, D J Wild, BMC bioinformatics. 111255Chen, B., Dong, X., Jiao, D., Wang, H., Zhu, Q., Ding, Y., Wild, D.J.: Chem2bio2rdf: a semantic framework for linking and data mining chemogenomic and systems chemical biology data. BMC bioinformatics 11(1), 255 (2010)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"W W W Consortium, Rdf 1.1 concepts and abstract syntax. Consortium, W.W.W., et al.: Rdf 1.1 concepts and abstract syntax (2014)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Sameas networks and beyond: analyzing deployment status and implications of owl: sameas in linked data. The Semantic Web-ISWC. L Ding, J Shinavier, Z Shangguan, D Mcguinness, Ding, L., Shinavier, J., Shangguan, Z., McGuinness, D.: Sameas networks and beyond: analyzing deployment status and implications of owl: sameas in linked data. The Semantic Web-ISWC 2010 pp. 145-160 (2010)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bio2rdf release 3: a larger connected network of linked data for the life sciences. M Dumontier, A Callahan, J Cruz-Toledo, P Ansell, V Emonet, F Belleau, A Droit, CEUR-WS. orgProceedings of the 2014 International Conference on Posters & Demonstrations Track. the 2014 International Conference on Posters & Demonstrations Track1272Dumontier, M., Callahan, A., Cruz-Toledo, J., Ansell, P., Emonet, V., Belleau, F., Droit, A.: Bio2rdf release 3: a larger connected network of linked data for the life sciences. In: Proceedings of the 2014 International Conference on Posters & Demonstrations Track-Volume 1272. pp. 401-404. CEUR-WS. org (2014)\",\n \"doi\": \"CEUR-WS. org\"\n },\n {\n \"text\": \"Unfolding physiological state: Mortality modelling in intensive care units. M Ghassemi, T Naumann, F Doshi-Velez, N Brimmer, R Joshi, A Rumshisky, P Szolovits, Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining. the 20th ACM SIGKDD international conference on Knowledge discovery and data miningACMGhassemi, M., Naumann, T., Doshi-Velez, F., Brimmer, N., Joshi, R., Rumshisky, A., Szolovits, P.: Unfolding physiological state: Mortality modelling in intensive care units. In: Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining. pp. 75-84. ACM (2014)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Mimic-iii, a freely accessible critical care database. A E Johnson, T J Pollard, L Shen, L W H Lehman, M Feng, M Ghassemi, B Moody, P Szolovits, L A Celi, R G Mark, 3Johnson, A.E., Pollard, T.J., Shen, L., Lehman, L.w.H., Feng, M., Ghassemi, M., Moody, B., Szolovits, P., Celi, L.A., Mark, R.G.: Mimic-iii, a freely accessible critical care database. Scientific data 3 (2016)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"V Law, C Knox, Y Djoumbou, T Jewison, A C Guo, Y Liu, A Maciejewski, D Arndt, M Wilson, V Neveu, Drugbank 4.0: shedding new light on drug metabolism. 42Law, V., Knox, C., Djoumbou, Y., Jewison, T., Guo, A.C., Liu, Y., Maciejewski, A., Arndt, D., Wilson, M., Neveu, V., et al.: Drugbank 4.0: shedding new light on drug metabolism. Nucleic acids research 42(D1), D1091-D1097 (2014)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Mining electronic health records using linked data. D J Odgers, M Dumontier, AMIA Summits on Translational Science Proceedings. 217Odgers, D.J., Dumontier, M.: Mining electronic health records using linked data. AMIA Summits on Translational Science Proceedings 2015, 217 (2015)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Applying linked data principles to represent patient's electronic health records at mayo clinic: a case report. J Pathak, R C Kiefer, C G Chute, Proceedings of the 2nd ACM SIGHIT International Health Informatics Symposium. the 2nd ACM SIGHIT International Health Informatics SymposiumACMPathak, J., Kiefer, R.C., Chute, C.G.: Applying linked data principles to represent patient's electronic health records at mayo clinic: a case report. In: Proceedings of the 2nd ACM SIGHIT International Health Informatics Symposium. pp. 455-464. ACM (2012)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Mayo clinical text analysis and knowledge extraction system (ctakes): architecture, component evaluation and applications. G K Savova, J J Masanz, P V Ogren, J Zheng, S Sohn, K C Kipper-Schuler, C G Chute, Journal of the American Medical Informatics Association. 175Savova, G.K., Masanz, J.J., Ogren, P.V., Zheng, J., Sohn, S., Kipper-Schuler, K.C., Chute, C.G.: Mayo clinical text analysis and knowledge extraction system (ctakes): architecture, component evaluation and applications. Journal of the American Medical Informatics Association 17(5), 507-513 (2010)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Disease ontology: a backbone for disease semantic integration. L M Schriml, C Arze, S Nadendla, Y W W Chang, M Mazaitis, V Felix, G Feng, W A Kibbe, Nucleic acids research. 40D1Schriml, L.M., Arze, C., Nadendla, S., Chang, Y.W.W., Mazaitis, M., Felix, V., Feng, G., Kibbe, W.A.: Disease ontology: a backbone for disease semantic integra- tion. Nucleic acids research 40(D1), D940-D946 (2012)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A cost-benefit analysis of electronic medical records in primary care. S J Wang, B Middleton, L A Prosser, C G Bardon, C D Spurr, P J Carchidi, A F Kittler, R C Goldszer, D G Fairchild, A J Sussman, The American journal of medicine. 1145Wang, S.J., Middleton, B., Prosser, L.A., Bardon, C.G., Spurr, C.D., Carchidi, P.J., Kittler, A.F., Goldszer, R.C., Fairchild, D.G., Sussman, A.J., et al.: A cost-benefit analysis of electronic medical records in primary care. The American journal of medicine 114(5), 397-403 (2003)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Diagnosis code assignment using sparsity-based disease correlation embedding. S Wang, X Chang, X Li, G Long, L Yao, Q Z Sheng, IEEE Transactions on Knowledge and Data Engineering. 2812Wang, S., Chang, X., Li, X., Long, G., Yao, L., Sheng, Q.Z.: Diagnosis code as- signment using sparsity-based disease correlation embedding. IEEE Transactions on Knowledge and Data Engineering 28(12), 3191-3202 (2016)\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":481,"cells":{"text":{"kind":"string","value":"\nFirst-principles study of the low-temperature charge density wave phase in the quasi-one-dimensional Weyl chiral compound (TaSe 4 ) 2 I\n\n\nYang Zhang \nDepartment of Physics and Astronomy\nUniversity of Tennessee\n37996KnoxvilleTNUSA\n\nSchool of Physics\nSoutheast University\n211189NanjingChina\n\nLing-Fang Lin \nDepartment of Physics and Astronomy\nUniversity of Tennessee\n37996KnoxvilleTNUSA\n\nSchool of Physics\nSoutheast University\n211189NanjingChina\n\nAdriana Moreo \nDepartment of Physics and Astronomy\nUniversity of Tennessee\n37996KnoxvilleTNUSA\n\nMaterials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n\nShuai Dong \nSchool of Physics\nSoutheast University\n211189NanjingChina\n\nElbio Dagotto \nDepartment of Physics and Astronomy\nUniversity of Tennessee\n37996KnoxvilleTNUSA\n\nMaterials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n\nFirst-principles study of the low-temperature charge density wave phase in the quasi-one-dimensional Weyl chiral compound (TaSe 4 ) 2 I\n(Dated: February 21, 2020)\nUsing ab initio density functional theory, we study the lattice phase transition of quasi-onedimensional (TaSe4)2I. In the undistorted state, the strongly anisotropic semimetal band structure presents two non-equivalent Weyl points. In previous efforts, two possible Ta-tetramerization patterns were proposed to be associated with the low-temperature structure. Our phonon calculations indicate that the orthorhombic F 222 CDW-I phase is the most likely ground state for this quasi-onedimensional system. In addition, the monoclinic C2 CDW-II phase may also be stable according to the phonon dispersion spectrum. Since these two phases have very similar energies in our DFT calculations, both these Ta-tetramerization distortions likely compete or coexist at low temperatures. The semimetal to insulator transition is induced by a Fermi-surface-driven instability that supports the Peierls scenario, which affects the Weyl physics developed above TCDW. Furthermore, the spin-orbit coupling generates Rashba-like band splittings in the insulating CDW phases.\n\nI. INTRODUCTION\n\nOne-dimensional (1D) systems continue attracting considerable attention due to their rich physical properties and reduced dimensional phase space. In particular, in low dimension systems the electron-electron, phononphonon, electron-phonon, and spin-phonon couplings are strongly enhanced by the interactions between transition metal ions [1,2]. Under some conditions, free carriers may form charge density wave (CDW) or spin density wave (SDW) states due to the partial or complete condensation of excitations, properties which are physically interesting and important for possible applications [3][4][5].\n\nSeveral 1D bulk compounds have been widely studied. For example, with focus on the electronic correlation effects the Cu-oxide 1D ladders were theoretically predicted and experimentally confirmed to have a spin gap and become superconducting [6][7][8][9]. Recently, an analogous behavior was shown to develop in iron ladders BaFe 2 X 3 (X=S or Se) that were reported to be superconducting at high pressure [10][11][12][13][14]. By considering phononic modes or spin-phonon instability, ferroelectric or multiferroelectric states were predicted in some 1D systems [15][16][17][18][19][20].\n\n(TaSe 4 ) 2 I is a typical paradigmatic quasi-onedimensional material that has been frequently studied for more than thirty years [21,22]. This system undergoes a Peierls phase transition at 263 K, the socalled CDW transition, accompanied by an incommensurate structural distortion at low temperatures [23][24][25]. Above the critical temperature T CDW = 263 K, the nearest neighbor (NN) Ta-Ta distances (d Ta−Ta = 3.206 A at room temperature [22,26]) are identical along each chain. As expected, the CDW instability breaks the symmetry of the isometric chains leading to Tatetramerization modes [25,26]. Related experiments re-vealed that the Ta-tetramerization periodicity may be c = 4d Ta−Ta [24,25,27]. In addition, this system is also considered to be a Weyl semimetal [30,31]. More recently, possible axion physics was proposed for this special chiral Weyl material [32,33]. It also should be noted that Rashba-like band splittings may also occur in a chiral system [34]. Since the spin-orbit coupling (SOC) for Ta's 5d orbitals can be robust, such Rashba splitting is expected. All these developments in (TaSe 4 ) 2 I provide a unique and promising platform to display different interesting physical properties in a single material.\n\nIn previous works, two Ta-tetramerization patterns were proposed for the low-temperature structure, corresponding to the one-dimensional B 1 and B 2 representations [26]. Since the Ta-tetramerization distortion is small, it can only be observed via weak reflections, and for this reason the lattice phase transition is still uncertain. For this reason, from the theoretical perspective ab initio phonon calculations for the superstructure could play an important role in clarifying the real mechanism of lattice structural transition in this compound.\n\nIn the present publication, we perform firstprinciples density functional theory (DFT) calculations for (TaSe 4 ) 2 I. First, our theoretical results indicate a strongly anisotropic electronic structure for the undistorted phase, corresponding to its quasi-one-dimensional geometry. Based on phononic dispersion calculations, we found that (TaSe 4 ) 2 I contains phonon softening instabilities in the Γ 4 mode, resulting into an orthorhombic distortion, which could correspond to the B 1 or B 2 representations. By considering the combined B 1 +B 2 symmetry breaking, the system would decrease to the monoclinic C2 phase (No. 5) with a pattern along chains involving four different NN Ta-Ta distances. Our DFT calculations suggest that the CDW-I lattice phase is the most likely ground state for this system. Since the energy difference between the two CDW lattices, B 1 or B 2 , can not be distinguished unambiguously, these two Ta-tetramerization phases are competitive at low temperatures. Our results also support that the semimetal-insulator transition is driven by the Fermi surface instability. Furthermore, for the heavy element Ta, moderate Rashba-like splitting bands are found in the low temperature CDW phases.\n\n\nII. METHODS AND MODEL SYSTEM\n\nIn this publication, DFT calculations were performed using the projector augmented wave (PAW) pseudopotentials with the Perdew-Burke-Ernzerhof format revised for solid exchange functional (PBEsol), as implemented in the Vienna ab initio Simulation Package (VASP) code [35][36][37][38]. The phonon spectra were calculated using the finite displacement approach and analyzed by the PHONONPY software [39,40]. Details are in the Supplemental Material (SM) [41].\n\nUnder ambient conditions, (TaSe 4 ) 2 I forms a quasione-dimensional body-centered tetragonal chiral crystal structure with the space group I422 (No. 97). The lattice parameters are a = 9.531Å and c = 12.824Å [22]. There are two adjacent TaSe 4 chains along the c-axis in the conventional cell, where iodine atoms are located between chains, as shown in Fig. 1(a). In each TaSe 4 chain, the Ta atoms are aligned equidistantly and the Se atoms form a \"screw\" arrangement along the c-axis [see Fig. 1(b)]. Previous experimental results suggested that the modulated structure of (TaSe 4 ) 2 I can be split into two parts [26] below the transition temperature T CDW = 263 K.\n\nSince the equal displacements of all atoms would not change the space-group symmetry, we will focus on the modulation of the Ta-tetramerized atoms, which corre- sponds to the parallel component of the k F vector (q CDW || ) along the Ta-chain direction. By considering the space group symmetry I422 (No. 97), the Ta atoms occupy two different Wyckoff sites (4c and 4d). Based on the Raman active modes analysis [42], the corresponding Raman active modes of the 4c and 4d sites have B 2 and B 1 symmetry, respectively. Since the CDW instability corresponds to the Ta-tetramerization modes [25,26], a long-longshort-short (LLSS) Ta-tetramerization CDW-I pattern is expected by only considering the lattice dynamical B 1 or B 2 Ta tetramerization, as shown in Fig. 1(c) [43]. If, instead, we consider the combined B 1 +B 2 Ta tetramerization, there would be four different nearest-neighbor Ta-Ta distances in the chain [see Fig. 1(d)]. In addition, the AMPLIMODES software was employed to perform the group theoretical analysis [44,45], indicating that the CDW-I phase would induce the orthorhombic distortion from the undistorted tetragonal phase with phonon mode Γ 4 . The same analysis for the CDW-II phase also indicates that a monoclinic distortion is obtained with phonon modes Γ 5 and Γ 4 corresponding to the undistorted tetragonal phase and the CDW-I orthorhombic phase, respectively.\n\n\nIII. NON-CDW PHASE\n\nBefore discussing the CDW phase, we will consider the electronic structure corresponding to the non-CDW phase of (TaSe 4 ) 2 I, which is displayed in Fig. 2. Here, we use the primitive cell instead of the conventional cell to calculate the electronic structure of (TaSe 4 ) 2 I. The volume of the primitive cell is half of the conventional cell with the primitive lattice: a 1 =(−a/2, a/2, c/2), a 2 =(a/2, a/2, c/2), and a 3 =(a/2, a/2, −c/2), where a and c are the conventional cell-lattice constants.\n\nFirst, we will present the band structure of the undistorted phase of (TaSe 4 ) 2 I without SOC effect. This band structure clearly displays strong anisotropic metallic behavior, as presented in Fig. 2(a). The band is more dispersive from Γ to Z along the chains than along other directions (i.e. N to P), which is compatible with the presence of quasi-one-dimensional chains along the k z axis. As shown in Fig. 2(a), the Fermi level is dominated by the Ta d 3z 2 −r 2 orbital along the chains, in agreement with previous studies [25,29]. The crossing point along the chain direction (Γ to Z) is not at π/c but at the incommensurate k F where 2k F =q CDW || . Next, we introduce the SOC effect to the undistorted state and now the bands begin to split as displayed in Fig. 2(b), which is consistent with other results [30,31]. In principle, as suggested by the band structure, (TaSe 4 ) 2 I should also display two interesting Weyl points with one along the Γ to Z path and another along the N to P path, although this aspect requires further detailed calculations and discussion beyond the scope of this publication. However, note that that the Weyl physics of this compound has already been studied in detail in recent publications [30,31]. For this reason, here we primarily focus on the phononic aspects.\n\n\nIV. CDW INSTABILITY\n\nIt is well known that the CDW instability is accompanied by an structural atomic rearrangement, related to the phonon mode instability. To understand the structural phase transition in (TaSe 4 ) 2 I, we performed phononic dispersion calculations for both the non-CDW and CDW phases. Figure 3(a) indicates the phonon dispersion spectrum that presents an imaginary frequency appearing at Γ for the undistorted structure. The Tatetramerization arising from this mode corresponds to the CDW-I phase. According to group theory analysis using the AMPLIMODES software [44,45], this spontaneous distortion mode is a Γ 4 mode resulting in a twice larger periodicity (4d Ta−Ta ) along the TaSe 4 chain [47]. In addition, the phonon softening mode also corresponds to the transverse component of the incommensurate CDW wavevector parallel to the chain direction (Γ to Z), as in X-ray diffraction and ARPES data Here, a 2 × 2 × 2 supercell was selected from the original primitive cell in our calculations. The coordinates of the high symmetry points in the bulk Brillouin zone (BZ) are obtained using the Seek-path software [46]. [26,28]. We fully relaxed the crystal lattice from the original tetragonal I422 phase along these mode displacements, and then obtained an orthorhombic F 222 state (No. 22). Based on the relaxed orthorhombic structure, we found the LLSS Ta-tetramerization configuration with the difference between the long NN and short NN Ta-Ta distances being about ∼ 0.083Å. Furthermore, we also calculated the phonon dispersion spectrum for the orthorhombic F 222 phase (No. 22), and in this case no imaginary frequency mode was found, as shown in Fig. 3(b). By extracting the lowest-energy phonon mode of the orthorhombic F 222 phase and applying to the F 222 phase, the symmetry of the crystal structure further decreases to the monoclinic C2 phase (No. 5) with a pattern of four different NN Ta-Ta distances along the chain, corresponding to the B 1 +B 2 representations. The AMPLIMODES software [44,45] indicates that this is the Γ 4 mode distortion for the orthorhombic F 222 phase. After full lattice relaxations for the monoclinic C2 phase (No. 5), we find that the difference of the two long or short Ta-Ta distances are 0.003Å and 0.002 A, respectively. Figure 3(c) indicates that the mono- clinic CDW-II phase is also dynamically stable because no imaginary frequency mode is obtained in the phonon dispersion calculation. Here, it should be noticed that the difference of energies of the orthorhombic and monoclinic cases is quite small (∼ 0.1 meV/Ta) which is beyond the accuracy of DFT. In this context, then those two phases cannot be distinguished by considering such small monoclinic distortion and almost negligible energy difference. Additional DFT results for those two phases can be found in the SM [41].\n\nNext, let us focus on the semimetal to insulator transition. It is well known that there is a very small overlap between the bottom of the conduction band and the top of the valence band in the semimetallic material, resulting in a negligible density of states (DOS) without SOC at the Fermi level, as shown in Fig. 4(a). The DOS of the undistorted phase (No. 97) is here presented in Fig. 4(b) indicating a semimetal with indeed quite a small density at the Fermi level. Furthermore, we also present in at the crossing point along Γ to Z corresponding to the Brillouin path of undistorted phases. Hence, our results are in agreement with the physical picture of a Fermisurface-driven instability resulting in the Peierls transition. Furthermore, strictly speaking the two Weyl points disappear in the low-temperature CDW phases due to the opening of a gap. This suggests that Weyl physics could be altered below T CDW . Because the phase transition seems to be of first order, as suggested by neutron scattering studies [26], the physics below and above T CDW could be qualitatively different with regards to the Weyl features.\n\nTo understand better the lattice phase transition, we simulated the switching \"path\" from the undistorted phase (no Ta-tetramerization) to the CDW-I LLSS Tatetramerization phase by a simple linear interpolation of the Γ 4 mode, as shown in Fig. 5(a). The transition energy wall between the two phases was computed to be ≈ 40 meV/Ta. However, note that we fixed the crystal constants to obtain the switching path and this would increase the energy barrier. In addition, we also compare the switching path between CDW-I and CDW-II phases and here the energy wall is much smaller ≈ 0.4 meV/Ta [see Fig. 5(b)], which corresponds to a small monoclinic distortion.\n\n\nV. ADDITIONAL DISCUSSION\n\nThe point groups of (TaSe 4 ) 2 I for the undistorted (space group No. 97), CDW-I (space group No. 22), and CDW-II (space group No. 5) phases are 422 (D 4 ), 222 (D 2 ), and 2 (C 2 ), respectively. In all these cases, optical isomerism is allowed in this quasi-one-dimensional system. For this reason, optical activity effects are expected by considering their chiral point group. We thus suggest that further experimental and theoretical works should focus on the optical isomerism of this material.\n\nFurthermore, for the heavy element Ta, the spin-orbit coupling for the 5d orbitals is expected to be large enough to induce a band splitting. Since the point groups of (TaSe 4 ) 2 I for both the non-CDW and the CDW phases are the non-centrosymmetric chiral group, then a Rashba-like band splitting is also expected in this system. We calculated the band structures for the two CDW phases with SOC incorporated, with results in Fig. 6. There it is shown that the splitted bands resemble Rashba splitting along the high symmetry directions, as shown in Figs. 6(a-b). Moreover, we also show the bands of CDW-I phases near the Γ and T points that indicate a clear Rashba-like band splitting, as in Fig. 6(c). Similar Rashba-like bands splittings, see Fig. 6(d), indicate that these effects can also occur in the CDW-II phase. This is also reasonable since the monoclinic distortion is too small to completely alter the dominant physical properties. In fact, Rashba-like bands also develop along the Γ to Z and N to P paths in the non-CDW parent phase. These issues deserve further studies in the future.\n\n\nVI. CONCLUSIONS\n\nIn summary, here the quasi-one-dimensional compound (TaSe 4 ) 2 I has been systematically studied using first-principles calculations. A strongly anisotropic metallic band structure was observed in the undistorted Weyl phase (non-CDW state), in agreement with previous experimental and theoretical works. In addition, using the group symmetry analysis and DFT calculations, two Ta-tetramerization CDW phases have been found here to become the stable low-temperature structure. The semimetal to insulator transition is induced by the Fermi-surface-driven instability, resulting in a Peierls transition that opens a gap (∼ 0.2 eV). In addition, based on the linear interpolation of the Γ 4 mode, the transition energy barrier is estimated to be ≈ 40 meV, corresponding to a high transition temperature. Furthermore, for the heavy element Ta, the spin-orbit coupling for its 5d orbitals should be robust, causing a delicate but notice- able Rashba splitting of the conducting bands around the Γ point in this polar system (chiral). Our results successfully reproduce the phase transition induced by the CDW instability and provide additional insight that may motivate further theoretical and experimental efforts. In future work we will explore the possibility of Rashba effects in these low-dimensional chiral materials. \n\nFIG. 1 .\n1Schematic crystal structure of the conventional cell of (TaSe4)2I with the convention: Blue = Ta; green = Se; purple = I. (a-b) The crystal structure is shown from the perspectives of the (a) bc plane [(100) direction] and (b) top [(001) direction]. (c-d) The two different Ta-tetramerization patterns along the chains.\n\nFIG. 2 .\n2Band structure of the undistorted (TaSe4)2I along a high symmetry path. (a) Without SOC. (b) With SOC.\n\nFIG. 3 .\n3Phonon spectrum of (TaSe4)2I for the undistorted (No. 97), CDW-I (No. 22), and CDW-II (No. 5) phases.\n\nFIG. 4 .\n4(a) Schematic diagram of the DOS semimetal to insulator transition. (b-d) DOS (without SOC) near the Fermi level for the undistorted parent (non-CDW), orthorhombic (CDW-I), and monoclinic (CDW-II) phases, respectively. Black = Total; red = Ta; blue = I; green = Se.\n\nFIG. 5 .\n5Figs. 4(c-d) the DOS for the distorted CDW phases showing that a gap opens in this case. By considering the SOC effect, the expected gap becomes approximately 0.(a) The energy well of the CDW-I and undistorted phases corresponding to the Γ4 mode vs distortion. (b) The energies well of the CDW-I and CDW-II phases.\n\nFIG. 6 .\n6Band structure of (TaSe4)2I with the effect of SOC shown along high symmetry paths. The coordinates of the high symmetry points in the bulk Brillouin zone (BZ) are obtained from the Seek-path software[46]. (a) and (c) are for the orthorhombic F 222 CWD-I phase. (b) and (d) are for the monoclinic C2 CDW-II phase.\n\n\nACKNOWLEDGMENTS E.D. and A.M. are supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Materials Sciences and Engineering Division. S.D., Y.Z., and L.F.L. are supported by the National Natural Science Foundation of China (Grant Nos. 11834002 and 11674055). L.F.L. and Y.Z. are supported by the China Scholarship Council. Y.Z. is also supported by the Scientific Research Foundation of Graduate School of Southeast University (Grant No. YBPY1826). All the calculations were carried out at the Advanced Computing Facility (ACF) of the University of Tennessee Knoxville.\n\nDensity Waves in Solids (Perseus. G Grüner, Cambridge, MAG. Grüner, Density Waves in Solids (Perseus, Cam- bridge, MA, 2000).\n\n. P Monceau, 10.1080/00018732.2012.719674Adv. Phys. 61325P. Monceau, Adv. Phys. 61, 325 (2012).\n\n. M Grioni, S Pons, E Frantzeskakis, 10.1088/0953-8984/21/2/023201J. Phys.: Condens. Matter. 2123201M. Grioni, S. Pons and E. Frantzeskakis, J. Phys.: Con- dens. Matter 21, 023201 (2009).\n\n. Z Wang, S.-C Zhang, https:/link.aps.org/doi/10.1103/PhysRevB.87.161107Phys. Rev. B. 87161107Z. Wang and S.-C. Zhang, Phys. Rev. B 87, 161107(R) (2013).\n\n. S M Soto, C P Slichter, A M Kini, H H Wang, U Geiser, J M Williams, https:/link.aps.org/doi/10.1103/PhysRevB.52.10364Phys. Rev. B. 5210364S. M. De Soto, C. P. Slichter, A. M. Kini, H. H. Wang, U. Geiser and J. M. Williams, Phys. Rev. B 52, 10364 (1995).\n\n. E Dagotto, J Riera, D Scalapino, 10.1103/PhysRevB.45.5744Phys. Rev. B. 455744E. Dagotto, J. Riera, and D. Scalapino, Phys. Rev. B 45, 5744(R) (1992).\n\n. E Dagotto, T M Rice, 10.1126/science.271.5249.618Science. 271618E. Dagotto and T. M. Rice, Science 271, 618 (1996).\n\n. E Dagotto, 10.1088/0034-4885/62/11/202Rep. Prog. Phys. 621525E. Dagotto, Rep. Prog. Phys. 62, 1525 (1999).\n\n. M Uehara, T Nagata, J Akimitsu, H Takahashi, N Mori, K Kinoshita, 10.1143/JPSJ.65.2764J. Phys. Soc. Jpn. 652764M. Uehara, T. Nagata, J. Akimitsu, H. Takahashi, N. Mori, and K. Kinoshita, J. Phys. Soc. Jpn. 65, 2764 (1996).\n\n. H Takahashi, A Sugimoto, Y Nambu, T Yamauchi, Y Hirata, T Kawakami, M Avdeev, K Matsubayashi, F Du, C Kawashima, H Soeda, S Nakano, Y Uwatoko, Y Ueda, T J Sato, K Ohgushi, 10.1038/nmat4351Nat. Mater. 141008H. Takahashi, A. Sugimoto, Y. Nambu, T. Yamauchi, Y. Hirata, T. Kawakami, M. Avdeev, K. Matsubayashi, F. Du, C. Kawashima, H. Soeda, S. Nakano, Y. Uwatoko, Y. Ueda, T. J. Sato and K. Ohgushi, Nat. Mater. 14, 1008 (2015).\n\n. Y Zhang, L F Lin, J J Zhang, E Dagotto, S Dong, 10.1103/PhysRevB.95.115154Phys. Rev. B. 95115154Y. Zhang, L. F. Lin, J. J. Zhang, E. Dagotto, and S. Dong, Phys. Rev. B 95, 115154 (2017).\n\n. J.-J Ying, H C Lei, C Petrovic, Y.-M Xiao, V.-V Struzhkin, 10.1103/PhysRevB.95.241109Phys. Rev. B. 95R241109J.-J. Ying, H. C. Lei, C. Petrovic, Y.-M. Xiao and V.-V. Struzhkin, Phys. Rev. B 95, 241109(R) (2017).\n\n. Y Zhang, L F Lin, J J Zhang, E Dagotto, S Dong, 10.1103/PhysRevB.97.045119Phys. Rev. B. 9745119Y. Zhang, L. F. Lin, J. J. Zhang, E. Dagotto, and S. Dong, Phys. Rev. B 97, 045119 (2018).\n\n. Y Zhang, L F Lin, A Moreo, S Dong, E Dagotto, 10.1103/PhysRevB.100.184419Phys. Rev. B. 100184419Y. Zhang, L.F. Lin, A. Moreo, S. Dong, and E. Dagotto, Phys. Rev. B 100, 184419 (2019).\n\n. L F Lin, Y Zhang, A Moreo, E Dagotto, S Dong, 10.1103/PhysRevMaterials.3.111401Phys. Rev. Mater. 3111401L. F. Lin, Y. Zhang, A. Moreo, E. Dagotto, and S. Dong, Phys. Rev. Mater. 3, 111401(R) (2019).\n\n. L F Lin, Y Zhang, A Moreo, E Dagotto, S Dong, 10.1103/PhysRevLett.123.067601Phys. Rev. Lett. 12367601L. F. Lin, Y. Zhang, A. Moreo, E. Dagotto, and S. Dong, Phys. Rev. Lett. 123, 067601 (2019).\n\n. M C Cross, D S Fisher, 10.1103/PhysRevB.19.402Phys. Rev. B. 19402M. C. Cross and D. S. Fisher, Phys. Rev. B 19, 402 (1979).\n\n. Y J Choi, H T Yi, S Lee, Q Huang, V Kiryukhin, S.-W Cheong, 10.1103/PhysRevLett.100.047601Phys. Rev. Lett. 10047601Y. J. Choi, H. T. Yi, S. Lee, Q. Huang, V. Kiryukhin, and S.-W. Cheong, Phys. Rev. Lett. 100, 047601 (2008).\n\n. Y Zhang, L F Lin, A Moreo, S Dong, E , arXiv:1912.07749Dagotto arXiv preprintY. Zhang, L. F. Lin, A. Moreo, S. Dong, and E. Dagotto arXiv preprint arXiv:1912.07749.\n\n. S Dong, J M Liu, E Dagotto, 10.1103/PhysRevLett.113.187204Phys. Rev. Lett. 113187204S. Dong, J. M. Liu, and E. Dagotto, Phys. Rev. Lett. 113, 187204 (2014).\n\n. P Gressier, A Meerschaut, L Guemas, J Rouxel, P Monceau, 10.1016/0022-4596(84)90327-XJournal of Solid State Chemistry. 51141P. Gressier, A. Meerschaut, L. Guemas, J. Rouxel and P. Monceau, Journal of Solid State Chemistry 51, 141 (1984).\n\n. P Gressier, L Guemas, A Meerschaut, 10.1107/S0567740882010176Acta Cryst. B. 382877P. Gressier and L. Guemas and A. Meerschaut, Acta Cryst. B 38, 2877 (1982).\n\nSolid State Commun. H Fujishita, M Sato, S Hoshino, 10.1016/0038-1098(84)90576-349313H. Fujishita, M. Sato and S. Hoshino, Solid State Com- mun. 49, 313 (1984).\n\n. S Van Smaalen, E J Lam, J Lüdecke, 10.1088/0953-8984/13/44/308J. Phys. Condens. Matter. 139923S. van Smaalen, E. J. Lam and J. Lüdecke, J. Phys. Con- dens. Matter 13, 9923 (2001).\n\n. V Favre-Nicolin, S Bos, J E Lorenzo, J.-L Hodeau, J.-F Berar, P Monceau, R Currat, F Levy, H Berger, https:/link.aps.org/doi/10.1103/PhysRevLett.87.015502Phys. Rev. Lett. 8715502V. Favre-Nicolin, S. Bos, J. E. Lorenzo, J.-L. Hodeau, J.- F. Berar, P. Monceau, R. Currat, F. Levy and H. Berger, Phys. Rev. Lett. 87, 015502 (2001).\n\n. J E Lorenzo, R Currat, P Monceau, B Hennion, H Berger, F Levy, 10.1088/0953-8984/10/23/010J. Phys. Condens. Matter. 105039J. E. Lorenzo, R. Currat, P. Monceau, B. Hennion, H. Berger and F. Levy, J. Phys. Condens. Matter 10, 5039 (1998).\n\n. J Voit, L Perfetti, F Zwick, H Berger, G Margaritondo, G Grüner, H Höchst, M Grioni, Science. 290501J. Voit, L. Perfetti, F. Zwick, H. Berger, G. Margari- tondo, G. Grüner, H. Höchst and M. Grioni, Science 290, 501 (2000).\n\n. C Tournier-Colletta, L Moreschini, G Autès, S Moser, A Crepaldi, H Berger, A L Walter, K S Kim, A Bostwick, P Monceau, E Rotenberg, O V Yazyev, M Grioni, https:/link.aps.org/doi/10.1103/PhysRevLett.110.236401Phys. Rev. Lett. 110236401C. Tournier-Colletta, L. Moreschini, G. Autès, S. Moser, A. Crepaldi, H. Berger, A. L. Walter, K. S. Kim, A. Bostwick, P. Monceau, E. Rotenberg, O. V. Yazyev and M. Grioni, Phys. Rev. Lett. 110, 236401 (2013).\n\n. P Gressier, M , H Whangbo, A Meerschaut, J Rouxel, 10.1021/ic00177a011Inorg. Chem. 231221P. Gressier, M, H. Whangbo, A. Meerschaut and J. Rouxel, Inorg. Chem. 23, 1221 (1984).\n\n. W Shi, B J Wieder, H L Meyerheim, Y Sun, Y Zhang, Y Li, L Shen, Y Qi, L Yang, K Jena, P Werner, K Koepernik, S Parkin, Y Chen, C Felser, B A Bernevig, Z Wang, arXiv:1909.04037arXiv preprintW. Shi, B. J. WiedeR, H.L. Meyerheim, Y. Sun, Y. Zhang, Y. Li, L. Shen, Y. Qi, L. Yang, K. Jena, P. Werner, K. Koepernik, S. Parkin, Y. Chen, C. Felser, B. A. Bernevig and Z. Wang, arXiv preprint arXiv:1909.04037.\n\n. X.-P Li, K Deng, B Fu, Y Li, D Ma, J Han, J Zhou, S Zhou, Y Yao, arXiv:1909.12178arXiv preprintX.-P. Li, K. Deng, B. Fu, Y. Li, D. Ma, J. Han, J. Zhou, S. Zhou and Y. Yao, arXiv preprint arXiv:1909.12178.\n\n. J Gooth, B Bradlyn, S Honnali, C Schindler, N Kumar, J Noky, Y Qi, C Shekhar, Y Sun, Z Wang, B A Bernevig, C Felser, 10.1038/s41586-019-1630-4Nature. 575315J. Gooth, B. Bradlyn, S. Honnali, C. Schindler, N. Ku- mar, J. Noky, Y. Qi, C. Shekhar, Y. Sun, Z. Wang, B. A. Bernevig and C. Felser , Nature 575, 315 (2019).\n\n. D Schmeltzer, arXiv:1910.13392arXiv preprintD. Schmeltzer, arXiv preprint arXiv:1910.13392.\n\n. D Di Sante, P Barone, A Stroppa, K F Garrity, D Vanderbilt, S Picozzi, https:/link.aps.org/doi/10.1103/PhysRevLett.117.076401Phys. Rev. Lett. 11776401D. Di Sante, P. Barone, A. Stroppa, K. F. Garrity, D. Vanderbilt and S. Picozzi, Phys. Rev. Lett. 117, 076401 (2016).\n\n. G Kresse, J Hafner, 10.1103/PhysRevB.47.558Phys. Rev. B. 47558G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).\n\n. G Kresse, J Furthmüller, 10.1103/PhysRevB.54.1169Phys. Rev. B. 5411169G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).\n\n. P E Blöchl, 10.1103/PhysRevB.50.17953Phys. Rev. B. 5017953P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).\n\n. J P Perdew, A Ruzsinszky, G I Csonka, O A Vydrov, G E Scuseria, L A Constantin, X Zhou, K Burke, 10.1103/PhysRevLett.100.136406Phys. Rev. Lett. 100136406J. P. Perdew and A. Ruzsinszky, and G. I. Csonka and O. A. Vydrov and G. E. Scuseria and L. A. Constantin and X. Zhou and K. Burke, Phys. Rev. Lett. 100, 136406 (2008).\n\n. L Chaput, A Togo, I Tanaka, G Hug, 10.1103/PhysRevB.84.094302Phys. Rev. B. 8494302L. Chaput, A. Togo, I. Tanaka, and G. Hug, Phys. Rev. B 84, 094302 (2011).\n\n. A Togo, I Tanaka, 10.1016/j.scriptamat.2015.07.021Scr. Mater. 1081A. Togo, I. Tanaka, Scr. Mater. 108, 1 (2015).\n\nFor more results. http:/link.aps.org/supplemental/10.1103/PhysRevB.xx/xxxxxxsee Supplemental Material atFor more results, see Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevB.xx/xxxxxx.\n\n. E Kroumova, M I Aroyo, J M Perez-Mato, A Kirov, C Capillas, S Ivantchev, H Wondratschek, 10.1080/0141159031000076110Phase Transitions. 76155E. Kroumova, M. I. Aroyo, J. M. Perez-Mato, A. Kirov, C. Capillas, S. Ivantchev, and H. Wondratschek, Phase Transitions 76, 155 (2003).\n\nIn the B1 representation, two Ta(1) atoms (4c site) move towards the connected Ta(2) atoms in chain-A and move away from the connected Ta(2) atoms in chain-B, resulting in a LLSS Ta-tetramerization pattern. Correspondingly, in the B2 representation, the Ta(2) atoms (4d site) move towards the connected Ta(1) atoms in chain-A and move away from the connected Ta(1) atoms in chain-B. Both the B1 and B2 representations would induce a simlar LLSS Ta-tetramerization CDW phase. The two Ta sites (4c and 4d) form a pattern Ta. Ta(1)-Ta(2) in the chain direction. as shown Fig. 1(cThe two Ta sites (4c and 4d) form a pattern Ta(1)-Ta(2)- Ta(1)-Ta(2) in the chain direction. In the B1 represen- tation, two Ta(1) atoms (4c site) move towards the con- nected Ta(2) atoms in chain-A and move away from the connected Ta(2) atoms in chain-B, resulting in a LLSS Ta-tetramerization pattern. Correspondingly, in the B2 representation, the Ta(2) atoms (4d site) move towards the connected Ta(1) atoms in chain-A and move away from the connected Ta(1) atoms in chain-B. Both the B1 and B2 representations would induce a simlar LLSS Ta-tetramerization CDW phase, as shown Fig. 1(c).\n\n. D Orobengoa, C Capillas, M I Aroyo, J M Perez-Mato, 10.1107/S0021889809028064J. Appl. Crystallogr. 42820D. Orobengoa, C. Capillas, M. I. Aroyo, and J. M. Perez- Mato, J. Appl. Crystallogr. 42, 820 (2009).\n\n. J Perez-Mato, D Orobengoa, M I Aroyo, 10.1107/S0108767310016247Acta Crystallogr. A. 66558J. Perez-Mato, D. Orobengoa, and M. I. Aroyo, Acta Crystallogr. A 66, 558 (2010).\n\n. Y Hinuma, G Pizzi, Y Kumagai, F Oba, I Tanaka, 10.1016/j.commatsci.2016.10.015Comput. Mater. Sci. 113221Y. Hinuma, G. Pizzi, Y. Kumagai, F. Oba and I. Tanaka, Comput. Mater. Sci. 113, 221 (2016).\n\nHere, for completeness we also used 1×1×2 and 1×1×4 supercells based on the conventional cell, but none of our physical conclusions changed, as shown in the SM. 41Here, for completeness we also used 1×1×2 and 1×1×4 supercells based on the conventional cell, but none of our physical conclusions changed, as shown in the SM [41].\n"},"corpus_id":{"kind":"number","value":211204724,"string":"211,204,724"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2002.08826\",\n \"mag\": 3024063761,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1103/physrevb.101.174106\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/2002.08826v2.pdf\"\n ],\n \"pdf_sha\": \"403523603f500908370ccf002e97635883606d01\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/2002.08826\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"First-principles study of the low-temperature charge density wave phase in the quasi-one-dimensional Weyl chiral compound (TaSe 4 ) 2 I"},"authors":{"kind":"list like","value":["Yang Zhang \nDepartment of Physics and Astronomy\nUniversity of Tennessee\n37996KnoxvilleTNUSA\n\nSchool of Physics\nSoutheast University\n211189NanjingChina","Ling-Fang Lin \nDepartment of Physics and Astronomy\nUniversity of Tennessee\n37996KnoxvilleTNUSA\n\nSchool of Physics\nSoutheast University\n211189NanjingChina","Adriana Moreo \nDepartment of Physics and Astronomy\nUniversity of Tennessee\n37996KnoxvilleTNUSA\n\nMaterials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA","Shuai Dong \nSchool of Physics\nSoutheast University\n211189NanjingChina","Elbio Dagotto \nDepartment of Physics and Astronomy\nUniversity of Tennessee\n37996KnoxvilleTNUSA\n\nMaterials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA"],"string":"[\n \"Yang Zhang \\nDepartment of Physics and Astronomy\\nUniversity of Tennessee\\n37996KnoxvilleTNUSA\\n\\nSchool of Physics\\nSoutheast University\\n211189NanjingChina\",\n \"Ling-Fang Lin \\nDepartment of Physics and Astronomy\\nUniversity of Tennessee\\n37996KnoxvilleTNUSA\\n\\nSchool of Physics\\nSoutheast University\\n211189NanjingChina\",\n \"Adriana Moreo \\nDepartment of Physics and Astronomy\\nUniversity of Tennessee\\n37996KnoxvilleTNUSA\\n\\nMaterials Science and Technology Division\\nOak Ridge National Laboratory\\n37831Oak RidgeTNUSA\",\n \"Shuai Dong \\nSchool of Physics\\nSoutheast University\\n211189NanjingChina\",\n \"Elbio Dagotto \\nDepartment of Physics and Astronomy\\nUniversity of Tennessee\\n37996KnoxvilleTNUSA\\n\\nMaterials Science and Technology Division\\nOak Ridge National Laboratory\\n37831Oak RidgeTNUSA\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Using ab initio density functional theory, we study the lattice phase transition of quasi-onedimensional (TaSe4)2I. In the undistorted state, the strongly anisotropic semimetal band structure presents two non-equivalent Weyl points. In previous efforts, two possible Ta-tetramerization patterns were proposed to be associated with the low-temperature structure. Our phonon calculations indicate that the orthorhombic F 222 CDW-I phase is the most likely ground state for this quasi-onedimensional system. In addition, the monoclinic C2 CDW-II phase may also be stable according to the phonon dispersion spectrum. Since these two phases have very similar energies in our DFT calculations, both these Ta-tetramerization distortions likely compete or coexist at low temperatures. The semimetal to insulator transition is induced by a Fermi-surface-driven instability that supports the Peierls scenario, which affects the Weyl physics developed above TCDW. Furthermore, the spin-orbit coupling generates Rashba-like band splittings in the insulating CDW phases."},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":"One-dimensional (1D) systems continue attracting considerable attention due to their rich physical properties and reduced dimensional phase space. In particular, in low dimension systems the electron-electron, phononphonon, electron-phonon, and spin-phonon couplings are strongly enhanced by the interactions between transition metal ions [1,2]. Under some conditions, free carriers may form charge density wave (CDW) or spin density wave (SDW) states due to the partial or complete condensation of excitations, properties which are physically interesting and important for possible applications [3][4][5].\nSeveral 1D bulk compounds have been widely studied. For example, with focus on the electronic correlation effects the Cu-oxide 1D ladders were theoretically predicted and experimentally confirmed to have a spin gap and become superconducting [6][7][8][9]. Recently, an analogous behavior was shown to develop in iron ladders BaFe 2 X 3 (X=S or Se) that were reported to be superconducting at high pressure [10][11][12][13][14]. By considering phononic modes or spin-phonon instability, ferroelectric or multiferroelectric states were predicted in some 1D systems [15][16][17][18][19][20].\n(TaSe 4 ) 2 I is a typical paradigmatic quasi-onedimensional material that has been frequently studied for more than thirty years [21,22]. This system undergoes a Peierls phase transition at 263 K, the socalled CDW transition, accompanied by an incommensurate structural distortion at low temperatures [23][24][25]. Above the critical temperature T CDW = 263 K, the nearest neighbor (NN) Ta-Ta distances (d Ta−Ta = 3.206 A at room temperature [22,26]) are identical along each chain. As expected, the CDW instability breaks the symmetry of the isometric chains leading to Tatetramerization modes [25,26]. Related experiments re-vealed that the Ta-tetramerization periodicity may be c = 4d Ta−Ta [24,25,27]. In addition, this system is also considered to be a Weyl semimetal [30,31]. More recently, possible axion physics was proposed for this special chiral Weyl material [32,33]. It also should be noted that Rashba-like band splittings may also occur in a chiral system [34]. Since the spin-orbit coupling (SOC) for Ta's 5d orbitals can be robust, such Rashba splitting is expected. All these developments in (TaSe 4 ) 2 I provide a unique and promising platform to display different interesting physical properties in a single material.\nIn previous works, two Ta-tetramerization patterns were proposed for the low-temperature structure, corresponding to the one-dimensional B 1 and B 2 representations [26]. Since the Ta-tetramerization distortion is small, it can only be observed via weak reflections, and for this reason the lattice phase transition is still uncertain. For this reason, from the theoretical perspective ab initio phonon calculations for the superstructure could play an important role in clarifying the real mechanism of lattice structural transition in this compound.\nIn the present publication, we perform firstprinciples density functional theory (DFT) calculations for (TaSe 4 ) 2 I. First, our theoretical results indicate a strongly anisotropic electronic structure for the undistorted phase, corresponding to its quasi-one-dimensional geometry. Based on phononic dispersion calculations, we found that (TaSe 4 ) 2 I contains phonon softening instabilities in the Γ 4 mode, resulting into an orthorhombic distortion, which could correspond to the B 1 or B 2 representations. By considering the combined B 1 +B 2 symmetry breaking, the system would decrease to the monoclinic C2 phase (No. 5) with a pattern along chains involving four different NN Ta-Ta distances. Our DFT calculations suggest that the CDW-I lattice phase is the most likely ground state for this system. Since the energy difference between the two CDW lattices, B 1 or B 2 , can not be distinguished unambiguously, these two Ta-tetramerization phases are competitive at low temperatures. Our results also support that the semimetal-insulator transition is driven by the Fermi surface instability. Furthermore, for the heavy element Ta, moderate Rashba-like splitting bands are found in the low temperature CDW phases."},{"title":"II. METHODS AND MODEL SYSTEM","content":"In this publication, DFT calculations were performed using the projector augmented wave (PAW) pseudopotentials with the Perdew-Burke-Ernzerhof format revised for solid exchange functional (PBEsol), as implemented in the Vienna ab initio Simulation Package (VASP) code [35][36][37][38]. The phonon spectra were calculated using the finite displacement approach and analyzed by the PHONONPY software [39,40]. Details are in the Supplemental Material (SM) [41].\nUnder ambient conditions, (TaSe 4 ) 2 I forms a quasione-dimensional body-centered tetragonal chiral crystal structure with the space group I422 (No. 97). The lattice parameters are a = 9.531Å and c = 12.824Å [22]. There are two adjacent TaSe 4 chains along the c-axis in the conventional cell, where iodine atoms are located between chains, as shown in Fig. 1(a). In each TaSe 4 chain, the Ta atoms are aligned equidistantly and the Se atoms form a \"screw\" arrangement along the c-axis [see Fig. 1(b)]. Previous experimental results suggested that the modulated structure of (TaSe 4 ) 2 I can be split into two parts [26] below the transition temperature T CDW = 263 K.\nSince the equal displacements of all atoms would not change the space-group symmetry, we will focus on the modulation of the Ta-tetramerized atoms, which corre- sponds to the parallel component of the k F vector (q CDW || ) along the Ta-chain direction. By considering the space group symmetry I422 (No. 97), the Ta atoms occupy two different Wyckoff sites (4c and 4d). Based on the Raman active modes analysis [42], the corresponding Raman active modes of the 4c and 4d sites have B 2 and B 1 symmetry, respectively. Since the CDW instability corresponds to the Ta-tetramerization modes [25,26], a long-longshort-short (LLSS) Ta-tetramerization CDW-I pattern is expected by only considering the lattice dynamical B 1 or B 2 Ta tetramerization, as shown in Fig. 1(c) [43]. If, instead, we consider the combined B 1 +B 2 Ta tetramerization, there would be four different nearest-neighbor Ta-Ta distances in the chain [see Fig. 1(d)]. In addition, the AMPLIMODES software was employed to perform the group theoretical analysis [44,45], indicating that the CDW-I phase would induce the orthorhombic distortion from the undistorted tetragonal phase with phonon mode Γ 4 . The same analysis for the CDW-II phase also indicates that a monoclinic distortion is obtained with phonon modes Γ 5 and Γ 4 corresponding to the undistorted tetragonal phase and the CDW-I orthorhombic phase, respectively."},{"title":"III. NON-CDW PHASE","content":"Before discussing the CDW phase, we will consider the electronic structure corresponding to the non-CDW phase of (TaSe 4 ) 2 I, which is displayed in Fig. 2. Here, we use the primitive cell instead of the conventional cell to calculate the electronic structure of (TaSe 4 ) 2 I. The volume of the primitive cell is half of the conventional cell with the primitive lattice: a 1 =(−a/2, a/2, c/2), a 2 =(a/2, a/2, c/2), and a 3 =(a/2, a/2, −c/2), where a and c are the conventional cell-lattice constants.\nFirst, we will present the band structure of the undistorted phase of (TaSe 4 ) 2 I without SOC effect. This band structure clearly displays strong anisotropic metallic behavior, as presented in Fig. 2(a). The band is more dispersive from Γ to Z along the chains than along other directions (i.e. N to P), which is compatible with the presence of quasi-one-dimensional chains along the k z axis. As shown in Fig. 2(a), the Fermi level is dominated by the Ta d 3z 2 −r 2 orbital along the chains, in agreement with previous studies [25,29]. The crossing point along the chain direction (Γ to Z) is not at π/c but at the incommensurate k F where 2k F =q CDW || . Next, we introduce the SOC effect to the undistorted state and now the bands begin to split as displayed in Fig. 2(b), which is consistent with other results [30,31]. In principle, as suggested by the band structure, (TaSe 4 ) 2 I should also display two interesting Weyl points with one along the Γ to Z path and another along the N to P path, although this aspect requires further detailed calculations and discussion beyond the scope of this publication. However, note that that the Weyl physics of this compound has already been studied in detail in recent publications [30,31]. For this reason, here we primarily focus on the phononic aspects."},{"title":"IV. CDW INSTABILITY","content":"It is well known that the CDW instability is accompanied by an structural atomic rearrangement, related to the phonon mode instability. To understand the structural phase transition in (TaSe 4 ) 2 I, we performed phononic dispersion calculations for both the non-CDW and CDW phases. Figure 3(a) indicates the phonon dispersion spectrum that presents an imaginary frequency appearing at Γ for the undistorted structure. The Tatetramerization arising from this mode corresponds to the CDW-I phase. According to group theory analysis using the AMPLIMODES software [44,45], this spontaneous distortion mode is a Γ 4 mode resulting in a twice larger periodicity (4d Ta−Ta ) along the TaSe 4 chain [47]. In addition, the phonon softening mode also corresponds to the transverse component of the incommensurate CDW wavevector parallel to the chain direction (Γ to Z), as in X-ray diffraction and ARPES data Here, a 2 × 2 × 2 supercell was selected from the original primitive cell in our calculations. The coordinates of the high symmetry points in the bulk Brillouin zone (BZ) are obtained using the Seek-path software [46]. [26,28]. We fully relaxed the crystal lattice from the original tetragonal I422 phase along these mode displacements, and then obtained an orthorhombic F 222 state (No. 22). Based on the relaxed orthorhombic structure, we found the LLSS Ta-tetramerization configuration with the difference between the long NN and short NN Ta-Ta distances being about ∼ 0.083Å. Furthermore, we also calculated the phonon dispersion spectrum for the orthorhombic F 222 phase (No. 22), and in this case no imaginary frequency mode was found, as shown in Fig. 3(b). By extracting the lowest-energy phonon mode of the orthorhombic F 222 phase and applying to the F 222 phase, the symmetry of the crystal structure further decreases to the monoclinic C2 phase (No. 5) with a pattern of four different NN Ta-Ta distances along the chain, corresponding to the B 1 +B 2 representations. The AMPLIMODES software [44,45] indicates that this is the Γ 4 mode distortion for the orthorhombic F 222 phase. After full lattice relaxations for the monoclinic C2 phase (No. 5), we find that the difference of the two long or short Ta-Ta distances are 0.003Å and 0.002 A, respectively. Figure 3(c) indicates that the mono- clinic CDW-II phase is also dynamically stable because no imaginary frequency mode is obtained in the phonon dispersion calculation. Here, it should be noticed that the difference of energies of the orthorhombic and monoclinic cases is quite small (∼ 0.1 meV/Ta) which is beyond the accuracy of DFT. In this context, then those two phases cannot be distinguished by considering such small monoclinic distortion and almost negligible energy difference. Additional DFT results for those two phases can be found in the SM [41].\nNext, let us focus on the semimetal to insulator transition. It is well known that there is a very small overlap between the bottom of the conduction band and the top of the valence band in the semimetallic material, resulting in a negligible density of states (DOS) without SOC at the Fermi level, as shown in Fig. 4(a). The DOS of the undistorted phase (No. 97) is here presented in Fig. 4(b) indicating a semimetal with indeed quite a small density at the Fermi level. Furthermore, we also present in at the crossing point along Γ to Z corresponding to the Brillouin path of undistorted phases. Hence, our results are in agreement with the physical picture of a Fermisurface-driven instability resulting in the Peierls transition. Furthermore, strictly speaking the two Weyl points disappear in the low-temperature CDW phases due to the opening of a gap. This suggests that Weyl physics could be altered below T CDW . Because the phase transition seems to be of first order, as suggested by neutron scattering studies [26], the physics below and above T CDW could be qualitatively different with regards to the Weyl features.\nTo understand better the lattice phase transition, we simulated the switching \"path\" from the undistorted phase (no Ta-tetramerization) to the CDW-I LLSS Tatetramerization phase by a simple linear interpolation of the Γ 4 mode, as shown in Fig. 5(a). The transition energy wall between the two phases was computed to be ≈ 40 meV/Ta. However, note that we fixed the crystal constants to obtain the switching path and this would increase the energy barrier. In addition, we also compare the switching path between CDW-I and CDW-II phases and here the energy wall is much smaller ≈ 0.4 meV/Ta [see Fig. 5(b)], which corresponds to a small monoclinic distortion."},{"title":"V. ADDITIONAL DISCUSSION","content":"The point groups of (TaSe 4 ) 2 I for the undistorted (space group No. 97), CDW-I (space group No. 22), and CDW-II (space group No. 5) phases are 422 (D 4 ), 222 (D 2 ), and 2 (C 2 ), respectively. In all these cases, optical isomerism is allowed in this quasi-one-dimensional system. For this reason, optical activity effects are expected by considering their chiral point group. We thus suggest that further experimental and theoretical works should focus on the optical isomerism of this material.\nFurthermore, for the heavy element Ta, the spin-orbit coupling for the 5d orbitals is expected to be large enough to induce a band splitting. Since the point groups of (TaSe 4 ) 2 I for both the non-CDW and the CDW phases are the non-centrosymmetric chiral group, then a Rashba-like band splitting is also expected in this system. We calculated the band structures for the two CDW phases with SOC incorporated, with results in Fig. 6. There it is shown that the splitted bands resemble Rashba splitting along the high symmetry directions, as shown in Figs. 6(a-b). Moreover, we also show the bands of CDW-I phases near the Γ and T points that indicate a clear Rashba-like band splitting, as in Fig. 6(c). Similar Rashba-like bands splittings, see Fig. 6(d), indicate that these effects can also occur in the CDW-II phase. This is also reasonable since the monoclinic distortion is too small to completely alter the dominant physical properties. In fact, Rashba-like bands also develop along the Γ to Z and N to P paths in the non-CDW parent phase. These issues deserve further studies in the future."},{"title":"VI. CONCLUSIONS","content":"In summary, here the quasi-one-dimensional compound (TaSe 4 ) 2 I has been systematically studied using first-principles calculations. A strongly anisotropic metallic band structure was observed in the undistorted Weyl phase (non-CDW state), in agreement with previous experimental and theoretical works. In addition, using the group symmetry analysis and DFT calculations, two Ta-tetramerization CDW phases have been found here to become the stable low-temperature structure. The semimetal to insulator transition is induced by the Fermi-surface-driven instability, resulting in a Peierls transition that opens a gap (∼ 0.2 eV). In addition, based on the linear interpolation of the Γ 4 mode, the transition energy barrier is estimated to be ≈ 40 meV, corresponding to a high transition temperature. Furthermore, for the heavy element Ta, the spin-orbit coupling for its 5d orbitals should be robust, causing a delicate but notice- able Rashba splitting of the conducting bands around the Γ point in this polar system (chiral). Our results successfully reproduce the phase transition induced by the CDW instability and provide additional insight that may motivate further theoretical and experimental efforts. In future work we will explore the possibility of Rashba effects in these low-dimensional chiral materials."},{"title":"FIG. 1 .","content":""},{"title":"FIG. 2 .","content":""},{"title":"FIG. 3 .","content":""},{"title":"FIG. 4 .","content":""},{"title":"FIG. 5 .","content":""},{"title":"FIG. 6 .","content":""}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"One-dimensional (1D) systems continue attracting considerable attention due to their rich physical properties and reduced dimensional phase space. In particular, in low dimension systems the electron-electron, phononphonon, electron-phonon, and spin-phonon couplings are strongly enhanced by the interactions between transition metal ions [1,2]. Under some conditions, free carriers may form charge density wave (CDW) or spin density wave (SDW) states due to the partial or complete condensation of excitations, properties which are physically interesting and important for possible applications [3][4][5].\\nSeveral 1D bulk compounds have been widely studied. For example, with focus on the electronic correlation effects the Cu-oxide 1D ladders were theoretically predicted and experimentally confirmed to have a spin gap and become superconducting [6][7][8][9]. Recently, an analogous behavior was shown to develop in iron ladders BaFe 2 X 3 (X=S or Se) that were reported to be superconducting at high pressure [10][11][12][13][14]. By considering phononic modes or spin-phonon instability, ferroelectric or multiferroelectric states were predicted in some 1D systems [15][16][17][18][19][20].\\n(TaSe 4 ) 2 I is a typical paradigmatic quasi-onedimensional material that has been frequently studied for more than thirty years [21,22]. This system undergoes a Peierls phase transition at 263 K, the socalled CDW transition, accompanied by an incommensurate structural distortion at low temperatures [23][24][25]. Above the critical temperature T CDW = 263 K, the nearest neighbor (NN) Ta-Ta distances (d Ta−Ta = 3.206 A at room temperature [22,26]) are identical along each chain. As expected, the CDW instability breaks the symmetry of the isometric chains leading to Tatetramerization modes [25,26]. Related experiments re-vealed that the Ta-tetramerization periodicity may be c = 4d Ta−Ta [24,25,27]. In addition, this system is also considered to be a Weyl semimetal [30,31]. More recently, possible axion physics was proposed for this special chiral Weyl material [32,33]. It also should be noted that Rashba-like band splittings may also occur in a chiral system [34]. Since the spin-orbit coupling (SOC) for Ta's 5d orbitals can be robust, such Rashba splitting is expected. All these developments in (TaSe 4 ) 2 I provide a unique and promising platform to display different interesting physical properties in a single material.\\nIn previous works, two Ta-tetramerization patterns were proposed for the low-temperature structure, corresponding to the one-dimensional B 1 and B 2 representations [26]. Since the Ta-tetramerization distortion is small, it can only be observed via weak reflections, and for this reason the lattice phase transition is still uncertain. For this reason, from the theoretical perspective ab initio phonon calculations for the superstructure could play an important role in clarifying the real mechanism of lattice structural transition in this compound.\\nIn the present publication, we perform firstprinciples density functional theory (DFT) calculations for (TaSe 4 ) 2 I. First, our theoretical results indicate a strongly anisotropic electronic structure for the undistorted phase, corresponding to its quasi-one-dimensional geometry. Based on phononic dispersion calculations, we found that (TaSe 4 ) 2 I contains phonon softening instabilities in the Γ 4 mode, resulting into an orthorhombic distortion, which could correspond to the B 1 or B 2 representations. By considering the combined B 1 +B 2 symmetry breaking, the system would decrease to the monoclinic C2 phase (No. 5) with a pattern along chains involving four different NN Ta-Ta distances. Our DFT calculations suggest that the CDW-I lattice phase is the most likely ground state for this system. Since the energy difference between the two CDW lattices, B 1 or B 2 , can not be distinguished unambiguously, these two Ta-tetramerization phases are competitive at low temperatures. Our results also support that the semimetal-insulator transition is driven by the Fermi surface instability. Furthermore, for the heavy element Ta, moderate Rashba-like splitting bands are found in the low temperature CDW phases.\"\n },\n {\n \"title\": \"II. METHODS AND MODEL SYSTEM\",\n \"content\": \"In this publication, DFT calculations were performed using the projector augmented wave (PAW) pseudopotentials with the Perdew-Burke-Ernzerhof format revised for solid exchange functional (PBEsol), as implemented in the Vienna ab initio Simulation Package (VASP) code [35][36][37][38]. The phonon spectra were calculated using the finite displacement approach and analyzed by the PHONONPY software [39,40]. Details are in the Supplemental Material (SM) [41].\\nUnder ambient conditions, (TaSe 4 ) 2 I forms a quasione-dimensional body-centered tetragonal chiral crystal structure with the space group I422 (No. 97). The lattice parameters are a = 9.531Å and c = 12.824Å [22]. There are two adjacent TaSe 4 chains along the c-axis in the conventional cell, where iodine atoms are located between chains, as shown in Fig. 1(a). In each TaSe 4 chain, the Ta atoms are aligned equidistantly and the Se atoms form a \\\"screw\\\" arrangement along the c-axis [see Fig. 1(b)]. Previous experimental results suggested that the modulated structure of (TaSe 4 ) 2 I can be split into two parts [26] below the transition temperature T CDW = 263 K.\\nSince the equal displacements of all atoms would not change the space-group symmetry, we will focus on the modulation of the Ta-tetramerized atoms, which corre- sponds to the parallel component of the k F vector (q CDW || ) along the Ta-chain direction. By considering the space group symmetry I422 (No. 97), the Ta atoms occupy two different Wyckoff sites (4c and 4d). Based on the Raman active modes analysis [42], the corresponding Raman active modes of the 4c and 4d sites have B 2 and B 1 symmetry, respectively. Since the CDW instability corresponds to the Ta-tetramerization modes [25,26], a long-longshort-short (LLSS) Ta-tetramerization CDW-I pattern is expected by only considering the lattice dynamical B 1 or B 2 Ta tetramerization, as shown in Fig. 1(c) [43]. If, instead, we consider the combined B 1 +B 2 Ta tetramerization, there would be four different nearest-neighbor Ta-Ta distances in the chain [see Fig. 1(d)]. In addition, the AMPLIMODES software was employed to perform the group theoretical analysis [44,45], indicating that the CDW-I phase would induce the orthorhombic distortion from the undistorted tetragonal phase with phonon mode Γ 4 . The same analysis for the CDW-II phase also indicates that a monoclinic distortion is obtained with phonon modes Γ 5 and Γ 4 corresponding to the undistorted tetragonal phase and the CDW-I orthorhombic phase, respectively.\"\n },\n {\n \"title\": \"III. NON-CDW PHASE\",\n \"content\": \"Before discussing the CDW phase, we will consider the electronic structure corresponding to the non-CDW phase of (TaSe 4 ) 2 I, which is displayed in Fig. 2. Here, we use the primitive cell instead of the conventional cell to calculate the electronic structure of (TaSe 4 ) 2 I. The volume of the primitive cell is half of the conventional cell with the primitive lattice: a 1 =(−a/2, a/2, c/2), a 2 =(a/2, a/2, c/2), and a 3 =(a/2, a/2, −c/2), where a and c are the conventional cell-lattice constants.\\nFirst, we will present the band structure of the undistorted phase of (TaSe 4 ) 2 I without SOC effect. This band structure clearly displays strong anisotropic metallic behavior, as presented in Fig. 2(a). The band is more dispersive from Γ to Z along the chains than along other directions (i.e. N to P), which is compatible with the presence of quasi-one-dimensional chains along the k z axis. As shown in Fig. 2(a), the Fermi level is dominated by the Ta d 3z 2 −r 2 orbital along the chains, in agreement with previous studies [25,29]. The crossing point along the chain direction (Γ to Z) is not at π/c but at the incommensurate k F where 2k F =q CDW || . Next, we introduce the SOC effect to the undistorted state and now the bands begin to split as displayed in Fig. 2(b), which is consistent with other results [30,31]. In principle, as suggested by the band structure, (TaSe 4 ) 2 I should also display two interesting Weyl points with one along the Γ to Z path and another along the N to P path, although this aspect requires further detailed calculations and discussion beyond the scope of this publication. However, note that that the Weyl physics of this compound has already been studied in detail in recent publications [30,31]. For this reason, here we primarily focus on the phononic aspects.\"\n },\n {\n \"title\": \"IV. CDW INSTABILITY\",\n \"content\": \"It is well known that the CDW instability is accompanied by an structural atomic rearrangement, related to the phonon mode instability. To understand the structural phase transition in (TaSe 4 ) 2 I, we performed phononic dispersion calculations for both the non-CDW and CDW phases. Figure 3(a) indicates the phonon dispersion spectrum that presents an imaginary frequency appearing at Γ for the undistorted structure. The Tatetramerization arising from this mode corresponds to the CDW-I phase. According to group theory analysis using the AMPLIMODES software [44,45], this spontaneous distortion mode is a Γ 4 mode resulting in a twice larger periodicity (4d Ta−Ta ) along the TaSe 4 chain [47]. In addition, the phonon softening mode also corresponds to the transverse component of the incommensurate CDW wavevector parallel to the chain direction (Γ to Z), as in X-ray diffraction and ARPES data Here, a 2 × 2 × 2 supercell was selected from the original primitive cell in our calculations. The coordinates of the high symmetry points in the bulk Brillouin zone (BZ) are obtained using the Seek-path software [46]. [26,28]. We fully relaxed the crystal lattice from the original tetragonal I422 phase along these mode displacements, and then obtained an orthorhombic F 222 state (No. 22). Based on the relaxed orthorhombic structure, we found the LLSS Ta-tetramerization configuration with the difference between the long NN and short NN Ta-Ta distances being about ∼ 0.083Å. Furthermore, we also calculated the phonon dispersion spectrum for the orthorhombic F 222 phase (No. 22), and in this case no imaginary frequency mode was found, as shown in Fig. 3(b). By extracting the lowest-energy phonon mode of the orthorhombic F 222 phase and applying to the F 222 phase, the symmetry of the crystal structure further decreases to the monoclinic C2 phase (No. 5) with a pattern of four different NN Ta-Ta distances along the chain, corresponding to the B 1 +B 2 representations. The AMPLIMODES software [44,45] indicates that this is the Γ 4 mode distortion for the orthorhombic F 222 phase. After full lattice relaxations for the monoclinic C2 phase (No. 5), we find that the difference of the two long or short Ta-Ta distances are 0.003Å and 0.002 A, respectively. Figure 3(c) indicates that the mono- clinic CDW-II phase is also dynamically stable because no imaginary frequency mode is obtained in the phonon dispersion calculation. Here, it should be noticed that the difference of energies of the orthorhombic and monoclinic cases is quite small (∼ 0.1 meV/Ta) which is beyond the accuracy of DFT. In this context, then those two phases cannot be distinguished by considering such small monoclinic distortion and almost negligible energy difference. Additional DFT results for those two phases can be found in the SM [41].\\nNext, let us focus on the semimetal to insulator transition. It is well known that there is a very small overlap between the bottom of the conduction band and the top of the valence band in the semimetallic material, resulting in a negligible density of states (DOS) without SOC at the Fermi level, as shown in Fig. 4(a). The DOS of the undistorted phase (No. 97) is here presented in Fig. 4(b) indicating a semimetal with indeed quite a small density at the Fermi level. Furthermore, we also present in at the crossing point along Γ to Z corresponding to the Brillouin path of undistorted phases. Hence, our results are in agreement with the physical picture of a Fermisurface-driven instability resulting in the Peierls transition. Furthermore, strictly speaking the two Weyl points disappear in the low-temperature CDW phases due to the opening of a gap. This suggests that Weyl physics could be altered below T CDW . Because the phase transition seems to be of first order, as suggested by neutron scattering studies [26], the physics below and above T CDW could be qualitatively different with regards to the Weyl features.\\nTo understand better the lattice phase transition, we simulated the switching \\\"path\\\" from the undistorted phase (no Ta-tetramerization) to the CDW-I LLSS Tatetramerization phase by a simple linear interpolation of the Γ 4 mode, as shown in Fig. 5(a). The transition energy wall between the two phases was computed to be ≈ 40 meV/Ta. However, note that we fixed the crystal constants to obtain the switching path and this would increase the energy barrier. In addition, we also compare the switching path between CDW-I and CDW-II phases and here the energy wall is much smaller ≈ 0.4 meV/Ta [see Fig. 5(b)], which corresponds to a small monoclinic distortion.\"\n },\n {\n \"title\": \"V. ADDITIONAL DISCUSSION\",\n \"content\": \"The point groups of (TaSe 4 ) 2 I for the undistorted (space group No. 97), CDW-I (space group No. 22), and CDW-II (space group No. 5) phases are 422 (D 4 ), 222 (D 2 ), and 2 (C 2 ), respectively. In all these cases, optical isomerism is allowed in this quasi-one-dimensional system. For this reason, optical activity effects are expected by considering their chiral point group. We thus suggest that further experimental and theoretical works should focus on the optical isomerism of this material.\\nFurthermore, for the heavy element Ta, the spin-orbit coupling for the 5d orbitals is expected to be large enough to induce a band splitting. Since the point groups of (TaSe 4 ) 2 I for both the non-CDW and the CDW phases are the non-centrosymmetric chiral group, then a Rashba-like band splitting is also expected in this system. We calculated the band structures for the two CDW phases with SOC incorporated, with results in Fig. 6. There it is shown that the splitted bands resemble Rashba splitting along the high symmetry directions, as shown in Figs. 6(a-b). Moreover, we also show the bands of CDW-I phases near the Γ and T points that indicate a clear Rashba-like band splitting, as in Fig. 6(c). Similar Rashba-like bands splittings, see Fig. 6(d), indicate that these effects can also occur in the CDW-II phase. This is also reasonable since the monoclinic distortion is too small to completely alter the dominant physical properties. In fact, Rashba-like bands also develop along the Γ to Z and N to P paths in the non-CDW parent phase. These issues deserve further studies in the future.\"\n },\n {\n \"title\": \"VI. CONCLUSIONS\",\n \"content\": \"In summary, here the quasi-one-dimensional compound (TaSe 4 ) 2 I has been systematically studied using first-principles calculations. A strongly anisotropic metallic band structure was observed in the undistorted Weyl phase (non-CDW state), in agreement with previous experimental and theoretical works. In addition, using the group symmetry analysis and DFT calculations, two Ta-tetramerization CDW phases have been found here to become the stable low-temperature structure. The semimetal to insulator transition is induced by the Fermi-surface-driven instability, resulting in a Peierls transition that opens a gap (∼ 0.2 eV). In addition, based on the linear interpolation of the Γ 4 mode, the transition energy barrier is estimated to be ≈ 40 meV, corresponding to a high transition temperature. Furthermore, for the heavy element Ta, the spin-orbit coupling for its 5d orbitals should be robust, causing a delicate but notice- able Rashba splitting of the conducting bands around the Γ point in this polar system (chiral). Our results successfully reproduce the phase transition induced by the CDW instability and provide additional insight that may motivate further theoretical and experimental efforts. In future work we will explore the possibility of Rashba effects in these low-dimensional chiral materials.\"\n },\n {\n \"title\": \"FIG. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 6 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Schematic crystal structure of the conventional cell of (TaSe4)2I with the convention: Blue = Ta; green = Se; purple = I. (a-b) The crystal structure is shown from the perspectives of the (a) bc plane [(100) direction] and (b) top [(001) direction]. (c-d) The two different Ta-tetramerization patterns along the chains.","content":"FIG. 1 .\n1Schematic crystal structure of the conventional cell of (TaSe4)2I with the convention: Blue = Ta; green = Se; purple = I. (a-b) The crystal structure is shown from the perspectives of the (a) bc plane [(100) direction] and (b) top [(001) direction]. (c-d) The two different Ta-tetramerization patterns along the chains."},{"id":"","caption":"Band structure of the undistorted (TaSe4)2I along a high symmetry path. (a) Without SOC. (b) With SOC.","content":"FIG. 2 .\n2Band structure of the undistorted (TaSe4)2I along a high symmetry path. (a) Without SOC. (b) With SOC."},{"id":"","caption":"Phonon spectrum of (TaSe4)2I for the undistorted (No. 97), CDW-I (No. 22), and CDW-II (No. 5) phases.","content":"FIG. 3 .\n3Phonon spectrum of (TaSe4)2I for the undistorted (No. 97), CDW-I (No. 22), and CDW-II (No. 5) phases."},{"id":"","caption":"(a) Schematic diagram of the DOS semimetal to insulator transition. (b-d) DOS (without SOC) near the Fermi level for the undistorted parent (non-CDW), orthorhombic (CDW-I), and monoclinic (CDW-II) phases, respectively. Black = Total; red = Ta; blue = I; green = Se.","content":"FIG. 4 .\n4(a) Schematic diagram of the DOS semimetal to insulator transition. (b-d) DOS (without SOC) near the Fermi level for the undistorted parent (non-CDW), orthorhombic (CDW-I), and monoclinic (CDW-II) phases, respectively. Black = Total; red = Ta; blue = I; green = Se."},{"id":"","caption":"Figs. 4(c-d) the DOS for the distorted CDW phases showing that a gap opens in this case. By considering the SOC effect, the expected gap becomes approximately 0.(a) The energy well of the CDW-I and undistorted phases corresponding to the Γ4 mode vs distortion. (b) The energies well of the CDW-I and CDW-II phases.","content":"FIG. 5 .\n5Figs. 4(c-d) the DOS for the distorted CDW phases showing that a gap opens in this case. By considering the SOC effect, the expected gap becomes approximately 0.(a) The energy well of the CDW-I and undistorted phases corresponding to the Γ4 mode vs distortion. (b) The energies well of the CDW-I and CDW-II phases."},{"id":"","caption":"Band structure of (TaSe4)2I with the effect of SOC shown along high symmetry paths. The coordinates of the high symmetry points in the bulk Brillouin zone (BZ) are obtained from the Seek-path software[46]. (a) and (c) are for the orthorhombic F 222 CWD-I phase. (b) and (d) are for the monoclinic C2 CDW-II phase.","content":"FIG. 6 .\n6Band structure of (TaSe4)2I with the effect of SOC shown along high symmetry paths. The coordinates of the high symmetry points in the bulk Brillouin zone (BZ) are obtained from the Seek-path software[46]. (a) and (c) are for the orthorhombic F 222 CWD-I phase. (b) and (d) are for the monoclinic C2 CDW-II phase."},{"id":"","caption":"ACKNOWLEDGMENTS E.D. and A.M. are supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Materials Sciences and Engineering Division. S.D., Y.Z., and L.F.L. are supported by the National Natural Science Foundation of China (Grant Nos. 11834002 and 11674055). L.F.L. and Y.Z. are supported by the China Scholarship Council. Y.Z. is also supported by the Scientific Research Foundation of Graduate School of Southeast University (Grant No. YBPY1826). All the calculations were carried out at the Advanced Computing Facility (ACF) of the University of Tennessee Knoxville.","content":"ACKNOWLEDGMENTS E.D. and A.M. are supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Materials Sciences and Engineering Division. S.D., Y.Z., and L.F.L. are supported by the National Natural Science Foundation of China (Grant Nos. 11834002 and 11674055). L.F.L. and Y.Z. are supported by the China Scholarship Council. Y.Z. is also supported by the Scientific Research Foundation of Graduate School of Southeast University (Grant No. YBPY1826). All the calculations were carried out at the Advanced Computing Facility (ACF) of the University of Tennessee Knoxville."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Schematic crystal structure of the conventional cell of (TaSe4)2I with the convention: Blue = Ta; green = Se; purple = I. (a-b) The crystal structure is shown from the perspectives of the (a) bc plane [(100) direction] and (b) top [(001) direction]. (c-d) The two different Ta-tetramerization patterns along the chains.\",\n \"content\": \"FIG. 1 .\\n1Schematic crystal structure of the conventional cell of (TaSe4)2I with the convention: Blue = Ta; green = Se; purple = I. (a-b) The crystal structure is shown from the perspectives of the (a) bc plane [(100) direction] and (b) top [(001) direction]. (c-d) The two different Ta-tetramerization patterns along the chains.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Band structure of the undistorted (TaSe4)2I along a high symmetry path. (a) Without SOC. (b) With SOC.\",\n \"content\": \"FIG. 2 .\\n2Band structure of the undistorted (TaSe4)2I along a high symmetry path. (a) Without SOC. (b) With SOC.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Phonon spectrum of (TaSe4)2I for the undistorted (No. 97), CDW-I (No. 22), and CDW-II (No. 5) phases.\",\n \"content\": \"FIG. 3 .\\n3Phonon spectrum of (TaSe4)2I for the undistorted (No. 97), CDW-I (No. 22), and CDW-II (No. 5) phases.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(a) Schematic diagram of the DOS semimetal to insulator transition. (b-d) DOS (without SOC) near the Fermi level for the undistorted parent (non-CDW), orthorhombic (CDW-I), and monoclinic (CDW-II) phases, respectively. Black = Total; red = Ta; blue = I; green = Se.\",\n \"content\": \"FIG. 4 .\\n4(a) Schematic diagram of the DOS semimetal to insulator transition. (b-d) DOS (without SOC) near the Fermi level for the undistorted parent (non-CDW), orthorhombic (CDW-I), and monoclinic (CDW-II) phases, respectively. Black = Total; red = Ta; blue = I; green = Se.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figs. 4(c-d) the DOS for the distorted CDW phases showing that a gap opens in this case. By considering the SOC effect, the expected gap becomes approximately 0.(a) The energy well of the CDW-I and undistorted phases corresponding to the Γ4 mode vs distortion. (b) The energies well of the CDW-I and CDW-II phases.\",\n \"content\": \"FIG. 5 .\\n5Figs. 4(c-d) the DOS for the distorted CDW phases showing that a gap opens in this case. By considering the SOC effect, the expected gap becomes approximately 0.(a) The energy well of the CDW-I and undistorted phases corresponding to the Γ4 mode vs distortion. (b) The energies well of the CDW-I and CDW-II phases.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Band structure of (TaSe4)2I with the effect of SOC shown along high symmetry paths. The coordinates of the high symmetry points in the bulk Brillouin zone (BZ) are obtained from the Seek-path software[46]. (a) and (c) are for the orthorhombic F 222 CWD-I phase. (b) and (d) are for the monoclinic C2 CDW-II phase.\",\n \"content\": \"FIG. 6 .\\n6Band structure of (TaSe4)2I with the effect of SOC shown along high symmetry paths. The coordinates of the high symmetry points in the bulk Brillouin zone (BZ) are obtained from the Seek-path software[46]. (a) and (c) are for the orthorhombic F 222 CWD-I phase. (b) and (d) are for the monoclinic C2 CDW-II phase.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"ACKNOWLEDGMENTS E.D. and A.M. are supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Materials Sciences and Engineering Division. S.D., Y.Z., and L.F.L. are supported by the National Natural Science Foundation of China (Grant Nos. 11834002 and 11674055). L.F.L. and Y.Z. are supported by the China Scholarship Council. Y.Z. is also supported by the Scientific Research Foundation of Graduate School of Southeast University (Grant No. YBPY1826). All the calculations were carried out at the Advanced Computing Facility (ACF) of the University of Tennessee Knoxville.\",\n \"content\": \"ACKNOWLEDGMENTS E.D. and A.M. are supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Materials Sciences and Engineering Division. S.D., Y.Z., and L.F.L. are supported by the National Natural Science Foundation of China (Grant Nos. 11834002 and 11674055). L.F.L. and Y.Z. are supported by the China Scholarship Council. Y.Z. is also supported by the Scientific Research Foundation of Graduate School of Southeast University (Grant No. YBPY1826). All the calculations were carried out at the Advanced Computing Facility (ACF) of the University of Tennessee Knoxville.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Density Waves in Solids (Perseus. G Grüner, Cambridge, MAG. Grüner, Density Waves in Solids (Perseus, Cam- bridge, MA, 2000).","doi":""},{"text":". P Monceau, 10.1080/00018732.2012.719674Adv. Phys. 61325P. Monceau, Adv. Phys. 61, 325 (2012).","doi":"10.1080/00018732.2012.719674"},{"text":". M Grioni, S Pons, E Frantzeskakis, 10.1088/0953-8984/21/2/023201J. Phys.: Condens. Matter. 2123201M. Grioni, S. Pons and E. Frantzeskakis, J. Phys.: Con- dens. Matter 21, 023201 (2009).","doi":"10.1088/0953-8984/21/2/023201"},{"text":". Z Wang, S.-C Zhang, https:/link.aps.org/doi/10.1103/PhysRevB.87.161107Phys. Rev. B. 87161107Z. Wang and S.-C. Zhang, Phys. Rev. B 87, 161107(R) (2013).","doi":"https:/link.aps.org/doi/10.1103/PhysRevB.87.161107"},{"text":". S M Soto, C P Slichter, A M Kini, H H Wang, U Geiser, J M Williams, https:/link.aps.org/doi/10.1103/PhysRevB.52.10364Phys. Rev. B. 5210364S. M. De Soto, C. P. Slichter, A. M. Kini, H. H. Wang, U. Geiser and J. M. Williams, Phys. Rev. B 52, 10364 (1995).","doi":"https:/link.aps.org/doi/10.1103/PhysRevB.52.10364"},{"text":". E Dagotto, J Riera, D Scalapino, 10.1103/PhysRevB.45.5744Phys. Rev. B. 455744E. Dagotto, J. Riera, and D. Scalapino, Phys. Rev. B 45, 5744(R) (1992).","doi":"10.1103/PhysRevB.45.5744"},{"text":". E Dagotto, T M Rice, 10.1126/science.271.5249.618Science. 271618E. Dagotto and T. M. Rice, Science 271, 618 (1996).","doi":"10.1126/science.271.5249.618"},{"text":". E Dagotto, 10.1088/0034-4885/62/11/202Rep. Prog. Phys. 621525E. Dagotto, Rep. Prog. Phys. 62, 1525 (1999).","doi":"10.1088/0034-4885/62/11/202"},{"text":". M Uehara, T Nagata, J Akimitsu, H Takahashi, N Mori, K Kinoshita, 10.1143/JPSJ.65.2764J. Phys. Soc. Jpn. 652764M. Uehara, T. Nagata, J. Akimitsu, H. Takahashi, N. Mori, and K. Kinoshita, J. Phys. Soc. Jpn. 65, 2764 (1996).","doi":"10.1143/JPSJ.65.2764"},{"text":". H Takahashi, A Sugimoto, Y Nambu, T Yamauchi, Y Hirata, T Kawakami, M Avdeev, K Matsubayashi, F Du, C Kawashima, H Soeda, S Nakano, Y Uwatoko, Y Ueda, T J Sato, K Ohgushi, 10.1038/nmat4351Nat. Mater. 141008H. Takahashi, A. Sugimoto, Y. Nambu, T. Yamauchi, Y. Hirata, T. Kawakami, M. Avdeev, K. Matsubayashi, F. Du, C. Kawashima, H. Soeda, S. Nakano, Y. Uwatoko, Y. Ueda, T. J. Sato and K. Ohgushi, Nat. Mater. 14, 1008 (2015).","doi":"10.1038/nmat4351"},{"text":". Y Zhang, L F Lin, J J Zhang, E Dagotto, S Dong, 10.1103/PhysRevB.95.115154Phys. Rev. B. 95115154Y. Zhang, L. F. Lin, J. J. Zhang, E. Dagotto, and S. Dong, Phys. Rev. B 95, 115154 (2017).","doi":"10.1103/PhysRevB.95.115154"},{"text":". J.-J Ying, H C Lei, C Petrovic, Y.-M Xiao, V.-V Struzhkin, 10.1103/PhysRevB.95.241109Phys. Rev. B. 95R241109J.-J. Ying, H. C. Lei, C. Petrovic, Y.-M. Xiao and V.-V. Struzhkin, Phys. Rev. B 95, 241109(R) (2017).","doi":"10.1103/PhysRevB.95.241109"},{"text":". Y Zhang, L F Lin, J J Zhang, E Dagotto, S Dong, 10.1103/PhysRevB.97.045119Phys. Rev. B. 9745119Y. Zhang, L. F. Lin, J. J. Zhang, E. Dagotto, and S. Dong, Phys. Rev. B 97, 045119 (2018).","doi":"10.1103/PhysRevB.97.045119"},{"text":". Y Zhang, L F Lin, A Moreo, S Dong, E Dagotto, 10.1103/PhysRevB.100.184419Phys. Rev. B. 100184419Y. Zhang, L.F. Lin, A. Moreo, S. Dong, and E. Dagotto, Phys. Rev. B 100, 184419 (2019).","doi":"10.1103/PhysRevB.100.184419"},{"text":". L F Lin, Y Zhang, A Moreo, E Dagotto, S Dong, 10.1103/PhysRevMaterials.3.111401Phys. Rev. Mater. 3111401L. F. Lin, Y. Zhang, A. Moreo, E. Dagotto, and S. Dong, Phys. Rev. Mater. 3, 111401(R) (2019).","doi":"10.1103/PhysRevMaterials.3.111401"},{"text":". L F Lin, Y Zhang, A Moreo, E Dagotto, S Dong, 10.1103/PhysRevLett.123.067601Phys. Rev. Lett. 12367601L. F. Lin, Y. Zhang, A. Moreo, E. Dagotto, and S. Dong, Phys. Rev. Lett. 123, 067601 (2019).","doi":"10.1103/PhysRevLett.123.067601"},{"text":". M C Cross, D S Fisher, 10.1103/PhysRevB.19.402Phys. Rev. B. 19402M. C. Cross and D. S. Fisher, Phys. Rev. B 19, 402 (1979).","doi":"10.1103/PhysRevB.19.402"},{"text":". Y J Choi, H T Yi, S Lee, Q Huang, V Kiryukhin, S.-W Cheong, 10.1103/PhysRevLett.100.047601Phys. Rev. Lett. 10047601Y. J. Choi, H. T. Yi, S. Lee, Q. Huang, V. Kiryukhin, and S.-W. Cheong, Phys. Rev. Lett. 100, 047601 (2008).","doi":"10.1103/PhysRevLett.100.047601"},{"text":". Y Zhang, L F Lin, A Moreo, S Dong, E , arXiv:1912.07749Dagotto arXiv preprintY. Zhang, L. F. Lin, A. Moreo, S. Dong, and E. Dagotto arXiv preprint arXiv:1912.07749.","doi":"arXiv:1912.07749"},{"text":". S Dong, J M Liu, E Dagotto, 10.1103/PhysRevLett.113.187204Phys. Rev. Lett. 113187204S. Dong, J. M. Liu, and E. Dagotto, Phys. Rev. Lett. 113, 187204 (2014).","doi":"10.1103/PhysRevLett.113.187204"},{"text":". P Gressier, A Meerschaut, L Guemas, J Rouxel, P Monceau, 10.1016/0022-4596(84)90327-XJournal of Solid State Chemistry. 51141P. Gressier, A. Meerschaut, L. Guemas, J. Rouxel and P. Monceau, Journal of Solid State Chemistry 51, 141 (1984).","doi":"10.1016/0022-4596(84)90327-X"},{"text":". P Gressier, L Guemas, A Meerschaut, 10.1107/S0567740882010176Acta Cryst. B. 382877P. Gressier and L. Guemas and A. Meerschaut, Acta Cryst. B 38, 2877 (1982).","doi":"10.1107/S0567740882010176"},{"text":"Solid State Commun. H Fujishita, M Sato, S Hoshino, 10.1016/0038-1098(84)90576-349313H. Fujishita, M. Sato and S. Hoshino, Solid State Com- mun. 49, 313 (1984).","doi":"10.1016/0038-1098(84)90576-3"},{"text":". S Van Smaalen, E J Lam, J Lüdecke, 10.1088/0953-8984/13/44/308J. Phys. Condens. Matter. 139923S. van Smaalen, E. J. Lam and J. Lüdecke, J. Phys. Con- dens. Matter 13, 9923 (2001).","doi":"10.1088/0953-8984/13/44/308"},{"text":". V Favre-Nicolin, S Bos, J E Lorenzo, J.-L Hodeau, J.-F Berar, P Monceau, R Currat, F Levy, H Berger, https:/link.aps.org/doi/10.1103/PhysRevLett.87.015502Phys. Rev. Lett. 8715502V. Favre-Nicolin, S. Bos, J. E. Lorenzo, J.-L. Hodeau, J.- F. Berar, P. Monceau, R. Currat, F. Levy and H. Berger, Phys. Rev. Lett. 87, 015502 (2001).","doi":"https:/link.aps.org/doi/10.1103/PhysRevLett.87.015502"},{"text":". J E Lorenzo, R Currat, P Monceau, B Hennion, H Berger, F Levy, 10.1088/0953-8984/10/23/010J. Phys. Condens. Matter. 105039J. E. Lorenzo, R. Currat, P. Monceau, B. Hennion, H. Berger and F. Levy, J. Phys. Condens. Matter 10, 5039 (1998).","doi":"10.1088/0953-8984/10/23/010"},{"text":". J Voit, L Perfetti, F Zwick, H Berger, G Margaritondo, G Grüner, H Höchst, M Grioni, Science. 290501J. Voit, L. Perfetti, F. Zwick, H. Berger, G. Margari- tondo, G. Grüner, H. Höchst and M. Grioni, Science 290, 501 (2000).","doi":""},{"text":". C Tournier-Colletta, L Moreschini, G Autès, S Moser, A Crepaldi, H Berger, A L Walter, K S Kim, A Bostwick, P Monceau, E Rotenberg, O V Yazyev, M Grioni, https:/link.aps.org/doi/10.1103/PhysRevLett.110.236401Phys. Rev. Lett. 110236401C. Tournier-Colletta, L. Moreschini, G. Autès, S. Moser, A. Crepaldi, H. Berger, A. L. Walter, K. S. Kim, A. Bostwick, P. Monceau, E. Rotenberg, O. V. Yazyev and M. Grioni, Phys. Rev. Lett. 110, 236401 (2013).","doi":"https:/link.aps.org/doi/10.1103/PhysRevLett.110.236401"},{"text":". P Gressier, M , H Whangbo, A Meerschaut, J Rouxel, 10.1021/ic00177a011Inorg. Chem. 231221P. Gressier, M, H. Whangbo, A. Meerschaut and J. Rouxel, Inorg. Chem. 23, 1221 (1984).","doi":"10.1021/ic00177a011"},{"text":". W Shi, B J Wieder, H L Meyerheim, Y Sun, Y Zhang, Y Li, L Shen, Y Qi, L Yang, K Jena, P Werner, K Koepernik, S Parkin, Y Chen, C Felser, B A Bernevig, Z Wang, arXiv:1909.04037arXiv preprintW. Shi, B. J. WiedeR, H.L. Meyerheim, Y. Sun, Y. Zhang, Y. Li, L. Shen, Y. Qi, L. Yang, K. Jena, P. Werner, K. Koepernik, S. Parkin, Y. Chen, C. Felser, B. A. Bernevig and Z. Wang, arXiv preprint arXiv:1909.04037.","doi":"arXiv:1909.04037"},{"text":". X.-P Li, K Deng, B Fu, Y Li, D Ma, J Han, J Zhou, S Zhou, Y Yao, arXiv:1909.12178arXiv preprintX.-P. Li, K. Deng, B. Fu, Y. Li, D. Ma, J. Han, J. Zhou, S. Zhou and Y. Yao, arXiv preprint arXiv:1909.12178.","doi":"arXiv:1909.12178"},{"text":". J Gooth, B Bradlyn, S Honnali, C Schindler, N Kumar, J Noky, Y Qi, C Shekhar, Y Sun, Z Wang, B A Bernevig, C Felser, 10.1038/s41586-019-1630-4Nature. 575315J. Gooth, B. Bradlyn, S. Honnali, C. Schindler, N. Ku- mar, J. Noky, Y. Qi, C. Shekhar, Y. Sun, Z. Wang, B. A. Bernevig and C. Felser , Nature 575, 315 (2019).","doi":"10.1038/s41586-019-1630-4"},{"text":". D Schmeltzer, arXiv:1910.13392arXiv preprintD. Schmeltzer, arXiv preprint arXiv:1910.13392.","doi":"arXiv:1910.13392"},{"text":". D Di Sante, P Barone, A Stroppa, K F Garrity, D Vanderbilt, S Picozzi, https:/link.aps.org/doi/10.1103/PhysRevLett.117.076401Phys. Rev. Lett. 11776401D. Di Sante, P. Barone, A. Stroppa, K. F. Garrity, D. Vanderbilt and S. Picozzi, Phys. Rev. Lett. 117, 076401 (2016).","doi":"https:/link.aps.org/doi/10.1103/PhysRevLett.117.076401"},{"text":". G Kresse, J Hafner, 10.1103/PhysRevB.47.558Phys. Rev. B. 47558G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).","doi":"10.1103/PhysRevB.47.558"},{"text":". G Kresse, J Furthmüller, 10.1103/PhysRevB.54.1169Phys. Rev. B. 5411169G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).","doi":"10.1103/PhysRevB.54.1169"},{"text":". P E Blöchl, 10.1103/PhysRevB.50.17953Phys. Rev. B. 5017953P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).","doi":"10.1103/PhysRevB.50.17953"},{"text":". J P Perdew, A Ruzsinszky, G I Csonka, O A Vydrov, G E Scuseria, L A Constantin, X Zhou, K Burke, 10.1103/PhysRevLett.100.136406Phys. Rev. Lett. 100136406J. P. Perdew and A. Ruzsinszky, and G. I. Csonka and O. A. Vydrov and G. E. Scuseria and L. A. Constantin and X. Zhou and K. Burke, Phys. Rev. Lett. 100, 136406 (2008).","doi":"10.1103/PhysRevLett.100.136406"},{"text":". L Chaput, A Togo, I Tanaka, G Hug, 10.1103/PhysRevB.84.094302Phys. Rev. B. 8494302L. Chaput, A. Togo, I. Tanaka, and G. Hug, Phys. Rev. B 84, 094302 (2011).","doi":"10.1103/PhysRevB.84.094302"},{"text":". A Togo, I Tanaka, 10.1016/j.scriptamat.2015.07.021Scr. Mater. 1081A. Togo, I. Tanaka, Scr. Mater. 108, 1 (2015).","doi":"10.1016/j.scriptamat.2015.07.021"},{"text":"For more results. http:/link.aps.org/supplemental/10.1103/PhysRevB.xx/xxxxxxsee Supplemental Material atFor more results, see Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevB.xx/xxxxxx.","doi":"http:/link.aps.org/supplemental/10.1103/PhysRevB.xx/xxxxxx"},{"text":". E Kroumova, M I Aroyo, J M Perez-Mato, A Kirov, C Capillas, S Ivantchev, H Wondratschek, 10.1080/0141159031000076110Phase Transitions. 76155E. Kroumova, M. I. Aroyo, J. M. Perez-Mato, A. Kirov, C. Capillas, S. Ivantchev, and H. Wondratschek, Phase Transitions 76, 155 (2003).","doi":"10.1080/0141159031000076110"},{"text":"In the B1 representation, two Ta(1) atoms (4c site) move towards the connected Ta(2) atoms in chain-A and move away from the connected Ta(2) atoms in chain-B, resulting in a LLSS Ta-tetramerization pattern. Correspondingly, in the B2 representation, the Ta(2) atoms (4d site) move towards the connected Ta(1) atoms in chain-A and move away from the connected Ta(1) atoms in chain-B. Both the B1 and B2 representations would induce a simlar LLSS Ta-tetramerization CDW phase. The two Ta sites (4c and 4d) form a pattern Ta. Ta(1)-Ta(2) in the chain direction. as shown Fig. 1(cThe two Ta sites (4c and 4d) form a pattern Ta(1)-Ta(2)- Ta(1)-Ta(2) in the chain direction. In the B1 represen- tation, two Ta(1) atoms (4c site) move towards the con- nected Ta(2) atoms in chain-A and move away from the connected Ta(2) atoms in chain-B, resulting in a LLSS Ta-tetramerization pattern. Correspondingly, in the B2 representation, the Ta(2) atoms (4d site) move towards the connected Ta(1) atoms in chain-A and move away from the connected Ta(1) atoms in chain-B. Both the B1 and B2 representations would induce a simlar LLSS Ta-tetramerization CDW phase, as shown Fig. 1(c).","doi":""},{"text":". D Orobengoa, C Capillas, M I Aroyo, J M Perez-Mato, 10.1107/S0021889809028064J. Appl. Crystallogr. 42820D. Orobengoa, C. Capillas, M. I. Aroyo, and J. M. Perez- Mato, J. Appl. Crystallogr. 42, 820 (2009).","doi":"10.1107/S0021889809028064"},{"text":". J Perez-Mato, D Orobengoa, M I Aroyo, 10.1107/S0108767310016247Acta Crystallogr. A. 66558J. Perez-Mato, D. Orobengoa, and M. I. Aroyo, Acta Crystallogr. A 66, 558 (2010).","doi":"10.1107/S0108767310016247"},{"text":". Y Hinuma, G Pizzi, Y Kumagai, F Oba, I Tanaka, 10.1016/j.commatsci.2016.10.015Comput. Mater. Sci. 113221Y. Hinuma, G. Pizzi, Y. Kumagai, F. Oba and I. Tanaka, Comput. Mater. Sci. 113, 221 (2016).","doi":"10.1016/j.commatsci.2016.10.015"},{"text":"Here, for completeness we also used 1×1×2 and 1×1×4 supercells based on the conventional cell, but none of our physical conclusions changed, as shown in the SM. 41Here, for completeness we also used 1×1×2 and 1×1×4 supercells based on the conventional cell, but none of our physical conclusions changed, as shown in the SM [41].","doi":""}],"string":"[\n {\n \"text\": \"Density Waves in Solids (Perseus. G Grüner, Cambridge, MAG. Grüner, Density Waves in Solids (Perseus, Cam- bridge, MA, 2000).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Monceau, 10.1080/00018732.2012.719674Adv. Phys. 61325P. Monceau, Adv. Phys. 61, 325 (2012).\",\n \"doi\": \"10.1080/00018732.2012.719674\"\n },\n {\n \"text\": \". M Grioni, S Pons, E Frantzeskakis, 10.1088/0953-8984/21/2/023201J. Phys.: Condens. Matter. 2123201M. Grioni, S. Pons and E. Frantzeskakis, J. Phys.: Con- dens. Matter 21, 023201 (2009).\",\n \"doi\": \"10.1088/0953-8984/21/2/023201\"\n },\n {\n \"text\": \". Z Wang, S.-C Zhang, https:/link.aps.org/doi/10.1103/PhysRevB.87.161107Phys. Rev. B. 87161107Z. Wang and S.-C. Zhang, Phys. Rev. B 87, 161107(R) (2013).\",\n \"doi\": \"https:/link.aps.org/doi/10.1103/PhysRevB.87.161107\"\n },\n {\n \"text\": \". S M Soto, C P Slichter, A M Kini, H H Wang, U Geiser, J M Williams, https:/link.aps.org/doi/10.1103/PhysRevB.52.10364Phys. Rev. B. 5210364S. M. De Soto, C. P. Slichter, A. M. Kini, H. H. Wang, U. Geiser and J. M. Williams, Phys. Rev. B 52, 10364 (1995).\",\n \"doi\": \"https:/link.aps.org/doi/10.1103/PhysRevB.52.10364\"\n },\n {\n \"text\": \". E Dagotto, J Riera, D Scalapino, 10.1103/PhysRevB.45.5744Phys. Rev. B. 455744E. Dagotto, J. Riera, and D. Scalapino, Phys. Rev. B 45, 5744(R) (1992).\",\n \"doi\": \"10.1103/PhysRevB.45.5744\"\n },\n {\n \"text\": \". E Dagotto, T M Rice, 10.1126/science.271.5249.618Science. 271618E. Dagotto and T. M. Rice, Science 271, 618 (1996).\",\n \"doi\": \"10.1126/science.271.5249.618\"\n },\n {\n \"text\": \". E Dagotto, 10.1088/0034-4885/62/11/202Rep. Prog. Phys. 621525E. Dagotto, Rep. Prog. Phys. 62, 1525 (1999).\",\n \"doi\": \"10.1088/0034-4885/62/11/202\"\n },\n {\n \"text\": \". M Uehara, T Nagata, J Akimitsu, H Takahashi, N Mori, K Kinoshita, 10.1143/JPSJ.65.2764J. Phys. Soc. Jpn. 652764M. Uehara, T. Nagata, J. Akimitsu, H. Takahashi, N. Mori, and K. Kinoshita, J. Phys. Soc. Jpn. 65, 2764 (1996).\",\n \"doi\": \"10.1143/JPSJ.65.2764\"\n },\n {\n \"text\": \". H Takahashi, A Sugimoto, Y Nambu, T Yamauchi, Y Hirata, T Kawakami, M Avdeev, K Matsubayashi, F Du, C Kawashima, H Soeda, S Nakano, Y Uwatoko, Y Ueda, T J Sato, K Ohgushi, 10.1038/nmat4351Nat. Mater. 141008H. Takahashi, A. Sugimoto, Y. Nambu, T. Yamauchi, Y. Hirata, T. Kawakami, M. Avdeev, K. Matsubayashi, F. Du, C. Kawashima, H. Soeda, S. Nakano, Y. Uwatoko, Y. Ueda, T. J. Sato and K. Ohgushi, Nat. Mater. 14, 1008 (2015).\",\n \"doi\": \"10.1038/nmat4351\"\n },\n {\n \"text\": \". Y Zhang, L F Lin, J J Zhang, E Dagotto, S Dong, 10.1103/PhysRevB.95.115154Phys. Rev. B. 95115154Y. Zhang, L. F. Lin, J. J. Zhang, E. Dagotto, and S. Dong, Phys. Rev. B 95, 115154 (2017).\",\n \"doi\": \"10.1103/PhysRevB.95.115154\"\n },\n {\n \"text\": \". J.-J Ying, H C Lei, C Petrovic, Y.-M Xiao, V.-V Struzhkin, 10.1103/PhysRevB.95.241109Phys. Rev. B. 95R241109J.-J. Ying, H. C. Lei, C. Petrovic, Y.-M. Xiao and V.-V. Struzhkin, Phys. Rev. B 95, 241109(R) (2017).\",\n \"doi\": \"10.1103/PhysRevB.95.241109\"\n },\n {\n \"text\": \". Y Zhang, L F Lin, J J Zhang, E Dagotto, S Dong, 10.1103/PhysRevB.97.045119Phys. Rev. B. 9745119Y. Zhang, L. F. Lin, J. J. Zhang, E. Dagotto, and S. Dong, Phys. Rev. B 97, 045119 (2018).\",\n \"doi\": \"10.1103/PhysRevB.97.045119\"\n },\n {\n \"text\": \". Y Zhang, L F Lin, A Moreo, S Dong, E Dagotto, 10.1103/PhysRevB.100.184419Phys. Rev. B. 100184419Y. Zhang, L.F. Lin, A. Moreo, S. Dong, and E. Dagotto, Phys. Rev. B 100, 184419 (2019).\",\n \"doi\": \"10.1103/PhysRevB.100.184419\"\n },\n {\n \"text\": \". L F Lin, Y Zhang, A Moreo, E Dagotto, S Dong, 10.1103/PhysRevMaterials.3.111401Phys. Rev. Mater. 3111401L. F. Lin, Y. Zhang, A. Moreo, E. Dagotto, and S. Dong, Phys. Rev. Mater. 3, 111401(R) (2019).\",\n \"doi\": \"10.1103/PhysRevMaterials.3.111401\"\n },\n {\n \"text\": \". L F Lin, Y Zhang, A Moreo, E Dagotto, S Dong, 10.1103/PhysRevLett.123.067601Phys. Rev. Lett. 12367601L. F. Lin, Y. Zhang, A. Moreo, E. Dagotto, and S. Dong, Phys. Rev. Lett. 123, 067601 (2019).\",\n \"doi\": \"10.1103/PhysRevLett.123.067601\"\n },\n {\n \"text\": \". M C Cross, D S Fisher, 10.1103/PhysRevB.19.402Phys. Rev. B. 19402M. C. Cross and D. S. Fisher, Phys. Rev. B 19, 402 (1979).\",\n \"doi\": \"10.1103/PhysRevB.19.402\"\n },\n {\n \"text\": \". Y J Choi, H T Yi, S Lee, Q Huang, V Kiryukhin, S.-W Cheong, 10.1103/PhysRevLett.100.047601Phys. Rev. Lett. 10047601Y. J. Choi, H. T. Yi, S. Lee, Q. Huang, V. Kiryukhin, and S.-W. Cheong, Phys. Rev. Lett. 100, 047601 (2008).\",\n \"doi\": \"10.1103/PhysRevLett.100.047601\"\n },\n {\n \"text\": \". Y Zhang, L F Lin, A Moreo, S Dong, E , arXiv:1912.07749Dagotto arXiv preprintY. Zhang, L. F. Lin, A. Moreo, S. Dong, and E. Dagotto arXiv preprint arXiv:1912.07749.\",\n \"doi\": \"arXiv:1912.07749\"\n },\n {\n \"text\": \". S Dong, J M Liu, E Dagotto, 10.1103/PhysRevLett.113.187204Phys. Rev. Lett. 113187204S. Dong, J. M. Liu, and E. Dagotto, Phys. Rev. Lett. 113, 187204 (2014).\",\n \"doi\": \"10.1103/PhysRevLett.113.187204\"\n },\n {\n \"text\": \". P Gressier, A Meerschaut, L Guemas, J Rouxel, P Monceau, 10.1016/0022-4596(84)90327-XJournal of Solid State Chemistry. 51141P. Gressier, A. Meerschaut, L. Guemas, J. Rouxel and P. Monceau, Journal of Solid State Chemistry 51, 141 (1984).\",\n \"doi\": \"10.1016/0022-4596(84)90327-X\"\n },\n {\n \"text\": \". P Gressier, L Guemas, A Meerschaut, 10.1107/S0567740882010176Acta Cryst. B. 382877P. Gressier and L. Guemas and A. Meerschaut, Acta Cryst. B 38, 2877 (1982).\",\n \"doi\": \"10.1107/S0567740882010176\"\n },\n {\n \"text\": \"Solid State Commun. H Fujishita, M Sato, S Hoshino, 10.1016/0038-1098(84)90576-349313H. Fujishita, M. Sato and S. Hoshino, Solid State Com- mun. 49, 313 (1984).\",\n \"doi\": \"10.1016/0038-1098(84)90576-3\"\n },\n {\n \"text\": \". S Van Smaalen, E J Lam, J Lüdecke, 10.1088/0953-8984/13/44/308J. Phys. Condens. Matter. 139923S. van Smaalen, E. J. Lam and J. Lüdecke, J. Phys. Con- dens. Matter 13, 9923 (2001).\",\n \"doi\": \"10.1088/0953-8984/13/44/308\"\n },\n {\n \"text\": \". V Favre-Nicolin, S Bos, J E Lorenzo, J.-L Hodeau, J.-F Berar, P Monceau, R Currat, F Levy, H Berger, https:/link.aps.org/doi/10.1103/PhysRevLett.87.015502Phys. Rev. Lett. 8715502V. Favre-Nicolin, S. Bos, J. E. Lorenzo, J.-L. Hodeau, J.- F. Berar, P. Monceau, R. Currat, F. Levy and H. Berger, Phys. Rev. Lett. 87, 015502 (2001).\",\n \"doi\": \"https:/link.aps.org/doi/10.1103/PhysRevLett.87.015502\"\n },\n {\n \"text\": \". J E Lorenzo, R Currat, P Monceau, B Hennion, H Berger, F Levy, 10.1088/0953-8984/10/23/010J. Phys. Condens. Matter. 105039J. E. Lorenzo, R. Currat, P. Monceau, B. Hennion, H. Berger and F. Levy, J. Phys. Condens. Matter 10, 5039 (1998).\",\n \"doi\": \"10.1088/0953-8984/10/23/010\"\n },\n {\n \"text\": \". J Voit, L Perfetti, F Zwick, H Berger, G Margaritondo, G Grüner, H Höchst, M Grioni, Science. 290501J. Voit, L. Perfetti, F. Zwick, H. Berger, G. Margari- tondo, G. Grüner, H. Höchst and M. Grioni, Science 290, 501 (2000).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Tournier-Colletta, L Moreschini, G Autès, S Moser, A Crepaldi, H Berger, A L Walter, K S Kim, A Bostwick, P Monceau, E Rotenberg, O V Yazyev, M Grioni, https:/link.aps.org/doi/10.1103/PhysRevLett.110.236401Phys. Rev. Lett. 110236401C. Tournier-Colletta, L. Moreschini, G. Autès, S. Moser, A. Crepaldi, H. Berger, A. L. Walter, K. S. Kim, A. Bostwick, P. Monceau, E. Rotenberg, O. V. Yazyev and M. Grioni, Phys. Rev. Lett. 110, 236401 (2013).\",\n \"doi\": \"https:/link.aps.org/doi/10.1103/PhysRevLett.110.236401\"\n },\n {\n \"text\": \". P Gressier, M , H Whangbo, A Meerschaut, J Rouxel, 10.1021/ic00177a011Inorg. Chem. 231221P. Gressier, M, H. Whangbo, A. Meerschaut and J. Rouxel, Inorg. Chem. 23, 1221 (1984).\",\n \"doi\": \"10.1021/ic00177a011\"\n },\n {\n \"text\": \". W Shi, B J Wieder, H L Meyerheim, Y Sun, Y Zhang, Y Li, L Shen, Y Qi, L Yang, K Jena, P Werner, K Koepernik, S Parkin, Y Chen, C Felser, B A Bernevig, Z Wang, arXiv:1909.04037arXiv preprintW. Shi, B. J. WiedeR, H.L. Meyerheim, Y. Sun, Y. Zhang, Y. Li, L. Shen, Y. Qi, L. Yang, K. Jena, P. Werner, K. Koepernik, S. Parkin, Y. Chen, C. Felser, B. A. Bernevig and Z. Wang, arXiv preprint arXiv:1909.04037.\",\n \"doi\": \"arXiv:1909.04037\"\n },\n {\n \"text\": \". X.-P Li, K Deng, B Fu, Y Li, D Ma, J Han, J Zhou, S Zhou, Y Yao, arXiv:1909.12178arXiv preprintX.-P. Li, K. Deng, B. Fu, Y. Li, D. Ma, J. Han, J. Zhou, S. Zhou and Y. Yao, arXiv preprint arXiv:1909.12178.\",\n \"doi\": \"arXiv:1909.12178\"\n },\n {\n \"text\": \". J Gooth, B Bradlyn, S Honnali, C Schindler, N Kumar, J Noky, Y Qi, C Shekhar, Y Sun, Z Wang, B A Bernevig, C Felser, 10.1038/s41586-019-1630-4Nature. 575315J. Gooth, B. Bradlyn, S. Honnali, C. Schindler, N. Ku- mar, J. Noky, Y. Qi, C. Shekhar, Y. Sun, Z. Wang, B. A. Bernevig and C. Felser , Nature 575, 315 (2019).\",\n \"doi\": \"10.1038/s41586-019-1630-4\"\n },\n {\n \"text\": \". D Schmeltzer, arXiv:1910.13392arXiv preprintD. Schmeltzer, arXiv preprint arXiv:1910.13392.\",\n \"doi\": \"arXiv:1910.13392\"\n },\n {\n \"text\": \". D Di Sante, P Barone, A Stroppa, K F Garrity, D Vanderbilt, S Picozzi, https:/link.aps.org/doi/10.1103/PhysRevLett.117.076401Phys. Rev. Lett. 11776401D. Di Sante, P. Barone, A. Stroppa, K. F. Garrity, D. Vanderbilt and S. Picozzi, Phys. Rev. Lett. 117, 076401 (2016).\",\n \"doi\": \"https:/link.aps.org/doi/10.1103/PhysRevLett.117.076401\"\n },\n {\n \"text\": \". G Kresse, J Hafner, 10.1103/PhysRevB.47.558Phys. Rev. B. 47558G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).\",\n \"doi\": \"10.1103/PhysRevB.47.558\"\n },\n {\n \"text\": \". G Kresse, J Furthmüller, 10.1103/PhysRevB.54.1169Phys. Rev. B. 5411169G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).\",\n \"doi\": \"10.1103/PhysRevB.54.1169\"\n },\n {\n \"text\": \". P E Blöchl, 10.1103/PhysRevB.50.17953Phys. Rev. B. 5017953P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).\",\n \"doi\": \"10.1103/PhysRevB.50.17953\"\n },\n {\n \"text\": \". J P Perdew, A Ruzsinszky, G I Csonka, O A Vydrov, G E Scuseria, L A Constantin, X Zhou, K Burke, 10.1103/PhysRevLett.100.136406Phys. Rev. Lett. 100136406J. P. Perdew and A. Ruzsinszky, and G. I. Csonka and O. A. Vydrov and G. E. Scuseria and L. A. Constantin and X. Zhou and K. Burke, Phys. Rev. Lett. 100, 136406 (2008).\",\n \"doi\": \"10.1103/PhysRevLett.100.136406\"\n },\n {\n \"text\": \". L Chaput, A Togo, I Tanaka, G Hug, 10.1103/PhysRevB.84.094302Phys. Rev. B. 8494302L. Chaput, A. Togo, I. Tanaka, and G. Hug, Phys. Rev. B 84, 094302 (2011).\",\n \"doi\": \"10.1103/PhysRevB.84.094302\"\n },\n {\n \"text\": \". A Togo, I Tanaka, 10.1016/j.scriptamat.2015.07.021Scr. Mater. 1081A. Togo, I. Tanaka, Scr. Mater. 108, 1 (2015).\",\n \"doi\": \"10.1016/j.scriptamat.2015.07.021\"\n },\n {\n \"text\": \"For more results. http:/link.aps.org/supplemental/10.1103/PhysRevB.xx/xxxxxxsee Supplemental Material atFor more results, see Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevB.xx/xxxxxx.\",\n \"doi\": \"http:/link.aps.org/supplemental/10.1103/PhysRevB.xx/xxxxxx\"\n },\n {\n \"text\": \". E Kroumova, M I Aroyo, J M Perez-Mato, A Kirov, C Capillas, S Ivantchev, H Wondratschek, 10.1080/0141159031000076110Phase Transitions. 76155E. Kroumova, M. I. Aroyo, J. M. Perez-Mato, A. Kirov, C. Capillas, S. Ivantchev, and H. Wondratschek, Phase Transitions 76, 155 (2003).\",\n \"doi\": \"10.1080/0141159031000076110\"\n },\n {\n \"text\": \"In the B1 representation, two Ta(1) atoms (4c site) move towards the connected Ta(2) atoms in chain-A and move away from the connected Ta(2) atoms in chain-B, resulting in a LLSS Ta-tetramerization pattern. Correspondingly, in the B2 representation, the Ta(2) atoms (4d site) move towards the connected Ta(1) atoms in chain-A and move away from the connected Ta(1) atoms in chain-B. Both the B1 and B2 representations would induce a simlar LLSS Ta-tetramerization CDW phase. The two Ta sites (4c and 4d) form a pattern Ta. Ta(1)-Ta(2) in the chain direction. as shown Fig. 1(cThe two Ta sites (4c and 4d) form a pattern Ta(1)-Ta(2)- Ta(1)-Ta(2) in the chain direction. In the B1 represen- tation, two Ta(1) atoms (4c site) move towards the con- nected Ta(2) atoms in chain-A and move away from the connected Ta(2) atoms in chain-B, resulting in a LLSS Ta-tetramerization pattern. Correspondingly, in the B2 representation, the Ta(2) atoms (4d site) move towards the connected Ta(1) atoms in chain-A and move away from the connected Ta(1) atoms in chain-B. Both the B1 and B2 representations would induce a simlar LLSS Ta-tetramerization CDW phase, as shown Fig. 1(c).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Orobengoa, C Capillas, M I Aroyo, J M Perez-Mato, 10.1107/S0021889809028064J. Appl. Crystallogr. 42820D. Orobengoa, C. Capillas, M. I. Aroyo, and J. M. Perez- Mato, J. Appl. Crystallogr. 42, 820 (2009).\",\n \"doi\": \"10.1107/S0021889809028064\"\n },\n {\n \"text\": \". J Perez-Mato, D Orobengoa, M I Aroyo, 10.1107/S0108767310016247Acta Crystallogr. A. 66558J. Perez-Mato, D. Orobengoa, and M. I. Aroyo, Acta Crystallogr. A 66, 558 (2010).\",\n \"doi\": \"10.1107/S0108767310016247\"\n },\n {\n \"text\": \". Y Hinuma, G Pizzi, Y Kumagai, F Oba, I Tanaka, 10.1016/j.commatsci.2016.10.015Comput. Mater. Sci. 113221Y. Hinuma, G. Pizzi, Y. Kumagai, F. Oba and I. Tanaka, Comput. Mater. Sci. 113, 221 (2016).\",\n \"doi\": \"10.1016/j.commatsci.2016.10.015\"\n },\n {\n \"text\": \"Here, for completeness we also used 1×1×2 and 1×1×4 supercells based on the conventional cell, but none of our physical conclusions changed, as shown in the SM. 41Here, for completeness we also used 1×1×2 and 1×1×4 supercells based on the conventional cell, but none of our physical conclusions changed, as shown in the SM [41].\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":482,"cells":{"text":{"kind":"string","value":"\nNegative modes of Schwarzschild black holes in Einstein-Gauss-Bonnet theory\n2008\n\nTakayuki Hirayama \nPhysics Division\nNational Center for Theoretical Sciences\nDepartment of Physics\nNational Taiwan Normal University\n300, 116Hsinchu, TaipeiTaiwan, Taiwan\n\nNegative modes of Schwarzschild black holes in Einstein-Gauss-Bonnet theory\n\nClass. Quantum Grav\n2516200810.1088/0264-9381/25/24/245006Received 7 May 2008, in final form 12 September 2008IOP PUBLISHING CLASSICAL AND QUANTUM GRAVITY Online at stacks.iop.org/CQG/25/245006PACS numbers: 0450Gh, 0470Dy\nWe study non-conformal negative modes of black holes in the five-dimensional Einstein-Gauss-Bonnet gravity theory with a zero or negative cosmological constant (which become Schwarzschild black holes in the Einstein limit), in the context of the semiclassical quantum gravity formulated by the path integral. We find a unique negative mode when the black hole has a negative heat capacity which is the same in Einstein theory. On the other hand, we still find negative modes inside a parameter region where a small black hole has a positive heat capacity. The number is one/two for the case of zero/negative cosmological constant, respectively. In the rest of the parameter region where the heat capacity is positive, we find no negative modes. We discuss the possible physical understanding of having one or two negative modes.\n\nIntroduction\n\nBlack holes are very important objects in classical and quantum gravity. One of the most important and profound properties in black hole physics is its thermodynamics. The thermodynamic quantities can be derived from the path integral formulation of quantum gravity, and thus have connections to the quantum nature of gravity.\n\nIn the case of four-dimensional Euclidean Einstein quantum gravity, the Schwarzschild black hole is an instanton to induce a decay of hot flat space by nucleation of black holes. This instability is suggested by a unique off-shell negative mode around the Schwarzschild instanton [1]. The existence of a negative mode is believed to have a connection with the local thermodynamic stability of Schwarzschild black hole. For example, the anti-de Sitter (AdS) Schwarzschild black hole has a unique negative mode when the black hole has a negative heat capacity, and the negative mode disappears exactly when the black hole becomes locally thermodynamically stable [2]. Reall [3] discussed in detail the connection between the local thermodynamic stability and the stability against small perturbations around a black hole solution (existence of negative modes) and showed they agree under some reasonable assumptions.\n\nThe negative mode has also been studied in connection with the Gregory-Laflamme instability [4], since the negative eigenvalue is related to a Kaluza-Klein mass square of graviton along the directions to which a black string extends [3]. Gubser and Mitra [5] conjectured that there is a negative mode iff the black hole has a thermodynamic instability (the correlated stability conjecture). This conjecture is known to be satisfied for many different black branes including the black branes discussed by Gregory and Laflamme [4], black p-brane solutions in string theory [6] and also black strings in the anti-de Sitter space [2,5,7]. Also more complicated systems such as the D0-D2 bound state [8] and the nonextremal smeared black branes [9] are discussed and the conjecture is satisfied. There are, on the other hand, counter examples [10], which are related to the non-uniqueness of a black string under fixed conserved charges. The correlated stability conjecture is checked to be true for all the known cases when a black string is uniquely determined by the conserved charges.\n\nSince the black hole thermodynamics has a deep connection with the quantum mechanics of gravity, we have a natural question of whether the properties related to the black hole thermodynamics are modified if we add higher dimensional corrections to the Einstein-Hilbert action which is expected to appear in an effective theory of quantum gravity, for example, string theory. The Gauss-Bonnet term is one of such terms and is the lowest dimensional term among them. The black hole solutions are known [11] and the thermodynamics is also calculated [12] in the Einstein-Gauss-Bonnet theory. We naively expect that the local thermodynamic stability and the stability against small perturbations still agree in the Einstein-Gauss-Bonnet theory. In this paper we study the negative mode of the Schwarzschild black hole in Gauss-Bonnet theory with and without a negative cosmological constant. Since the Gauss-Bonnet term becomes topological in four dimensions, we study the five dimensions.\n\nWe find there is a unique negative mode when the black hole has a negative heat capacity. As the black hole becomes larger, the negative eigenvalue goes up and becomes zero exactly when the sign of heat capacity changes from negative to positive. This behavior is exactly the same as that in Einstein gravity and consistent with the fact that the effect of the Gauss-Bonnet term is negligible when a black hole has a large size and the spacetime is weakly curved. Of course, the point where the negative eigenvalue crosses zero slightly differs from that in Einstein gravity due to the Gauss-Bonnet term. However, as the five-dimensional black hole becomes smaller and starts having a positive heat capacity due to the Gauss-Bonnet term, we find a new behavior. We still find negative modes and the numbers of negative modes are one or two in the case of zero or negative cosmological constant, respectively. We also find one or two positive modes for the case of negative cosmological constant. As the black hole becomes further smaller, the negative modes disappear. These results lead us to reconsider the connection between the thermodynamic stability and the existence of negative modes.\n\nIn the next section, we review the thermodynamics of the Schwarzschild black hole in the Gauss-Bonnet theory. Then in section 3, we carry out the S-mode perturbations around the black hole and study the non-conformal negative modes. We only study a simple case, i.e. the five-dimensional Gauss-Bonnet theory. The detailed procedure of the numerical analysis is summarized in appendix A. We summarize our numerical results in figures 1-5 (and also figures B1-B2 in appendix B). The real numerical data are listed in appendix B. In section 4, we interpret our numerical results and conclude in section 5.\n\n\nBlack hole thermodynamics\n\nIn this section, we summarize the setup and thermodynamic quantities of a black hole in the Gauss-Bonnet theory.\n\nThe path integral formalism of Euclidean quantum gravity with a finite temperature T is schematically written as\nZ = Dg exp[−I (g)], F = −T ln Z, (2.1) I = 1 16πG n M d n x √ g R − 2 + c 2 L GB + I b , (2.2) L GB = R 2 − 4R ab R ab + R abcd R abcd , (2.3)\nwhere F is the free energy, G n is the Newton constant, is the cosmological constant, c is a coupling constant of mass dimension −2 in front of the Gauss-Bonnet term L GB , h is the induced metric on the boundary, the surface term I g denotes a generalized Gibbons-Hawking surface term [13] and n is the spacetime dimensions. We also need gauge fixing terms. In this section, we consider < 0. The manifold is asymptotically S 1 × R n−1 and the periodicity for S 1 along the Euclidean time τ is given τ = τ + β, β = 1/T . The partition function Z is semiclassically approximated by the sum of classical paths (saddle point approximation). We are then interested in a black hole solution and the small perturbations about the solution. The black hole solution in the n dimensions is known as [11] \nds 2 =ḡ ab dx a dx b = f (r) dτ 2 + dr 2 f (r) + r 2 d 2 n−2 , (2.4) f (r) = 1 + r 2 c(n − 3)(n − 4) 1 + 1 + 4c(n − 3)(n − 4) (n − 2)(n − 1) (n − 1)μ r n−1 + , (2.5)\nwhere μ is a parameter and is related to the black hole mass M = μA n−2 /(8πG n ), where A n−2 is the area of unit (n − 2) sphere. There are two solutions = ±1 and the solution with = −1 becomes the Schwarzschild black hole in the Einstein gravity in the limit of c → 0. Thus, we only discuss the case of = −1 in this paper. The temperature T is determined to avoid a conical singularity at the horizon and then given by the surface gravitational force at the horizon r = r h , (f (r h ) = 0),\nT = f (r h ) 4π = (n − 2)(n − 3) 2r 2 h + c(n − 4)(n − 5) − 4r 4 h 8πr h (n − 2) r 2 h + c(n − 3)(n − 4) (2.6) for r 2 h > −c(n − 3)(n − 4)\n. Using the horizon radius r h , the black hole mass is written as\nM = A n−2 8πG n n − 2 2 r n−3 h + c(n − 2)(n − 3)(n − 4) 4 r n−5 h − n − 1 r n−1 h . (2.7)\nThe entropy which satisfies the first law in thermodynamic mechanics is computed [12] from\nI (ḡ) = 1 T M(ḡ) − S(ḡ), S = A n−2 4G n r n−2 h + (n − 2)(n − 3)cr n−4 h , (2.8)\nwhere the entropy is no longer simply proportional to the area of the horizon. The local stability as a thermodynamic system is determined from the sign of heat capacity C V ,\nC V = ∂M ∂T . (2.9)\nIf we denote a one-parameter family of geometries which contains the black hole solution as x, the second derivative of the action along the path x is\n∂ 2 I ∂x 2 T = T dx dT −2 dM dT at ∂I ∂x T = 0. (2.10)\nThus if the specific heat is negative, the perturbations around the black hole where (∂I /∂x) T = 0 must have at least one negative mode [3]. This is a basic observation that the local thermodynamic stability is equivalent to the stability against the perturbations about the black hole solution in the semiclassical quantum gravity.\n\n\nBlack hole in five dimensions\n\nThe heat capacity depends on various parameters. Hereafter we study n = 5, since the five dimension is the lowest dimensions where the Gauss-Bonnet term nontrivially contributes to the local dynamics of gravity. In five dimensions (n = 5), the mass is a monotonically increasing function in terms of r h and then the sign of heat capacity is equal to the sign of dT /dr h . Since the temperature is\nT = 3 − r 2 h 6π(r h + (2c/r h )) , (2.11) for r 2 h > −c/2, we divide three regions (1) c 0, (2) 0 < c < 1 −6 and (3) c 1 −6\ndepending on the behavior of heat capacity in terms of the black hole radius r h .\n\n(1) c 0: the temperature behaves similarly to the case of AdS Schwarzschild black hole in Einstein gravity, but goes to infinity at r h = √ −2c before r h = 0. See figure 1. The temperature takes the lowest value at r h = r c ,\nr c = 1 −2 [3 + 6c + 3(1 + 6c )(3 + 2c )]. (2.12)\nThus, C V < 0 for r h < r c and C V > 0 for r h > r c . (2) 0 < c < 1 −6 : the temperature is largely modified at a small black hole region and there is a region where there are three different black holes at a same temperature. See figure 3. The temperature starts from zero and increases up to the black hole radius r a and starts decreasing until r h = r c . After that, the temperature increases again as the black hole becomes larger, r c is same as (2.12) and r a is\nr a = 1 −2 [3 + 6c − 3(1 + 6c )(3 + 2c )]. (2.13)\nAs c → 1 −6 , r a approaches to r c , i.e. r a → r c . r c does not change much by changing c. C V < 0 for r a < r h < r c , and C V > 0 for r h < r a and r h > r c . For c > 0, the temperature goes to zero as the size of the event horizon r h goes to zero due to 2c/r h in (2.11) instead that the temperature goes to infinity for more than five dimensions, see (2.6). Because of this behavior of the temperature in five dimensions, the thermodynamics has a more rich structure and we can study the relation between the thermodynamics and negative modes in a wide situation.\n\n\nS-mode perturbations and negative modes\n\nWe have described the thermodynamic properties of a black hole in the Gauss-Bonnet theory. We now study small perturbations around the black hole solution in five dimensions. The action around the soliton up to the quadratic orders in the perturbations δg ab , (g ab =ḡ ab +δg ab ), becomes\nI = I 0 (ḡ) + I 2 (ḡ, δg), I 2 (ḡ, δg) = d n xδg a b a c b d (ḡ)δg c d . (3.1)\nIn the case that the theory is Einstein gravity (i.e. c = 0) and δg c d is traceless, the operator (ḡ) is called the Euclidean Lichnerowicz operator. Thus, the one loop quantum collections around the solution are proportional to √ det (ḡ), and if we have one or more negative eigenvalues in this operator (ḡ) the black hole solution is not a local minimum, but a saddle point.\n\nWe in this paper study the time-independent S-mode perturbations around the black hole solution. This is because this mode is known to induce the unique negative mode of the Schwarzschild black hole in Einstein gravity and we expect that negative modes appear only in the S-mode perturbations. The time-independent S-mode perturbations are given as\nδg a b = diag(H tt (r), H rr (r), K(r), K(r), K(r)), (3.2)\nwhere the Lorentz indices are made up and down by the background metricḡ ab . In the case of Einstein gravity, these fluctuations have two independent modes, where one mode is a trace mode and the other mode is a traceless mode. The trace mode is related to a conformal rescaling and has a ghost-like kinetic term. The traceless mode gives the unique negative mode. In the Gauss-Bonnet theory, there are also two modes and we study the mode which becomes a traceless mode in the limit to Einstein gravity. We name this mode the 'traceless' mode even though it is no longer traceless and study the negative eigenmodes in this mode. Similarly, we name the other mode which becomes a trace mode in the limit to Einstein gravity as the 'trace' mode.\n\nIn order to compute the eigenvalues of the operator, we have to solve the equation of motion for the eigenmodes\na c b d (ḡ)δg c d = λδg a b ,(3.3)\nwith appropriate boundary conditions at the infinity and the horizon. We note that a eigenmode with λ = 0 is an off-shell mode and a mode with λ = 0 is an on-shell classical perturbation. This eigenequation is complicated and we do not separate the two modes in the equation level, but we put the 'trace' mode zero by giving the appropriate boundary condition in the numerical analysis. We give the detail procedure of numerical analysis in appendix A.\n\nAsymptotic solutions. We first study the asymptotic behaviors of S-mode perturbations at the horizon and the infinity in r. The asymptotic solutions of the 'traceless' mode at the horizon\nare ( = r − r h ) H tt (r) ∝ 1 + O( ), H rr (r) ∝ −1 + O( ), K tt (r) ∝ 2 3 + O( ). (3.4) H tt (r) ∝ 1 + O(ln ), H rr (r) ∝ − 1 + O(ln ), K tt (r) ∝ 2 3 ln + O(1). (3.5)\nAt the leading order, the asymptotic solutions are the same as those in the Einstein theory and they are traceless. The first solution is a normalizable mode which becomes clear if we use the Kruskal coordinate. Similarly, the asymptotic solutions for = 0 at the infinity are\nH tt (r) ∝ r a + +2 + O(r a + ), H rr (r) ∝ r a + + O(r a + −2 ), K tt (r) ∝ 1 3 r a + +2 + O(r a + ). (3.6) H tt (r) ∝ r a − +2 + O(r a − ), H rr (r) ∝ r a − + O(r a − −2 ), K tt (r) ∝ 1 3 r a − +2 + O(r a − ). (3.7)\nHere\na ± = −4 ± 2 1 − λ (1 − c −˜ 3 ) −˜ 3 ,˜ = 3 c 1 + 2c 3 − 1 ,(3.8)\nwhere˜ is an effective cosmological constant computed from the AdS black hole metric in the infinity, i.e.\nf (r) = 1 + r 2 2c 1 − 1 + 2c 3 4μ r 4 + = 1 −˜ 6 r 2 + O(r −2 ). (3.9)\nThe asymptotic solutions are traceless at the leading order and the second solution (3.7) is a normalizable mode. Therefore, a small perturbation must have the asymptotic solutions (3.4) and (3.7) at the horizon and infinity. In order to obtain the eigenvalues and eigenfunctions, we rely on a numerical analysis. In the numerical analysis, we start from a point near the horizon with the normalizable mode (3.4) and solve the equations of motion toward the infinity. We change the value of λ and find the eigenmode by the shooting method. See appendix A for more details.\n\n\nNegative modes\n\nWe now show our numerical results of the eigenvalue λ. Here we only describe the results, and in the next section we will discuss how to understand the results.\n\nAs discussed before, the heat capacity depends on the parameter c in front of the Gauss-Bonnet term and we divide three regions depending on how the heat capacity behaves: (1) c 0, (2) 0 < c < 1 −6 and (3) c 1 −6 . Since the equations of motion are invariant under the following rescaling\nr → αr, M → α 2 M, c → α 2 c, → α −2 , λ → α −2 λ,(3.10)\nwith α > 0, we can always take = −1 in the numerical analysis. The numerical data used in figures 1-5 are given in appendix B. We show our numerical result in figure 1 with (c, ) = (−0.1, −1) in region (1). The behavior for a general value c in region (1) is essentially the same. We can see that the behavior is similar to that in Einstein gravity [2,7] (see figure 2) except that the eigenvalue goes to negative infinity at r h = √ −2c. We find a unique negative mode when the heat capacity is negative. As the horizon size becomes larger, the value becomes larger and goes to zero exactly when the heat capacity changes its sign. The eigenvalue is always positive when the heat capacity is positive.\n\nWe show our numerical result in figure 3 with (c, ) = (0.01, −1) in region (2). The left part of figure 3 is for a small black hole where the small black hole has a positive heat capacity and is thus stable as a thermodynamic system. There are two other larger black holes which have the same temperature of a small black hole with the horizon size r h ∈ [0.024, r a = 0.143]. The behavior of the eigenvalue λ is similar to that in region (1) when a black hole has a large size. This is simply because a large black hole is weakly curved and the effect of the Gauss-Bonnet term is negligible in this region. We find no negative mode and one positive mode when a large black hole has a positive heat capacity and the eigenvalue decreases as a black hole becomes smaller and goes to zero exactly when the heat capacity changes its sign (r h = 1.71). Then, the eigenvalue becomes negative when a black hole has a negative heat capacity. This behavior is the same as in the previous case. However, we find a new behavior in a small black hole case. As the black hole becomes more small, the heat capacity becomes positive again due to the Gauss-Bonnet term (r h = 0.143). Although the negative eigenvalue quickly goes to zero, we still find negative modes until r h = 0.094. Moreover, we find two negative modes for r h ∈ [0.094, 0.143] and the small negative eigenvalue goes exactly to zero at r h = 0.143. We also find one or two positive modes in r h ∈ [0, 0.3]. We show our numerical results for (c, ) = (1, −1) in figure 4 in region (3). In this region, the heat capacity is always positive and we find no negative mode and only find a positive mode whose eigenvalue is essentially constant and does not depend on the black hole size.\n\nFinally, we show the numerical results for the case of the zero cosmological constant in figure 5. We chose c = 0.01. Note that the equations are invariant under the rescaling (3.10) with = 0 and we can further fix one parameter, say c, without losing generality. The heat capacity is positive for small black holes (r h √ 2c) in contrast to the case of Einstein gravity where a Schwarzschild black hole always has a negative heat capacity. We find that a negative mode still exists even if a small black hole has a positive heat capacity as long as r h 0.0612 for c = 0.01.\n\nWe now explain how the behavior of eigenmodes in region (2) continuously reaches to the behavior in regions (1) modes become non-normalizable in the limit → 0. This is clear in figure B2 in appendix B. In order to discuss this transition, it is more proper to introduce a finite cavity. If we introduce a finite cavity at r = r B for the case of zero cosmological constant, we have a large black hole r h ∝ r B with the same temperature and this large black hole has a positive heat capacity. Then the positive eigenmodes λ ∝ 1 r 2 B are expected and become non-normalizable as r B → ∞. In the following section, we discuss how we can understand these results. We are especially interested in the negative modes of small black holes with a positive heat capacity. It is interesting to note that the number of negative modes in this region is two or one and there are two or one larger black holes which have the same temperature for the negative or zero cosmological constant, respectively.\n\n\nLocal dynamics versus local thermodynamic versus global stability\n\nIn region (2), we find the discrepancy between the sign of heat capacity and the existence of negative modes. This leads us to reconsider the connection between the stability against small perturbations and the local and global thermodynamic stabilities of black hole. As mentioned in the introduction, the argument on the connection between the local thermodynamic stability and the stability against small perturbations is given in detail by Reall [3], and they are shown to agree under some reasonable assumptions.\n\nIt is good to start with looking at the counterexamples [10] for the so-called correlated stability conjecture by Gubser and Mitra [5]. This conjecture is stated for a black string. Let us denote the direction to which a black string extends by z. The classical gravitational S-mode perturbations having the wavefunction e ikz along z satisfy the Lorentzian Einstein Lichnerowicz equation (3.1) of a black hole in a lower dimensions with a negative eigenvalue λ = −k 2 . Gubser and Mitra conjecture that a black string with translational symmetry and infinite extent shows a Gregory-Laflamme instability if and only if it has a local thermodynamic instability. In many cases, this conjecture is checked to be true. However, the counter examples are known and those black strings show a Gregory-Laflamme instability even they have the local thermodynamic stability. This discrepancy is understood as follows [10]. The black string is not uniquely identified by the conserved charges and has moduli along which the conserved charges do not change. Since the local thermodynamic of the black hole cannot discuss the stability along the moduli, the Gregory-Laflamme instability can exist along the moduli and this happens in the counterexamples.\n\nLet us discuss whether a similar argument can apply for our numerical results where we found there exist two negative modes in a small black hole in region (2). First of all, there are two known black holes, = 1 or −1 in (2.5). However they have different values of the asymptotic AdS curvature in the infinity and they are not connected by a continuous parameter. A simple product of the n-dimensional Schwarzschild black hole in the Gauss-Bonnet theory and S 1 is not a solution of the n + 1-dimensional Gauss-Bonnet theory and then we cannot simply apply their argument on a black string instability here. If we can embed the Schwarzschild black hole metric in the Gauss-Bonnet theory into a black string metric as a sub-manifold, we may find that a black string is not uniquely identified by the conserved charges.\n\nThe non-uniqueness of a black hole is discussed in [14] and the Reissner-Nordstrom black hole in four dimensions can be made classically unstable by developing a massive scalar hair under the mass and charge fixed. There is a massive scalar mode around the flat space in the Gauss-Bonnet theory [15], but the negative mode is a off-shell mode and we do not see a connection between them.\n\nThus, it is not clear that the existence of negative modes in a small black hole has a connection to the non-uniqueness of a black hole. On the other hand, we have an observation that having two negative modes seems to have a relation with the fact that there are two larger black holes in addition to the smallest black hole at the same temperature, which leads us to reconsider the global stability. (The number of the negative mode is 1 for the case of zero cosmological constant and there are two different size black holes which have the same temperature in this region. ) We then refer to the original interpretation of having a negative mode as the tunneling process which induces the decay of a metastable hot flat space into nucleation of black holes in [1]. At a temperature when the smallest black hole has two negative modes, the free energy of the smallest black hole is largest among that of three black holes and the free energy of the largest black hole is smallest. Then we may speculate that the large negative mode indicates the tunneling process of a hot AdS space into the nucleation of the largest black holes by passing through the smallest black hole solution. The small negative mode may indicate a more complicated tunneling process of a hot AdS space into the nucleation of the largest black holes by first passing through the small black hole solution and next passing through a middle size black hole ( figure 6). We note that we found positive eigenmodes around the middle size black hole. In the entropy comparison between a black string and a black hole in the context of the Gregory-Laflamme instability, the instability mode is a classical (on-shell) mode and thus we have to compare the entropy with a fixed mass when we discuss about the global thermodynamic stability. However the negative modes are off-shell modes in quantum gravity and a tunneling process is discussed at a fixed temperature.\n\nIn this section, we discussed some speculations from our numerical results, but nothing is at all conclusive.\n\n\nSummary and discussions\n\nWe studied non-conformal negative modes around the Schwarzschild black hole in fivedimensional Gauss-Bonnet theory with and without a negative cosmological constant. We found a unique negative mode when a black hole has a large radius and a negative heat capacity. The negative value goes to zero exactly when the sign of heat capacity changes its sign from negative to positive as the black hole radius becomes larger. This behavior is similar to the case of Einstein gravity and is consistent with the fact that the effect of the Gauss-Bonnet term is negligible for a large black hole. On the other hand, we found that negative modes still exist even when a small black hole has a positive heat capacity and the number is 1 or 2 for zero or non-zero negative cosmological constant, respectively. We do not fully understand why this happens and this result may indicate that the black hole is not uniquely parametrized by conserved charges. Even if we could refine the thermodynamic stability to be consistent with the existence of negative modes, it would be still difficult to predict the number of negative modes. Following the original argument in [1], we speculated that the negative modes may indicate the tunneling processes of the decay of a metastable hot AdS space into the nucleation of black holes. However these are speculations and further study is necessary.\n\nThere is a report [16] that a Schwarzschild black hole in five-dimensional Gauss-Bonnet theory without the cosmological constant shows a classical instability against the perturbations which have an arbitrary high-angular momentum along S 3 for the range r h ∈ [0, √ 2(1 + √ 2)c] with c > 0. This instability is expected to exist, at least, for the small negative cosmological constant − r 2 h 1 . This result is also counterintuitive since an angular momentum effectively behaves like a mass and we do not expect that there exist an instability by heavy massive modes. Therefore, the black hole thermodynamics does not capture many gravitational dynamics.\n\nIt is interesting to search for a possibility to embed a Schwarzschild black hole in the Gauss-Bonnet theory into a black string in higher dimensions in order to discuss the correlated stability conjecture. In this direction, it is interesting to study the Gregory-Laflamme instability for a black string solution in the Gauss-Bonnet theory constructed in [17].\n\nThe quantum effects become important when the spacetime curvature becomes strong and we expect that a part of such quantum effects can be captured from studying the effects by higher dimensional operators like Gauss-Bonnet terms. It would be important to understand our results in terms of quantum gravity.\n\nparameters are c, r h and λ. We fix c and r h and change λ to find eigenmodes by the shooting method.\n\nWe analytically solve the equations of motion near the horizon up to 4 ( = r − r h ), for H tt and H rr and up to 5 for K(r) and compute the boundary condition at a small value , i.e. H tt (r h + ), H rr (r h + ), K(r h + ) and K (r h + ). We took = 0.001 for the cases = −1, c = −0.1, 0, 0.01, 1 and r h ∈ [0, 10] in figures 1-4. We then solve the equations of motion numerically using a Fehlberg fourth-fifth order Runge-Kutta method with a degree four interpolant in Maple 11 up to a large value r f in r, r f = 10 3 in figures 1-4.\n\nWe then fit the numerical values H tt (r f ), H rr (r f ), K(r f ) and K (r f ) by the asymptotic solutions at infinity. We analytically compute the asymptotic solutions up to the second leading orders in r. There are four solutions since there are normalizable and non-normalizable solutions of the 'traceless' and 'trace' modes. We fit the numerical values by using all four of these solutions, since even we start the boundary condition near the horizon such that the 'trace' mode is set to be zero, there always exist numerical errors and the 'trace' modes are included in the numerical values. We then obtain the coefficient of the non-normalizable solution of 'traceless' mode in the numerical values and study when the coefficient becomes zero by changing λ using the shooting method.\n\nWe checked that our numerical analysis has a good accuracy from various approaches. Our results approach that in Einstein gravity by taking of smaller values c. We changed the value of and checked that the invariance under the rescaling is satisfied with a good accuracy. We changed the value of and r f and studied that the results do not change.\n\nIn the case of a zero cosmological constant, the procedure is the same except that r f = 100. Since the asymptotic solutions are exponential (∝ e ± √ −2λr ) for the 'traceless' mode and sine (∝ sin( √ ±2λ/3)) for the 'trace' mode, the exponential growing is already manifest at r f = 100.\n\n\nAppendix B. Numerical results\n\nWe show our numerical results. These values are used to plot the figures. We employ the shooting method and change λ by 0.01 to find the eigenvalue. Thus for example the first value (0.8, −2.925) means that the real value λ is located between −2.92 and −2.93. \n\n\nthe region where the black hole has a negative heat capacity disappears and the black hole is always stable as a thermodynamic system. Seefigure 4. C V > 0 for all r h .\n\nFigure 1 .Figure 2 .\n12The eigenvalue λ in terms of r h for c = −0.1 with = −1. The value on the vertical line is for λ (not for T). r c = 1.94.λ → −∞, (T → ∞), at r h The eigenvalue λ in terms of r h for c = 0 (Einstein theory) with = −1.\n\nFigure 3 .\n3The eigenvalue λ for (c, ) = (0.01, −1) in terms of r h ∈ [0, 0.4] (left) and r h ∈ [0, 6.5] (right). r a = 0.143, r c = 1.71. There are two other larger black holes which have the same temperature of a small black hole with the horizon size r h ∈ [0.024, r a = 0.143].(Note that the temperature goes to zero as r h → 0 as explained in the previous section, see(2.11).)\n\n1 − 6 .Figure 4 .\n164,(3) or the case for = 0, by changing c or . Region(2) approaches to the region (1) by decreasing c with fixed. Since r a = √ 2c + O(c 3/2 ) for a small c, the small black hole region simply disappears as c → 0 and region (2) smoothly reaches the region (1). Region (3) can be approached by increasing c from region(2). The region where the black hole has a negative heat capacity (r a r h r c ) shrinks as c gets larger and only the most positive eigenmode remains in region(3). The other two modes are getting close and become no longer normalizable modes when c reaches region (3) c = We can see this tendency infigure B1in appendix B. The limit to = 0 with c fixed is similarly discussed. As approaches zero, the value of positive eigenmodes approaches zero (∝ − ) The eigenvalue λ in terms of r h for c = 1 with = −1.\n\nFigure 5 .\n5The eigenvalue λ in terms of r h ∈ [0, 1] for c = 0.01 with = 0. r c = √ 2c.\n\nFigure 6 .\n6The one-loop effective potential around the AdS space (A), smallest (B) and middle size (C) black holes in region(2). The tunneling processes are shown by dotted lines.\n\n( 1 )( 2 )Figure B1 .Figure( 6 )\n12B16(c, ) = (−0.1, (c, ) = (0.01, λ for (c, ) = (0.1, −1). The dotted line is λ for (c, ) = (0.01, B2. λ for (c, ) = (0.01, −0.1). The dotted line is λ for (c, ) = (0.01, (c, ) = (0.01,\nAcknowledgmentsThe author would like to thank Kazuyuki Furuuchi, Gungwon Kang, Makoto Narita and Dan Tomino. He would also like to thank the Korea Institute for Advanced Study for warm hospitality during his visit where the work was finished. This work is supported by the National Center for Theoretical Sciences, Taiwan (No NSC 97-2119-M-002-001).Appendix A. Numerical analysisWe explain the procedure of numerical analysis. We solve the equations of motion We can see that most of the equations are solved with the above ansatz and finally have three nontrivial equations which are schematically writtenwhere F i (i = 1, 2, 3) are functions of variables written inside the bracket and H tt (r) = dH tt (r)/dr etc. From counting the number of variables and equations, it is easy to see that there are two independent modes in these perturbations which we refer to as 'traceless' and 'trace' modes since they are traceless and trace modes in Einstein gravity limit. We solve the equations of motion from the horizon to the infinity. Because of the invariance under the rescaling (3.10), we can take = −1 in the numerical analysis. The\n. D J Gross, M J Perry, Yaffe L G , 10.1103/PhysRevD.25.330Phys. Rev. D. 25330Gross D J, Perry M J and Yaffe L G 1982 Phys. Rev. D 25 330\n\n. T Prestidge, 10.1103/PhysRevD.61.084002arXiv:hep-th/9907163Phys. Rev. D. 6184002Prestidge T 2000 Phys. Rev. D 61 084002 (arXiv:hep-th/9907163)\n\n. H Reall, 10.1103/PhysRevD.64.044005arXiv:hep-th/0104071Phys. Rev. D. 6444005Reall H S 2001 Phys. Rev. D 64 044005 (arXiv:hep-th/0104071)\n\n. R Gregory, R Laflamme, 10.1103/PhysRevLett.70.2837arXiv:hep-th/9301052Phys. Rev. Lett. 702837Gregory R and Laflamme R 1993 Phys. Rev. Lett. 70 2837 (arXiv:hep-th/9301052)\n\n. R Gregory, R Laflamme, 10.1016/0550-3213(94)90206-2arXiv:hep-th/9404071Nucl. Phys. B. 428399Gregory R and Laflamme R 1994 Nucl. Phys. B 428 399 (arXiv:hep-th/9404071)\n\n. R Gregory, R Laflamme, 10.1103/PhysRevD.51.R305arXiv:hep-th/9410050Phys. Rev. D. 51305Gregory R and Laflamme R 1995 Phys. Rev. D 51 305 (arXiv:hep-th/9410050)\n\n. S S Gubser, Mitra I , arXiv:hep-th/0009126Gubser S S and Mitra I 2000 arXiv:hep-th/0009126\n\n. S S Gubser, Mitra I , 10.1088/1126-6708/2001/08/018arXiv:hep-th/0011127J. High Energy Phys. 0818Gubser S S and Mitra I 2001 J. High Energy Phys. JHEP08(2001)018 (arXiv:hep-th/0011127)\n\n. T Hirayama, G Kang, Y O Lee, 10.1103/PhysRevD.67.024007arXiv:hep-th/0209181Phys. Rev. D. 6724007Hirayama T, Kang G W and Lee Y O 2003 Phys. Rev. D 67 024007 (arXiv:hep-th/0209181)\n\n. T Hirayama, G Kang, 10.1103/PhysRevD.64.064010arXiv:hep-th/0104213Phys. Rev. D. 6464010Hirayama T and Kang G 2001 Phys. Rev. D 64 064010 (arXiv:hep-th/0104213)\n\n. S S ; Gubser, S F Ross, T Wiseman, 10.1088/0264-9381/22/14/006arXiv:hep-th/0411257arXiv:hep-th/0503152J. High Energy Phys. 022933Class. Quantum Grav.Gubser S S 2005 J. High Energy Phys. JHEP02(2005)040 (arXiv:hep-th/0411257) Ross S F and Wiseman T 2005 Class. Quantum Grav. 22 2933 (arXiv:hep-th/0503152)\n\n. T Harmark, Niarchos V Obers, N A , 10.1088/1126-6708/2005/10/045arXiv:hep-th/0509011J. High Energy Phys. 1045Harmark T, Niarchos V and Obers N A 2005 J. High Energy Phys. JHEP10(2005)045 (arXiv:hep-th/0509011)\n\n. D Marolf, B C Palmer, 10.1103/PhysRevD.70.084045arXiv:hep-th/0404139Phys. Rev. D. 7084045Marolf D and Palmer B C 2004 Phys. Rev. D 70 084045 (arXiv:hep-th/0404139)\n\n. J J Friess, S Gubser, Mitra I , 10.1103/PhysRevD.72.104019arXiv:hep-th/0508220Phys. Rev. D. 72104019Friess J J, Gubser S S and Mitra I 2005 Phys. Rev. D 72 104019 (arXiv:hep-th/0508220)\n\n. D G Boulware, S Deser, 10.1103/PhysRevLett.55.2656Phys. Rev. Lett. 552656Boulware D G and Deser S 1985 Phys. Rev. Lett. 55 2656\n\n. R Myers, J Z Simon, 10.1103/PhysRevD.38.2434Phys. Rev. D. 382434Myers R C and Simon J Z 1988 Phys. Rev. D 38 2434\n\n. R Myers, 10.1103/PhysRevD.36.392Phys. Rev. D. 36392Myers R C 1987 Phys. Rev. D 36 392\n\n. S Gubser, 10.1088/0264-9381/22/23/013arXiv:hep-th/0505189Class. Quantum Grav. 225121Gubser S S 2005 Class. Quantum Grav. 22 5121 (arXiv:hep-th/0505189)\n\n. K Stelle, 10.1007/BF00760427Gen. Rel. Grav. 9353Stelle K S 1978 Gen. Rel. Grav. 9 353\n\n. R J Gleiser, Dotti G ; Beroiz, M , Dotti G Gleiser, R J ; Konoplya, R A Zhidenko, A , 10.1103/PhysRevD.76.024012arXiv:gr-qc/0510069arXiv:0802.0267Phys. Rev. D. 7224012Phys. Rev. DGleiser R J and Dotti G 2005 Phys. Rev. D 72 124002 (arXiv:gr-qc/0510069) Beroiz M, Dotti G and Gleiser R J 2007 Phys. Rev. D 76 024012 (arXiv:hep-th/0703074) Konoplya R A and Zhidenko A 2008 arXiv:0802.0267\n\n. T Kobayashi, T Tanaka, 10.1103/PhysRevD.71.084005arXiv:gr-qc/0412139Phys. Rev. D. 7184005Kobayashi T and Tanaka T 2005 Phys. Rev. D 71 084005 (arXiv:gr-qc/0412139)\n"},"corpus_id":{"kind":"number","value":18991817,"string":"18,991,817"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"0804.3694\",\n \"mag\": 2049028784,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1088/0264-9381/25/24/245006\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [],\n \"pdf_sha\": \"21a714fe1824430c075bb6d30810eb128ff659db\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/0804.3694\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Negative modes of Schwarzschild black holes in Einstein-Gauss-Bonnet theory"},"authors":{"kind":"list like","value":["Takayuki Hirayama \nPhysics Division\nNational Center for Theoretical Sciences\nDepartment of Physics\nNational Taiwan Normal University\n300, 116Hsinchu, TaipeiTaiwan, Taiwan"],"string":"[\n \"Takayuki Hirayama \\nPhysics Division\\nNational Center for Theoretical Sciences\\nDepartment of Physics\\nNational Taiwan Normal University\\n300, 116Hsinchu, TaipeiTaiwan, Taiwan\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We study non-conformal negative modes of black holes in the five-dimensional Einstein-Gauss-Bonnet gravity theory with a zero or negative cosmological constant (which become Schwarzschild black holes in the Einstein limit), in the context of the semiclassical quantum gravity formulated by the path integral. We find a unique negative mode when the black hole has a negative heat capacity which is the same in Einstein theory. On the other hand, we still find negative modes inside a parameter region where a small black hole has a positive heat capacity. The number is one/two for the case of zero/negative cosmological constant, respectively. In the rest of the parameter region where the heat capacity is positive, we find no negative modes. We discuss the possible physical understanding of having one or two negative modes."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Black holes are very important objects in classical and quantum gravity. One of the most important and profound properties in black hole physics is its thermodynamics. The thermodynamic quantities can be derived from the path integral formulation of quantum gravity, and thus have connections to the quantum nature of gravity.\nIn the case of four-dimensional Euclidean Einstein quantum gravity, the Schwarzschild black hole is an instanton to induce a decay of hot flat space by nucleation of black holes. This instability is suggested by a unique off-shell negative mode around the Schwarzschild instanton [1]. The existence of a negative mode is believed to have a connection with the local thermodynamic stability of Schwarzschild black hole. For example, the anti-de Sitter (AdS) Schwarzschild black hole has a unique negative mode when the black hole has a negative heat capacity, and the negative mode disappears exactly when the black hole becomes locally thermodynamically stable [2]. Reall [3] discussed in detail the connection between the local thermodynamic stability and the stability against small perturbations around a black hole solution (existence of negative modes) and showed they agree under some reasonable assumptions.\nThe negative mode has also been studied in connection with the Gregory-Laflamme instability [4], since the negative eigenvalue is related to a Kaluza-Klein mass square of graviton along the directions to which a black string extends [3]. Gubser and Mitra [5] conjectured that there is a negative mode iff the black hole has a thermodynamic instability (the correlated stability conjecture). This conjecture is known to be satisfied for many different black branes including the black branes discussed by Gregory and Laflamme [4], black p-brane solutions in string theory [6] and also black strings in the anti-de Sitter space [2,5,7]. Also more complicated systems such as the D0-D2 bound state [8] and the nonextremal smeared black branes [9] are discussed and the conjecture is satisfied. There are, on the other hand, counter examples [10], which are related to the non-uniqueness of a black string under fixed conserved charges. The correlated stability conjecture is checked to be true for all the known cases when a black string is uniquely determined by the conserved charges.\nSince the black hole thermodynamics has a deep connection with the quantum mechanics of gravity, we have a natural question of whether the properties related to the black hole thermodynamics are modified if we add higher dimensional corrections to the Einstein-Hilbert action which is expected to appear in an effective theory of quantum gravity, for example, string theory. The Gauss-Bonnet term is one of such terms and is the lowest dimensional term among them. The black hole solutions are known [11] and the thermodynamics is also calculated [12] in the Einstein-Gauss-Bonnet theory. We naively expect that the local thermodynamic stability and the stability against small perturbations still agree in the Einstein-Gauss-Bonnet theory. In this paper we study the negative mode of the Schwarzschild black hole in Gauss-Bonnet theory with and without a negative cosmological constant. Since the Gauss-Bonnet term becomes topological in four dimensions, we study the five dimensions.\nWe find there is a unique negative mode when the black hole has a negative heat capacity. As the black hole becomes larger, the negative eigenvalue goes up and becomes zero exactly when the sign of heat capacity changes from negative to positive. This behavior is exactly the same as that in Einstein gravity and consistent with the fact that the effect of the Gauss-Bonnet term is negligible when a black hole has a large size and the spacetime is weakly curved. Of course, the point where the negative eigenvalue crosses zero slightly differs from that in Einstein gravity due to the Gauss-Bonnet term. However, as the five-dimensional black hole becomes smaller and starts having a positive heat capacity due to the Gauss-Bonnet term, we find a new behavior. We still find negative modes and the numbers of negative modes are one or two in the case of zero or negative cosmological constant, respectively. We also find one or two positive modes for the case of negative cosmological constant. As the black hole becomes further smaller, the negative modes disappear. These results lead us to reconsider the connection between the thermodynamic stability and the existence of negative modes.\nIn the next section, we review the thermodynamics of the Schwarzschild black hole in the Gauss-Bonnet theory. Then in section 3, we carry out the S-mode perturbations around the black hole and study the non-conformal negative modes. We only study a simple case, i.e. the five-dimensional Gauss-Bonnet theory. The detailed procedure of the numerical analysis is summarized in appendix A. We summarize our numerical results in figures 1-5 (and also figures B1-B2 in appendix B). The real numerical data are listed in appendix B. In section 4, we interpret our numerical results and conclude in section 5."},{"title":"Black hole thermodynamics","content":"In this section, we summarize the setup and thermodynamic quantities of a black hole in the Gauss-Bonnet theory.\nThe path integral formalism of Euclidean quantum gravity with a finite temperature T is schematically written as\nwhere F is the free energy, G n is the Newton constant, is the cosmological constant, c is a coupling constant of mass dimension −2 in front of the Gauss-Bonnet term L GB , h is the induced metric on the boundary, the surface term I g denotes a generalized Gibbons-Hawking surface term [13] and n is the spacetime dimensions. We also need gauge fixing terms. In this section, we consider < 0. The manifold is asymptotically S 1 × R n−1 and the periodicity for S 1 along the Euclidean time τ is given τ = τ + β, β = 1/T . The partition function Z is semiclassically approximated by the sum of classical paths (saddle point approximation). We are then interested in a black hole solution and the small perturbations about the solution. The black hole solution in the n dimensions is known as [11]\nwhere μ is a parameter and is related to the black hole mass M = μA n−2 /(8πG n ), where A n−2 is the area of unit (n − 2) sphere. There are two solutions = ±1 and the solution with = −1 becomes the Schwarzschild black hole in the Einstein gravity in the limit of c → 0. Thus, we only discuss the case of = −1 in this paper. The temperature T is determined to avoid a conical singularity at the horizon and then given by the surface gravitational force at the horizon r = r h , (f (r h ) = 0),\n. Using the horizon radius r h , the black hole mass is written as\nThe entropy which satisfies the first law in thermodynamic mechanics is computed [12] from\nwhere the entropy is no longer simply proportional to the area of the horizon. The local stability as a thermodynamic system is determined from the sign of heat capacity C V ,\nIf we denote a one-parameter family of geometries which contains the black hole solution as x, the second derivative of the action along the path x is\nThus if the specific heat is negative, the perturbations around the black hole where (∂I /∂x) T = 0 must have at least one negative mode [3]. This is a basic observation that the local thermodynamic stability is equivalent to the stability against the perturbations about the black hole solution in the semiclassical quantum gravity."},{"title":"Black hole in five dimensions","content":"The heat capacity depends on various parameters. Hereafter we study n = 5, since the five dimension is the lowest dimensions where the Gauss-Bonnet term nontrivially contributes to the local dynamics of gravity. In five dimensions (n = 5), the mass is a monotonically increasing function in terms of r h and then the sign of heat capacity is equal to the sign of dT /dr h . Since the temperature is\ndepending on the behavior of heat capacity in terms of the black hole radius r h .\n(1) c 0: the temperature behaves similarly to the case of AdS Schwarzschild black hole in Einstein gravity, but goes to infinity at r h = √ −2c before r h = 0. See figure 1. The temperature takes the lowest value at r h = r c ,\nThus, C V < 0 for r h < r c and C V > 0 for r h > r c . (2) 0 < c < 1 −6 : the temperature is largely modified at a small black hole region and there is a region where there are three different black holes at a same temperature. See figure 3. The temperature starts from zero and increases up to the black hole radius r a and starts decreasing until r h = r c . After that, the temperature increases again as the black hole becomes larger, r c is same as (2.12) and r a is\nAs c → 1 −6 , r a approaches to r c , i.e. r a → r c . r c does not change much by changing c. C V < 0 for r a < r h < r c , and C V > 0 for r h < r a and r h > r c . For c > 0, the temperature goes to zero as the size of the event horizon r h goes to zero due to 2c/r h in (2.11) instead that the temperature goes to infinity for more than five dimensions, see (2.6). Because of this behavior of the temperature in five dimensions, the thermodynamics has a more rich structure and we can study the relation between the thermodynamics and negative modes in a wide situation."},{"title":"S-mode perturbations and negative modes","content":"We have described the thermodynamic properties of a black hole in the Gauss-Bonnet theory. We now study small perturbations around the black hole solution in five dimensions. The action around the soliton up to the quadratic orders in the perturbations δg ab , (g ab =ḡ ab +δg ab ), becomes\nIn the case that the theory is Einstein gravity (i.e. c = 0) and δg c d is traceless, the operator (ḡ) is called the Euclidean Lichnerowicz operator. Thus, the one loop quantum collections around the solution are proportional to √ det (ḡ), and if we have one or more negative eigenvalues in this operator (ḡ) the black hole solution is not a local minimum, but a saddle point.\nWe in this paper study the time-independent S-mode perturbations around the black hole solution. This is because this mode is known to induce the unique negative mode of the Schwarzschild black hole in Einstein gravity and we expect that negative modes appear only in the S-mode perturbations. The time-independent S-mode perturbations are given as\nwhere the Lorentz indices are made up and down by the background metricḡ ab . In the case of Einstein gravity, these fluctuations have two independent modes, where one mode is a trace mode and the other mode is a traceless mode. The trace mode is related to a conformal rescaling and has a ghost-like kinetic term. The traceless mode gives the unique negative mode. In the Gauss-Bonnet theory, there are also two modes and we study the mode which becomes a traceless mode in the limit to Einstein gravity. We name this mode the 'traceless' mode even though it is no longer traceless and study the negative eigenmodes in this mode. Similarly, we name the other mode which becomes a trace mode in the limit to Einstein gravity as the 'trace' mode.\nIn order to compute the eigenvalues of the operator, we have to solve the equation of motion for the eigenmodes\nwith appropriate boundary conditions at the infinity and the horizon. We note that a eigenmode with λ = 0 is an off-shell mode and a mode with λ = 0 is an on-shell classical perturbation. This eigenequation is complicated and we do not separate the two modes in the equation level, but we put the 'trace' mode zero by giving the appropriate boundary condition in the numerical analysis. We give the detail procedure of numerical analysis in appendix A.\nAsymptotic solutions. We first study the asymptotic behaviors of S-mode perturbations at the horizon and the infinity in r. The asymptotic solutions of the 'traceless' mode at the horizon\nAt the leading order, the asymptotic solutions are the same as those in the Einstein theory and they are traceless. The first solution is a normalizable mode which becomes clear if we use the Kruskal coordinate. Similarly, the asymptotic solutions for = 0 at the infinity are\nHere\nwhere˜ is an effective cosmological constant computed from the AdS black hole metric in the infinity, i.e.\nThe asymptotic solutions are traceless at the leading order and the second solution (3.7) is a normalizable mode. Therefore, a small perturbation must have the asymptotic solutions (3.4) and (3.7) at the horizon and infinity. In order to obtain the eigenvalues and eigenfunctions, we rely on a numerical analysis. In the numerical analysis, we start from a point near the horizon with the normalizable mode (3.4) and solve the equations of motion toward the infinity. We change the value of λ and find the eigenmode by the shooting method. See appendix A for more details."},{"title":"Negative modes","content":"We now show our numerical results of the eigenvalue λ. Here we only describe the results, and in the next section we will discuss how to understand the results.\nAs discussed before, the heat capacity depends on the parameter c in front of the Gauss-Bonnet term and we divide three regions depending on how the heat capacity behaves: (1) c 0, (2) 0 < c < 1 −6 and (3) c 1 −6 . Since the equations of motion are invariant under the following rescaling\nwith α > 0, we can always take = −1 in the numerical analysis. The numerical data used in figures 1-5 are given in appendix B. We show our numerical result in figure 1 with (c, ) = (−0.1, −1) in region (1). The behavior for a general value c in region (1) is essentially the same. We can see that the behavior is similar to that in Einstein gravity [2,7] (see figure 2) except that the eigenvalue goes to negative infinity at r h = √ −2c. We find a unique negative mode when the heat capacity is negative. As the horizon size becomes larger, the value becomes larger and goes to zero exactly when the heat capacity changes its sign. The eigenvalue is always positive when the heat capacity is positive.\nWe show our numerical result in figure 3 with (c, ) = (0.01, −1) in region (2). The left part of figure 3 is for a small black hole where the small black hole has a positive heat capacity and is thus stable as a thermodynamic system. There are two other larger black holes which have the same temperature of a small black hole with the horizon size r h ∈ [0.024, r a = 0.143]. The behavior of the eigenvalue λ is similar to that in region (1) when a black hole has a large size. This is simply because a large black hole is weakly curved and the effect of the Gauss-Bonnet term is negligible in this region. We find no negative mode and one positive mode when a large black hole has a positive heat capacity and the eigenvalue decreases as a black hole becomes smaller and goes to zero exactly when the heat capacity changes its sign (r h = 1.71). Then, the eigenvalue becomes negative when a black hole has a negative heat capacity. This behavior is the same as in the previous case. However, we find a new behavior in a small black hole case. As the black hole becomes more small, the heat capacity becomes positive again due to the Gauss-Bonnet term (r h = 0.143). Although the negative eigenvalue quickly goes to zero, we still find negative modes until r h = 0.094. Moreover, we find two negative modes for r h ∈ [0.094, 0.143] and the small negative eigenvalue goes exactly to zero at r h = 0.143. We also find one or two positive modes in r h ∈ [0, 0.3]. We show our numerical results for (c, ) = (1, −1) in figure 4 in region (3). In this region, the heat capacity is always positive and we find no negative mode and only find a positive mode whose eigenvalue is essentially constant and does not depend on the black hole size.\nFinally, we show the numerical results for the case of the zero cosmological constant in figure 5. We chose c = 0.01. Note that the equations are invariant under the rescaling (3.10) with = 0 and we can further fix one parameter, say c, without losing generality. The heat capacity is positive for small black holes (r h √ 2c) in contrast to the case of Einstein gravity where a Schwarzschild black hole always has a negative heat capacity. We find that a negative mode still exists even if a small black hole has a positive heat capacity as long as r h 0.0612 for c = 0.01.\nWe now explain how the behavior of eigenmodes in region (2) continuously reaches to the behavior in regions (1) modes become non-normalizable in the limit → 0. This is clear in figure B2 in appendix B. In order to discuss this transition, it is more proper to introduce a finite cavity. If we introduce a finite cavity at r = r B for the case of zero cosmological constant, we have a large black hole r h ∝ r B with the same temperature and this large black hole has a positive heat capacity. Then the positive eigenmodes λ ∝ 1 r 2 B are expected and become non-normalizable as r B → ∞. In the following section, we discuss how we can understand these results. We are especially interested in the negative modes of small black holes with a positive heat capacity. It is interesting to note that the number of negative modes in this region is two or one and there are two or one larger black holes which have the same temperature for the negative or zero cosmological constant, respectively."},{"title":"Local dynamics versus local thermodynamic versus global stability","content":"In region (2), we find the discrepancy between the sign of heat capacity and the existence of negative modes. This leads us to reconsider the connection between the stability against small perturbations and the local and global thermodynamic stabilities of black hole. As mentioned in the introduction, the argument on the connection between the local thermodynamic stability and the stability against small perturbations is given in detail by Reall [3], and they are shown to agree under some reasonable assumptions.\nIt is good to start with looking at the counterexamples [10] for the so-called correlated stability conjecture by Gubser and Mitra [5]. This conjecture is stated for a black string. Let us denote the direction to which a black string extends by z. The classical gravitational S-mode perturbations having the wavefunction e ikz along z satisfy the Lorentzian Einstein Lichnerowicz equation (3.1) of a black hole in a lower dimensions with a negative eigenvalue λ = −k 2 . Gubser and Mitra conjecture that a black string with translational symmetry and infinite extent shows a Gregory-Laflamme instability if and only if it has a local thermodynamic instability. In many cases, this conjecture is checked to be true. However, the counter examples are known and those black strings show a Gregory-Laflamme instability even they have the local thermodynamic stability. This discrepancy is understood as follows [10]. The black string is not uniquely identified by the conserved charges and has moduli along which the conserved charges do not change. Since the local thermodynamic of the black hole cannot discuss the stability along the moduli, the Gregory-Laflamme instability can exist along the moduli and this happens in the counterexamples.\nLet us discuss whether a similar argument can apply for our numerical results where we found there exist two negative modes in a small black hole in region (2). First of all, there are two known black holes, = 1 or −1 in (2.5). However they have different values of the asymptotic AdS curvature in the infinity and they are not connected by a continuous parameter. A simple product of the n-dimensional Schwarzschild black hole in the Gauss-Bonnet theory and S 1 is not a solution of the n + 1-dimensional Gauss-Bonnet theory and then we cannot simply apply their argument on a black string instability here. If we can embed the Schwarzschild black hole metric in the Gauss-Bonnet theory into a black string metric as a sub-manifold, we may find that a black string is not uniquely identified by the conserved charges.\nThe non-uniqueness of a black hole is discussed in [14] and the Reissner-Nordstrom black hole in four dimensions can be made classically unstable by developing a massive scalar hair under the mass and charge fixed. There is a massive scalar mode around the flat space in the Gauss-Bonnet theory [15], but the negative mode is a off-shell mode and we do not see a connection between them.\nThus, it is not clear that the existence of negative modes in a small black hole has a connection to the non-uniqueness of a black hole. On the other hand, we have an observation that having two negative modes seems to have a relation with the fact that there are two larger black holes in addition to the smallest black hole at the same temperature, which leads us to reconsider the global stability. (The number of the negative mode is 1 for the case of zero cosmological constant and there are two different size black holes which have the same temperature in this region. ) We then refer to the original interpretation of having a negative mode as the tunneling process which induces the decay of a metastable hot flat space into nucleation of black holes in [1]. At a temperature when the smallest black hole has two negative modes, the free energy of the smallest black hole is largest among that of three black holes and the free energy of the largest black hole is smallest. Then we may speculate that the large negative mode indicates the tunneling process of a hot AdS space into the nucleation of the largest black holes by passing through the smallest black hole solution. The small negative mode may indicate a more complicated tunneling process of a hot AdS space into the nucleation of the largest black holes by first passing through the small black hole solution and next passing through a middle size black hole ( figure 6). We note that we found positive eigenmodes around the middle size black hole. In the entropy comparison between a black string and a black hole in the context of the Gregory-Laflamme instability, the instability mode is a classical (on-shell) mode and thus we have to compare the entropy with a fixed mass when we discuss about the global thermodynamic stability. However the negative modes are off-shell modes in quantum gravity and a tunneling process is discussed at a fixed temperature.\nIn this section, we discussed some speculations from our numerical results, but nothing is at all conclusive."},{"title":"Summary and discussions","content":"We studied non-conformal negative modes around the Schwarzschild black hole in fivedimensional Gauss-Bonnet theory with and without a negative cosmological constant. We found a unique negative mode when a black hole has a large radius and a negative heat capacity. The negative value goes to zero exactly when the sign of heat capacity changes its sign from negative to positive as the black hole radius becomes larger. This behavior is similar to the case of Einstein gravity and is consistent with the fact that the effect of the Gauss-Bonnet term is negligible for a large black hole. On the other hand, we found that negative modes still exist even when a small black hole has a positive heat capacity and the number is 1 or 2 for zero or non-zero negative cosmological constant, respectively. We do not fully understand why this happens and this result may indicate that the black hole is not uniquely parametrized by conserved charges. Even if we could refine the thermodynamic stability to be consistent with the existence of negative modes, it would be still difficult to predict the number of negative modes. Following the original argument in [1], we speculated that the negative modes may indicate the tunneling processes of the decay of a metastable hot AdS space into the nucleation of black holes. However these are speculations and further study is necessary.\nThere is a report [16] that a Schwarzschild black hole in five-dimensional Gauss-Bonnet theory without the cosmological constant shows a classical instability against the perturbations which have an arbitrary high-angular momentum along S 3 for the range r h ∈ [0, √ 2(1 + √ 2)c] with c > 0. This instability is expected to exist, at least, for the small negative cosmological constant − r 2 h 1 . This result is also counterintuitive since an angular momentum effectively behaves like a mass and we do not expect that there exist an instability by heavy massive modes. Therefore, the black hole thermodynamics does not capture many gravitational dynamics.\nIt is interesting to search for a possibility to embed a Schwarzschild black hole in the Gauss-Bonnet theory into a black string in higher dimensions in order to discuss the correlated stability conjecture. In this direction, it is interesting to study the Gregory-Laflamme instability for a black string solution in the Gauss-Bonnet theory constructed in [17].\nThe quantum effects become important when the spacetime curvature becomes strong and we expect that a part of such quantum effects can be captured from studying the effects by higher dimensional operators like Gauss-Bonnet terms. It would be important to understand our results in terms of quantum gravity.\nparameters are c, r h and λ. We fix c and r h and change λ to find eigenmodes by the shooting method.\nWe analytically solve the equations of motion near the horizon up to 4 ( = r − r h ), for H tt and H rr and up to 5 for K(r) and compute the boundary condition at a small value , i.e. H tt (r h + ), H rr (r h + ), K(r h + ) and K (r h + ). We took = 0.001 for the cases = −1, c = −0.1, 0, 0.01, 1 and r h ∈ [0, 10] in figures 1-4. We then solve the equations of motion numerically using a Fehlberg fourth-fifth order Runge-Kutta method with a degree four interpolant in Maple 11 up to a large value r f in r, r f = 10 3 in figures 1-4.\nWe then fit the numerical values H tt (r f ), H rr (r f ), K(r f ) and K (r f ) by the asymptotic solutions at infinity. We analytically compute the asymptotic solutions up to the second leading orders in r. There are four solutions since there are normalizable and non-normalizable solutions of the 'traceless' and 'trace' modes. We fit the numerical values by using all four of these solutions, since even we start the boundary condition near the horizon such that the 'trace' mode is set to be zero, there always exist numerical errors and the 'trace' modes are included in the numerical values. We then obtain the coefficient of the non-normalizable solution of 'traceless' mode in the numerical values and study when the coefficient becomes zero by changing λ using the shooting method.\nWe checked that our numerical analysis has a good accuracy from various approaches. Our results approach that in Einstein gravity by taking of smaller values c. We changed the value of and checked that the invariance under the rescaling is satisfied with a good accuracy. We changed the value of and r f and studied that the results do not change.\nIn the case of a zero cosmological constant, the procedure is the same except that r f = 100. Since the asymptotic solutions are exponential (∝ e ± √ −2λr ) for the 'traceless' mode and sine (∝ sin( √ ±2λ/3)) for the 'trace' mode, the exponential growing is already manifest at r f = 100."},{"title":"Appendix B. Numerical results","content":"We show our numerical results. These values are used to plot the figures. We employ the shooting method and change λ by 0.01 to find the eigenvalue. Thus for example the first value (0.8, −2.925) means that the real value λ is located between −2.92 and −2.93."},{"title":"Figure 1 .Figure 2 .","content":""},{"title":"Figure 3 .","content":""},{"title":"1 − 6 .Figure 4 .","content":""},{"title":"Figure 5 .","content":""},{"title":"Figure 6 .","content":""},{"title":"( 1 )( 2 )Figure B1 .Figure( 6 )","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Black holes are very important objects in classical and quantum gravity. One of the most important and profound properties in black hole physics is its thermodynamics. The thermodynamic quantities can be derived from the path integral formulation of quantum gravity, and thus have connections to the quantum nature of gravity.\\nIn the case of four-dimensional Euclidean Einstein quantum gravity, the Schwarzschild black hole is an instanton to induce a decay of hot flat space by nucleation of black holes. This instability is suggested by a unique off-shell negative mode around the Schwarzschild instanton [1]. The existence of a negative mode is believed to have a connection with the local thermodynamic stability of Schwarzschild black hole. For example, the anti-de Sitter (AdS) Schwarzschild black hole has a unique negative mode when the black hole has a negative heat capacity, and the negative mode disappears exactly when the black hole becomes locally thermodynamically stable [2]. Reall [3] discussed in detail the connection between the local thermodynamic stability and the stability against small perturbations around a black hole solution (existence of negative modes) and showed they agree under some reasonable assumptions.\\nThe negative mode has also been studied in connection with the Gregory-Laflamme instability [4], since the negative eigenvalue is related to a Kaluza-Klein mass square of graviton along the directions to which a black string extends [3]. Gubser and Mitra [5] conjectured that there is a negative mode iff the black hole has a thermodynamic instability (the correlated stability conjecture). This conjecture is known to be satisfied for many different black branes including the black branes discussed by Gregory and Laflamme [4], black p-brane solutions in string theory [6] and also black strings in the anti-de Sitter space [2,5,7]. Also more complicated systems such as the D0-D2 bound state [8] and the nonextremal smeared black branes [9] are discussed and the conjecture is satisfied. There are, on the other hand, counter examples [10], which are related to the non-uniqueness of a black string under fixed conserved charges. The correlated stability conjecture is checked to be true for all the known cases when a black string is uniquely determined by the conserved charges.\\nSince the black hole thermodynamics has a deep connection with the quantum mechanics of gravity, we have a natural question of whether the properties related to the black hole thermodynamics are modified if we add higher dimensional corrections to the Einstein-Hilbert action which is expected to appear in an effective theory of quantum gravity, for example, string theory. The Gauss-Bonnet term is one of such terms and is the lowest dimensional term among them. The black hole solutions are known [11] and the thermodynamics is also calculated [12] in the Einstein-Gauss-Bonnet theory. We naively expect that the local thermodynamic stability and the stability against small perturbations still agree in the Einstein-Gauss-Bonnet theory. In this paper we study the negative mode of the Schwarzschild black hole in Gauss-Bonnet theory with and without a negative cosmological constant. Since the Gauss-Bonnet term becomes topological in four dimensions, we study the five dimensions.\\nWe find there is a unique negative mode when the black hole has a negative heat capacity. As the black hole becomes larger, the negative eigenvalue goes up and becomes zero exactly when the sign of heat capacity changes from negative to positive. This behavior is exactly the same as that in Einstein gravity and consistent with the fact that the effect of the Gauss-Bonnet term is negligible when a black hole has a large size and the spacetime is weakly curved. Of course, the point where the negative eigenvalue crosses zero slightly differs from that in Einstein gravity due to the Gauss-Bonnet term. However, as the five-dimensional black hole becomes smaller and starts having a positive heat capacity due to the Gauss-Bonnet term, we find a new behavior. We still find negative modes and the numbers of negative modes are one or two in the case of zero or negative cosmological constant, respectively. We also find one or two positive modes for the case of negative cosmological constant. As the black hole becomes further smaller, the negative modes disappear. These results lead us to reconsider the connection between the thermodynamic stability and the existence of negative modes.\\nIn the next section, we review the thermodynamics of the Schwarzschild black hole in the Gauss-Bonnet theory. Then in section 3, we carry out the S-mode perturbations around the black hole and study the non-conformal negative modes. We only study a simple case, i.e. the five-dimensional Gauss-Bonnet theory. The detailed procedure of the numerical analysis is summarized in appendix A. We summarize our numerical results in figures 1-5 (and also figures B1-B2 in appendix B). The real numerical data are listed in appendix B. In section 4, we interpret our numerical results and conclude in section 5.\"\n },\n {\n \"title\": \"Black hole thermodynamics\",\n \"content\": \"In this section, we summarize the setup and thermodynamic quantities of a black hole in the Gauss-Bonnet theory.\\nThe path integral formalism of Euclidean quantum gravity with a finite temperature T is schematically written as\\nwhere F is the free energy, G n is the Newton constant, is the cosmological constant, c is a coupling constant of mass dimension −2 in front of the Gauss-Bonnet term L GB , h is the induced metric on the boundary, the surface term I g denotes a generalized Gibbons-Hawking surface term [13] and n is the spacetime dimensions. We also need gauge fixing terms. In this section, we consider < 0. The manifold is asymptotically S 1 × R n−1 and the periodicity for S 1 along the Euclidean time τ is given τ = τ + β, β = 1/T . The partition function Z is semiclassically approximated by the sum of classical paths (saddle point approximation). We are then interested in a black hole solution and the small perturbations about the solution. The black hole solution in the n dimensions is known as [11]\\nwhere μ is a parameter and is related to the black hole mass M = μA n−2 /(8πG n ), where A n−2 is the area of unit (n − 2) sphere. There are two solutions = ±1 and the solution with = −1 becomes the Schwarzschild black hole in the Einstein gravity in the limit of c → 0. Thus, we only discuss the case of = −1 in this paper. The temperature T is determined to avoid a conical singularity at the horizon and then given by the surface gravitational force at the horizon r = r h , (f (r h ) = 0),\\n. Using the horizon radius r h , the black hole mass is written as\\nThe entropy which satisfies the first law in thermodynamic mechanics is computed [12] from\\nwhere the entropy is no longer simply proportional to the area of the horizon. The local stability as a thermodynamic system is determined from the sign of heat capacity C V ,\\nIf we denote a one-parameter family of geometries which contains the black hole solution as x, the second derivative of the action along the path x is\\nThus if the specific heat is negative, the perturbations around the black hole where (∂I /∂x) T = 0 must have at least one negative mode [3]. This is a basic observation that the local thermodynamic stability is equivalent to the stability against the perturbations about the black hole solution in the semiclassical quantum gravity.\"\n },\n {\n \"title\": \"Black hole in five dimensions\",\n \"content\": \"The heat capacity depends on various parameters. Hereafter we study n = 5, since the five dimension is the lowest dimensions where the Gauss-Bonnet term nontrivially contributes to the local dynamics of gravity. In five dimensions (n = 5), the mass is a monotonically increasing function in terms of r h and then the sign of heat capacity is equal to the sign of dT /dr h . Since the temperature is\\ndepending on the behavior of heat capacity in terms of the black hole radius r h .\\n(1) c 0: the temperature behaves similarly to the case of AdS Schwarzschild black hole in Einstein gravity, but goes to infinity at r h = √ −2c before r h = 0. See figure 1. The temperature takes the lowest value at r h = r c ,\\nThus, C V < 0 for r h < r c and C V > 0 for r h > r c . (2) 0 < c < 1 −6 : the temperature is largely modified at a small black hole region and there is a region where there are three different black holes at a same temperature. See figure 3. The temperature starts from zero and increases up to the black hole radius r a and starts decreasing until r h = r c . After that, the temperature increases again as the black hole becomes larger, r c is same as (2.12) and r a is\\nAs c → 1 −6 , r a approaches to r c , i.e. r a → r c . r c does not change much by changing c. C V < 0 for r a < r h < r c , and C V > 0 for r h < r a and r h > r c . For c > 0, the temperature goes to zero as the size of the event horizon r h goes to zero due to 2c/r h in (2.11) instead that the temperature goes to infinity for more than five dimensions, see (2.6). Because of this behavior of the temperature in five dimensions, the thermodynamics has a more rich structure and we can study the relation between the thermodynamics and negative modes in a wide situation.\"\n },\n {\n \"title\": \"S-mode perturbations and negative modes\",\n \"content\": \"We have described the thermodynamic properties of a black hole in the Gauss-Bonnet theory. We now study small perturbations around the black hole solution in five dimensions. The action around the soliton up to the quadratic orders in the perturbations δg ab , (g ab =ḡ ab +δg ab ), becomes\\nIn the case that the theory is Einstein gravity (i.e. c = 0) and δg c d is traceless, the operator (ḡ) is called the Euclidean Lichnerowicz operator. Thus, the one loop quantum collections around the solution are proportional to √ det (ḡ), and if we have one or more negative eigenvalues in this operator (ḡ) the black hole solution is not a local minimum, but a saddle point.\\nWe in this paper study the time-independent S-mode perturbations around the black hole solution. This is because this mode is known to induce the unique negative mode of the Schwarzschild black hole in Einstein gravity and we expect that negative modes appear only in the S-mode perturbations. The time-independent S-mode perturbations are given as\\nwhere the Lorentz indices are made up and down by the background metricḡ ab . In the case of Einstein gravity, these fluctuations have two independent modes, where one mode is a trace mode and the other mode is a traceless mode. The trace mode is related to a conformal rescaling and has a ghost-like kinetic term. The traceless mode gives the unique negative mode. In the Gauss-Bonnet theory, there are also two modes and we study the mode which becomes a traceless mode in the limit to Einstein gravity. We name this mode the 'traceless' mode even though it is no longer traceless and study the negative eigenmodes in this mode. Similarly, we name the other mode which becomes a trace mode in the limit to Einstein gravity as the 'trace' mode.\\nIn order to compute the eigenvalues of the operator, we have to solve the equation of motion for the eigenmodes\\nwith appropriate boundary conditions at the infinity and the horizon. We note that a eigenmode with λ = 0 is an off-shell mode and a mode with λ = 0 is an on-shell classical perturbation. This eigenequation is complicated and we do not separate the two modes in the equation level, but we put the 'trace' mode zero by giving the appropriate boundary condition in the numerical analysis. We give the detail procedure of numerical analysis in appendix A.\\nAsymptotic solutions. We first study the asymptotic behaviors of S-mode perturbations at the horizon and the infinity in r. The asymptotic solutions of the 'traceless' mode at the horizon\\nAt the leading order, the asymptotic solutions are the same as those in the Einstein theory and they are traceless. The first solution is a normalizable mode which becomes clear if we use the Kruskal coordinate. Similarly, the asymptotic solutions for = 0 at the infinity are\\nHere\\nwhere˜ is an effective cosmological constant computed from the AdS black hole metric in the infinity, i.e.\\nThe asymptotic solutions are traceless at the leading order and the second solution (3.7) is a normalizable mode. Therefore, a small perturbation must have the asymptotic solutions (3.4) and (3.7) at the horizon and infinity. In order to obtain the eigenvalues and eigenfunctions, we rely on a numerical analysis. In the numerical analysis, we start from a point near the horizon with the normalizable mode (3.4) and solve the equations of motion toward the infinity. We change the value of λ and find the eigenmode by the shooting method. See appendix A for more details.\"\n },\n {\n \"title\": \"Negative modes\",\n \"content\": \"We now show our numerical results of the eigenvalue λ. Here we only describe the results, and in the next section we will discuss how to understand the results.\\nAs discussed before, the heat capacity depends on the parameter c in front of the Gauss-Bonnet term and we divide three regions depending on how the heat capacity behaves: (1) c 0, (2) 0 < c < 1 −6 and (3) c 1 −6 . Since the equations of motion are invariant under the following rescaling\\nwith α > 0, we can always take = −1 in the numerical analysis. The numerical data used in figures 1-5 are given in appendix B. We show our numerical result in figure 1 with (c, ) = (−0.1, −1) in region (1). The behavior for a general value c in region (1) is essentially the same. We can see that the behavior is similar to that in Einstein gravity [2,7] (see figure 2) except that the eigenvalue goes to negative infinity at r h = √ −2c. We find a unique negative mode when the heat capacity is negative. As the horizon size becomes larger, the value becomes larger and goes to zero exactly when the heat capacity changes its sign. The eigenvalue is always positive when the heat capacity is positive.\\nWe show our numerical result in figure 3 with (c, ) = (0.01, −1) in region (2). The left part of figure 3 is for a small black hole where the small black hole has a positive heat capacity and is thus stable as a thermodynamic system. There are two other larger black holes which have the same temperature of a small black hole with the horizon size r h ∈ [0.024, r a = 0.143]. The behavior of the eigenvalue λ is similar to that in region (1) when a black hole has a large size. This is simply because a large black hole is weakly curved and the effect of the Gauss-Bonnet term is negligible in this region. We find no negative mode and one positive mode when a large black hole has a positive heat capacity and the eigenvalue decreases as a black hole becomes smaller and goes to zero exactly when the heat capacity changes its sign (r h = 1.71). Then, the eigenvalue becomes negative when a black hole has a negative heat capacity. This behavior is the same as in the previous case. However, we find a new behavior in a small black hole case. As the black hole becomes more small, the heat capacity becomes positive again due to the Gauss-Bonnet term (r h = 0.143). Although the negative eigenvalue quickly goes to zero, we still find negative modes until r h = 0.094. Moreover, we find two negative modes for r h ∈ [0.094, 0.143] and the small negative eigenvalue goes exactly to zero at r h = 0.143. We also find one or two positive modes in r h ∈ [0, 0.3]. We show our numerical results for (c, ) = (1, −1) in figure 4 in region (3). In this region, the heat capacity is always positive and we find no negative mode and only find a positive mode whose eigenvalue is essentially constant and does not depend on the black hole size.\\nFinally, we show the numerical results for the case of the zero cosmological constant in figure 5. We chose c = 0.01. Note that the equations are invariant under the rescaling (3.10) with = 0 and we can further fix one parameter, say c, without losing generality. The heat capacity is positive for small black holes (r h √ 2c) in contrast to the case of Einstein gravity where a Schwarzschild black hole always has a negative heat capacity. We find that a negative mode still exists even if a small black hole has a positive heat capacity as long as r h 0.0612 for c = 0.01.\\nWe now explain how the behavior of eigenmodes in region (2) continuously reaches to the behavior in regions (1) modes become non-normalizable in the limit → 0. This is clear in figure B2 in appendix B. In order to discuss this transition, it is more proper to introduce a finite cavity. If we introduce a finite cavity at r = r B for the case of zero cosmological constant, we have a large black hole r h ∝ r B with the same temperature and this large black hole has a positive heat capacity. Then the positive eigenmodes λ ∝ 1 r 2 B are expected and become non-normalizable as r B → ∞. In the following section, we discuss how we can understand these results. We are especially interested in the negative modes of small black holes with a positive heat capacity. It is interesting to note that the number of negative modes in this region is two or one and there are two or one larger black holes which have the same temperature for the negative or zero cosmological constant, respectively.\"\n },\n {\n \"title\": \"Local dynamics versus local thermodynamic versus global stability\",\n \"content\": \"In region (2), we find the discrepancy between the sign of heat capacity and the existence of negative modes. This leads us to reconsider the connection between the stability against small perturbations and the local and global thermodynamic stabilities of black hole. As mentioned in the introduction, the argument on the connection between the local thermodynamic stability and the stability against small perturbations is given in detail by Reall [3], and they are shown to agree under some reasonable assumptions.\\nIt is good to start with looking at the counterexamples [10] for the so-called correlated stability conjecture by Gubser and Mitra [5]. This conjecture is stated for a black string. Let us denote the direction to which a black string extends by z. The classical gravitational S-mode perturbations having the wavefunction e ikz along z satisfy the Lorentzian Einstein Lichnerowicz equation (3.1) of a black hole in a lower dimensions with a negative eigenvalue λ = −k 2 . Gubser and Mitra conjecture that a black string with translational symmetry and infinite extent shows a Gregory-Laflamme instability if and only if it has a local thermodynamic instability. In many cases, this conjecture is checked to be true. However, the counter examples are known and those black strings show a Gregory-Laflamme instability even they have the local thermodynamic stability. This discrepancy is understood as follows [10]. The black string is not uniquely identified by the conserved charges and has moduli along which the conserved charges do not change. Since the local thermodynamic of the black hole cannot discuss the stability along the moduli, the Gregory-Laflamme instability can exist along the moduli and this happens in the counterexamples.\\nLet us discuss whether a similar argument can apply for our numerical results where we found there exist two negative modes in a small black hole in region (2). First of all, there are two known black holes, = 1 or −1 in (2.5). However they have different values of the asymptotic AdS curvature in the infinity and they are not connected by a continuous parameter. A simple product of the n-dimensional Schwarzschild black hole in the Gauss-Bonnet theory and S 1 is not a solution of the n + 1-dimensional Gauss-Bonnet theory and then we cannot simply apply their argument on a black string instability here. If we can embed the Schwarzschild black hole metric in the Gauss-Bonnet theory into a black string metric as a sub-manifold, we may find that a black string is not uniquely identified by the conserved charges.\\nThe non-uniqueness of a black hole is discussed in [14] and the Reissner-Nordstrom black hole in four dimensions can be made classically unstable by developing a massive scalar hair under the mass and charge fixed. There is a massive scalar mode around the flat space in the Gauss-Bonnet theory [15], but the negative mode is a off-shell mode and we do not see a connection between them.\\nThus, it is not clear that the existence of negative modes in a small black hole has a connection to the non-uniqueness of a black hole. On the other hand, we have an observation that having two negative modes seems to have a relation with the fact that there are two larger black holes in addition to the smallest black hole at the same temperature, which leads us to reconsider the global stability. (The number of the negative mode is 1 for the case of zero cosmological constant and there are two different size black holes which have the same temperature in this region. ) We then refer to the original interpretation of having a negative mode as the tunneling process which induces the decay of a metastable hot flat space into nucleation of black holes in [1]. At a temperature when the smallest black hole has two negative modes, the free energy of the smallest black hole is largest among that of three black holes and the free energy of the largest black hole is smallest. Then we may speculate that the large negative mode indicates the tunneling process of a hot AdS space into the nucleation of the largest black holes by passing through the smallest black hole solution. The small negative mode may indicate a more complicated tunneling process of a hot AdS space into the nucleation of the largest black holes by first passing through the small black hole solution and next passing through a middle size black hole ( figure 6). We note that we found positive eigenmodes around the middle size black hole. In the entropy comparison between a black string and a black hole in the context of the Gregory-Laflamme instability, the instability mode is a classical (on-shell) mode and thus we have to compare the entropy with a fixed mass when we discuss about the global thermodynamic stability. However the negative modes are off-shell modes in quantum gravity and a tunneling process is discussed at a fixed temperature.\\nIn this section, we discussed some speculations from our numerical results, but nothing is at all conclusive.\"\n },\n {\n \"title\": \"Summary and discussions\",\n \"content\": \"We studied non-conformal negative modes around the Schwarzschild black hole in fivedimensional Gauss-Bonnet theory with and without a negative cosmological constant. We found a unique negative mode when a black hole has a large radius and a negative heat capacity. The negative value goes to zero exactly when the sign of heat capacity changes its sign from negative to positive as the black hole radius becomes larger. This behavior is similar to the case of Einstein gravity and is consistent with the fact that the effect of the Gauss-Bonnet term is negligible for a large black hole. On the other hand, we found that negative modes still exist even when a small black hole has a positive heat capacity and the number is 1 or 2 for zero or non-zero negative cosmological constant, respectively. We do not fully understand why this happens and this result may indicate that the black hole is not uniquely parametrized by conserved charges. Even if we could refine the thermodynamic stability to be consistent with the existence of negative modes, it would be still difficult to predict the number of negative modes. Following the original argument in [1], we speculated that the negative modes may indicate the tunneling processes of the decay of a metastable hot AdS space into the nucleation of black holes. However these are speculations and further study is necessary.\\nThere is a report [16] that a Schwarzschild black hole in five-dimensional Gauss-Bonnet theory without the cosmological constant shows a classical instability against the perturbations which have an arbitrary high-angular momentum along S 3 for the range r h ∈ [0, √ 2(1 + √ 2)c] with c > 0. This instability is expected to exist, at least, for the small negative cosmological constant − r 2 h 1 . This result is also counterintuitive since an angular momentum effectively behaves like a mass and we do not expect that there exist an instability by heavy massive modes. Therefore, the black hole thermodynamics does not capture many gravitational dynamics.\\nIt is interesting to search for a possibility to embed a Schwarzschild black hole in the Gauss-Bonnet theory into a black string in higher dimensions in order to discuss the correlated stability conjecture. In this direction, it is interesting to study the Gregory-Laflamme instability for a black string solution in the Gauss-Bonnet theory constructed in [17].\\nThe quantum effects become important when the spacetime curvature becomes strong and we expect that a part of such quantum effects can be captured from studying the effects by higher dimensional operators like Gauss-Bonnet terms. It would be important to understand our results in terms of quantum gravity.\\nparameters are c, r h and λ. We fix c and r h and change λ to find eigenmodes by the shooting method.\\nWe analytically solve the equations of motion near the horizon up to 4 ( = r − r h ), for H tt and H rr and up to 5 for K(r) and compute the boundary condition at a small value , i.e. H tt (r h + ), H rr (r h + ), K(r h + ) and K (r h + ). We took = 0.001 for the cases = −1, c = −0.1, 0, 0.01, 1 and r h ∈ [0, 10] in figures 1-4. We then solve the equations of motion numerically using a Fehlberg fourth-fifth order Runge-Kutta method with a degree four interpolant in Maple 11 up to a large value r f in r, r f = 10 3 in figures 1-4.\\nWe then fit the numerical values H tt (r f ), H rr (r f ), K(r f ) and K (r f ) by the asymptotic solutions at infinity. We analytically compute the asymptotic solutions up to the second leading orders in r. There are four solutions since there are normalizable and non-normalizable solutions of the 'traceless' and 'trace' modes. We fit the numerical values by using all four of these solutions, since even we start the boundary condition near the horizon such that the 'trace' mode is set to be zero, there always exist numerical errors and the 'trace' modes are included in the numerical values. We then obtain the coefficient of the non-normalizable solution of 'traceless' mode in the numerical values and study when the coefficient becomes zero by changing λ using the shooting method.\\nWe checked that our numerical analysis has a good accuracy from various approaches. Our results approach that in Einstein gravity by taking of smaller values c. We changed the value of and checked that the invariance under the rescaling is satisfied with a good accuracy. We changed the value of and r f and studied that the results do not change.\\nIn the case of a zero cosmological constant, the procedure is the same except that r f = 100. Since the asymptotic solutions are exponential (∝ e ± √ −2λr ) for the 'traceless' mode and sine (∝ sin( √ ±2λ/3)) for the 'trace' mode, the exponential growing is already manifest at r f = 100.\"\n },\n {\n \"title\": \"Appendix B. Numerical results\",\n \"content\": \"We show our numerical results. These values are used to plot the figures. We employ the shooting method and change λ by 0.01 to find the eigenvalue. Thus for example the first value (0.8, −2.925) means that the real value λ is located between −2.92 and −2.93.\"\n },\n {\n \"title\": \"Figure 1 .Figure 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"1 − 6 .Figure 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 6 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"( 1 )( 2 )Figure B1 .Figure( 6 )\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"the region where the black hole has a negative heat capacity disappears and the black hole is always stable as a thermodynamic system. Seefigure 4. C V > 0 for all r h .","content":"the region where the black hole has a negative heat capacity disappears and the black hole is always stable as a thermodynamic system. Seefigure 4. C V > 0 for all r h ."},{"id":"","caption":"The eigenvalue λ in terms of r h for c = −0.1 with = −1. The value on the vertical line is for λ (not for T). r c = 1.94.λ → −∞, (T → ∞), at r h The eigenvalue λ in terms of r h for c = 0 (Einstein theory) with = −1.","content":"Figure 1 .Figure 2 .\n12The eigenvalue λ in terms of r h for c = −0.1 with = −1. The value on the vertical line is for λ (not for T). r c = 1.94.λ → −∞, (T → ∞), at r h The eigenvalue λ in terms of r h for c = 0 (Einstein theory) with = −1."},{"id":"","caption":"The eigenvalue λ for (c, ) = (0.01, −1) in terms of r h ∈ [0, 0.4] (left) and r h ∈ [0, 6.5] (right). r a = 0.143, r c = 1.71. There are two other larger black holes which have the same temperature of a small black hole with the horizon size r h ∈ [0.024, r a = 0.143].(Note that the temperature goes to zero as r h → 0 as explained in the previous section, see(2.11).)","content":"Figure 3 .\n3The eigenvalue λ for (c, ) = (0.01, −1) in terms of r h ∈ [0, 0.4] (left) and r h ∈ [0, 6.5] (right). r a = 0.143, r c = 1.71. There are two other larger black holes which have the same temperature of a small black hole with the horizon size r h ∈ [0.024, r a = 0.143].(Note that the temperature goes to zero as r h → 0 as explained in the previous section, see(2.11).)"},{"id":"","caption":",(3) or the case for = 0, by changing c or . Region(2) approaches to the region (1) by decreasing c with fixed. Since r a = √ 2c + O(c 3/2 ) for a small c, the small black hole region simply disappears as c → 0 and region (2) smoothly reaches the region (1). Region (3) can be approached by increasing c from region(2). The region where the black hole has a negative heat capacity (r a r h r c ) shrinks as c gets larger and only the most positive eigenmode remains in region(3). The other two modes are getting close and become no longer normalizable modes when c reaches region (3) c = We can see this tendency infigure B1in appendix B. The limit to = 0 with c fixed is similarly discussed. As approaches zero, the value of positive eigenmodes approaches zero (∝ − ) The eigenvalue λ in terms of r h for c = 1 with = −1.","content":"1 − 6 .Figure 4 .\n164,(3) or the case for = 0, by changing c or . Region(2) approaches to the region (1) by decreasing c with fixed. Since r a = √ 2c + O(c 3/2 ) for a small c, the small black hole region simply disappears as c → 0 and region (2) smoothly reaches the region (1). Region (3) can be approached by increasing c from region(2). The region where the black hole has a negative heat capacity (r a r h r c ) shrinks as c gets larger and only the most positive eigenmode remains in region(3). The other two modes are getting close and become no longer normalizable modes when c reaches region (3) c = We can see this tendency infigure B1in appendix B. The limit to = 0 with c fixed is similarly discussed. As approaches zero, the value of positive eigenmodes approaches zero (∝ − ) The eigenvalue λ in terms of r h for c = 1 with = −1."},{"id":"","caption":"The eigenvalue λ in terms of r h ∈ [0, 1] for c = 0.01 with = 0. r c = √ 2c.","content":"Figure 5 .\n5The eigenvalue λ in terms of r h ∈ [0, 1] for c = 0.01 with = 0. r c = √ 2c."},{"id":"","caption":"The one-loop effective potential around the AdS space (A), smallest (B) and middle size (C) black holes in region(2). The tunneling processes are shown by dotted lines.","content":"Figure 6 .\n6The one-loop effective potential around the AdS space (A), smallest (B) and middle size (C) black holes in region(2). The tunneling processes are shown by dotted lines."},{"id":"","caption":"(c, ) = (−0.1, (c, ) = (0.01, λ for (c, ) = (0.1, −1). The dotted line is λ for (c, ) = (0.01, B2. λ for (c, ) = (0.01, −0.1). The dotted line is λ for (c, ) = (0.01, (c, ) = (0.01,","content":"( 1 )( 2 )Figure B1 .Figure( 6 )\n12B16(c, ) = (−0.1, (c, ) = (0.01, λ for (c, ) = (0.1, −1). The dotted line is λ for (c, ) = (0.01, B2. λ for (c, ) = (0.01, −0.1). The dotted line is λ for (c, ) = (0.01, (c, ) = (0.01,"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"the region where the black hole has a negative heat capacity disappears and the black hole is always stable as a thermodynamic system. Seefigure 4. C V > 0 for all r h .\",\n \"content\": \"the region where the black hole has a negative heat capacity disappears and the black hole is always stable as a thermodynamic system. Seefigure 4. C V > 0 for all r h .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The eigenvalue λ in terms of r h for c = −0.1 with = −1. The value on the vertical line is for λ (not for T). r c = 1.94.λ → −∞, (T → ∞), at r h The eigenvalue λ in terms of r h for c = 0 (Einstein theory) with = −1.\",\n \"content\": \"Figure 1 .Figure 2 .\\n12The eigenvalue λ in terms of r h for c = −0.1 with = −1. The value on the vertical line is for λ (not for T). r c = 1.94.λ → −∞, (T → ∞), at r h The eigenvalue λ in terms of r h for c = 0 (Einstein theory) with = −1.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The eigenvalue λ for (c, ) = (0.01, −1) in terms of r h ∈ [0, 0.4] (left) and r h ∈ [0, 6.5] (right). r a = 0.143, r c = 1.71. There are two other larger black holes which have the same temperature of a small black hole with the horizon size r h ∈ [0.024, r a = 0.143].(Note that the temperature goes to zero as r h → 0 as explained in the previous section, see(2.11).)\",\n \"content\": \"Figure 3 .\\n3The eigenvalue λ for (c, ) = (0.01, −1) in terms of r h ∈ [0, 0.4] (left) and r h ∈ [0, 6.5] (right). r a = 0.143, r c = 1.71. There are two other larger black holes which have the same temperature of a small black hole with the horizon size r h ∈ [0.024, r a = 0.143].(Note that the temperature goes to zero as r h → 0 as explained in the previous section, see(2.11).)\"\n },\n {\n \"id\": \"\",\n \"caption\": \",(3) or the case for = 0, by changing c or . Region(2) approaches to the region (1) by decreasing c with fixed. Since r a = √ 2c + O(c 3/2 ) for a small c, the small black hole region simply disappears as c → 0 and region (2) smoothly reaches the region (1). Region (3) can be approached by increasing c from region(2). The region where the black hole has a negative heat capacity (r a r h r c ) shrinks as c gets larger and only the most positive eigenmode remains in region(3). The other two modes are getting close and become no longer normalizable modes when c reaches region (3) c = We can see this tendency infigure B1in appendix B. The limit to = 0 with c fixed is similarly discussed. As approaches zero, the value of positive eigenmodes approaches zero (∝ − ) The eigenvalue λ in terms of r h for c = 1 with = −1.\",\n \"content\": \"1 − 6 .Figure 4 .\\n164,(3) or the case for = 0, by changing c or . Region(2) approaches to the region (1) by decreasing c with fixed. Since r a = √ 2c + O(c 3/2 ) for a small c, the small black hole region simply disappears as c → 0 and region (2) smoothly reaches the region (1). Region (3) can be approached by increasing c from region(2). The region where the black hole has a negative heat capacity (r a r h r c ) shrinks as c gets larger and only the most positive eigenmode remains in region(3). The other two modes are getting close and become no longer normalizable modes when c reaches region (3) c = We can see this tendency infigure B1in appendix B. The limit to = 0 with c fixed is similarly discussed. As approaches zero, the value of positive eigenmodes approaches zero (∝ − ) The eigenvalue λ in terms of r h for c = 1 with = −1.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The eigenvalue λ in terms of r h ∈ [0, 1] for c = 0.01 with = 0. r c = √ 2c.\",\n \"content\": \"Figure 5 .\\n5The eigenvalue λ in terms of r h ∈ [0, 1] for c = 0.01 with = 0. r c = √ 2c.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The one-loop effective potential around the AdS space (A), smallest (B) and middle size (C) black holes in region(2). The tunneling processes are shown by dotted lines.\",\n \"content\": \"Figure 6 .\\n6The one-loop effective potential around the AdS space (A), smallest (B) and middle size (C) black holes in region(2). The tunneling processes are shown by dotted lines.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(c, ) = (−0.1, (c, ) = (0.01, λ for (c, ) = (0.1, −1). The dotted line is λ for (c, ) = (0.01, B2. λ for (c, ) = (0.01, −0.1). The dotted line is λ for (c, ) = (0.01, (c, ) = (0.01,\",\n \"content\": \"( 1 )( 2 )Figure B1 .Figure( 6 )\\n12B16(c, ) = (−0.1, (c, ) = (0.01, λ for (c, ) = (0.1, −1). The dotted line is λ for (c, ) = (0.01, B2. λ for (c, ) = (0.01, −0.1). The dotted line is λ for (c, ) = (0.01, (c, ) = (0.01,\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". D J Gross, M J Perry, Yaffe L G , 10.1103/PhysRevD.25.330Phys. Rev. D. 25330Gross D J, Perry M J and Yaffe L G 1982 Phys. Rev. D 25 330","doi":""},{"text":". T Prestidge, 10.1103/PhysRevD.61.084002arXiv:hep-th/9907163Phys. Rev. D. 6184002Prestidge T 2000 Phys. Rev. D 61 084002 (arXiv:hep-th/9907163)","doi":""},{"text":". H Reall, 10.1103/PhysRevD.64.044005arXiv:hep-th/0104071Phys. Rev. D. 6444005Reall H S 2001 Phys. Rev. D 64 044005 (arXiv:hep-th/0104071)","doi":""},{"text":". R Gregory, R Laflamme, 10.1103/PhysRevLett.70.2837arXiv:hep-th/9301052Phys. Rev. Lett. 702837Gregory R and Laflamme R 1993 Phys. Rev. Lett. 70 2837 (arXiv:hep-th/9301052)","doi":""},{"text":". R Gregory, R Laflamme, 10.1016/0550-3213(94)90206-2arXiv:hep-th/9404071Nucl. Phys. B. 428399Gregory R and Laflamme R 1994 Nucl. Phys. B 428 399 (arXiv:hep-th/9404071)","doi":""},{"text":". R Gregory, R Laflamme, 10.1103/PhysRevD.51.R305arXiv:hep-th/9410050Phys. Rev. D. 51305Gregory R and Laflamme R 1995 Phys. Rev. D 51 305 (arXiv:hep-th/9410050)","doi":""},{"text":". S S Gubser, Mitra I , arXiv:hep-th/0009126Gubser S S and Mitra I 2000 arXiv:hep-th/0009126","doi":""},{"text":". S S Gubser, Mitra I , 10.1088/1126-6708/2001/08/018arXiv:hep-th/0011127J. High Energy Phys. 0818Gubser S S and Mitra I 2001 J. High Energy Phys. JHEP08(2001)018 (arXiv:hep-th/0011127)","doi":""},{"text":". T Hirayama, G Kang, Y O Lee, 10.1103/PhysRevD.67.024007arXiv:hep-th/0209181Phys. Rev. D. 6724007Hirayama T, Kang G W and Lee Y O 2003 Phys. Rev. D 67 024007 (arXiv:hep-th/0209181)","doi":""},{"text":". T Hirayama, G Kang, 10.1103/PhysRevD.64.064010arXiv:hep-th/0104213Phys. Rev. D. 6464010Hirayama T and Kang G 2001 Phys. Rev. D 64 064010 (arXiv:hep-th/0104213)","doi":""},{"text":". S S ; Gubser, S F Ross, T Wiseman, 10.1088/0264-9381/22/14/006arXiv:hep-th/0411257arXiv:hep-th/0503152J. High Energy Phys. 022933Class. Quantum Grav.Gubser S S 2005 J. High Energy Phys. JHEP02(2005)040 (arXiv:hep-th/0411257) Ross S F and Wiseman T 2005 Class. Quantum Grav. 22 2933 (arXiv:hep-th/0503152)","doi":""},{"text":". T Harmark, Niarchos V Obers, N A , 10.1088/1126-6708/2005/10/045arXiv:hep-th/0509011J. High Energy Phys. 1045Harmark T, Niarchos V and Obers N A 2005 J. High Energy Phys. JHEP10(2005)045 (arXiv:hep-th/0509011)","doi":""},{"text":". D Marolf, B C Palmer, 10.1103/PhysRevD.70.084045arXiv:hep-th/0404139Phys. Rev. D. 7084045Marolf D and Palmer B C 2004 Phys. Rev. D 70 084045 (arXiv:hep-th/0404139)","doi":""},{"text":". J J Friess, S Gubser, Mitra I , 10.1103/PhysRevD.72.104019arXiv:hep-th/0508220Phys. Rev. D. 72104019Friess J J, Gubser S S and Mitra I 2005 Phys. Rev. D 72 104019 (arXiv:hep-th/0508220)","doi":""},{"text":". D G Boulware, S Deser, 10.1103/PhysRevLett.55.2656Phys. Rev. Lett. 552656Boulware D G and Deser S 1985 Phys. Rev. Lett. 55 2656","doi":""},{"text":". R Myers, J Z Simon, 10.1103/PhysRevD.38.2434Phys. Rev. D. 382434Myers R C and Simon J Z 1988 Phys. Rev. D 38 2434","doi":""},{"text":". R Myers, 10.1103/PhysRevD.36.392Phys. Rev. D. 36392Myers R C 1987 Phys. Rev. D 36 392","doi":""},{"text":". S Gubser, 10.1088/0264-9381/22/23/013arXiv:hep-th/0505189Class. Quantum Grav. 225121Gubser S S 2005 Class. Quantum Grav. 22 5121 (arXiv:hep-th/0505189)","doi":""},{"text":". K Stelle, 10.1007/BF00760427Gen. Rel. Grav. 9353Stelle K S 1978 Gen. Rel. Grav. 9 353","doi":""},{"text":". R J Gleiser, Dotti G ; Beroiz, M , Dotti G Gleiser, R J ; Konoplya, R A Zhidenko, A , 10.1103/PhysRevD.76.024012arXiv:gr-qc/0510069arXiv:0802.0267Phys. Rev. D. 7224012Phys. Rev. DGleiser R J and Dotti G 2005 Phys. Rev. D 72 124002 (arXiv:gr-qc/0510069) Beroiz M, Dotti G and Gleiser R J 2007 Phys. Rev. D 76 024012 (arXiv:hep-th/0703074) Konoplya R A and Zhidenko A 2008 arXiv:0802.0267","doi":""},{"text":". T Kobayashi, T Tanaka, 10.1103/PhysRevD.71.084005arXiv:gr-qc/0412139Phys. Rev. D. 7184005Kobayashi T and Tanaka T 2005 Phys. Rev. D 71 084005 (arXiv:gr-qc/0412139)","doi":""}],"string":"[\n {\n \"text\": \". D J Gross, M J Perry, Yaffe L G , 10.1103/PhysRevD.25.330Phys. Rev. D. 25330Gross D J, Perry M J and Yaffe L G 1982 Phys. Rev. D 25 330\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Prestidge, 10.1103/PhysRevD.61.084002arXiv:hep-th/9907163Phys. Rev. D. 6184002Prestidge T 2000 Phys. Rev. D 61 084002 (arXiv:hep-th/9907163)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Reall, 10.1103/PhysRevD.64.044005arXiv:hep-th/0104071Phys. Rev. D. 6444005Reall H S 2001 Phys. Rev. D 64 044005 (arXiv:hep-th/0104071)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Gregory, R Laflamme, 10.1103/PhysRevLett.70.2837arXiv:hep-th/9301052Phys. Rev. Lett. 702837Gregory R and Laflamme R 1993 Phys. Rev. Lett. 70 2837 (arXiv:hep-th/9301052)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Gregory, R Laflamme, 10.1016/0550-3213(94)90206-2arXiv:hep-th/9404071Nucl. Phys. B. 428399Gregory R and Laflamme R 1994 Nucl. Phys. B 428 399 (arXiv:hep-th/9404071)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Gregory, R Laflamme, 10.1103/PhysRevD.51.R305arXiv:hep-th/9410050Phys. Rev. D. 51305Gregory R and Laflamme R 1995 Phys. Rev. D 51 305 (arXiv:hep-th/9410050)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S S Gubser, Mitra I , arXiv:hep-th/0009126Gubser S S and Mitra I 2000 arXiv:hep-th/0009126\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S S Gubser, Mitra I , 10.1088/1126-6708/2001/08/018arXiv:hep-th/0011127J. High Energy Phys. 0818Gubser S S and Mitra I 2001 J. High Energy Phys. JHEP08(2001)018 (arXiv:hep-th/0011127)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Hirayama, G Kang, Y O Lee, 10.1103/PhysRevD.67.024007arXiv:hep-th/0209181Phys. Rev. D. 6724007Hirayama T, Kang G W and Lee Y O 2003 Phys. Rev. D 67 024007 (arXiv:hep-th/0209181)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Hirayama, G Kang, 10.1103/PhysRevD.64.064010arXiv:hep-th/0104213Phys. Rev. D. 6464010Hirayama T and Kang G 2001 Phys. Rev. D 64 064010 (arXiv:hep-th/0104213)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S S ; Gubser, S F Ross, T Wiseman, 10.1088/0264-9381/22/14/006arXiv:hep-th/0411257arXiv:hep-th/0503152J. High Energy Phys. 022933Class. Quantum Grav.Gubser S S 2005 J. High Energy Phys. JHEP02(2005)040 (arXiv:hep-th/0411257) Ross S F and Wiseman T 2005 Class. Quantum Grav. 22 2933 (arXiv:hep-th/0503152)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Harmark, Niarchos V Obers, N A , 10.1088/1126-6708/2005/10/045arXiv:hep-th/0509011J. High Energy Phys. 1045Harmark T, Niarchos V and Obers N A 2005 J. High Energy Phys. JHEP10(2005)045 (arXiv:hep-th/0509011)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Marolf, B C Palmer, 10.1103/PhysRevD.70.084045arXiv:hep-th/0404139Phys. Rev. D. 7084045Marolf D and Palmer B C 2004 Phys. Rev. D 70 084045 (arXiv:hep-th/0404139)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J J Friess, S Gubser, Mitra I , 10.1103/PhysRevD.72.104019arXiv:hep-th/0508220Phys. Rev. D. 72104019Friess J J, Gubser S S and Mitra I 2005 Phys. Rev. D 72 104019 (arXiv:hep-th/0508220)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D G Boulware, S Deser, 10.1103/PhysRevLett.55.2656Phys. Rev. Lett. 552656Boulware D G and Deser S 1985 Phys. Rev. Lett. 55 2656\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Myers, J Z Simon, 10.1103/PhysRevD.38.2434Phys. Rev. D. 382434Myers R C and Simon J Z 1988 Phys. Rev. D 38 2434\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Myers, 10.1103/PhysRevD.36.392Phys. Rev. D. 36392Myers R C 1987 Phys. Rev. D 36 392\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Gubser, 10.1088/0264-9381/22/23/013arXiv:hep-th/0505189Class. Quantum Grav. 225121Gubser S S 2005 Class. Quantum Grav. 22 5121 (arXiv:hep-th/0505189)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Stelle, 10.1007/BF00760427Gen. Rel. Grav. 9353Stelle K S 1978 Gen. Rel. Grav. 9 353\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R J Gleiser, Dotti G ; Beroiz, M , Dotti G Gleiser, R J ; Konoplya, R A Zhidenko, A , 10.1103/PhysRevD.76.024012arXiv:gr-qc/0510069arXiv:0802.0267Phys. Rev. D. 7224012Phys. Rev. DGleiser R J and Dotti G 2005 Phys. Rev. D 72 124002 (arXiv:gr-qc/0510069) Beroiz M, Dotti G and Gleiser R J 2007 Phys. Rev. D 76 024012 (arXiv:hep-th/0703074) Konoplya R A and Zhidenko A 2008 arXiv:0802.0267\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Kobayashi, T Tanaka, 10.1103/PhysRevD.71.084005arXiv:gr-qc/0412139Phys. Rev. D. 7184005Kobayashi T and Tanaka T 2005 Phys. Rev. D 71 084005 (arXiv:gr-qc/0412139)\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":483,"cells":{"text":{"kind":"string","value":"\nApproximate Task Tree Retrieval in a Knowledge Network for Robotic Cooking\n8 Jul 2022\n\nMd Sadman Sakib \nDavid Paulius \nYu Sun \nApproximate Task Tree Retrieval in a Knowledge Network for Robotic Cooking\n8 Jul 2022B2A168BBE8E57E0B15A27CAF981EDE90arXiv:2207.03693v1[cs.RO]Service RoboticsTask PlanningPlanning under Uncertainty\nFlexible task planning continues to pose a difficult challenge for robots, where a robot is unable to creatively adapt their task plans to new or unseen problems, which is mainly due to the limited knowledge it has about its actions and world.Motivated by a human's ability to adapt, we explore how task plans from a knowledge graph, known as the Functional Object-Oriented Network (FOON), can be generated for novel problems requiring concepts that are not readily available to the robot in its knowledge base.Knowledge from 140 cooking recipes are structured in a FOON knowledge graph, which is used for acquiring task plan sequences known as task trees.Task trees can be modified to replicate recipes in a FOON knowledge graph format, which can be useful for enriching FOON with new recipes containing unknown object and state combinations, by relying upon semantic similarity.We demonstrate the power of task tree generation to create task trees with never-before-seen ingredient and state combinations as seen in recipes from the Recipe1M+ dataset, with which we evaluate the quality of the trees based on how accurately they depict newly added ingredients.Our experimental results show that our system is able to provide task sequences with 76% correctness.\n\nI. INTRODUCTION\n\nM AJOR efforts in robotics research have been devoted to the development of intelligent agents with the ability to understand human intentions and to perform actions for problems in human-centered domains.Such domains include, but are not limited to, the application of robots for assisting the elderly and disabled, food delivery, and cooking activities.However, the main difficulty in designing robots for humancentered domains lies in the variety of tasks and the dynamic nature of environments in which these robots will operate.\n\nIn terms of robotic cooking, ingredients or objects may come in different forms, shapes or sizes, and there are many states to account for when executing recipes [1].Additionally, there may be times where a robot is unable to execute an entire recipe due to the unavailability of certain ingredients or objects in its surroundings; this may happen when a robot is required Manuscript received: February 24, 2022; Revised May 20, 2022; Accepted June 25, 2022.This paper was recommended for publication by Jingang Yi upon evaluation of the Associate Editor and Reviewers' comments.This work was supported by the National Science Foundation under Grants 1910040 and 1812933. 1 Md Sadman Sakib and Yu Sun are members of the Robot Perception and Action Lab (RPAL), which is part of the Department of Computer Science & Engineering at the University of South Florida, Tampa, FL, USA.mdsadman@usf.edu,yusun@usf.edu 2 David Paulius is a postdoctoral researcher in the Intelligent Robot Lab at Brown University, Providence, RI, USA.dpaulius@cs.brown.eduDigital Object Identifier (DOI): see top of this page.\n\n\nFOON Reference Task Tree\n\nFinal Task Tree\n\n\nSearch Modify\n\nDish: Salad Ingredients: q (onion, peeled) q (cucumber, diced) q (oregano, dried) q (prunes, dried) q (feta cheese, cubed)\n\nInput: Ingredient list in the from of (object, state) pairs\n\n\nWord Embedding\n\nFig. 1.Overview of our task tree generation procedure for novel problems using FOON.As input to our task tree generation system, a set of ingredients and their states are given, from which a novel task tree can be generated.\n\nto prepare meal variations.In the latter case, a robot would need to adapt its knowledge to include or disregard ingredients, which may or may not be known by the robot, in its task plan, which is not often considered in modern robot applications.Motivated by a human's ability to creatively adapt to novel scenarios, we explore the problem of generating task plans for novel scenarios that would be presented to a robot, given the limited knowledge available to it as a knowledge graph.\n\nIn this work, we use a knowledge representation known as the Functional Object-Oriented Network (FOON) [2], which builds upon previous work on joint object-action representation [3]- [5].In prior work, we introduced how a FOON can be created from video annotations and used for task planning [2].However, as with other works, task planning is limited to the knowledge found in a FOON, as it only contains knowledge for a limited number of recipe and ingredient variations.For example, if a robot were to prepare a salad with a certain combination of ingredients that have never been encountered together before in FOON, then this would pose a problem since there is no concept of that type of salad.Previously, we investigated how the knowledge in FOON can be extended so as to generalize concepts across object types [6].Based on this intuition, we propose that existing knowledge of similar recipes can be used to derive alternative solutions (as graphs) that did not exist before.Therefore, in this work, we introduce a procedure known as task tree generation (described in Figure 1), where a reference task tree is extracted from FOON and then modified to match a required set of ingredients for a given recipe.This may require the addition of possibly neverbefore-seen ingredients and the removal of ingredients in the reference task tree that are not valid for the new recipe.To show the effectiveness of our approach, we apply task tree generation on recipes from the Recipe1M+ dataset [7].This is particularly useful for creating new FOONs and task plans for never-before-seen recipes without manual effort.\n\nOur contributions and outcomes in this paper are as follows: (i) We develop a recipe generation pipeline that takes a set of ingredients as input and outputs a task plan in the form of a task tree; (ii) We design a heuristic-based search algorithm to find a recipe from the knowledge graph, which closely matches with the required ingredients, without exploring all possible paths and thus minimizing time and space cost; (iii) We present a task tree visualization method that highlights how each ingredient's state changes in a recipe known as the progress line; and (iv) We perform an evaluation on the generalizability of FOON for Recipe1M+ recipes.\n\n\nII. BACKGROUND\n\n\nA. Functional Object-Oriented Network\n\nA FOON is a bipartite network with two types of nodes: object nodes and motion nodes.Affordances [8] are depicted via edges that connect objects to actions and enforce action sequencing.A fundamental structure known as a functional unit represents actions in FOON by describing the state change of objects before and after execution, and it has input object nodes, output object nodes, and a motion node.It is akin to a planning operator in PDDL (short for Planning Domain Definition Language) [9], where input and output nodes describe preconditions and effects respectively.Figure 2 depicts examples of functional units for the action sequence of: 1) picking and placing a whole tomato on a cutting board, 2) slicing the tomato with a knife, and 3) pouring the sliced tomato from the cutting board into a bowl.\n\nTypically, we create FOONs by annotating video demonstrations.A FOON representing a single activity is referred to as a subgraph; a subgraph contains functional units in sequence to describe objects' states before and after each action occurs, and what objects are being manipulated.Presently, this annotation process is done manually, but previous work has investigated how graphs can be annotated in a semi-automatic manner [10].Two or more subgraphs can be merged together to form what we call a universal FOON.A universal FOON can contain variations of recipes once it has been merged with several sources of knowledge.This merging procedure is simply a union operation applied to all functional units from each subgraph we wish to combine; as a result, duplicate functional units are eliminated in the merged network [2].Presently, FOON comprises of 140 subgraph annotations of videos from YouTube, Activity-Net [11], and EPIC-KITCHENS [12], which are all available on our website [13].\n\n1) Task Planning with FOON: Task planning involves retrieving a task tree [2] (a high-level task plan) that satisfies a goal defined as an object node.Finding a task tree requires knowledge about the state of a robot's environment, specifically what objects are available and the state(s) in which they are in.Starting from the goal, the algorithm searches for candidate functional units in a depth-wise manner, and then, for each candidate unit, it searches its input nodes in a breadthwise manner to see if they are available to the robot.More recently, we introduced a modified version of the algorithm to find the optimal path to reach a goal node, which is reflected by weights given to each functional unit corresponding to its success rate of execution by a robot [14].However, these algorithms are not suited to generate task trees for recipes not found in FOON; for this task, we use semantic similarity [6] while searching to modify task trees as required for a given problem and ingredient set.\n\n\nB. Related Works\n\nDeveloping a robotic agent that understands human instructions like \"cut the onion in half\" and executing them precisely in a real world setting is a very difficult task.The most noteworthy work to address this challenge is the knowledge processing framework KNOWROB [15].They created this knowledge base by collecting information through observations in a sensor-equipped environment.Although the robots in [16] were successful in retrieving a pancake recipe and executing it using the KNOWROB reasoning framework, the authors did not show whether the robots are able to modify the recipe to meet users' preference.On the contrary, our work focuses on this degree of flexibility so that one can obtain a task plan for any combination of ingredients.\n\nFurniture assembly and cooking are similar in a sense that both of these task domains begin with raw materials and end with a finished product in the form of an assembled furniture piece or a dish.Given geometric details of parts, IkeaBot [17] deduces a task plan to assemble furniture by using geometric reasoning and checking possible sub-assemblies, which is computationally expensive.On the other hand, we explore the FOON knowledge base by using an efficient heuristic-based graph search.[18] relied on deep learning whereas [19] created an ontology based on the product description found on IKEA website to retrieve information about an object.Manipulation action tree banks [20] were proposed to organize manipulation actions in multiple levels of abstraction, which a robot can use for understanding and executing a task.Unlike our work, they did not handle different object states and are unable to produce a plan for complex manipulation tasks.More recently, the authors in [21] presented a method for robot assembly planning by recognizing graphical instruction manuals.They identify objects from a series of images, find the relationship between the previous and current image, and create a task sequence graph for robot execution.\n\nEfforts have been made to dynamically modify a recipe to meet certain constraints.[22] gathered various datasets to understand existing recipes, which flavors people like, and chemical structure of ingredients to invent novel recipes based on big data approaches.[23] proposed a new approach to generate novel recipes based on ingredients, preparation time, and cooking steps, where a user can specify those parameters.They used a taxonomy to define the semantics of ingredients and cooking steps where the inner nodes refer to generalized items (e.g., meat) and the items close to the leaf nodes are more specific (e.g., beef, pork).EvoChef [24] and AutoChef [25], created novel recipes using natural language processing and evolutionary algorithms but without considering user preference.[26] optimize a recipe according to consumers' taste by tuning parameters such as cooking time, amount of salt.However, they maintain the same ingredients in their plans, whereas our work shows more flexibility by making the same dish using different ingredient combinations based on the consumers' dietary preferences.\n\nSome works focused on finding a reasonable substitution for an ingredient.[27] collected contextual information from the MyFitnessPal mobile app and used it to search for an equivalent ingredient.For example, chicken and beef can substitute each other if they are often consumed with rice and salad.[28] addressed the issue of specific dietary requirements.Similar to our work, they combined the semantic information from a knowledge graph of food and a word embedding model to substitute ingredients to improve a dish's nutritional value.Similar works were done by [29] using adaptation rules and by [30] using the transformer-based model BERT.However, these works solely explored ingredient substitution, whereas we incorporate this idea for robotic execution of a recipe.Conceptually, our work is similar to [31], [32] that use ontology and domain knowledge to derive missing information.In addition to the knowledge required to manipulate unknown objects, a task tree shows how they can be integrated with other known or unknown items in a complex task like cooking.\n\n\nIII. PROPOSED METHOD\n\nOur objective in this work is to generate FOON task plans for never-before-seen recipes through a method that uses FOON as a knowledge base.This method requires a recipe's ingredient set I and associated dish type D as input and produces a task tree outlining how to prepare the dish.Due to the limited recipe variety in FOON, the likelihood of encountering recipes with different ingredient combinations is very high.Specifically, FOON may not contain references to the desired recipe, or, more likely, many ingredients will be absent from FOON.Despite this shortcoming, our idea is to use the limited knowledge of a FOON to produce task trees for novel recipes, so as to take advantage of existing recipe knowledge bases like Recipe1M+ [7].The main idea is to retrieve a reference task tree from FOON, which closely resembles the desired recipe, and then modify it to produce a final task tree as shown in Figure 1.The three main steps of our method are discussed in the following subsections.\n\n\nA. Identifying a Reference Goal Node\n\nFrom a universal FOON, we extract a recipe's task tree by retrieving all functional units related to the final state of a dish, which we refer to as a goal node.It is likely that a goal node for a dish does not exist in FOON; in this case, we try to find a suitable node that closely matches the required dish, which we refer to as a reference goal node (denoted as G).Such a node is used to extract a reference task tree T , which is simply a task tree that creates the reference goal object.To facilitate the selection of the ideal reference goal node G, we created a recipe classification with 30 dish classes (e.g., salad, pizza), with which we can categorize recipes in the form of FOON subgraphs.The purpose of this classification is to find similar recipes known to us given a dish type.score ← Find similarity between I and S 5: end for 6: G ← End product of subgraph with maximum score Output: G Algorithm 1 uses this classification to find candidate goal nodes.For example, if D = soup, candidates can be {corn soup, potato soup, ...} depending on the soup recipes that exist in FOON.We store the list of ingredients required to prepare a dish in its goal node so that we can easily identify the required ingredients to make a dish.Next, the algorithm checks each candidate goal node to select the one that has maximum Word2Vec similarity with the ingredient set I.\n\nWe use a word embedding model provided by spaCy [33], an open-source library for natural language processing, to get a vectorized representation of each word.Similar words are grouped together in the embedding space.For instance, cucumber and zucchini will be close to each other but far away from salt.Similarity is given as a score ranging from 0 to 1, where 1 indicates that two items are semantically similar.In this work, we deem two items similar if the score exceeds a predefined threshold value of 0.90.\n\n\nB. Extracting a Reference Task Tree\n\nA dish can be prepared in many ways depending on its ingredients and recipe instructions.Similarly, in FOON, there are different paths to reach a goal node.We aim to find the path that closely resembles the desired recipe to use it as a base of reference for our final task tree.We assume that all utensils and ingredients needed for a recipe are available in the kitchen, although it is possible that the ingredients are not available in the desired state.For example, a recipe may require sliced carrot, but the kitchen contains whole carrot.\n\nIn a typically large FOON, exploring all paths can be timeconsuming and memory intensive.Hence, unlike the previous search algorithm that aims to find the most optimal task tree for making a certain object [14], we use Best First Search.At each level, this method evaluates all candidate nodes based on the heuristic function g(n) = |A n ∩ I|, where A n and I denote the set of ingredients stored in node n and ingredients to be added to the dish respectively, and g(n) computes a similarity score checking the overlap between the two sets of ingredients.We use Word2Vec similarity instead of word matching so that equivalent ingredients are considered as an overlap.The similarity score indicates how close we are to the end product (i.e., final dish).The node with the maximum score is considered as the closest to the goal and thus will be favored over the other candidates as shown in Figure 3.\n\nAlgorithm 2 Retrieve the task tree of the reference goal node Input: reference goal node G and given ingredients I 1: T ← A list of functional units in the task tree 2: K ← A list of items available in the kitchen 3: Q ← A queue of items to search 4: Q.push(G) 5: while Q is not empty do 6:\n\nL ← Q.dequeue()\n7:\nif L does not exist in K then 8:\n\nC ← Find all functional units that create L 9:\n\nC ← Select one from C using heuristic function 10:\n\nT .append(C ) 11:\n\nfor each input n in C do 12:\n\nif n is not visited then 13:\n\nQ.enqueue(n)\n\n14:\n\nMark n as visited 15:\n\nend if\n16:\nend for 17:\n\nend if 18: end while 19: T .reverse()Output: T\n\nThe retrieval algorithm (presented as Algorithm 2) works as follows.Let the initial task tree T be an empty list.T will be populated with the functional units required to prepare the dish defined by the goal node G. Let K be the list of items in the kitchen and Q be a queue for keeping track of items that need to be explored.At each iteration, we remove the first item L from Q and check if it is found in K in its desired state.If it does not contain L, we search for the functional units C that create it (i.e., functional units that contain L as an output node).For instance, while searching for cake batter, we may find that it is not available in the kitchen, but there are a few functional units in FOON that show how to make different types of cake batter made up of varying sets of ingredients.These units are added to C as potential candidates.We choose the best candidate C from C based on our heuristic function and add it to the task tree.To execute C , we have to make sure that we know how to prepare all of its input ingredients.This is why all of its input ingredients are added to Q to be searched next.The search continues until Q is empty, meaning that all of the required items are discovered.At the end, T is reversed to provide the functional units in the correct sequence that a robot can follow to prepare the dish at G.\n\n\nC. Task Tree Modification\n\nThe reference tree T gives a blueprint about how the tasks should be executed, but it may not contain all the ingredients required for a new recipe.Therefore, we need to modify T to include missing ingredients.The modification procedure first integrates all required ingredients and then removes those that were not given as input.If an ingredient does not exist in FOON, or if it exists in a different state, we find a semantically similar ingredient in the knowledge base using a multi-level search.First, we try to find a suitable match for similar ingredients in T because we already know how the ingredients in T are integrated together to produce the final dish.This gives us high confidence to incorporate missing ingredients.Otherwise, we search in the universal FOON for an equivalent ingredient.As in Section III-A, we use spaCy's word embedding model to find the closest object in the embedding space.Each ingredient contains an object name and its state.When checking if an ingredient exists in T , there are four possible cases depending on the match with the object and state (also presented in Table I):\n\nCase 1: If the given ingredient exactly matches a leaf node (i.e., a starting node in the state required by the recipe) in T in terms of object and state name, no adjustment is required.\n\nCase 2: The exact object is not found in T , but there is an equivalent ingredient with the exact same state.In that case, we can simply substitute the equivalent object name with that of the required object.For instance, if chopped chili pepper is required in the recipe but its equivalent in T is chopped jalapeño, we can replace jalapeño with chili pepper.If jalapeño is also required, a copy of the functional units required for processing jalapeño is created, after which substitution is performed to preserve jalapeño in T .\n\nCase 3: If the object is found in T but not in the desired state, we search FOON for functional units needed to get the object in the required state.We do this with another retrieval query using Algorithm 2, where the goal node will be an object node in the leaf of T .We then add the retrieved functional units to that leaf.For example, if whole tomato is given as the starting state but there is diced tomato in the leaf of the tree, we will search using the goal node diced tomato as the criteria and the added functional units will reflect how diced tomato can be made from its whole state.\n\nCase 4: The object and state of the equivalent ingredient do not match with the given ingredient.We can handle this case by combining the solution from the previous two cases.\n\nFirst, we substitute the object name and then add the necessary functional units to achieve its desired state.\n\n(a) FOON-generated task tree for greek salad\n\n\nProgress line of recipe: 00a7b36a48 onion peeled,whole ---> pick-and-place ---> peeled,whole(cutting board) ---> slice ---> sliced(cutting board) ---> pick-and-place ---> sliced(mixing bowl) ---> mix\n\n\nEnd product: greek salad cucumber diced(bowl) ---> pick-and-place ---> diced(mixing bowl) ---> mix\n\n\nEnd product: greek salad oregano dried(bowl) ---> pour ---> dried(mixing bowl) ---> mix\n\nEnd product: greek salad prunes (substituted from raisin, confidence: 62.9%)\n\n\ndried(bowl) ---> pour ---> dried(mixing bowl) ---> mix\n\n\nEnd product: greek salad feta cheese cubed(container) ---> scoop ---> cubed(spoon) ---> pour\n\nEnd product: greek salad (b) Progress line for greek salad Fig. 4. Illustration of: a) a task tree generated with our approach, and b) its corresponding progress line visualization.\n\nIf the equivalent ingredient does not exist in the retrieved task tree but is featured in other recipes in FOON, it poses some extra challenges.Let us assume that the recipe requires roasted turkey but the closest object to turkey available in FOON is chicken.Normally, chicken can have states such as raw, fried, baked, and marinated across recipes.This makes it difficult to choose the state best suited for the task tree.To aid in selecting an appropriate state, we used a state taxonomy [1] with 12 different categories, such as raw, finely separated, and liquid.Similar states, such as sliced and diced, are grouped together such that existing states can be used as a blueprint for other states of the same category.If there are multiple candidates to choose from, we break the tie by favoring the most frequently used state in the recipes of the given dish type.\n\nWhen substituting a state, we also have to select an appropriate manipulation for which the state is valid.For instance, with the states diced and sliced, we treat the appropriate action with different motion verbs dice and slice.For this, we find the most frequently used verb associated with the required state and use it as the motion node label.Once we retrieve the functional units to prepare an ingredient, the challenge is to connect them to the reference tree.To do this, we find functional units in T that can easily be modified, which are those that new ingredients can be added as input nodes.For example, any unit that involve mixing or adding ingredients to a container can accept any number of inputs, whereas a robot cannot cut or chop multiple items at the same time.Therefore, for each missing ingredient, we connect the subtrees to these types of functional units, resulting in a connected acyclic graph.We do not need to create this connection if the equivalent ingredient already exists in T .Finally, the modified task tree will have a list of functional units containing the input ingredients that, if executed sequentially, will produce the desired dish.\n\n\nIV. EXPERIMENTS\n\nIt is difficult to quantitatively compare a knowledge graphbased method with others, as the benefit we get from it lies in its flexibility and adaptability to answer a query.Therefore, in our experiments, we used Recipe1M+ to create a diverse set of queries for ingredient combinations from several dish classes as shown in Table II.Our focus is on generating a correct task tree for recipes in Recipe1M+.To make our evaluation more robust, we excluded dish classes for which we had less than 4 recipes in FOON, as it is difficult to gain a general picture of a dish class with too few recipes.Among the selected dish types, we randomly selected 5 classes for our evaluation: salad, cake, soup, omelette and drinks.Next, 100 recipes of each category were randomly selected from Recipe1M+ for a total of 500 recipes.We ran these experiments on a MacBook Pro equipped with 2.6 GHz 6-Core Intel Core i7 processor and 16GB RAM.\n\n\nA. Evaluation Strategy\n\nOur objective is to evaluate the quality of a task tree for unseen ingredient combinations from the Recipe1M+ dataset.It is impossible to judge a recipe by solely relying on automated testing such as Intersection over Union (IoU) due to there being many ways to prepare a meal.Furthermore, despite having the same ingredients, one recipe can differ from another recipe in terms of cooking steps while both are valid or correct.Hence, we chose to evaluate the FOON-generated task trees by manually checking each task plan.Although it requires a lot of manual effort, we believe that human judgement is necessary to evaluate recipes.To make the process easier, instead of checking the functional units in the task tree, we check each ingredient by reviewing its progress line.\n\nProgress Line: In cooking, ingredients undergo several changes in their physical state (e.g., from whole to sliced) and/or location.We refer to this sequence of changes as an ingredient's progress line, which is another visualization of a task tree.We developed a tool to view the progress line of each ingredient to manually evaluate generated task trees.Figures 4a and 4b shows an example of a generated task tree and its progress line respectively, where objects, states and motions are colored in black, green and red respectively.\n\n\nB. Results\n\nTask trees generated for the 500 recipes were thoroughly evaluated using the progress line visualizer.When checking a recipe, we review each of its ingredients and indicate whether a given ingredient is incorrect, partially incorrect or correct by assigning it a score from {0,1,2}, where '0' means that an ingredient's progress line is incorrect, '1' means partially incorrect, and '2' means correct.A label of partially incorrect means that a state label assigned to the ingredient is not contextually or physically relevant to it.Without modifying the reference task tree, we would only see higher scores for ingredients that already exist within the unmodified tree, while missing ingredients (i.e., items that need to be added to the tree) or extra ingredients (i.e., items that need to be pruned from the tree) will be assigned an incorrect score.However, our proposed algorithm will be able to generate task plans for unseen ingredients in addition to already existing items while removing extra ingredients.In Figure 5, we present a comparison of these two approaches by reporting the percentage of correctly generated recipes with varying degrees of thresholds.The result shows that the proposed algorithm can generate 76% of the recipes correctly on average if the threshold of correctness for each recipe is set to 100%.When the threshold is 100%, a task tree must have correct progress lines for each ingredient.It is evident from the experiments that using similarity-based ingredient substitution greatly improves the performance and correctness thresholds across all dish types.\n\n\nC. User Study\n\nTo verify correctness and completeness, we conducted a user study to see whether there is any significant difference between the FOON generated and Recipe1M+ recipes, following the same approach and questionnaires (see Figure 6) from a previous study [34].Since a task tree in its original format is not easy to read and verify for individuals not familiar with FOON, we translated each tree into recipe-like sentences.In this study, there were 9 participants, where each was given 10 recipes randomly selected from Recipe1M+ and its equivalent generated recipe from our method to assign ratings.This survey can be accessed by the provided link. 1 Our hypothesis H o is that there is no significant difference between FOONgenerated and Recipe1M+ recipes.With the ratings collected from users for questions 4-7, we performed a null hypothesis test with significance level α = 0.05 and obtained the pvalues 0.12, 0.34, 0.20, 0.64, 0.68, 0.73 and 0.35 respectively.Since all the p-values are fairly greater than α, we cannot reject H o ; hence, we can conclude that the generated recipes are equivalent to Recipe1M+ recipes in terms of correctness, completeness and clarity.\n\nV. DISCUSSION Recipe1M+ has a significantly higher quantity of recipes and ingredients than FOON, making it very challenging to produce a task plan that perfectly matches the original textual recipe.Nevertheless, even with the limited concepts available in FOON, the flexibility of the task tree generation process allows us to achieve correct results in 76% of the cases.This mainly depends on the selection of equivalent objects for unseen ingredients, as an incorrect substitution may introduce errors in an ingredient's progress line.For example, the equivalent of apricot in FOON is lemon according to the word embedding model we used.However, we cannot squeeze an apricot to extract its juice as we can with a lemon.Therefore, the progress line of apricot does not correctly reflect how it should change its state after each manipulation action.I have made this exact dish Yes, but I left out some of the ingredients listed Yes, but I added some ingredients not listed No 4) Does this recipe seem correct to you? 5) Do the steps appear to be in the right order? 6) Are all steps in the recipe correct?7) Does each recipe step give enough information in order to complete it?8) Does the recipe skip steps that are obvious to you? 9) If attempting to follow the given steps, how confident are you that you could make this dish?10) Do the steps in this recipe appear to be clear and easy to follow?Fig. 6.Overview of the survey questions used in our user study.An ingredient may be used differently across recipes; for example, pineapple is mixed with other fruits while preparing a fruit salad, but added as a topping in a pizza.For this reason, when we add a branch to the task tree, we follow different rules to decide where the missing ingredient should be added.For instance, in drink recipes, branches can be added when objects are being poured into a container whereas for salad, ingredients are typically mixed in a bowl.These concepts are usually not captured in the word embedding model, and rulebased actions are not often sufficient to accurately infer the task plan.Hence, we will focus on improving our methodology through generalization in the future.We also noted that it is not always possible to include the user-provided state of an ingredient if the kitchen contains it in a non-reversible state, such as melted cheese when diced cheese is required.We do not mark those as errors since a new batch of ingredients is required to follow user input.Overall, we noted the following errors in our experiments as incorrect substitution: 39%, incorrect state: 27%, incorrect motion: 14% and incorrect integration: 20%.\n\nSuch errors can be minimized with correction from users, where instead of selecting the top suggestion from the embedding model for each unseen ingredient, users can decide on the ideal substitute among the top 5 candidates.In the case of Figure 7, corrections can be made for the ingredients cool whip and pistachio using the second (whipped cream) and fourth (peanut) ranked candidates respectively.Furthermore, we can enrich the knowledge base by adding recipes with diverse set of ingredients so that we can make substitutions with higher confidence and with minimal user correction.\n\nFinally, up to this point, recipes are added to FOON by manually labelling cooking videos, which is a very timeconsuming and error-prone process.If we can automate the process of creating subgraphs from video or text, we could easily expand FOON by adding recipes from various sources.Using a method such as the task tree generation process can help to reduce the amount of effort required by constructing graphs in a semi-automatic manner, in addition to computer vision approaches for extracting graphs from video [10].\n\nA task tree can be used for bootstrapping robot task execution, as it provides a high-level description of a sequence of actions needed to be executed by a robot free from domainspecific robot details or properties (e.g. the number of endeffectors, which end-effector to use for certain actions, motion primitives, etc.).For instance, a task tree extracted from a knowledge graph containing instructions of furniture assembly can represent how the raw materials should be connected to build the furnished product.Although a task tree cannot be executed in its original state, it can however be connected to existing frameworks for manipulation planning, such as PDDL where functional units can be treated as planning operators and can then be used to retrieve plans corresponding to robot low-level motion primitives [35].\n\n\nA. Complexity Analysis of the Algorithm\n\nA FOON is represented as an adjacency list, which allows us to retrieve functional unit candidates in O(1) time.In the worst case, the number of candidates is equal to the total number of FOON recipes.However, this number is very small on average and thus is negligible.Therefore, for n functional units in the task tree, the required time is O(n).At the tree modification stage, if an equivalent ingredient exists in the reference tree, substitution can be done in O(1) time.However, if no suitable equivalent is present, we have to search other recipes for equivalents, which takes O(k) time, where k is the number of unique ingredients in FOON.We accelerate this process by computing a mapping of equivalent ingredient in the pre-processing phase to find an equivalent ingredient in O(1) time.Hence, the overall time complexity of the search algorithm is O(n), where n is the length of the retrieved path.The space complexity is also O(n) since we store only one path.With the addition of new recipes, FOON increases in size, which in turn can potentially create alternative paths to a goal node.Exploring all paths will incur a larger time and space complexity.To avoid that, we chose the heuristic-based search algorithm where the heuristic was designed using the given ingredients.Overall, we found the average computation time required to solve a problem instance was 80 milliseconds.\n\n\nVI. CONCLUSION AND FUTURE WORK\n\nIn this work, our goal was to create a system that will enable a robot to derive task plans in novel scenarios not found in its knowledge base.It is almost impossible to equip a robot with knowledge of every possible recipe with countless combinations of ingredients.To address this, we explored recipe generation using FOON as a knowledge base.Similar to how humans can modify a recipe dynamically by using their previous cooking knowledge to determine how items available in the kitchen can be used, we demonstrate how an unseen ingredient can be integrated to an existing recipe by learning from an equivalent ingredient based on semantic similarity metrics.In our experiments, we generated task plans for different ingredient combinations using our approach and showed that plans were accurate in most cases.\n\nThe evaluation method used in this work is subjective.In the future, we will explore how generated task trees can be executed with a robot to concretely evaluate our method.Furthermore, we aim to enrich our FOON knowledge base with a more diverse array of recipes by semi-automatically creating subgraphs from instructional videos or text-based sources such as cookbooks.Furthermore, additional work is required to learn a new embedding model that would not only measure the similarity between two words but also the similarity of two functional units or two task plans that considers the context of the ingredients.\n\nFig. 2 .\n2\nFig. 2. Illustration of a universal FOON made of 140 recipe subgraphs, and three (3) connected functional units, where object nodes are denoted by green circles and motion nodes are denoted by red squares.\n\n\nAlgorithm 1\n1\nFind a reference goal node Input: given ingredients I, dish type D 1: candidates ← Retrieve recipes of type D 2: for each subgraph in candidates do 3: S ← Ingredients of subgraph 4:\n\n\nFig. 3 .\n3\nFig. 3. Overview of our reference task tree retrieval method.Each node's value indicate a score returned by the heuristic function; empty nodes are not reached.Red arrows denote the chosen path based on maximum score.\n\n\nFig. 5 .\n5\nFig.5.Graph showing the relationship between the threshold of correctness (X-axis) of each recipe and the number of recipes (Y-axis) we can generate with that threshold.The result of recipe generation with and without using task tree modification are presented as solid and dashed lines respectively; the same colored lines indicate the same dish type.\n\n\nFig. 7 .\n7\nFig. 7. Top 5 equivalent ingredients in FOON for some randomly chosen unseen ingredients from Recipe1M+.\n\n\nTABLE I RULES\nI\nTO MODIFY THE REFERENCE TASK TREE BASED ON THE DIFFERENCE BETWEEN A GIVEN INGREDIENT AND ITS EQUIVALENT IN THE TREE AS DISCUSSED IN SECTION III-C.THE TERM FU DENOTES FUNCTIONAL UNIT.\nCase End leaf in refer-Given ingredientObjectStateSubstitutionAdd branchence task treematchedmatched1sliced carrotsliced carrot--2chopped jalapeñochopped chili peppersubstitute jalapeño with-chili pepper3diced tomatowhole tomato-add FUs to make dicedtomato from whole tomato4minced shallotpeeled onionsubstitute shallot withadd FUs to make mincedoniononion from peeled onion\n\nTABLE II OVERVIEW\nII\nOF THE TEST DATASET CREATED FROM RECIPE1M+\nDish TypeNo. of Recipes inAverage No. of Ingre-Recipe1M+dients per Recipesalad58,8699cake29,88710soup35,35710omelette6867drinks2,7325\nLink to survey -https://bit.ly/3wAK909\n\nJoint Object and State Recognition using Language Knowledge. A B Jelodar, Y Sun, 2019 IEEE International Conference on Image Processing (ICIP). IEEE2019\n\nFunctional Object-Oriented Network for Manipulation Learning. D Paulius, Y Huang, R Milton, W D Buchanan, J Sam, Y Sun, Intelligent Robots and Systems (IROS). IEEE2016. 2016\n\nHuman-object-object-interaction affordance. S Ren, Y Sun, Workshop on Robot Vision. 2013\n\nObject-object interaction affordance learning. Y Sun, S Ren, Y Lin, Robotics and Autonomous Systems. 2013\n\nRobot grasp planning based on demonstrated grasp strategies. Y Lin, Y Sun, Intl. Journal of Robotics Research. 3412015\n\nFunctional Object-Oriented Network: Construction & Expansion. D Paulius, A B Jelodar, Y Sun, 2018 IEEE International Conference on Robotics and Automation (ICRA). IEEE2018\n\nRecipe1M+: A Dataset for Learning Cross-Modal Embeddings for Cooking Recipes and Food Images. J Marin, A Biswas, F Ofli, N Hynes, A Salvador, Y Aytar, I Weber, A Torralba, IEEE Trans. Pattern Anal. Mach. Intell. 2019\n\nThe theory of affordances. J Gibson, Perceiving, Acting and Knowing, R. Shaw and J. Bransford1977ErlbaumHillsdale, NJ\n\nPDDL -The Planning Domain Definition Language. D Mcdermott, M Ghallab, A Howe, C Knoblock, A Ram, M Veloso, D Weld, D Wilkins, CVC TR-98-003/DCS TR-11651998Yale Center for Computational Vision and ControlTech. Rep.\n\nLong Activity Video Understanding Using Functional Object-Oriented Network. A B Jelodar, D Paulius, Y Sun, IEEE Transactions on Multimedia. 217July 2019\n\nActivitynet: A large-scale video benchmark for human activity understanding. B G Fabian Caba Heilbron, Victor Escorcia, J C Niebles, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern Recognition2015\n\nScaling Egocentric Vision: The EPIC-KITCHENS Dataset. D Damen, H Doughty, G M Farinella, S Fidler, A Furnari, E Kazakos, D Moltisanti, J Munro, T Perrett, W Price, M Wray, European Conference on Computer Vision (ECCV). 2018\n\nFOON Website: Graph Viewer and Videos. 2021\n\nTask Planning with a Weighted Functional Object-Oriented Network. D Paulius, K S P Dong, Y Sun, 2021 IEEE International Conference on Robotics and Automation (ICRA). IEEE2021\n\nKNOWROB -knowledge processing for autonomous personal robots. M Tenorth, M Beetz, 2009. 2009\n\nRobotic roommates making pancakes. M Beetz, U Klank, I Kresse, A Maldonado, L Mösenlechner, D Pangercic, T Rühr, M Tenorth, 2011 11th IEEE-RAS International Conference on Humanoid Robots. 2011\n\nIkeaBot: An autonomous multi-robot coordinated furniture assembly system. R A Knepper, T Layton, J Romanishin, D Rus, 2013 IEEE International Conference on Robotics and Automation. 2013\n\nIKEA Furniture Assembly Environment for Long-Horizon Complex Manipulation Tasks. Y Lee, E Hu, Z Yang, A C Yin, J J Lim, 2021 IEEE International Conference on Robotics and Automation (ICRA). 2021\n\nOnto-IKEA: A Knowledge Retrieval Framework Based on IKEA Ontology. A Vassiliades, N Zarkadas, N Bassiliades, T Patkos, JOWO. 2021\n\nManipulation action tree bank: A knowledge resource for humanoids. Y Yang, A Guha, C Fermuller, Y Aloimonos, Humanoid Robots (Humanoids). IEEE2014 14th. 2014\n\nAssembly planning by recognizing a graphical instruction manual. I Sera, N Yamanobe, I G Ramirez-Alpizar, Z Wang, W Wan, K Harada, 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). 2021\n\nA big data approach to computational creativity. L R Varshney, F Pinel, K R Varshney, D Bhattacharjya, A Schörgendorfer, Y.-M Chee, abs/1311.1213ArXiv. 2013\n\nCooking Made Easy: On a Novel Approach to Complexity-Aware Recipe Generation. G Müller, R Bergmann, ICCBR2017\n\nEvoChef: Show Me What to Cook! Artificial Evolution of Culinary Arts. H Jabeen, N Tahara, J Lehmann, EvoMUSART2019\n\nAutoChef: Automated Generation of Cooking Recipes. H Jabeen, J Weinz, J Lehmann, 2020 IEEE Congress on Evolutionary Computation (CEC). 2020\n\nImproving robotic cooking using batch bayesian optimization. K Junge, J Hughes, T George Thuruthel, F Iida, IEEE Robotics and Automation Letters. 012020\n\nExtracting Food Substitutes From Food Diary via Distributional Similarity. P Achananuparp, I Weber, abs/1607.08807CoRR. 2016\n\nIdentifying ingredient substitutions using a knowledge graph of food. S Shirai, O W Seneviratne, M Gordon, C.-H Chen, D L Mcguinness, Frontiers in Artificial Intelligence. 32020\n\nImproving Ingredient Substitution using Formal Concept Analysis and Adaptation of Ingredient Quantities with Mixed Linear Optimization. E Gaillard, J Lieber, E Nauer, ICCBR2015\n\nExploiting food embeddings for ingredient substitution. C Pellegrini, E Özsoy, M Wintergerst, G Groh, HEALTHINF2021\n\nTowards robust one-shot task execution using knowledge graph embeddings. A A Daruna, L Nair, W Liu, S Chernova, 2021 IEEE International Conference on Robotics and Automation (ICRA). 2021\n\nOntology-Assisted Generalisation of Robot Action Execution Knowledge. A Mitrevski, P.-G Plöger, G Lakemeyer, 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). 2021\n\nspaCy: Industrial-strength Natural Language Processing in Python. M Honnibal, I Montani, S Van Landeghem, A Boyd, 10.5281/zenodo.12123032020\n\nEvaluating Recipes Generated from Functional Object-Oriented Network. M S Sakib, H Baez, D Paulius, Y Sun, arXiv:2106.00728Featured in 18th International Conference on Ubiquitous Robots. UR2021. 2021arXiv preprint\n\nA Road-map to Robot Task Execution with the Functional Object-Oriented Network. D Paulius, A Agostini, Y Sun, D Lee, arXiv:2106.00158Featured in 18th International Conference on Ubiquitous Robots. UR2021. 2021arXiv preprint\n"},"corpus_id":{"kind":"number","value":250407773,"string":"250,407,773"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2207.03693\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"journals/ral/SakibPS22\",\n \"doi\": \"10.1109/lra.2022.3191068\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/2207.03693v1.pdf\"\n ],\n \"pdf_sha\": \"badc46d8915e9adce6859456b78036f6b1edec5c\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"https://arxiv.org/pdf/2207.03693\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Approximate Task Tree Retrieval in a Knowledge Network for Robotic Cooking"},"authors":{"kind":"list like","value":["Md Sadman Sakib","David Paulius","Yu Sun"],"string":"[\n \"Md Sadman Sakib\",\n \"David Paulius\",\n \"Yu Sun\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Flexible task planning continues to pose a difficult challenge for robots, where a robot is unable to creatively adapt their task plans to new or unseen problems, which is mainly due to the limited knowledge it has about its actions and world.Motivated by a human's ability to adapt, we explore how task plans from a knowledge graph, known as the Functional Object-Oriented Network (FOON), can be generated for novel problems requiring concepts that are not readily available to the robot in its knowledge base.Knowledge from 140 cooking recipes are structured in a FOON knowledge graph, which is used for acquiring task plan sequences known as task trees.Task trees can be modified to replicate recipes in a FOON knowledge graph format, which can be useful for enriching FOON with new recipes containing unknown object and state combinations, by relying upon semantic similarity.We demonstrate the power of task tree generation to create task trees with never-before-seen ingredient and state combinations as seen in recipes from the Recipe1M+ dataset, with which we evaluate the quality of the trees based on how accurately they depict newly added ingredients.Our experimental results show that our system is able to provide task sequences with 76% correctness."},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":"M AJOR efforts in robotics research have been devoted to the development of intelligent agents with the ability to understand human intentions and to perform actions for problems in human-centered domains.Such domains include, but are not limited to, the application of robots for assisting the elderly and disabled, food delivery, and cooking activities.However, the main difficulty in designing robots for humancentered domains lies in the variety of tasks and the dynamic nature of environments in which these robots will operate.\nIn terms of robotic cooking, ingredients or objects may come in different forms, shapes or sizes, and there are many states to account for when executing recipes [1].Additionally, there may be times where a robot is unable to execute an entire recipe due to the unavailability of certain ingredients or objects in its surroundings; this may happen when a robot is required Manuscript received: February 24, 2022; Revised May 20, 2022; Accepted June 25, 2022.This paper was recommended for publication by Jingang Yi upon evaluation of the Associate Editor and Reviewers' comments.This work was supported by the National Science Foundation under Grants 1910040 and 1812933. 1 Md Sadman Sakib and Yu Sun are members of the Robot Perception and Action Lab (RPAL), which is part of the Department of Computer Science & Engineering at the University of South Florida, Tampa, FL, USA.mdsadman@usf.edu,yusun@usf.edu 2 David Paulius is a postdoctoral researcher in the Intelligent Robot Lab at Brown University, Providence, RI, USA.dpaulius@cs.brown.eduDigital Object Identifier (DOI): see top of this page."},{"title":"FOON Reference Task Tree","content":"Final Task Tree"},{"title":"Search Modify","content":"Dish: Salad Ingredients: q (onion, peeled) q (cucumber, diced) q (oregano, dried) q (prunes, dried) q (feta cheese, cubed)\nInput: Ingredient list in the from of (object, state) pairs"},{"title":"Word Embedding","content":"Fig. 1.Overview of our task tree generation procedure for novel problems using FOON.As input to our task tree generation system, a set of ingredients and their states are given, from which a novel task tree can be generated.\nto prepare meal variations.In the latter case, a robot would need to adapt its knowledge to include or disregard ingredients, which may or may not be known by the robot, in its task plan, which is not often considered in modern robot applications.Motivated by a human's ability to creatively adapt to novel scenarios, we explore the problem of generating task plans for novel scenarios that would be presented to a robot, given the limited knowledge available to it as a knowledge graph.\nIn this work, we use a knowledge representation known as the Functional Object-Oriented Network (FOON) [2], which builds upon previous work on joint object-action representation [3]- [5].In prior work, we introduced how a FOON can be created from video annotations and used for task planning [2].However, as with other works, task planning is limited to the knowledge found in a FOON, as it only contains knowledge for a limited number of recipe and ingredient variations.For example, if a robot were to prepare a salad with a certain combination of ingredients that have never been encountered together before in FOON, then this would pose a problem since there is no concept of that type of salad.Previously, we investigated how the knowledge in FOON can be extended so as to generalize concepts across object types [6].Based on this intuition, we propose that existing knowledge of similar recipes can be used to derive alternative solutions (as graphs) that did not exist before.Therefore, in this work, we introduce a procedure known as task tree generation (described in Figure 1), where a reference task tree is extracted from FOON and then modified to match a required set of ingredients for a given recipe.This may require the addition of possibly neverbefore-seen ingredients and the removal of ingredients in the reference task tree that are not valid for the new recipe.To show the effectiveness of our approach, we apply task tree generation on recipes from the Recipe1M+ dataset [7].This is particularly useful for creating new FOONs and task plans for never-before-seen recipes without manual effort.\nOur contributions and outcomes in this paper are as follows: (i) We develop a recipe generation pipeline that takes a set of ingredients as input and outputs a task plan in the form of a task tree; (ii) We design a heuristic-based search algorithm to find a recipe from the knowledge graph, which closely matches with the required ingredients, without exploring all possible paths and thus minimizing time and space cost; (iii) We present a task tree visualization method that highlights how each ingredient's state changes in a recipe known as the progress line; and (iv) We perform an evaluation on the generalizability of FOON for Recipe1M+ recipes."},{"title":"II. BACKGROUND","content":""},{"title":"A. Functional Object-Oriented Network","content":"A FOON is a bipartite network with two types of nodes: object nodes and motion nodes.Affordances [8] are depicted via edges that connect objects to actions and enforce action sequencing.A fundamental structure known as a functional unit represents actions in FOON by describing the state change of objects before and after execution, and it has input object nodes, output object nodes, and a motion node.It is akin to a planning operator in PDDL (short for Planning Domain Definition Language) [9], where input and output nodes describe preconditions and effects respectively.Figure 2 depicts examples of functional units for the action sequence of: 1) picking and placing a whole tomato on a cutting board, 2) slicing the tomato with a knife, and 3) pouring the sliced tomato from the cutting board into a bowl.\nTypically, we create FOONs by annotating video demonstrations.A FOON representing a single activity is referred to as a subgraph; a subgraph contains functional units in sequence to describe objects' states before and after each action occurs, and what objects are being manipulated.Presently, this annotation process is done manually, but previous work has investigated how graphs can be annotated in a semi-automatic manner [10].Two or more subgraphs can be merged together to form what we call a universal FOON.A universal FOON can contain variations of recipes once it has been merged with several sources of knowledge.This merging procedure is simply a union operation applied to all functional units from each subgraph we wish to combine; as a result, duplicate functional units are eliminated in the merged network [2].Presently, FOON comprises of 140 subgraph annotations of videos from YouTube, Activity-Net [11], and EPIC-KITCHENS [12], which are all available on our website [13].\n1) Task Planning with FOON: Task planning involves retrieving a task tree [2] (a high-level task plan) that satisfies a goal defined as an object node.Finding a task tree requires knowledge about the state of a robot's environment, specifically what objects are available and the state(s) in which they are in.Starting from the goal, the algorithm searches for candidate functional units in a depth-wise manner, and then, for each candidate unit, it searches its input nodes in a breadthwise manner to see if they are available to the robot.More recently, we introduced a modified version of the algorithm to find the optimal path to reach a goal node, which is reflected by weights given to each functional unit corresponding to its success rate of execution by a robot [14].However, these algorithms are not suited to generate task trees for recipes not found in FOON; for this task, we use semantic similarity [6] while searching to modify task trees as required for a given problem and ingredient set."},{"title":"B. Related Works","content":"Developing a robotic agent that understands human instructions like \"cut the onion in half\" and executing them precisely in a real world setting is a very difficult task.The most noteworthy work to address this challenge is the knowledge processing framework KNOWROB [15].They created this knowledge base by collecting information through observations in a sensor-equipped environment.Although the robots in [16] were successful in retrieving a pancake recipe and executing it using the KNOWROB reasoning framework, the authors did not show whether the robots are able to modify the recipe to meet users' preference.On the contrary, our work focuses on this degree of flexibility so that one can obtain a task plan for any combination of ingredients.\nFurniture assembly and cooking are similar in a sense that both of these task domains begin with raw materials and end with a finished product in the form of an assembled furniture piece or a dish.Given geometric details of parts, IkeaBot [17] deduces a task plan to assemble furniture by using geometric reasoning and checking possible sub-assemblies, which is computationally expensive.On the other hand, we explore the FOON knowledge base by using an efficient heuristic-based graph search.[18] relied on deep learning whereas [19] created an ontology based on the product description found on IKEA website to retrieve information about an object.Manipulation action tree banks [20] were proposed to organize manipulation actions in multiple levels of abstraction, which a robot can use for understanding and executing a task.Unlike our work, they did not handle different object states and are unable to produce a plan for complex manipulation tasks.More recently, the authors in [21] presented a method for robot assembly planning by recognizing graphical instruction manuals.They identify objects from a series of images, find the relationship between the previous and current image, and create a task sequence graph for robot execution.\nEfforts have been made to dynamically modify a recipe to meet certain constraints.[22] gathered various datasets to understand existing recipes, which flavors people like, and chemical structure of ingredients to invent novel recipes based on big data approaches.[23] proposed a new approach to generate novel recipes based on ingredients, preparation time, and cooking steps, where a user can specify those parameters.They used a taxonomy to define the semantics of ingredients and cooking steps where the inner nodes refer to generalized items (e.g., meat) and the items close to the leaf nodes are more specific (e.g., beef, pork).EvoChef [24] and AutoChef [25], created novel recipes using natural language processing and evolutionary algorithms but without considering user preference.[26] optimize a recipe according to consumers' taste by tuning parameters such as cooking time, amount of salt.However, they maintain the same ingredients in their plans, whereas our work shows more flexibility by making the same dish using different ingredient combinations based on the consumers' dietary preferences.\nSome works focused on finding a reasonable substitution for an ingredient.[27] collected contextual information from the MyFitnessPal mobile app and used it to search for an equivalent ingredient.For example, chicken and beef can substitute each other if they are often consumed with rice and salad.[28] addressed the issue of specific dietary requirements.Similar to our work, they combined the semantic information from a knowledge graph of food and a word embedding model to substitute ingredients to improve a dish's nutritional value.Similar works were done by [29] using adaptation rules and by [30] using the transformer-based model BERT.However, these works solely explored ingredient substitution, whereas we incorporate this idea for robotic execution of a recipe.Conceptually, our work is similar to [31], [32] that use ontology and domain knowledge to derive missing information.In addition to the knowledge required to manipulate unknown objects, a task tree shows how they can be integrated with other known or unknown items in a complex task like cooking."},{"title":"III. PROPOSED METHOD","content":"Our objective in this work is to generate FOON task plans for never-before-seen recipes through a method that uses FOON as a knowledge base.This method requires a recipe's ingredient set I and associated dish type D as input and produces a task tree outlining how to prepare the dish.Due to the limited recipe variety in FOON, the likelihood of encountering recipes with different ingredient combinations is very high.Specifically, FOON may not contain references to the desired recipe, or, more likely, many ingredients will be absent from FOON.Despite this shortcoming, our idea is to use the limited knowledge of a FOON to produce task trees for novel recipes, so as to take advantage of existing recipe knowledge bases like Recipe1M+ [7].The main idea is to retrieve a reference task tree from FOON, which closely resembles the desired recipe, and then modify it to produce a final task tree as shown in Figure 1.The three main steps of our method are discussed in the following subsections."},{"title":"A. Identifying a Reference Goal Node","content":"From a universal FOON, we extract a recipe's task tree by retrieving all functional units related to the final state of a dish, which we refer to as a goal node.It is likely that a goal node for a dish does not exist in FOON; in this case, we try to find a suitable node that closely matches the required dish, which we refer to as a reference goal node (denoted as G).Such a node is used to extract a reference task tree T , which is simply a task tree that creates the reference goal object.To facilitate the selection of the ideal reference goal node G, we created a recipe classification with 30 dish classes (e.g., salad, pizza), with which we can categorize recipes in the form of FOON subgraphs.The purpose of this classification is to find similar recipes known to us given a dish type.score ← Find similarity between I and S 5: end for 6: G ← End product of subgraph with maximum score Output: G Algorithm 1 uses this classification to find candidate goal nodes.For example, if D = soup, candidates can be {corn soup, potato soup, ...} depending on the soup recipes that exist in FOON.We store the list of ingredients required to prepare a dish in its goal node so that we can easily identify the required ingredients to make a dish.Next, the algorithm checks each candidate goal node to select the one that has maximum Word2Vec similarity with the ingredient set I.\nWe use a word embedding model provided by spaCy [33], an open-source library for natural language processing, to get a vectorized representation of each word.Similar words are grouped together in the embedding space.For instance, cucumber and zucchini will be close to each other but far away from salt.Similarity is given as a score ranging from 0 to 1, where 1 indicates that two items are semantically similar.In this work, we deem two items similar if the score exceeds a predefined threshold value of 0.90."},{"title":"B. Extracting a Reference Task Tree","content":"A dish can be prepared in many ways depending on its ingredients and recipe instructions.Similarly, in FOON, there are different paths to reach a goal node.We aim to find the path that closely resembles the desired recipe to use it as a base of reference for our final task tree.We assume that all utensils and ingredients needed for a recipe are available in the kitchen, although it is possible that the ingredients are not available in the desired state.For example, a recipe may require sliced carrot, but the kitchen contains whole carrot.\nIn a typically large FOON, exploring all paths can be timeconsuming and memory intensive.Hence, unlike the previous search algorithm that aims to find the most optimal task tree for making a certain object [14], we use Best First Search.At each level, this method evaluates all candidate nodes based on the heuristic function g(n) = |A n ∩ I|, where A n and I denote the set of ingredients stored in node n and ingredients to be added to the dish respectively, and g(n) computes a similarity score checking the overlap between the two sets of ingredients.We use Word2Vec similarity instead of word matching so that equivalent ingredients are considered as an overlap.The similarity score indicates how close we are to the end product (i.e., final dish).The node with the maximum score is considered as the closest to the goal and thus will be favored over the other candidates as shown in Figure 3.\nAlgorithm 2 Retrieve the task tree of the reference goal node Input: reference goal node G and given ingredients I 1: T ← A list of functional units in the task tree 2: K ← A list of items available in the kitchen 3: Q ← A queue of items to search 4: Q.push(G) 5: while Q is not empty do 6:\nL ← Q.dequeue()\nif L does not exist in K then 8:\nC ← Find all functional units that create L 9:\nC ← Select one from C using heuristic function 10:\nT .append(C ) 11:\nfor each input n in C do 12:\nif n is not visited then 13:\nQ.enqueue(n)\n14:\nMark n as visited 15:\nend if\nend for 17:\nend if 18: end while 19: T .reverse()Output: T\nThe retrieval algorithm (presented as Algorithm 2) works as follows.Let the initial task tree T be an empty list.T will be populated with the functional units required to prepare the dish defined by the goal node G. Let K be the list of items in the kitchen and Q be a queue for keeping track of items that need to be explored.At each iteration, we remove the first item L from Q and check if it is found in K in its desired state.If it does not contain L, we search for the functional units C that create it (i.e., functional units that contain L as an output node).For instance, while searching for cake batter, we may find that it is not available in the kitchen, but there are a few functional units in FOON that show how to make different types of cake batter made up of varying sets of ingredients.These units are added to C as potential candidates.We choose the best candidate C from C based on our heuristic function and add it to the task tree.To execute C , we have to make sure that we know how to prepare all of its input ingredients.This is why all of its input ingredients are added to Q to be searched next.The search continues until Q is empty, meaning that all of the required items are discovered.At the end, T is reversed to provide the functional units in the correct sequence that a robot can follow to prepare the dish at G."},{"title":"C. Task Tree Modification","content":"The reference tree T gives a blueprint about how the tasks should be executed, but it may not contain all the ingredients required for a new recipe.Therefore, we need to modify T to include missing ingredients.The modification procedure first integrates all required ingredients and then removes those that were not given as input.If an ingredient does not exist in FOON, or if it exists in a different state, we find a semantically similar ingredient in the knowledge base using a multi-level search.First, we try to find a suitable match for similar ingredients in T because we already know how the ingredients in T are integrated together to produce the final dish.This gives us high confidence to incorporate missing ingredients.Otherwise, we search in the universal FOON for an equivalent ingredient.As in Section III-A, we use spaCy's word embedding model to find the closest object in the embedding space.Each ingredient contains an object name and its state.When checking if an ingredient exists in T , there are four possible cases depending on the match with the object and state (also presented in Table I):\nCase 1: If the given ingredient exactly matches a leaf node (i.e., a starting node in the state required by the recipe) in T in terms of object and state name, no adjustment is required.\nCase 2: The exact object is not found in T , but there is an equivalent ingredient with the exact same state.In that case, we can simply substitute the equivalent object name with that of the required object.For instance, if chopped chili pepper is required in the recipe but its equivalent in T is chopped jalapeño, we can replace jalapeño with chili pepper.If jalapeño is also required, a copy of the functional units required for processing jalapeño is created, after which substitution is performed to preserve jalapeño in T .\nCase 3: If the object is found in T but not in the desired state, we search FOON for functional units needed to get the object in the required state.We do this with another retrieval query using Algorithm 2, where the goal node will be an object node in the leaf of T .We then add the retrieved functional units to that leaf.For example, if whole tomato is given as the starting state but there is diced tomato in the leaf of the tree, we will search using the goal node diced tomato as the criteria and the added functional units will reflect how diced tomato can be made from its whole state.\nCase 4: The object and state of the equivalent ingredient do not match with the given ingredient.We can handle this case by combining the solution from the previous two cases.\nFirst, we substitute the object name and then add the necessary functional units to achieve its desired state.\n(a) FOON-generated task tree for greek salad"},{"title":"Progress line of recipe: 00a7b36a48 onion peeled,whole ---> pick-and-place ---> peeled,whole(cutting board) ---> slice ---> sliced(cutting board) ---> pick-and-place ---> sliced(mixing bowl) ---> mix","content":""},{"title":"End product: greek salad cucumber diced(bowl) ---> pick-and-place ---> diced(mixing bowl) ---> mix","content":""},{"title":"End product: greek salad oregano dried(bowl) ---> pour ---> dried(mixing bowl) ---> mix","content":"End product: greek salad prunes (substituted from raisin, confidence: 62.9%)"},{"title":"dried(bowl) ---> pour ---> dried(mixing bowl) ---> mix","content":""},{"title":"End product: greek salad feta cheese cubed(container) ---> scoop ---> cubed(spoon) ---> pour","content":"End product: greek salad (b) Progress line for greek salad Fig. 4. Illustration of: a) a task tree generated with our approach, and b) its corresponding progress line visualization.\nIf the equivalent ingredient does not exist in the retrieved task tree but is featured in other recipes in FOON, it poses some extra challenges.Let us assume that the recipe requires roasted turkey but the closest object to turkey available in FOON is chicken.Normally, chicken can have states such as raw, fried, baked, and marinated across recipes.This makes it difficult to choose the state best suited for the task tree.To aid in selecting an appropriate state, we used a state taxonomy [1] with 12 different categories, such as raw, finely separated, and liquid.Similar states, such as sliced and diced, are grouped together such that existing states can be used as a blueprint for other states of the same category.If there are multiple candidates to choose from, we break the tie by favoring the most frequently used state in the recipes of the given dish type.\nWhen substituting a state, we also have to select an appropriate manipulation for which the state is valid.For instance, with the states diced and sliced, we treat the appropriate action with different motion verbs dice and slice.For this, we find the most frequently used verb associated with the required state and use it as the motion node label.Once we retrieve the functional units to prepare an ingredient, the challenge is to connect them to the reference tree.To do this, we find functional units in T that can easily be modified, which are those that new ingredients can be added as input nodes.For example, any unit that involve mixing or adding ingredients to a container can accept any number of inputs, whereas a robot cannot cut or chop multiple items at the same time.Therefore, for each missing ingredient, we connect the subtrees to these types of functional units, resulting in a connected acyclic graph.We do not need to create this connection if the equivalent ingredient already exists in T .Finally, the modified task tree will have a list of functional units containing the input ingredients that, if executed sequentially, will produce the desired dish."},{"title":"IV. EXPERIMENTS","content":"It is difficult to quantitatively compare a knowledge graphbased method with others, as the benefit we get from it lies in its flexibility and adaptability to answer a query.Therefore, in our experiments, we used Recipe1M+ to create a diverse set of queries for ingredient combinations from several dish classes as shown in Table II.Our focus is on generating a correct task tree for recipes in Recipe1M+.To make our evaluation more robust, we excluded dish classes for which we had less than 4 recipes in FOON, as it is difficult to gain a general picture of a dish class with too few recipes.Among the selected dish types, we randomly selected 5 classes for our evaluation: salad, cake, soup, omelette and drinks.Next, 100 recipes of each category were randomly selected from Recipe1M+ for a total of 500 recipes.We ran these experiments on a MacBook Pro equipped with 2.6 GHz 6-Core Intel Core i7 processor and 16GB RAM."},{"title":"A. Evaluation Strategy","content":"Our objective is to evaluate the quality of a task tree for unseen ingredient combinations from the Recipe1M+ dataset.It is impossible to judge a recipe by solely relying on automated testing such as Intersection over Union (IoU) due to there being many ways to prepare a meal.Furthermore, despite having the same ingredients, one recipe can differ from another recipe in terms of cooking steps while both are valid or correct.Hence, we chose to evaluate the FOON-generated task trees by manually checking each task plan.Although it requires a lot of manual effort, we believe that human judgement is necessary to evaluate recipes.To make the process easier, instead of checking the functional units in the task tree, we check each ingredient by reviewing its progress line.\nProgress Line: In cooking, ingredients undergo several changes in their physical state (e.g., from whole to sliced) and/or location.We refer to this sequence of changes as an ingredient's progress line, which is another visualization of a task tree.We developed a tool to view the progress line of each ingredient to manually evaluate generated task trees.Figures 4a and 4b shows an example of a generated task tree and its progress line respectively, where objects, states and motions are colored in black, green and red respectively."},{"title":"B. Results","content":"Task trees generated for the 500 recipes were thoroughly evaluated using the progress line visualizer.When checking a recipe, we review each of its ingredients and indicate whether a given ingredient is incorrect, partially incorrect or correct by assigning it a score from {0,1,2}, where '0' means that an ingredient's progress line is incorrect, '1' means partially incorrect, and '2' means correct.A label of partially incorrect means that a state label assigned to the ingredient is not contextually or physically relevant to it.Without modifying the reference task tree, we would only see higher scores for ingredients that already exist within the unmodified tree, while missing ingredients (i.e., items that need to be added to the tree) or extra ingredients (i.e., items that need to be pruned from the tree) will be assigned an incorrect score.However, our proposed algorithm will be able to generate task plans for unseen ingredients in addition to already existing items while removing extra ingredients.In Figure 5, we present a comparison of these two approaches by reporting the percentage of correctly generated recipes with varying degrees of thresholds.The result shows that the proposed algorithm can generate 76% of the recipes correctly on average if the threshold of correctness for each recipe is set to 100%.When the threshold is 100%, a task tree must have correct progress lines for each ingredient.It is evident from the experiments that using similarity-based ingredient substitution greatly improves the performance and correctness thresholds across all dish types."},{"title":"C. User Study","content":"To verify correctness and completeness, we conducted a user study to see whether there is any significant difference between the FOON generated and Recipe1M+ recipes, following the same approach and questionnaires (see Figure 6) from a previous study [34].Since a task tree in its original format is not easy to read and verify for individuals not familiar with FOON, we translated each tree into recipe-like sentences.In this study, there were 9 participants, where each was given 10 recipes randomly selected from Recipe1M+ and its equivalent generated recipe from our method to assign ratings.This survey can be accessed by the provided link. 1 Our hypothesis H o is that there is no significant difference between FOONgenerated and Recipe1M+ recipes.With the ratings collected from users for questions 4-7, we performed a null hypothesis test with significance level α = 0.05 and obtained the pvalues 0.12, 0.34, 0.20, 0.64, 0.68, 0.73 and 0.35 respectively.Since all the p-values are fairly greater than α, we cannot reject H o ; hence, we can conclude that the generated recipes are equivalent to Recipe1M+ recipes in terms of correctness, completeness and clarity.\nV. DISCUSSION Recipe1M+ has a significantly higher quantity of recipes and ingredients than FOON, making it very challenging to produce a task plan that perfectly matches the original textual recipe.Nevertheless, even with the limited concepts available in FOON, the flexibility of the task tree generation process allows us to achieve correct results in 76% of the cases.This mainly depends on the selection of equivalent objects for unseen ingredients, as an incorrect substitution may introduce errors in an ingredient's progress line.For example, the equivalent of apricot in FOON is lemon according to the word embedding model we used.However, we cannot squeeze an apricot to extract its juice as we can with a lemon.Therefore, the progress line of apricot does not correctly reflect how it should change its state after each manipulation action.I have made this exact dish Yes, but I left out some of the ingredients listed Yes, but I added some ingredients not listed No 4) Does this recipe seem correct to you? 5) Do the steps appear to be in the right order? 6) Are all steps in the recipe correct?7) Does each recipe step give enough information in order to complete it?8) Does the recipe skip steps that are obvious to you? 9) If attempting to follow the given steps, how confident are you that you could make this dish?10) Do the steps in this recipe appear to be clear and easy to follow?Fig. 6.Overview of the survey questions used in our user study.An ingredient may be used differently across recipes; for example, pineapple is mixed with other fruits while preparing a fruit salad, but added as a topping in a pizza.For this reason, when we add a branch to the task tree, we follow different rules to decide where the missing ingredient should be added.For instance, in drink recipes, branches can be added when objects are being poured into a container whereas for salad, ingredients are typically mixed in a bowl.These concepts are usually not captured in the word embedding model, and rulebased actions are not often sufficient to accurately infer the task plan.Hence, we will focus on improving our methodology through generalization in the future.We also noted that it is not always possible to include the user-provided state of an ingredient if the kitchen contains it in a non-reversible state, such as melted cheese when diced cheese is required.We do not mark those as errors since a new batch of ingredients is required to follow user input.Overall, we noted the following errors in our experiments as incorrect substitution: 39%, incorrect state: 27%, incorrect motion: 14% and incorrect integration: 20%.\nSuch errors can be minimized with correction from users, where instead of selecting the top suggestion from the embedding model for each unseen ingredient, users can decide on the ideal substitute among the top 5 candidates.In the case of Figure 7, corrections can be made for the ingredients cool whip and pistachio using the second (whipped cream) and fourth (peanut) ranked candidates respectively.Furthermore, we can enrich the knowledge base by adding recipes with diverse set of ingredients so that we can make substitutions with higher confidence and with minimal user correction.\nFinally, up to this point, recipes are added to FOON by manually labelling cooking videos, which is a very timeconsuming and error-prone process.If we can automate the process of creating subgraphs from video or text, we could easily expand FOON by adding recipes from various sources.Using a method such as the task tree generation process can help to reduce the amount of effort required by constructing graphs in a semi-automatic manner, in addition to computer vision approaches for extracting graphs from video [10].\nA task tree can be used for bootstrapping robot task execution, as it provides a high-level description of a sequence of actions needed to be executed by a robot free from domainspecific robot details or properties (e.g. the number of endeffectors, which end-effector to use for certain actions, motion primitives, etc.).For instance, a task tree extracted from a knowledge graph containing instructions of furniture assembly can represent how the raw materials should be connected to build the furnished product.Although a task tree cannot be executed in its original state, it can however be connected to existing frameworks for manipulation planning, such as PDDL where functional units can be treated as planning operators and can then be used to retrieve plans corresponding to robot low-level motion primitives [35]."},{"title":"A. Complexity Analysis of the Algorithm","content":"A FOON is represented as an adjacency list, which allows us to retrieve functional unit candidates in O(1) time.In the worst case, the number of candidates is equal to the total number of FOON recipes.However, this number is very small on average and thus is negligible.Therefore, for n functional units in the task tree, the required time is O(n).At the tree modification stage, if an equivalent ingredient exists in the reference tree, substitution can be done in O(1) time.However, if no suitable equivalent is present, we have to search other recipes for equivalents, which takes O(k) time, where k is the number of unique ingredients in FOON.We accelerate this process by computing a mapping of equivalent ingredient in the pre-processing phase to find an equivalent ingredient in O(1) time.Hence, the overall time complexity of the search algorithm is O(n), where n is the length of the retrieved path.The space complexity is also O(n) since we store only one path.With the addition of new recipes, FOON increases in size, which in turn can potentially create alternative paths to a goal node.Exploring all paths will incur a larger time and space complexity.To avoid that, we chose the heuristic-based search algorithm where the heuristic was designed using the given ingredients.Overall, we found the average computation time required to solve a problem instance was 80 milliseconds."},{"title":"VI. CONCLUSION AND FUTURE WORK","content":"In this work, our goal was to create a system that will enable a robot to derive task plans in novel scenarios not found in its knowledge base.It is almost impossible to equip a robot with knowledge of every possible recipe with countless combinations of ingredients.To address this, we explored recipe generation using FOON as a knowledge base.Similar to how humans can modify a recipe dynamically by using their previous cooking knowledge to determine how items available in the kitchen can be used, we demonstrate how an unseen ingredient can be integrated to an existing recipe by learning from an equivalent ingredient based on semantic similarity metrics.In our experiments, we generated task plans for different ingredient combinations using our approach and showed that plans were accurate in most cases.\nThe evaluation method used in this work is subjective.In the future, we will explore how generated task trees can be executed with a robot to concretely evaluate our method.Furthermore, we aim to enrich our FOON knowledge base with a more diverse array of recipes by semi-automatically creating subgraphs from instructional videos or text-based sources such as cookbooks.Furthermore, additional work is required to learn a new embedding model that would not only measure the similarity between two words but also the similarity of two functional units or two task plans that considers the context of the ingredients."},{"title":"Fig. 2 .","content":"Fig. 2. Illustration of a universal FOON made of 140 recipe subgraphs, and three (3) connected functional units, where object nodes are denoted by green circles and motion nodes are denoted by red squares."},{"title":"Algorithm 1","content":"Find a reference goal node Input: given ingredients I, dish type D 1: candidates ← Retrieve recipes of type D 2: for each subgraph in candidates do 3: S ← Ingredients of subgraph 4:"},{"title":"Fig. 3 .","content":"Fig. 3. Overview of our reference task tree retrieval method.Each node's value indicate a score returned by the heuristic function; empty nodes are not reached.Red arrows denote the chosen path based on maximum score."},{"title":"Fig. 5 .","content":"Fig.5.Graph showing the relationship between the threshold of correctness (X-axis) of each recipe and the number of recipes (Y-axis) we can generate with that threshold.The result of recipe generation with and without using task tree modification are presented as solid and dashed lines respectively; the same colored lines indicate the same dish type."},{"title":"Fig. 7 .","content":"Fig. 7. Top 5 equivalent ingredients in FOON for some randomly chosen unseen ingredients from Recipe1M+."},{"title":"TABLE I RULES","content":"TO MODIFY THE REFERENCE TASK TREE BASED ON THE DIFFERENCE BETWEEN A GIVEN INGREDIENT AND ITS EQUIVALENT IN THE TREE AS DISCUSSED IN SECTION III-C.THE TERM FU DENOTES FUNCTIONAL UNIT."},{"title":"TABLE II OVERVIEW","content":"OF THE TEST DATASET CREATED FROM RECIPE1M+"}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"M AJOR efforts in robotics research have been devoted to the development of intelligent agents with the ability to understand human intentions and to perform actions for problems in human-centered domains.Such domains include, but are not limited to, the application of robots for assisting the elderly and disabled, food delivery, and cooking activities.However, the main difficulty in designing robots for humancentered domains lies in the variety of tasks and the dynamic nature of environments in which these robots will operate.\\nIn terms of robotic cooking, ingredients or objects may come in different forms, shapes or sizes, and there are many states to account for when executing recipes [1].Additionally, there may be times where a robot is unable to execute an entire recipe due to the unavailability of certain ingredients or objects in its surroundings; this may happen when a robot is required Manuscript received: February 24, 2022; Revised May 20, 2022; Accepted June 25, 2022.This paper was recommended for publication by Jingang Yi upon evaluation of the Associate Editor and Reviewers' comments.This work was supported by the National Science Foundation under Grants 1910040 and 1812933. 1 Md Sadman Sakib and Yu Sun are members of the Robot Perception and Action Lab (RPAL), which is part of the Department of Computer Science & Engineering at the University of South Florida, Tampa, FL, USA.mdsadman@usf.edu,yusun@usf.edu 2 David Paulius is a postdoctoral researcher in the Intelligent Robot Lab at Brown University, Providence, RI, USA.dpaulius@cs.brown.eduDigital Object Identifier (DOI): see top of this page.\"\n },\n {\n \"title\": \"FOON Reference Task Tree\",\n \"content\": \"Final Task Tree\"\n },\n {\n \"title\": \"Search Modify\",\n \"content\": \"Dish: Salad Ingredients: q (onion, peeled) q (cucumber, diced) q (oregano, dried) q (prunes, dried) q (feta cheese, cubed)\\nInput: Ingredient list in the from of (object, state) pairs\"\n },\n {\n \"title\": \"Word Embedding\",\n \"content\": \"Fig. 1.Overview of our task tree generation procedure for novel problems using FOON.As input to our task tree generation system, a set of ingredients and their states are given, from which a novel task tree can be generated.\\nto prepare meal variations.In the latter case, a robot would need to adapt its knowledge to include or disregard ingredients, which may or may not be known by the robot, in its task plan, which is not often considered in modern robot applications.Motivated by a human's ability to creatively adapt to novel scenarios, we explore the problem of generating task plans for novel scenarios that would be presented to a robot, given the limited knowledge available to it as a knowledge graph.\\nIn this work, we use a knowledge representation known as the Functional Object-Oriented Network (FOON) [2], which builds upon previous work on joint object-action representation [3]- [5].In prior work, we introduced how a FOON can be created from video annotations and used for task planning [2].However, as with other works, task planning is limited to the knowledge found in a FOON, as it only contains knowledge for a limited number of recipe and ingredient variations.For example, if a robot were to prepare a salad with a certain combination of ingredients that have never been encountered together before in FOON, then this would pose a problem since there is no concept of that type of salad.Previously, we investigated how the knowledge in FOON can be extended so as to generalize concepts across object types [6].Based on this intuition, we propose that existing knowledge of similar recipes can be used to derive alternative solutions (as graphs) that did not exist before.Therefore, in this work, we introduce a procedure known as task tree generation (described in Figure 1), where a reference task tree is extracted from FOON and then modified to match a required set of ingredients for a given recipe.This may require the addition of possibly neverbefore-seen ingredients and the removal of ingredients in the reference task tree that are not valid for the new recipe.To show the effectiveness of our approach, we apply task tree generation on recipes from the Recipe1M+ dataset [7].This is particularly useful for creating new FOONs and task plans for never-before-seen recipes without manual effort.\\nOur contributions and outcomes in this paper are as follows: (i) We develop a recipe generation pipeline that takes a set of ingredients as input and outputs a task plan in the form of a task tree; (ii) We design a heuristic-based search algorithm to find a recipe from the knowledge graph, which closely matches with the required ingredients, without exploring all possible paths and thus minimizing time and space cost; (iii) We present a task tree visualization method that highlights how each ingredient's state changes in a recipe known as the progress line; and (iv) We perform an evaluation on the generalizability of FOON for Recipe1M+ recipes.\"\n },\n {\n \"title\": \"II. BACKGROUND\",\n \"content\": \"\"\n },\n {\n \"title\": \"A. Functional Object-Oriented Network\",\n \"content\": \"A FOON is a bipartite network with two types of nodes: object nodes and motion nodes.Affordances [8] are depicted via edges that connect objects to actions and enforce action sequencing.A fundamental structure known as a functional unit represents actions in FOON by describing the state change of objects before and after execution, and it has input object nodes, output object nodes, and a motion node.It is akin to a planning operator in PDDL (short for Planning Domain Definition Language) [9], where input and output nodes describe preconditions and effects respectively.Figure 2 depicts examples of functional units for the action sequence of: 1) picking and placing a whole tomato on a cutting board, 2) slicing the tomato with a knife, and 3) pouring the sliced tomato from the cutting board into a bowl.\\nTypically, we create FOONs by annotating video demonstrations.A FOON representing a single activity is referred to as a subgraph; a subgraph contains functional units in sequence to describe objects' states before and after each action occurs, and what objects are being manipulated.Presently, this annotation process is done manually, but previous work has investigated how graphs can be annotated in a semi-automatic manner [10].Two or more subgraphs can be merged together to form what we call a universal FOON.A universal FOON can contain variations of recipes once it has been merged with several sources of knowledge.This merging procedure is simply a union operation applied to all functional units from each subgraph we wish to combine; as a result, duplicate functional units are eliminated in the merged network [2].Presently, FOON comprises of 140 subgraph annotations of videos from YouTube, Activity-Net [11], and EPIC-KITCHENS [12], which are all available on our website [13].\\n1) Task Planning with FOON: Task planning involves retrieving a task tree [2] (a high-level task plan) that satisfies a goal defined as an object node.Finding a task tree requires knowledge about the state of a robot's environment, specifically what objects are available and the state(s) in which they are in.Starting from the goal, the algorithm searches for candidate functional units in a depth-wise manner, and then, for each candidate unit, it searches its input nodes in a breadthwise manner to see if they are available to the robot.More recently, we introduced a modified version of the algorithm to find the optimal path to reach a goal node, which is reflected by weights given to each functional unit corresponding to its success rate of execution by a robot [14].However, these algorithms are not suited to generate task trees for recipes not found in FOON; for this task, we use semantic similarity [6] while searching to modify task trees as required for a given problem and ingredient set.\"\n },\n {\n \"title\": \"B. Related Works\",\n \"content\": \"Developing a robotic agent that understands human instructions like \\\"cut the onion in half\\\" and executing them precisely in a real world setting is a very difficult task.The most noteworthy work to address this challenge is the knowledge processing framework KNOWROB [15].They created this knowledge base by collecting information through observations in a sensor-equipped environment.Although the robots in [16] were successful in retrieving a pancake recipe and executing it using the KNOWROB reasoning framework, the authors did not show whether the robots are able to modify the recipe to meet users' preference.On the contrary, our work focuses on this degree of flexibility so that one can obtain a task plan for any combination of ingredients.\\nFurniture assembly and cooking are similar in a sense that both of these task domains begin with raw materials and end with a finished product in the form of an assembled furniture piece or a dish.Given geometric details of parts, IkeaBot [17] deduces a task plan to assemble furniture by using geometric reasoning and checking possible sub-assemblies, which is computationally expensive.On the other hand, we explore the FOON knowledge base by using an efficient heuristic-based graph search.[18] relied on deep learning whereas [19] created an ontology based on the product description found on IKEA website to retrieve information about an object.Manipulation action tree banks [20] were proposed to organize manipulation actions in multiple levels of abstraction, which a robot can use for understanding and executing a task.Unlike our work, they did not handle different object states and are unable to produce a plan for complex manipulation tasks.More recently, the authors in [21] presented a method for robot assembly planning by recognizing graphical instruction manuals.They identify objects from a series of images, find the relationship between the previous and current image, and create a task sequence graph for robot execution.\\nEfforts have been made to dynamically modify a recipe to meet certain constraints.[22] gathered various datasets to understand existing recipes, which flavors people like, and chemical structure of ingredients to invent novel recipes based on big data approaches.[23] proposed a new approach to generate novel recipes based on ingredients, preparation time, and cooking steps, where a user can specify those parameters.They used a taxonomy to define the semantics of ingredients and cooking steps where the inner nodes refer to generalized items (e.g., meat) and the items close to the leaf nodes are more specific (e.g., beef, pork).EvoChef [24] and AutoChef [25], created novel recipes using natural language processing and evolutionary algorithms but without considering user preference.[26] optimize a recipe according to consumers' taste by tuning parameters such as cooking time, amount of salt.However, they maintain the same ingredients in their plans, whereas our work shows more flexibility by making the same dish using different ingredient combinations based on the consumers' dietary preferences.\\nSome works focused on finding a reasonable substitution for an ingredient.[27] collected contextual information from the MyFitnessPal mobile app and used it to search for an equivalent ingredient.For example, chicken and beef can substitute each other if they are often consumed with rice and salad.[28] addressed the issue of specific dietary requirements.Similar to our work, they combined the semantic information from a knowledge graph of food and a word embedding model to substitute ingredients to improve a dish's nutritional value.Similar works were done by [29] using adaptation rules and by [30] using the transformer-based model BERT.However, these works solely explored ingredient substitution, whereas we incorporate this idea for robotic execution of a recipe.Conceptually, our work is similar to [31], [32] that use ontology and domain knowledge to derive missing information.In addition to the knowledge required to manipulate unknown objects, a task tree shows how they can be integrated with other known or unknown items in a complex task like cooking.\"\n },\n {\n \"title\": \"III. PROPOSED METHOD\",\n \"content\": \"Our objective in this work is to generate FOON task plans for never-before-seen recipes through a method that uses FOON as a knowledge base.This method requires a recipe's ingredient set I and associated dish type D as input and produces a task tree outlining how to prepare the dish.Due to the limited recipe variety in FOON, the likelihood of encountering recipes with different ingredient combinations is very high.Specifically, FOON may not contain references to the desired recipe, or, more likely, many ingredients will be absent from FOON.Despite this shortcoming, our idea is to use the limited knowledge of a FOON to produce task trees for novel recipes, so as to take advantage of existing recipe knowledge bases like Recipe1M+ [7].The main idea is to retrieve a reference task tree from FOON, which closely resembles the desired recipe, and then modify it to produce a final task tree as shown in Figure 1.The three main steps of our method are discussed in the following subsections.\"\n },\n {\n \"title\": \"A. Identifying a Reference Goal Node\",\n \"content\": \"From a universal FOON, we extract a recipe's task tree by retrieving all functional units related to the final state of a dish, which we refer to as a goal node.It is likely that a goal node for a dish does not exist in FOON; in this case, we try to find a suitable node that closely matches the required dish, which we refer to as a reference goal node (denoted as G).Such a node is used to extract a reference task tree T , which is simply a task tree that creates the reference goal object.To facilitate the selection of the ideal reference goal node G, we created a recipe classification with 30 dish classes (e.g., salad, pizza), with which we can categorize recipes in the form of FOON subgraphs.The purpose of this classification is to find similar recipes known to us given a dish type.score ← Find similarity between I and S 5: end for 6: G ← End product of subgraph with maximum score Output: G Algorithm 1 uses this classification to find candidate goal nodes.For example, if D = soup, candidates can be {corn soup, potato soup, ...} depending on the soup recipes that exist in FOON.We store the list of ingredients required to prepare a dish in its goal node so that we can easily identify the required ingredients to make a dish.Next, the algorithm checks each candidate goal node to select the one that has maximum Word2Vec similarity with the ingredient set I.\\nWe use a word embedding model provided by spaCy [33], an open-source library for natural language processing, to get a vectorized representation of each word.Similar words are grouped together in the embedding space.For instance, cucumber and zucchini will be close to each other but far away from salt.Similarity is given as a score ranging from 0 to 1, where 1 indicates that two items are semantically similar.In this work, we deem two items similar if the score exceeds a predefined threshold value of 0.90.\"\n },\n {\n \"title\": \"B. Extracting a Reference Task Tree\",\n \"content\": \"A dish can be prepared in many ways depending on its ingredients and recipe instructions.Similarly, in FOON, there are different paths to reach a goal node.We aim to find the path that closely resembles the desired recipe to use it as a base of reference for our final task tree.We assume that all utensils and ingredients needed for a recipe are available in the kitchen, although it is possible that the ingredients are not available in the desired state.For example, a recipe may require sliced carrot, but the kitchen contains whole carrot.\\nIn a typically large FOON, exploring all paths can be timeconsuming and memory intensive.Hence, unlike the previous search algorithm that aims to find the most optimal task tree for making a certain object [14], we use Best First Search.At each level, this method evaluates all candidate nodes based on the heuristic function g(n) = |A n ∩ I|, where A n and I denote the set of ingredients stored in node n and ingredients to be added to the dish respectively, and g(n) computes a similarity score checking the overlap between the two sets of ingredients.We use Word2Vec similarity instead of word matching so that equivalent ingredients are considered as an overlap.The similarity score indicates how close we are to the end product (i.e., final dish).The node with the maximum score is considered as the closest to the goal and thus will be favored over the other candidates as shown in Figure 3.\\nAlgorithm 2 Retrieve the task tree of the reference goal node Input: reference goal node G and given ingredients I 1: T ← A list of functional units in the task tree 2: K ← A list of items available in the kitchen 3: Q ← A queue of items to search 4: Q.push(G) 5: while Q is not empty do 6:\\nL ← Q.dequeue()\\nif L does not exist in K then 8:\\nC ← Find all functional units that create L 9:\\nC ← Select one from C using heuristic function 10:\\nT .append(C ) 11:\\nfor each input n in C do 12:\\nif n is not visited then 13:\\nQ.enqueue(n)\\n14:\\nMark n as visited 15:\\nend if\\nend for 17:\\nend if 18: end while 19: T .reverse()Output: T\\nThe retrieval algorithm (presented as Algorithm 2) works as follows.Let the initial task tree T be an empty list.T will be populated with the functional units required to prepare the dish defined by the goal node G. Let K be the list of items in the kitchen and Q be a queue for keeping track of items that need to be explored.At each iteration, we remove the first item L from Q and check if it is found in K in its desired state.If it does not contain L, we search for the functional units C that create it (i.e., functional units that contain L as an output node).For instance, while searching for cake batter, we may find that it is not available in the kitchen, but there are a few functional units in FOON that show how to make different types of cake batter made up of varying sets of ingredients.These units are added to C as potential candidates.We choose the best candidate C from C based on our heuristic function and add it to the task tree.To execute C , we have to make sure that we know how to prepare all of its input ingredients.This is why all of its input ingredients are added to Q to be searched next.The search continues until Q is empty, meaning that all of the required items are discovered.At the end, T is reversed to provide the functional units in the correct sequence that a robot can follow to prepare the dish at G.\"\n },\n {\n \"title\": \"C. Task Tree Modification\",\n \"content\": \"The reference tree T gives a blueprint about how the tasks should be executed, but it may not contain all the ingredients required for a new recipe.Therefore, we need to modify T to include missing ingredients.The modification procedure first integrates all required ingredients and then removes those that were not given as input.If an ingredient does not exist in FOON, or if it exists in a different state, we find a semantically similar ingredient in the knowledge base using a multi-level search.First, we try to find a suitable match for similar ingredients in T because we already know how the ingredients in T are integrated together to produce the final dish.This gives us high confidence to incorporate missing ingredients.Otherwise, we search in the universal FOON for an equivalent ingredient.As in Section III-A, we use spaCy's word embedding model to find the closest object in the embedding space.Each ingredient contains an object name and its state.When checking if an ingredient exists in T , there are four possible cases depending on the match with the object and state (also presented in Table I):\\nCase 1: If the given ingredient exactly matches a leaf node (i.e., a starting node in the state required by the recipe) in T in terms of object and state name, no adjustment is required.\\nCase 2: The exact object is not found in T , but there is an equivalent ingredient with the exact same state.In that case, we can simply substitute the equivalent object name with that of the required object.For instance, if chopped chili pepper is required in the recipe but its equivalent in T is chopped jalapeño, we can replace jalapeño with chili pepper.If jalapeño is also required, a copy of the functional units required for processing jalapeño is created, after which substitution is performed to preserve jalapeño in T .\\nCase 3: If the object is found in T but not in the desired state, we search FOON for functional units needed to get the object in the required state.We do this with another retrieval query using Algorithm 2, where the goal node will be an object node in the leaf of T .We then add the retrieved functional units to that leaf.For example, if whole tomato is given as the starting state but there is diced tomato in the leaf of the tree, we will search using the goal node diced tomato as the criteria and the added functional units will reflect how diced tomato can be made from its whole state.\\nCase 4: The object and state of the equivalent ingredient do not match with the given ingredient.We can handle this case by combining the solution from the previous two cases.\\nFirst, we substitute the object name and then add the necessary functional units to achieve its desired state.\\n(a) FOON-generated task tree for greek salad\"\n },\n {\n \"title\": \"Progress line of recipe: 00a7b36a48 onion peeled,whole ---> pick-and-place ---> peeled,whole(cutting board) ---> slice ---> sliced(cutting board) ---> pick-and-place ---> sliced(mixing bowl) ---> mix\",\n \"content\": \"\"\n },\n {\n \"title\": \"End product: greek salad cucumber diced(bowl) ---> pick-and-place ---> diced(mixing bowl) ---> mix\",\n \"content\": \"\"\n },\n {\n \"title\": \"End product: greek salad oregano dried(bowl) ---> pour ---> dried(mixing bowl) ---> mix\",\n \"content\": \"End product: greek salad prunes (substituted from raisin, confidence: 62.9%)\"\n },\n {\n \"title\": \"dried(bowl) ---> pour ---> dried(mixing bowl) ---> mix\",\n \"content\": \"\"\n },\n {\n \"title\": \"End product: greek salad feta cheese cubed(container) ---> scoop ---> cubed(spoon) ---> pour\",\n \"content\": \"End product: greek salad (b) Progress line for greek salad Fig. 4. Illustration of: a) a task tree generated with our approach, and b) its corresponding progress line visualization.\\nIf the equivalent ingredient does not exist in the retrieved task tree but is featured in other recipes in FOON, it poses some extra challenges.Let us assume that the recipe requires roasted turkey but the closest object to turkey available in FOON is chicken.Normally, chicken can have states such as raw, fried, baked, and marinated across recipes.This makes it difficult to choose the state best suited for the task tree.To aid in selecting an appropriate state, we used a state taxonomy [1] with 12 different categories, such as raw, finely separated, and liquid.Similar states, such as sliced and diced, are grouped together such that existing states can be used as a blueprint for other states of the same category.If there are multiple candidates to choose from, we break the tie by favoring the most frequently used state in the recipes of the given dish type.\\nWhen substituting a state, we also have to select an appropriate manipulation for which the state is valid.For instance, with the states diced and sliced, we treat the appropriate action with different motion verbs dice and slice.For this, we find the most frequently used verb associated with the required state and use it as the motion node label.Once we retrieve the functional units to prepare an ingredient, the challenge is to connect them to the reference tree.To do this, we find functional units in T that can easily be modified, which are those that new ingredients can be added as input nodes.For example, any unit that involve mixing or adding ingredients to a container can accept any number of inputs, whereas a robot cannot cut or chop multiple items at the same time.Therefore, for each missing ingredient, we connect the subtrees to these types of functional units, resulting in a connected acyclic graph.We do not need to create this connection if the equivalent ingredient already exists in T .Finally, the modified task tree will have a list of functional units containing the input ingredients that, if executed sequentially, will produce the desired dish.\"\n },\n {\n \"title\": \"IV. EXPERIMENTS\",\n \"content\": \"It is difficult to quantitatively compare a knowledge graphbased method with others, as the benefit we get from it lies in its flexibility and adaptability to answer a query.Therefore, in our experiments, we used Recipe1M+ to create a diverse set of queries for ingredient combinations from several dish classes as shown in Table II.Our focus is on generating a correct task tree for recipes in Recipe1M+.To make our evaluation more robust, we excluded dish classes for which we had less than 4 recipes in FOON, as it is difficult to gain a general picture of a dish class with too few recipes.Among the selected dish types, we randomly selected 5 classes for our evaluation: salad, cake, soup, omelette and drinks.Next, 100 recipes of each category were randomly selected from Recipe1M+ for a total of 500 recipes.We ran these experiments on a MacBook Pro equipped with 2.6 GHz 6-Core Intel Core i7 processor and 16GB RAM.\"\n },\n {\n \"title\": \"A. Evaluation Strategy\",\n \"content\": \"Our objective is to evaluate the quality of a task tree for unseen ingredient combinations from the Recipe1M+ dataset.It is impossible to judge a recipe by solely relying on automated testing such as Intersection over Union (IoU) due to there being many ways to prepare a meal.Furthermore, despite having the same ingredients, one recipe can differ from another recipe in terms of cooking steps while both are valid or correct.Hence, we chose to evaluate the FOON-generated task trees by manually checking each task plan.Although it requires a lot of manual effort, we believe that human judgement is necessary to evaluate recipes.To make the process easier, instead of checking the functional units in the task tree, we check each ingredient by reviewing its progress line.\\nProgress Line: In cooking, ingredients undergo several changes in their physical state (e.g., from whole to sliced) and/or location.We refer to this sequence of changes as an ingredient's progress line, which is another visualization of a task tree.We developed a tool to view the progress line of each ingredient to manually evaluate generated task trees.Figures 4a and 4b shows an example of a generated task tree and its progress line respectively, where objects, states and motions are colored in black, green and red respectively.\"\n },\n {\n \"title\": \"B. Results\",\n \"content\": \"Task trees generated for the 500 recipes were thoroughly evaluated using the progress line visualizer.When checking a recipe, we review each of its ingredients and indicate whether a given ingredient is incorrect, partially incorrect or correct by assigning it a score from {0,1,2}, where '0' means that an ingredient's progress line is incorrect, '1' means partially incorrect, and '2' means correct.A label of partially incorrect means that a state label assigned to the ingredient is not contextually or physically relevant to it.Without modifying the reference task tree, we would only see higher scores for ingredients that already exist within the unmodified tree, while missing ingredients (i.e., items that need to be added to the tree) or extra ingredients (i.e., items that need to be pruned from the tree) will be assigned an incorrect score.However, our proposed algorithm will be able to generate task plans for unseen ingredients in addition to already existing items while removing extra ingredients.In Figure 5, we present a comparison of these two approaches by reporting the percentage of correctly generated recipes with varying degrees of thresholds.The result shows that the proposed algorithm can generate 76% of the recipes correctly on average if the threshold of correctness for each recipe is set to 100%.When the threshold is 100%, a task tree must have correct progress lines for each ingredient.It is evident from the experiments that using similarity-based ingredient substitution greatly improves the performance and correctness thresholds across all dish types.\"\n },\n {\n \"title\": \"C. User Study\",\n \"content\": \"To verify correctness and completeness, we conducted a user study to see whether there is any significant difference between the FOON generated and Recipe1M+ recipes, following the same approach and questionnaires (see Figure 6) from a previous study [34].Since a task tree in its original format is not easy to read and verify for individuals not familiar with FOON, we translated each tree into recipe-like sentences.In this study, there were 9 participants, where each was given 10 recipes randomly selected from Recipe1M+ and its equivalent generated recipe from our method to assign ratings.This survey can be accessed by the provided link. 1 Our hypothesis H o is that there is no significant difference between FOONgenerated and Recipe1M+ recipes.With the ratings collected from users for questions 4-7, we performed a null hypothesis test with significance level α = 0.05 and obtained the pvalues 0.12, 0.34, 0.20, 0.64, 0.68, 0.73 and 0.35 respectively.Since all the p-values are fairly greater than α, we cannot reject H o ; hence, we can conclude that the generated recipes are equivalent to Recipe1M+ recipes in terms of correctness, completeness and clarity.\\nV. DISCUSSION Recipe1M+ has a significantly higher quantity of recipes and ingredients than FOON, making it very challenging to produce a task plan that perfectly matches the original textual recipe.Nevertheless, even with the limited concepts available in FOON, the flexibility of the task tree generation process allows us to achieve correct results in 76% of the cases.This mainly depends on the selection of equivalent objects for unseen ingredients, as an incorrect substitution may introduce errors in an ingredient's progress line.For example, the equivalent of apricot in FOON is lemon according to the word embedding model we used.However, we cannot squeeze an apricot to extract its juice as we can with a lemon.Therefore, the progress line of apricot does not correctly reflect how it should change its state after each manipulation action.I have made this exact dish Yes, but I left out some of the ingredients listed Yes, but I added some ingredients not listed No 4) Does this recipe seem correct to you? 5) Do the steps appear to be in the right order? 6) Are all steps in the recipe correct?7) Does each recipe step give enough information in order to complete it?8) Does the recipe skip steps that are obvious to you? 9) If attempting to follow the given steps, how confident are you that you could make this dish?10) Do the steps in this recipe appear to be clear and easy to follow?Fig. 6.Overview of the survey questions used in our user study.An ingredient may be used differently across recipes; for example, pineapple is mixed with other fruits while preparing a fruit salad, but added as a topping in a pizza.For this reason, when we add a branch to the task tree, we follow different rules to decide where the missing ingredient should be added.For instance, in drink recipes, branches can be added when objects are being poured into a container whereas for salad, ingredients are typically mixed in a bowl.These concepts are usually not captured in the word embedding model, and rulebased actions are not often sufficient to accurately infer the task plan.Hence, we will focus on improving our methodology through generalization in the future.We also noted that it is not always possible to include the user-provided state of an ingredient if the kitchen contains it in a non-reversible state, such as melted cheese when diced cheese is required.We do not mark those as errors since a new batch of ingredients is required to follow user input.Overall, we noted the following errors in our experiments as incorrect substitution: 39%, incorrect state: 27%, incorrect motion: 14% and incorrect integration: 20%.\\nSuch errors can be minimized with correction from users, where instead of selecting the top suggestion from the embedding model for each unseen ingredient, users can decide on the ideal substitute among the top 5 candidates.In the case of Figure 7, corrections can be made for the ingredients cool whip and pistachio using the second (whipped cream) and fourth (peanut) ranked candidates respectively.Furthermore, we can enrich the knowledge base by adding recipes with diverse set of ingredients so that we can make substitutions with higher confidence and with minimal user correction.\\nFinally, up to this point, recipes are added to FOON by manually labelling cooking videos, which is a very timeconsuming and error-prone process.If we can automate the process of creating subgraphs from video or text, we could easily expand FOON by adding recipes from various sources.Using a method such as the task tree generation process can help to reduce the amount of effort required by constructing graphs in a semi-automatic manner, in addition to computer vision approaches for extracting graphs from video [10].\\nA task tree can be used for bootstrapping robot task execution, as it provides a high-level description of a sequence of actions needed to be executed by a robot free from domainspecific robot details or properties (e.g. the number of endeffectors, which end-effector to use for certain actions, motion primitives, etc.).For instance, a task tree extracted from a knowledge graph containing instructions of furniture assembly can represent how the raw materials should be connected to build the furnished product.Although a task tree cannot be executed in its original state, it can however be connected to existing frameworks for manipulation planning, such as PDDL where functional units can be treated as planning operators and can then be used to retrieve plans corresponding to robot low-level motion primitives [35].\"\n },\n {\n \"title\": \"A. Complexity Analysis of the Algorithm\",\n \"content\": \"A FOON is represented as an adjacency list, which allows us to retrieve functional unit candidates in O(1) time.In the worst case, the number of candidates is equal to the total number of FOON recipes.However, this number is very small on average and thus is negligible.Therefore, for n functional units in the task tree, the required time is O(n).At the tree modification stage, if an equivalent ingredient exists in the reference tree, substitution can be done in O(1) time.However, if no suitable equivalent is present, we have to search other recipes for equivalents, which takes O(k) time, where k is the number of unique ingredients in FOON.We accelerate this process by computing a mapping of equivalent ingredient in the pre-processing phase to find an equivalent ingredient in O(1) time.Hence, the overall time complexity of the search algorithm is O(n), where n is the length of the retrieved path.The space complexity is also O(n) since we store only one path.With the addition of new recipes, FOON increases in size, which in turn can potentially create alternative paths to a goal node.Exploring all paths will incur a larger time and space complexity.To avoid that, we chose the heuristic-based search algorithm where the heuristic was designed using the given ingredients.Overall, we found the average computation time required to solve a problem instance was 80 milliseconds.\"\n },\n {\n \"title\": \"VI. CONCLUSION AND FUTURE WORK\",\n \"content\": \"In this work, our goal was to create a system that will enable a robot to derive task plans in novel scenarios not found in its knowledge base.It is almost impossible to equip a robot with knowledge of every possible recipe with countless combinations of ingredients.To address this, we explored recipe generation using FOON as a knowledge base.Similar to how humans can modify a recipe dynamically by using their previous cooking knowledge to determine how items available in the kitchen can be used, we demonstrate how an unseen ingredient can be integrated to an existing recipe by learning from an equivalent ingredient based on semantic similarity metrics.In our experiments, we generated task plans for different ingredient combinations using our approach and showed that plans were accurate in most cases.\\nThe evaluation method used in this work is subjective.In the future, we will explore how generated task trees can be executed with a robot to concretely evaluate our method.Furthermore, we aim to enrich our FOON knowledge base with a more diverse array of recipes by semi-automatically creating subgraphs from instructional videos or text-based sources such as cookbooks.Furthermore, additional work is required to learn a new embedding model that would not only measure the similarity between two words but also the similarity of two functional units or two task plans that considers the context of the ingredients.\"\n },\n {\n \"title\": \"Fig. 2 .\",\n \"content\": \"Fig. 2. Illustration of a universal FOON made of 140 recipe subgraphs, and three (3) connected functional units, where object nodes are denoted by green circles and motion nodes are denoted by red squares.\"\n },\n {\n \"title\": \"Algorithm 1\",\n \"content\": \"Find a reference goal node Input: given ingredients I, dish type D 1: candidates ← Retrieve recipes of type D 2: for each subgraph in candidates do 3: S ← Ingredients of subgraph 4:\"\n },\n {\n \"title\": \"Fig. 3 .\",\n \"content\": \"Fig. 3. Overview of our reference task tree retrieval method.Each node's value indicate a score returned by the heuristic function; empty nodes are not reached.Red arrows denote the chosen path based on maximum score.\"\n },\n {\n \"title\": \"Fig. 5 .\",\n \"content\": \"Fig.5.Graph showing the relationship between the threshold of correctness (X-axis) of each recipe and the number of recipes (Y-axis) we can generate with that threshold.The result of recipe generation with and without using task tree modification are presented as solid and dashed lines respectively; the same colored lines indicate the same dish type.\"\n },\n {\n \"title\": \"Fig. 7 .\",\n \"content\": \"Fig. 7. Top 5 equivalent ingredients in FOON for some randomly chosen unseen ingredients from Recipe1M+.\"\n },\n {\n \"title\": \"TABLE I RULES\",\n \"content\": \"TO MODIFY THE REFERENCE TASK TREE BASED ON THE DIFFERENCE BETWEEN A GIVEN INGREDIENT AND ITS EQUIVALENT IN THE TREE AS DISCUSSED IN SECTION III-C.THE TERM FU DENOTES FUNCTIONAL UNIT.\"\n },\n {\n \"title\": \"TABLE II OVERVIEW\",\n \"content\": \"OF THE TEST DATASET CREATED FROM RECIPE1M+\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Fig. 2. Illustration of a universal FOON made of 140 recipe subgraphs, and three (3) connected functional units, where object nodes are denoted by green circles and motion nodes are denoted by red squares.","content":"Fig. 2 .\n2\nFig. 2. Illustration of a universal FOON made of 140 recipe subgraphs, and three (3) connected functional units, where object nodes are denoted by green circles and motion nodes are denoted by red squares."},{"id":"","caption":"Find a reference goal node Input: given ingredients I, dish type D 1: candidates ← Retrieve recipes of type D 2: for each subgraph in candidates do 3: S ← Ingredients of subgraph 4:","content":"Algorithm 1\n1\nFind a reference goal node Input: given ingredients I, dish type D 1: candidates ← Retrieve recipes of type D 2: for each subgraph in candidates do 3: S ← Ingredients of subgraph 4:"},{"id":"","caption":"Fig. 3. Overview of our reference task tree retrieval method.Each node's value indicate a score returned by the heuristic function; empty nodes are not reached.Red arrows denote the chosen path based on maximum score.","content":"Fig. 3 .\n3\nFig. 3. Overview of our reference task tree retrieval method.Each node's value indicate a score returned by the heuristic function; empty nodes are not reached.Red arrows denote the chosen path based on maximum score."},{"id":"","caption":"Fig.5.Graph showing the relationship between the threshold of correctness (X-axis) of each recipe and the number of recipes (Y-axis) we can generate with that threshold.The result of recipe generation with and without using task tree modification are presented as solid and dashed lines respectively; the same colored lines indicate the same dish type.","content":"Fig. 5 .\n5\nFig.5.Graph showing the relationship between the threshold of correctness (X-axis) of each recipe and the number of recipes (Y-axis) we can generate with that threshold.The result of recipe generation with and without using task tree modification are presented as solid and dashed lines respectively; the same colored lines indicate the same dish type."},{"id":"","caption":"Fig. 7. Top 5 equivalent ingredients in FOON for some randomly chosen unseen ingredients from Recipe1M+.","content":"Fig. 7 .\n7\nFig. 7. Top 5 equivalent ingredients in FOON for some randomly chosen unseen ingredients from Recipe1M+."},{"id":"","caption":"TO MODIFY THE REFERENCE TASK TREE BASED ON THE DIFFERENCE BETWEEN A GIVEN INGREDIENT AND ITS EQUIVALENT IN THE TREE AS DISCUSSED IN SECTION III-C.THE TERM FU DENOTES FUNCTIONAL UNIT.","content":"TABLE I RULES\nI\nTO MODIFY THE REFERENCE TASK TREE BASED ON THE DIFFERENCE BETWEEN A GIVEN INGREDIENT AND ITS EQUIVALENT IN THE TREE AS DISCUSSED IN SECTION III-C.THE TERM FU DENOTES FUNCTIONAL UNIT.\nCase End leaf in refer-Given ingredientObjectStateSubstitutionAdd branchence task treematchedmatched1sliced carrotsliced carrot--2chopped jalapeñochopped chili peppersubstitute jalapeño with-chili pepper3diced tomatowhole tomato-add FUs to make dicedtomato from whole tomato4minced shallotpeeled onionsubstitute shallot withadd FUs to make mincedoniononion from peeled onion"},{"id":"","caption":"OF THE TEST DATASET CREATED FROM RECIPE1M+","content":"TABLE II OVERVIEW\nII\nOF THE TEST DATASET CREATED FROM RECIPE1M+\nDish TypeNo. of Recipes inAverage No. of Ingre-Recipe1M+dients per Recipesalad58,8699cake29,88710soup35,35710omelette6867drinks2,7325"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Fig. 2. Illustration of a universal FOON made of 140 recipe subgraphs, and three (3) connected functional units, where object nodes are denoted by green circles and motion nodes are denoted by red squares.\",\n \"content\": \"Fig. 2 .\\n2\\nFig. 2. Illustration of a universal FOON made of 140 recipe subgraphs, and three (3) connected functional units, where object nodes are denoted by green circles and motion nodes are denoted by red squares.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Find a reference goal node Input: given ingredients I, dish type D 1: candidates ← Retrieve recipes of type D 2: for each subgraph in candidates do 3: S ← Ingredients of subgraph 4:\",\n \"content\": \"Algorithm 1\\n1\\nFind a reference goal node Input: given ingredients I, dish type D 1: candidates ← Retrieve recipes of type D 2: for each subgraph in candidates do 3: S ← Ingredients of subgraph 4:\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Fig. 3. Overview of our reference task tree retrieval method.Each node's value indicate a score returned by the heuristic function; empty nodes are not reached.Red arrows denote the chosen path based on maximum score.\",\n \"content\": \"Fig. 3 .\\n3\\nFig. 3. Overview of our reference task tree retrieval method.Each node's value indicate a score returned by the heuristic function; empty nodes are not reached.Red arrows denote the chosen path based on maximum score.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Fig.5.Graph showing the relationship between the threshold of correctness (X-axis) of each recipe and the number of recipes (Y-axis) we can generate with that threshold.The result of recipe generation with and without using task tree modification are presented as solid and dashed lines respectively; the same colored lines indicate the same dish type.\",\n \"content\": \"Fig. 5 .\\n5\\nFig.5.Graph showing the relationship between the threshold of correctness (X-axis) of each recipe and the number of recipes (Y-axis) we can generate with that threshold.The result of recipe generation with and without using task tree modification are presented as solid and dashed lines respectively; the same colored lines indicate the same dish type.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Fig. 7. Top 5 equivalent ingredients in FOON for some randomly chosen unseen ingredients from Recipe1M+.\",\n \"content\": \"Fig. 7 .\\n7\\nFig. 7. Top 5 equivalent ingredients in FOON for some randomly chosen unseen ingredients from Recipe1M+.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"TO MODIFY THE REFERENCE TASK TREE BASED ON THE DIFFERENCE BETWEEN A GIVEN INGREDIENT AND ITS EQUIVALENT IN THE TREE AS DISCUSSED IN SECTION III-C.THE TERM FU DENOTES FUNCTIONAL UNIT.\",\n \"content\": \"TABLE I RULES\\nI\\nTO MODIFY THE REFERENCE TASK TREE BASED ON THE DIFFERENCE BETWEEN A GIVEN INGREDIENT AND ITS EQUIVALENT IN THE TREE AS DISCUSSED IN SECTION III-C.THE TERM FU DENOTES FUNCTIONAL UNIT.\\nCase End leaf in refer-Given ingredientObjectStateSubstitutionAdd branchence task treematchedmatched1sliced carrotsliced carrot--2chopped jalapeñochopped chili peppersubstitute jalapeño with-chili pepper3diced tomatowhole tomato-add FUs to make dicedtomato from whole tomato4minced shallotpeeled onionsubstitute shallot withadd FUs to make mincedoniononion from peeled onion\"\n },\n {\n \"id\": \"\",\n \"caption\": \"OF THE TEST DATASET CREATED FROM RECIPE1M+\",\n \"content\": \"TABLE II OVERVIEW\\nII\\nOF THE TEST DATASET CREATED FROM RECIPE1M+\\nDish TypeNo. of Recipes inAverage No. of Ingre-Recipe1M+dients per Recipesalad58,8699cake29,88710soup35,35710omelette6867drinks2,7325\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Joint Object and State Recognition using Language Knowledge. A B Jelodar, Y Sun, 2019 IEEE International Conference on Image Processing (ICIP). IEEE2019","doi":""},{"text":"Functional Object-Oriented Network for Manipulation Learning. D Paulius, Y Huang, R Milton, W D Buchanan, J Sam, Y Sun, Intelligent Robots and Systems (IROS). IEEE2016. 2016","doi":""},{"text":"Human-object-object-interaction affordance. S Ren, Y Sun, Workshop on Robot Vision. 2013","doi":""},{"text":"Object-object interaction affordance learning. Y Sun, S Ren, Y Lin, Robotics and Autonomous Systems. 2013","doi":""},{"text":"Robot grasp planning based on demonstrated grasp strategies. Y Lin, Y Sun, Intl. Journal of Robotics Research. 3412015","doi":""},{"text":"Functional Object-Oriented Network: Construction & Expansion. D Paulius, A B Jelodar, Y Sun, 2018 IEEE International Conference on Robotics and Automation (ICRA). IEEE2018","doi":""},{"text":"Recipe1M+: A Dataset for Learning Cross-Modal Embeddings for Cooking Recipes and Food Images. J Marin, A Biswas, F Ofli, N Hynes, A Salvador, Y Aytar, I Weber, A Torralba, IEEE Trans. Pattern Anal. Mach. Intell. 2019","doi":""},{"text":"The theory of affordances. J Gibson, Perceiving, Acting and Knowing, R. Shaw and J. Bransford1977ErlbaumHillsdale, NJ","doi":""},{"text":"PDDL -The Planning Domain Definition Language. D Mcdermott, M Ghallab, A Howe, C Knoblock, A Ram, M Veloso, D Weld, D Wilkins, CVC TR-98-003/DCS TR-11651998Yale Center for Computational Vision and ControlTech. Rep.","doi":""},{"text":"Long Activity Video Understanding Using Functional Object-Oriented Network. A B Jelodar, D Paulius, Y Sun, IEEE Transactions on Multimedia. 217July 2019","doi":""},{"text":"Activitynet: A large-scale video benchmark for human activity understanding. B G Fabian Caba Heilbron, Victor Escorcia, J C Niebles, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern Recognition2015","doi":""},{"text":"Scaling Egocentric Vision: The EPIC-KITCHENS Dataset. D Damen, H Doughty, G M Farinella, S Fidler, A Furnari, E Kazakos, D Moltisanti, J Munro, T Perrett, W Price, M Wray, European Conference on Computer Vision (ECCV). 2018","doi":""},{"text":"FOON Website: Graph Viewer and Videos. 2021","doi":""},{"text":"Task Planning with a Weighted Functional Object-Oriented Network. D Paulius, K S P Dong, Y Sun, 2021 IEEE International Conference on Robotics and Automation (ICRA). IEEE2021","doi":""},{"text":"KNOWROB -knowledge processing for autonomous personal robots. M Tenorth, M Beetz, 2009. 2009","doi":""},{"text":"Robotic roommates making pancakes. M Beetz, U Klank, I Kresse, A Maldonado, L Mösenlechner, D Pangercic, T Rühr, M Tenorth, 2011 11th IEEE-RAS International Conference on Humanoid Robots. 2011","doi":""},{"text":"IkeaBot: An autonomous multi-robot coordinated furniture assembly system. R A Knepper, T Layton, J Romanishin, D Rus, 2013 IEEE International Conference on Robotics and Automation. 2013","doi":""},{"text":"IKEA Furniture Assembly Environment for Long-Horizon Complex Manipulation Tasks. Y Lee, E Hu, Z Yang, A C Yin, J J Lim, 2021 IEEE International Conference on Robotics and Automation (ICRA). 2021","doi":""},{"text":"Onto-IKEA: A Knowledge Retrieval Framework Based on IKEA Ontology. A Vassiliades, N Zarkadas, N Bassiliades, T Patkos, JOWO. 2021","doi":""},{"text":"Manipulation action tree bank: A knowledge resource for humanoids. Y Yang, A Guha, C Fermuller, Y Aloimonos, Humanoid Robots (Humanoids). IEEE2014 14th. 2014","doi":""},{"text":"Assembly planning by recognizing a graphical instruction manual. I Sera, N Yamanobe, I G Ramirez-Alpizar, Z Wang, W Wan, K Harada, 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). 2021","doi":""},{"text":"A big data approach to computational creativity. L R Varshney, F Pinel, K R Varshney, D Bhattacharjya, A Schörgendorfer, Y.-M Chee, abs/1311.1213ArXiv. 2013","doi":""},{"text":"Cooking Made Easy: On a Novel Approach to Complexity-Aware Recipe Generation. G Müller, R Bergmann, ICCBR2017","doi":""},{"text":"EvoChef: Show Me What to Cook! Artificial Evolution of Culinary Arts. H Jabeen, N Tahara, J Lehmann, EvoMUSART2019","doi":""},{"text":"AutoChef: Automated Generation of Cooking Recipes. H Jabeen, J Weinz, J Lehmann, 2020 IEEE Congress on Evolutionary Computation (CEC). 2020","doi":""},{"text":"Improving robotic cooking using batch bayesian optimization. K Junge, J Hughes, T George Thuruthel, F Iida, IEEE Robotics and Automation Letters. 012020","doi":""},{"text":"Extracting Food Substitutes From Food Diary via Distributional Similarity. P Achananuparp, I Weber, abs/1607.08807CoRR. 2016","doi":""},{"text":"Identifying ingredient substitutions using a knowledge graph of food. S Shirai, O W Seneviratne, M Gordon, C.-H Chen, D L Mcguinness, Frontiers in Artificial Intelligence. 32020","doi":""},{"text":"Improving Ingredient Substitution using Formal Concept Analysis and Adaptation of Ingredient Quantities with Mixed Linear Optimization. E Gaillard, J Lieber, E Nauer, ICCBR2015","doi":""},{"text":"Exploiting food embeddings for ingredient substitution. C Pellegrini, E Özsoy, M Wintergerst, G Groh, HEALTHINF2021","doi":""},{"text":"Towards robust one-shot task execution using knowledge graph embeddings. A A Daruna, L Nair, W Liu, S Chernova, 2021 IEEE International Conference on Robotics and Automation (ICRA). 2021","doi":""},{"text":"Ontology-Assisted Generalisation of Robot Action Execution Knowledge. A Mitrevski, P.-G Plöger, G Lakemeyer, 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). 2021","doi":""},{"text":"spaCy: Industrial-strength Natural Language Processing in Python. M Honnibal, I Montani, S Van Landeghem, A Boyd, 10.5281/zenodo.12123032020","doi":""},{"text":"Evaluating Recipes Generated from Functional Object-Oriented Network. M S Sakib, H Baez, D Paulius, Y Sun, arXiv:2106.00728Featured in 18th International Conference on Ubiquitous Robots. UR2021. 2021arXiv preprint","doi":""},{"text":"A Road-map to Robot Task Execution with the Functional Object-Oriented Network. D Paulius, A Agostini, Y Sun, D Lee, arXiv:2106.00158Featured in 18th International Conference on Ubiquitous Robots. UR2021. 2021arXiv preprint","doi":""}],"string":"[\n {\n \"text\": \"Joint Object and State Recognition using Language Knowledge. A B Jelodar, Y Sun, 2019 IEEE International Conference on Image Processing (ICIP). IEEE2019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Functional Object-Oriented Network for Manipulation Learning. D Paulius, Y Huang, R Milton, W D Buchanan, J Sam, Y Sun, Intelligent Robots and Systems (IROS). IEEE2016. 2016\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Human-object-object-interaction affordance. S Ren, Y Sun, Workshop on Robot Vision. 2013\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Object-object interaction affordance learning. Y Sun, S Ren, Y Lin, Robotics and Autonomous Systems. 2013\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Robot grasp planning based on demonstrated grasp strategies. Y Lin, Y Sun, Intl. Journal of Robotics Research. 3412015\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Functional Object-Oriented Network: Construction & Expansion. D Paulius, A B Jelodar, Y Sun, 2018 IEEE International Conference on Robotics and Automation (ICRA). IEEE2018\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Recipe1M+: A Dataset for Learning Cross-Modal Embeddings for Cooking Recipes and Food Images. J Marin, A Biswas, F Ofli, N Hynes, A Salvador, Y Aytar, I Weber, A Torralba, IEEE Trans. Pattern Anal. Mach. Intell. 2019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The theory of affordances. J Gibson, Perceiving, Acting and Knowing, R. Shaw and J. Bransford1977ErlbaumHillsdale, NJ\",\n \"doi\": \"\"\n },\n {\n \"text\": \"PDDL -The Planning Domain Definition Language. D Mcdermott, M Ghallab, A Howe, C Knoblock, A Ram, M Veloso, D Weld, D Wilkins, CVC TR-98-003/DCS TR-11651998Yale Center for Computational Vision and ControlTech. Rep.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Long Activity Video Understanding Using Functional Object-Oriented Network. A B Jelodar, D Paulius, Y Sun, IEEE Transactions on Multimedia. 217July 2019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Activitynet: A large-scale video benchmark for human activity understanding. B G Fabian Caba Heilbron, Victor Escorcia, J C Niebles, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern Recognition2015\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Scaling Egocentric Vision: The EPIC-KITCHENS Dataset. D Damen, H Doughty, G M Farinella, S Fidler, A Furnari, E Kazakos, D Moltisanti, J Munro, T Perrett, W Price, M Wray, European Conference on Computer Vision (ECCV). 2018\",\n \"doi\": \"\"\n },\n {\n \"text\": \"FOON Website: Graph Viewer and Videos. 2021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Task Planning with a Weighted Functional Object-Oriented Network. D Paulius, K S P Dong, Y Sun, 2021 IEEE International Conference on Robotics and Automation (ICRA). IEEE2021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"KNOWROB -knowledge processing for autonomous personal robots. M Tenorth, M Beetz, 2009. 2009\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Robotic roommates making pancakes. M Beetz, U Klank, I Kresse, A Maldonado, L Mösenlechner, D Pangercic, T Rühr, M Tenorth, 2011 11th IEEE-RAS International Conference on Humanoid Robots. 2011\",\n \"doi\": \"\"\n },\n {\n \"text\": \"IkeaBot: An autonomous multi-robot coordinated furniture assembly system. R A Knepper, T Layton, J Romanishin, D Rus, 2013 IEEE International Conference on Robotics and Automation. 2013\",\n \"doi\": \"\"\n },\n {\n \"text\": \"IKEA Furniture Assembly Environment for Long-Horizon Complex Manipulation Tasks. Y Lee, E Hu, Z Yang, A C Yin, J J Lim, 2021 IEEE International Conference on Robotics and Automation (ICRA). 2021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Onto-IKEA: A Knowledge Retrieval Framework Based on IKEA Ontology. A Vassiliades, N Zarkadas, N Bassiliades, T Patkos, JOWO. 2021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Manipulation action tree bank: A knowledge resource for humanoids. Y Yang, A Guha, C Fermuller, Y Aloimonos, Humanoid Robots (Humanoids). IEEE2014 14th. 2014\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Assembly planning by recognizing a graphical instruction manual. I Sera, N Yamanobe, I G Ramirez-Alpizar, Z Wang, W Wan, K Harada, 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). 2021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A big data approach to computational creativity. L R Varshney, F Pinel, K R Varshney, D Bhattacharjya, A Schörgendorfer, Y.-M Chee, abs/1311.1213ArXiv. 2013\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cooking Made Easy: On a Novel Approach to Complexity-Aware Recipe Generation. G Müller, R Bergmann, ICCBR2017\",\n \"doi\": \"\"\n },\n {\n \"text\": \"EvoChef: Show Me What to Cook! Artificial Evolution of Culinary Arts. H Jabeen, N Tahara, J Lehmann, EvoMUSART2019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"AutoChef: Automated Generation of Cooking Recipes. H Jabeen, J Weinz, J Lehmann, 2020 IEEE Congress on Evolutionary Computation (CEC). 2020\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Improving robotic cooking using batch bayesian optimization. K Junge, J Hughes, T George Thuruthel, F Iida, IEEE Robotics and Automation Letters. 012020\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Extracting Food Substitutes From Food Diary via Distributional Similarity. P Achananuparp, I Weber, abs/1607.08807CoRR. 2016\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Identifying ingredient substitutions using a knowledge graph of food. S Shirai, O W Seneviratne, M Gordon, C.-H Chen, D L Mcguinness, Frontiers in Artificial Intelligence. 32020\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Improving Ingredient Substitution using Formal Concept Analysis and Adaptation of Ingredient Quantities with Mixed Linear Optimization. E Gaillard, J Lieber, E Nauer, ICCBR2015\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Exploiting food embeddings for ingredient substitution. C Pellegrini, E Özsoy, M Wintergerst, G Groh, HEALTHINF2021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Towards robust one-shot task execution using knowledge graph embeddings. A A Daruna, L Nair, W Liu, S Chernova, 2021 IEEE International Conference on Robotics and Automation (ICRA). 2021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Ontology-Assisted Generalisation of Robot Action Execution Knowledge. A Mitrevski, P.-G Plöger, G Lakemeyer, 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). 2021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"spaCy: Industrial-strength Natural Language Processing in Python. M Honnibal, I Montani, S Van Landeghem, A Boyd, 10.5281/zenodo.12123032020\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Evaluating Recipes Generated from Functional Object-Oriented Network. M S Sakib, H Baez, D Paulius, Y Sun, arXiv:2106.00728Featured in 18th International Conference on Ubiquitous Robots. UR2021. 2021arXiv preprint\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Road-map to Robot Task Execution with the Functional Object-Oriented Network. D Paulius, A Agostini, Y Sun, D Lee, arXiv:2106.00158Featured in 18th International Conference on Ubiquitous Robots. UR2021. 2021arXiv preprint\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":484,"cells":{"text":{"kind":"string","value":"\nEntanglement Entropies in Conformal Systems with Boundaries\n(Dated: September 2, 2013)\n\nL Taddia \nDipartimento di Fisica e Astronomia dell'Università di Bologna and INFN\nvia Irnerio 4640126BolognaItaly\n\nInstituto de Física Teórica UAM/CSIC\nUniversidad Autónoma de Madrid\n28049Cantoblanco, MadridSpain\n\nJ C Xavier \nInstituto de Física\nUniversidade Federal de Uberlândia\nCaixa Postal 59338400-902UberlândiaMGBrazil\n\nF C Alcaraz \nInstituto de Física de São Carlos\nUniversidade de São Paulo\nCaixa Postal 369São CarlosSPBrazil\n\nG Sierra \nInstituto de Física Teórica UAM/CSIC\nUniversidad Autónoma de Madrid\n28049Cantoblanco, MadridSpain\n\nEntanglement Entropies in Conformal Systems with Boundaries\n(Dated: September 2, 2013)\nWe study the entanglement entropies in one-dimensional open critical systems, whose effective description is given by a conformal field theory with boundaries. We show that for pure-state systems formed by the ground state or by the excited states associated to primary fields, the entanglement entropies have a finite-size behavior that depends on the correlation of the underlying field theory. The analytical results are checked numerically, finding excellent agreement for the quantum chains ruled by the theories with central charge c = 1/2 and c = 1.PACS numbers: 03.67. 03.67.Mn, 64.60.an, 11.25.Hf In recent years there has been a flurry of activity devoted to characterize quantum many body systems and quantum field theories using the concept of entanglement (see[1]and references therein for a review). Under certain conditions, the low energy states of local Hamiltonians satisfy an entropic area law according to which the entanglement entropy (EE) of a subsystem is proportional to the area of its boundary[2]. In one spatial dimension, violations to this area law appear if the system is gapless and described by a conformal field theory (CFT)[3,4]: in this case, the ground-state (GS) EE grows with the log of the subsystem size[5][6][7]. On the other hand, it is well known that the dominant finite-size correction to the GS energy of critical systems is related to the central charge c [8]. A similar correspondence also exists between the central charge and the Rényi entanglement entropies (REE's) of the GS [5-7]. In the same vein, we would expect that the scaling dimensions of the primary operators are related with the REE's of the excited states (ES's), since the finite mass gaps of critical lattice Hamiltonians are also related with them[8]. Indeed, this latter correspondence was proven recently for conformal systems with periodic boundary conditions [9, 10]. The main result of these works is that the n th REE is given in terms of the 2n-point correlator of the corresponding primary field placed at special positions depending on the subsystem size.In this letter we shall generalize this result to open lattice Hamiltonians, effectively described by boundary conformal field theory (BCFT)[11]. BCFT has a wide range of applications to impurity problems [12], string theory [13], etc, which justifies the generalization pursued here. Moreover, the generalization is, as we shall see, non trivial, since BCFT has a very rich mathematical structure, which is worth to analyze from the entanglement point of view. Apart from these many reasons of interest, it is worth to cite that the problem is quite long-standing[14,15], and our work represents its final solution.\n\nWe study the entanglement entropies in one-dimensional open critical systems, whose effective description is given by a conformal field theory with boundaries. We show that for pure-state systems formed by the ground state or by the excited states associated to primary fields, the entanglement entropies have a finite-size behavior that depends on the correlation of the underlying field theory. The analytical results are checked numerically, finding excellent agreement for the quantum chains ruled by the theories with central charge c = 1/2 and c = 1. In recent years there has been a flurry of activity devoted to characterize quantum many body systems and quantum field theories using the concept of entanglement (see [1] and references therein for a review). Under certain conditions, the low energy states of local Hamiltonians satisfy an entropic area law according to which the entanglement entropy (EE) of a subsystem is proportional to the area of its boundary [2]. In one spatial dimension, violations to this area law appear if the system is gapless and described by a conformal field theory (CFT) [3,4]: in this case, the ground-state (GS) EE grows with the log of the subsystem size [5][6][7]. On the other hand, it is well known that the dominant finite-size correction to the GS energy of critical systems is related to the central charge c [8]. A similar correspondence also exists between the central charge and the Rényi entanglement entropies (REE's) of the GS [5][6][7]. In the same vein, we would expect that the scaling dimensions of the primary operators are related with the REE's of the excited states (ES's), since the finite mass gaps of critical lattice Hamiltonians are also related with them [8]. Indeed, this latter correspondence was proven recently for conformal systems with periodic boundary conditions [9,10]. The main result of these works is that the n th REE is given in terms of the 2n-point correlator of the corresponding primary field placed at special positions depending on the subsystem size.\n\nIn this letter we shall generalize this result to open lattice Hamiltonians, effectively described by boundary conformal field theory (BCFT) [11]. BCFT has a wide range of applications to impurity problems [12], string theory [13], etc, which justifies the generalization pursued here. Moreover, the generalization is, as we shall see, non trivial, since BCFT has a very rich mathematical structure, which is worth to analyze from the entanglement point of view. Apart from these many reasons of interest, it is worth to cite that the problem is quite long-standing [14,15], and our work represents its final solution.\n\nWe shall start with some basic definitions. The REE's are defined as follows:\nS n ≡ 1 1 − n ln Tr A ρ n A ,(1)\nwhere n is a positive real, ρ A is the reduced density matrix of a subsystem A and the trace is over the A's Hilbert space. If ρ A is constructed from the GS of a CFT, one has [7]:\nS CF T n (l, L) = c η n + c 3η 1 + 1 n ln ηL π sin πl L ,(2)\nwhere l is the size of A, L is the total-system size and η = 1, 2 for periodic/free boundary conditions (PBC/FBC); c is the central charge and c η n is a constant that depends on the BC's [4]. For open systems, l is measured from the left edge. Equation (2) may have significant corrections which carry useful information about the underlying CFT [16][17][18]. Different kind of corrections arise when, instead of considering the GS, one looks at the ES's [9,10,19,20], or when the system satisfies general BC's preserving its conformal invariance (FBC are just one of them) [14,15]. In particular, the corrections in this last case have, up to now, not have been derived analytically, and their knowledge would be a great advance, both from a practical and a conceptual point of view. In the following, we present the general CFT framework allowing the analytical computation of such corrections. These results are then verified with DMRG calculations in two examples: the c = 1/2 minimal CFT and the c = 1 massless compactified free boson.\n\nLet us consider a 1D conformal system, defined on a strip of width L. Cardy [11] showed the existence of BC's, denoted as {α}, that preserve the conformal invariance, and such that the partition function takes the form where q ≡ e −πβ/L (being β the inverse temperature), h are the conformal dimensions of the primary fields, and χ h are the associated characters. The integers N h αβ are the fusion coefficients of the theory [3,4]. The derivation of the REE's of ES's, originating from the primary operator Υ(w) for open conformal systems follows closely the one of periodic boundary conditions given in [9]. The REE's are given by\nZαβ(q) = h N h αβ χ h (q),(3)S n (l, L) = S CF T n (l, L) + 1 1 − n ln F (n) Υ ,(4)F (n) Υ (x) = Tr A ρ n A,Υ Tr A ρ n A,GS ,\nwhere ρ A,Υ and ρ A,GS are the reduced density matrices obtained from the ES produced by the primary field Υ and from the GS, respectively. The ratio F (n) Υ can be computed using path integral methods [9]. We explain below the main steps of the derivation.\n\nLet us first split the strip into the subsystems A = [0, l], B = [l, L]. This strip is parameterized by the com-\nplex coordinate w = σ + iτ , σ ∈ [0, L], τ ∈ (−∞, ∞). We make the conformal transformation ζ = sin π(w−l) 2L sin π(w+l) 2L ,(5)\nwhich maps the strip into the unit disc D = {ζ, |ζ| ≤ 1}. The intervals A and B are mapped into the segments (−1, 0) and (0, 1) respectively. The boundaries of the strip, σ = 0, L are mapped into the boundary of the disc (|ζ| = 1), and the infinite past (w − ∞ = −i∞) and infinite future (w + ∞ = i∞) are mapped into\nw − ∞ → ζ − ∞ = e −iπx , w + ∞ → ζ + ∞ = e iπx , x ≡ l L .(6)\nNext, we make n copies of the unit disc and sew them along the cut (−1, 0), obtaining the Riemann surface R n , which can also be mapped into the unit disc by the conformal transformation\nz = ζ 1/n = sin π(w−l) 2L sin π(w+l) 2L 1/n .(7)\nThe points ζ ± ∞ give rise to 2n points on R n z ± k,n = e iπ n (±x+2k) , k = 0, 1, . . . , n − 1,\n\nwhere the primary fields Υ and its conjugate Υ † are inserted. Repeating the same steps as in [9], we arrive at an expression for Eq. (4)\nF (n) Υ (x) = e i2π(n−1)h n 2nh n−1 k=0 Υ(z − n,k )Υ † (z + n,k ) Υ(z − 1,0 )Υ † (z + 1,0 ) n ,(9)\nwhere we have assumed that Υ is a chiral primary field with conformal dimension h. A similar formula holds for non chiral fields. The correlators in Eq.(9) are computed on the unit disk for fields inserted at its boundary. These primary fields change the boundary conditions a and b on each edge of the strip: Υ changes them from a to b, while Υ † from b to a. Hence the numerator of Eq. (9) is proportional to the partition function of a disc with 2n segments where the boundary conditions a and b alternate [4]. Equation (9) constitutes the main result of the work, which we shall verify below for two models.\n\nThe c = 1/2 minimal CFT. This CFT contains three primary fields: the identity I (h I = 0), a Majorana fermion χ (h χ = 1/2) and a spin field σ (h σ = 1/16), whose fusion rules are\nN h 00 = N h 1 2 1 2 = δ h 0 , N h 0 1 2 = δ h 1 2 , N h 0 1 16 = N h 1 16 1 2 = δ h 1 16 , N h 1 16 1 16 = δ h 0 + δ h 1 2 .(10)\nThis CFT describes the long distance properties of the critical Ising model in a transverse field whose lattice Hamiltonian is\nH I = − 1 2 L−1 j=1 σ x j σ x j+1 − 1 2 L j=1 σ z j ,(11)\nwhere σ x,y,z are the Pauli matrices. The correspondence between the conformal BC's {α} and the lattice BC's is the following:0 (1 2 ), corresponds to fix σ x 1,L to +1 (-1), while1 16 corresponds to free BC's [11,14]. For simplicity, we shall denote0,1 2 ,1 16 by +, −, F .\n\nWe begin by considering the REE's of the GS with F F BC's (the two F 's refer to free BC's on both edges). The spin Hamiltonian (11) can be mapped into a free fermion Hamiltonian [21], which we use to compute, by using the method of [22], the von Neumann entropy, i.e., the n = 1 REE, for chains with lengths 60-180 in multiples of 20 (this will be the sizes considered in the rest of the work). Using Eq. 14], is satisfied in this case, because the boundary entropy (BE) ln g vanishes for F F BC's [23]. Moreover, as a different reliability check, we verified that the REE's in this case are exactly one half of the ones of an XX spin-1/2 chain with F F BC's (see below), according to reference [24].\n1 − c η=1 1 /2 = ln g [7,\nWe next study the ++BC case (that is equivalent to −−BC). The fusion rule N h 00 = δ h 0 , lead us to consider only the case where h = 0, for which no corrections arise (apart, as we shall see, from constant BE contributions). We compute the REE's for the Hamiltonian (11) using the DMRG method [25] with up to 800 states per block, and 3 sweeps, which yields a truncation error of 10 −12 or less. The Figs. 1(a), (b) display the results for n = 2, 3 REE's: the data progressively flattens to the theoretical value − 1 2 ln 2 [14,26]. The convergence to the CFT predictions is of order 10 −3 or less, as confirmed by a FSS analysis (Fig. 1(i)). This behavior can be ascribed to the presence of a slowly L-depending finite-size correction [24], previously observed, for the Ising model, in [10].\n\nWe now consider the F F BC case. The fusion rule N χ σ,σ = 1 implies that the first ES is generated by the primary field χ. The associated F function is given by\nF (n) χ = F (n)\ni∂φ , where φ is a massless free boson with c = 1 [10]. Quite interestingly, F We then consider the +F BC case. The fusion rule N σ I,σ = 1 implies that the spectrum contains just the conformal tower of the σ field, so the F function must be F (n) σ . The nontrivial fusion rule of the field σ yields different chiral correlators, which were computed for general n in [28]. In our case though, only a combination of them yields the appropriate F functions, which for n = 2, 3 are given by\nF (2) σ (x) = cos πx 4 , F (3) σ (x) = cos πx 3 .(13)\nThe Figs. 1(e), (f) (also see Fig. 1(j)) display the DMRG results and the CFT predictions: the agreement, for L → ∞ is excellent, up to the constant term − 1 2 ln 2. Finally, we consider the +−BC case (equivalent to the −+BC case). The spectrum contains just the conformal tower of the χ operator, and therefore the REE's of the GS shall be the same of the ones of the first ES in the F F BC case (see Figs. 1(g), (h)). To conclude, we have shown that the CFT predictions for the Ising model with all possible conformal BC's agree with the numerical results, up to the constant contribution due to the BE.\n\nThe c = 1 compactified free boson. We shall next consider a massless free boson with a compactification radius R = 1. This CFT is rational, meaning that the chiral symmetry is enhanced in such a way that the number of primary fields is finite and have conformal dimensions h = 0, 1/8, 1/2, 1/8 [4]. The corresponding conformal characters are given by setting λ = 0, 1, 2, 3 mod 4 (h λ = λ 2 /8) in where η(q) is the Dedekind function. The conformal BC's are Dirichlet (D) and Neumann (N ), and the corresponding boundary entropies are 0 and − 1 2 ln 2 [12].\nK λ (q) ≡ 1 η(q) n∈Z q 1 8 (4n+λ) 2 ,(14)\nThe lattice realization of this CFT is given by a spin-1/2 XX chain, with boundary couplings [29] \nH B = − L−1 j=1 σ x j σ x j+1 + σ x j σ x j+1 + − 1 2 (α − σ − 1 + α + σ + 1 + α z σ z 1 + β − σ − L + β + σ + L + β z σ z L ),(15)\nwhere the DBC, on a given edge, is realized by setting all the boundary couplings to zero, while the N BC is realized, say on the left side, by choosing α z = 0 and α + = α − = 2 (moreover, in the sermonic picture of the DDBC case, one has to work in the half-filled sector). The operator content of the various models described by the Hamiltonian (15) can be expressed in terms of the partition functions of a free boson with different BC's [12,29]. The DD and N N partition functions can be written in terms of the characters (14)\nZ DD (q) = K 0 (q) + K 2 (q),(16)Z N N (q) = K 0 (q).(17)\nThe absence of corrections for the GS in the DDBC case, with the exception of the usual oscillating ones [18], has already been observed [30] and we confirm it numerically. We expect the same feature for the GS in the N N BC case, which we analyze with DMRG for system size L = 100, keeping up to 1100 states, using 3 sweeps and achieving a truncation error of 10 −10 . The results are shown in Figs. 2(a), (b), for n = 2, 3 : up to oscillating corrections, typical of c = 1 systems, and the BE − 1 2 ln 2, as expected from Eq. (9) we do not see any non-constant correction.\n\nWe then consider the first ES with DDBC which, according to Eq. (16), is generated by a vertex operator with conformal dimension 1/2. The F (n) Υ functions of vertex operators Υ are equal to 1, for all n, so the REE's receive no corrections [9]. We verify this result by identifying this excitation in the fermionic version of the Hamiltonian (15) with the addition (or subtraction) of a fermion at half filling. The Figs. 2(c), (d) show the n = 1, 2 REE's for these states, which confirmed the absence of corrections.\n\nFor DD and N N BC, there is another ES, with conformal weight h = 1, associated to the primary field i∂φ. For DDBC it corresponds to the lowest particlehole state created from the half-filled ground state [9,10]; in the N N BC case, there is no particle number conservation, so it is simply the first ES. In the former case, we use the method of [22], and in the latter the multi-target DMRG [31], to compute the relative REE's, shown in Figs 2(e), (f), (g), (h). Up to oscillations (and the BE in the N N case), we find excellent agreement with Eq. (12).\n\nFinally, we study the N DBC case, whose partition function [12] cannot be written in terms of the characters (14). However, it can be shown that Z N D (q) = χ 1/16 (q)[χ 0 (q) + χ 1/2 (q)],\n\nwhere χ 0,1/2,1/16 are the characters of the primary fields of the Ising model. Equation (18) implies that the operator content corresponding to N DBC is given by the tensor product of two Ising models. In particular, the GS is associated with the operator I ⊗ σ, resulting in the correction F (n) σ , and the first ES with σ ⊗ χ, resulting in the correction F (n) σ F (n) χ (plus a BE contribution − 1 2 ln 2): we show in Fig. 3 the results for the n = 2, 3, obtained with multi-target DMRG, finding, even in this case, a remarkable agreement between CFT and numerics.\n\nWe conclude that for the critical Ising and XX quantum spin chains under any conformal BC's, the REE's of the low-lying states are obtained, apart from the BE's and the oscillations, from the correlators of the primary operators of the underlying CFT: the finite-size behavior \n\n\nPACS numbers: 03.67.-a, 03.67.Mn, 64.60.an, 11.25.Hf\n\n\n(2), and finite-size scaling (FSS) techniques, we obtain the asymptotic values c = 0.499 and c η=2 1 = 0.241. The value of c agrees to high precision with the central charge of the critical Ising model. Moreover, the value of c η=2 1 is very close to c η=1 1 /2, that can be obtained with PBC. The relation c η=2\n\n\nhas a general expression valid for any value of n > 0[27] \n\n\n. 1(c) ,(d) show the convergence of the numerical results to the CFT prediction obtained with (12), for n = 1, 2.\n\nFIG. 1 :\n1REE's in the 1D critical Ising model. Panels (a)-(h): black to turquoise (dotted lines): L = 60 to 180 numerical data; purple (solid line) + arrow: CFT predictions (the BE's are added to the CFT formulas when necessary); (i), (j): FSS relative to panels (a) and (e) (best fits use the 5-parameters formula y = a0 +a1x a 2 +a3x a 4 ; dots are numerical data, solid lines best fits and squares the CFT predictions).\n\nFIG. 2 :\n2REE's in the model (15) (I). Black dotted lines: L = 100 numerical data; red solid line: CFT predictions (the BE's are added to the CFT formulas when necessary).\n\nFIG. 3 :\n3REE's in the model (15) (II). See the caption of figure 2. of the REE's of quantum chains with open boundaries identify the primary operators in the CFT, providing, in principle, a tool for deepening the understanding of the operator content of a CFT.\n\n. P Calabrese, J Cardy, B Doyon, J. Phys. A.: Math. Theor. 42500301P. Calabrese, J. Cardy, B. Doyon, J. Phys. A.: Math. Theor. 42, 500301 (2009).\n\n. J Eisert, M Cramer, M B Plenio, Rev. Mod. Phys. 82277J. Eisert, M. Cramer, M. B. Plenio, Rev. Mod. Phys. 82, 277 (2010).\n\n. A A Belavin, A M Polyakov, A B Zamolodchikov, Nuclear Physics. 241333A. A. Belavin, A. M. Polyakov, A. B. Zamolodchikov, Nuclear Physics B241, 333 (1984).\n\nP Di Francesco, P Mathieu, D Sénéchal, Conformal Field Theory. SpringerP. Di Francesco, P. Mathieu, D. Sénéchal, Conformal Field Theory, Springer (1997).\n\n. C Holzhey, F Larsen, F Wilczek, Nucl. Phys. B. 424443C. Holzhey, F. Larsen, F. Wilczek, Nucl. Phys. B 424, 443 (1994).\n\n. G Vidal, Phys. Rev. Lett. 90227902G. Vidal et al., Phys. Rev. Lett. 90 227902 (2003).\n\n. P Calabrese, J Cardy, J. Stat. Mech. 42504005J. Phys. AP. Calabrese, J. Cardy, J. Stat. Mech. (2004), P06002; J. Phys. A 42, 504005 (2009).\n\n. H W J Blöte, J L Cardy, M M Nightingale, Phys. Rev. Lett. 56742H. W. J. Blöte, J. L. Cardy, M. M. Nightingale, Phys. Rev. Lett. 56 742 (1986);\n\n. I Affleck, Phys. Rev. Lett. 56746I. Affleck, Phys. Rev. Lett. 56 746 (1986).\n\n. F C Alcaraz, M I Berganza, G Sierra, Phys. Rev. Lett. 106201601F. C. Alcaraz, M. I. Berganza, G. Sierra, Phys. Rev. Lett. 106, 201601 (2011).\n\n. M I Berganza, F C Alcaraz, G Sierra, J. Stat. Mech. 1016M. I. Berganza, F. C. Alcaraz, G. Sierra, J. Stat. Mech. (2012), P01016.\n\n. J L Cardy, Nucl. Phys. 275200J. L. Cardy, Nucl. Phys. B275, 200 (1986).\n\nTopological aspects of low dimensional systems. arXiv:cond-mat/9812110H. Saleur in A. Comtet et al.SpringerH. Saleur in A. Comtet et al. (ed.), Topological aspects of low dimensional systems, Springer (1998); arXiv:cond- mat/9812110.\n\nJ Polchinski, String Theory Vols. I. New YorkCambridge Univ. PressIIJ. Polchinski, String Theory Vols. I, II, Cambridge Univ. Press, New York (1998).\n\n. H.-Q Zhou, Phys. Rev. A. 7450305H.-Q. Zhou et al., Phys. Rev. A 74, 050305 (2006).\n\n. I Affleck, N Laflorencie, E S Sørensen, J. Phys. A: Math. Theor. 42504009I. Affleck, N. Laflorencie, E. S. Sørensen, J. Phys. A: Math. Theor. 42, 504009 (2009).\n\n. J C Xavier, F C Alcaraz, Phys. Rev. B. 8524418J. C. Xavier, F. C. Alcaraz, Phys. Rev. B 85, 024418 (2012);\n\n. J C Xavier, F C Alcaraz, Phys. Rev. B. 8494410J. C. Xavier, F. C. Alcaraz, Phys. Rev. B 84, 094410 (2011).\n\n. N Laflorencie, Phys. Rev. Lett. 96100603N. Laflorencie et al., Phys. Rev. Lett. 96, 100603 (2006).\n\n. P Calabrese, Phys. Rev. Lett. 10495701P. Calabrese et al., Phys. Rev. Lett. 104, 095701 (2010).\n\n. M Dalmonte, E Ercolessi, L Taddia, Phys. Rev. B. 85165112M. Dalmonte, E. Ercolessi, L. Taddia, Phys. Rev. B 85, 165112 (2012).\n\n. D Eloy, J C Xavier, Phys. Rev. B. 8664421D. Eloy, J. C. Xavier, Phys. Rev. B 86, 064421 (2012).\n\n. E Lieb, T Schultz, D Mattis, Ann. Phys. 16407E. Lieb, T. Schultz, D. Mattis, Ann. Phys. 16, 407 (1961).\n\n. I Peschel, J. Phys. A: Math. Gen. 36205I. Peschel, J. Phys. A: Math. Gen. 36 (2003), L205.\n\n. I Affleck, A W W Ludwig, Phys. Rev. Lett. 67161I. Affleck, A. W. W. Ludwig, Phys. Rev. Lett. 67, 161 (1991).\n\n. F Iglói, R Juhász, EPL. 8157003F. Iglói, R. Juhász, EPL 81, 57003 (2008).\n\n. S R White, Phys. Rev. Lett. 692863S. R. White, Phys. Rev. Lett. 69, 2863 (1992);\n\n. Phys. Rev. B. 4810345Phys. Rev. B 48, 10345 (1993).\n\n. J L Cardy, Nucl. Phys. 324581J. L. Cardy, Nucl. Phys. B324, 581 (1989).\n\n. F H L Essler, A M Läuchli, P Calabrese, arXiv:1211.2474F. H. L. Essler, A. M. Läuchli, P. Calabrese, arXiv:1211.2474.\n\n. E Ardonne, G Sierra, J. Phys. A: Math. Theor. 43505402E. Ardonne, G. Sierra, J. Phys. A: Math. Theor. 43 505402 (2010).\n\n. U Bilstein, J. Phys. A: Math. Gen. 334437U. Bilstein, J. Phys. A: Math. Gen. 33, 4437 (2000).\n\n. M Dalmonte, E Ercolessi, L Taddia, Phys. Rev. B. 8485110M. Dalmonte, E. Ercolessi, L. Taddia, Phys. Rev. B 84, 085110 (2011).\n\n. C. Degli Esposti Boschi, F Ortolani, Eur , J. Phys. B. 41503C. Degli Esposti Boschi, F. Ortolani, Eur. J. Phys. B 41, 503 (2004).\n"},"corpus_id":{"kind":"number","value":53462122,"string":"53,462,122"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1302.6222\",\n \"mag\": 2327779959,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1103/physrevb.88.075112\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1302.6222v2.pdf\"\n ],\n \"pdf_sha\": \"dfe1c34eb6db35475e00401506c4418b1ba62c73\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1302.6222\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Entanglement Entropies in Conformal Systems with Boundaries"},"authors":{"kind":"list like","value":["L Taddia \nDipartimento di Fisica e Astronomia dell'Università di Bologna and INFN\nvia Irnerio 4640126BolognaItaly\n\nInstituto de Física Teórica UAM/CSIC\nUniversidad Autónoma de Madrid\n28049Cantoblanco, MadridSpain","J C Xavier \nInstituto de Física\nUniversidade Federal de Uberlândia\nCaixa Postal 59338400-902UberlândiaMGBrazil","F C Alcaraz \nInstituto de Física de São Carlos\nUniversidade de São Paulo\nCaixa Postal 369São CarlosSPBrazil","G Sierra \nInstituto de Física Teórica UAM/CSIC\nUniversidad Autónoma de Madrid\n28049Cantoblanco, MadridSpain"],"string":"[\n \"L Taddia \\nDipartimento di Fisica e Astronomia dell'Università di Bologna and INFN\\nvia Irnerio 4640126BolognaItaly\\n\\nInstituto de Física Teórica UAM/CSIC\\nUniversidad Autónoma de Madrid\\n28049Cantoblanco, MadridSpain\",\n \"J C Xavier \\nInstituto de Física\\nUniversidade Federal de Uberlândia\\nCaixa Postal 59338400-902UberlândiaMGBrazil\",\n \"F C Alcaraz \\nInstituto de Física de São Carlos\\nUniversidade de São Paulo\\nCaixa Postal 369São CarlosSPBrazil\",\n \"G Sierra \\nInstituto de Física Teórica UAM/CSIC\\nUniversidad Autónoma de Madrid\\n28049Cantoblanco, MadridSpain\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We study the entanglement entropies in one-dimensional open critical systems, whose effective description is given by a conformal field theory with boundaries. We show that for pure-state systems formed by the ground state or by the excited states associated to primary fields, the entanglement entropies have a finite-size behavior that depends on the correlation of the underlying field theory. The analytical results are checked numerically, finding excellent agreement for the quantum chains ruled by the theories with central charge c = 1/2 and c = 1.PACS numbers: 03.67. 03.67.Mn, 64.60.an, 11.25.Hf In recent years there has been a flurry of activity devoted to characterize quantum many body systems and quantum field theories using the concept of entanglement (see[1]and references therein for a review). Under certain conditions, the low energy states of local Hamiltonians satisfy an entropic area law according to which the entanglement entropy (EE) of a subsystem is proportional to the area of its boundary[2]. In one spatial dimension, violations to this area law appear if the system is gapless and described by a conformal field theory (CFT)[3,4]: in this case, the ground-state (GS) EE grows with the log of the subsystem size[5][6][7]. On the other hand, it is well known that the dominant finite-size correction to the GS energy of critical systems is related to the central charge c [8]. A similar correspondence also exists between the central charge and the Rényi entanglement entropies (REE's) of the GS [5-7]. In the same vein, we would expect that the scaling dimensions of the primary operators are related with the REE's of the excited states (ES's), since the finite mass gaps of critical lattice Hamiltonians are also related with them[8]. Indeed, this latter correspondence was proven recently for conformal systems with periodic boundary conditions [9, 10]. The main result of these works is that the n th REE is given in terms of the 2n-point correlator of the corresponding primary field placed at special positions depending on the subsystem size.In this letter we shall generalize this result to open lattice Hamiltonians, effectively described by boundary conformal field theory (BCFT)[11]. BCFT has a wide range of applications to impurity problems [12], string theory [13], etc, which justifies the generalization pursued here. Moreover, the generalization is, as we shall see, non trivial, since BCFT has a very rich mathematical structure, which is worth to analyze from the entanglement point of view. Apart from these many reasons of interest, it is worth to cite that the problem is quite long-standing[14,15], and our work represents its final solution."},"sections":{"kind":"list like","value":[{"title":"FIG. 1 :","content":""},{"title":"FIG. 2 :","content":""},{"title":"FIG. 3 :","content":""}],"string":"[\n {\n \"title\": \"FIG. 1 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 2 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 3 :\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"PACS numbers: 03.67.-a, 03.67.Mn, 64.60.an, 11.25.Hf","content":"PACS numbers: 03.67.-a, 03.67.Mn, 64.60.an, 11.25.Hf"},{"id":"","caption":"(2), and finite-size scaling (FSS) techniques, we obtain the asymptotic values c = 0.499 and c η=2 1 = 0.241. The value of c agrees to high precision with the central charge of the critical Ising model. Moreover, the value of c η=2 1 is very close to c η=1 1 /2, that can be obtained with PBC. The relation c η=2","content":"(2), and finite-size scaling (FSS) techniques, we obtain the asymptotic values c = 0.499 and c η=2 1 = 0.241. The value of c agrees to high precision with the central charge of the critical Ising model. Moreover, the value of c η=2 1 is very close to c η=1 1 /2, that can be obtained with PBC. The relation c η=2"},{"id":"","caption":"has a general expression valid for any value of n > 0[27]","content":"has a general expression valid for any value of n > 0[27]"},{"id":"","caption":". 1(c) ,(d) show the convergence of the numerical results to the CFT prediction obtained with (12), for n = 1, 2.","content":". 1(c) ,(d) show the convergence of the numerical results to the CFT prediction obtained with (12), for n = 1, 2."},{"id":"","caption":"REE's in the 1D critical Ising model. Panels (a)-(h): black to turquoise (dotted lines): L = 60 to 180 numerical data; purple (solid line) + arrow: CFT predictions (the BE's are added to the CFT formulas when necessary); (i), (j): FSS relative to panels (a) and (e) (best fits use the 5-parameters formula y = a0 +a1x a 2 +a3x a 4 ; dots are numerical data, solid lines best fits and squares the CFT predictions).","content":"FIG. 1 :\n1REE's in the 1D critical Ising model. Panels (a)-(h): black to turquoise (dotted lines): L = 60 to 180 numerical data; purple (solid line) + arrow: CFT predictions (the BE's are added to the CFT formulas when necessary); (i), (j): FSS relative to panels (a) and (e) (best fits use the 5-parameters formula y = a0 +a1x a 2 +a3x a 4 ; dots are numerical data, solid lines best fits and squares the CFT predictions)."},{"id":"","caption":"REE's in the model (15) (I). Black dotted lines: L = 100 numerical data; red solid line: CFT predictions (the BE's are added to the CFT formulas when necessary).","content":"FIG. 2 :\n2REE's in the model (15) (I). Black dotted lines: L = 100 numerical data; red solid line: CFT predictions (the BE's are added to the CFT formulas when necessary)."},{"id":"","caption":"REE's in the model (15) (II). See the caption of figure 2. of the REE's of quantum chains with open boundaries identify the primary operators in the CFT, providing, in principle, a tool for deepening the understanding of the operator content of a CFT.","content":"FIG. 3 :\n3REE's in the model (15) (II). See the caption of figure 2. of the REE's of quantum chains with open boundaries identify the primary operators in the CFT, providing, in principle, a tool for deepening the understanding of the operator content of a CFT."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"PACS numbers: 03.67.-a, 03.67.Mn, 64.60.an, 11.25.Hf\",\n \"content\": \"PACS numbers: 03.67.-a, 03.67.Mn, 64.60.an, 11.25.Hf\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(2), and finite-size scaling (FSS) techniques, we obtain the asymptotic values c = 0.499 and c η=2 1 = 0.241. The value of c agrees to high precision with the central charge of the critical Ising model. Moreover, the value of c η=2 1 is very close to c η=1 1 /2, that can be obtained with PBC. The relation c η=2\",\n \"content\": \"(2), and finite-size scaling (FSS) techniques, we obtain the asymptotic values c = 0.499 and c η=2 1 = 0.241. The value of c agrees to high precision with the central charge of the critical Ising model. Moreover, the value of c η=2 1 is very close to c η=1 1 /2, that can be obtained with PBC. The relation c η=2\"\n },\n {\n \"id\": \"\",\n \"caption\": \"has a general expression valid for any value of n > 0[27]\",\n \"content\": \"has a general expression valid for any value of n > 0[27]\"\n },\n {\n \"id\": \"\",\n \"caption\": \". 1(c) ,(d) show the convergence of the numerical results to the CFT prediction obtained with (12), for n = 1, 2.\",\n \"content\": \". 1(c) ,(d) show the convergence of the numerical results to the CFT prediction obtained with (12), for n = 1, 2.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"REE's in the 1D critical Ising model. Panels (a)-(h): black to turquoise (dotted lines): L = 60 to 180 numerical data; purple (solid line) + arrow: CFT predictions (the BE's are added to the CFT formulas when necessary); (i), (j): FSS relative to panels (a) and (e) (best fits use the 5-parameters formula y = a0 +a1x a 2 +a3x a 4 ; dots are numerical data, solid lines best fits and squares the CFT predictions).\",\n \"content\": \"FIG. 1 :\\n1REE's in the 1D critical Ising model. Panels (a)-(h): black to turquoise (dotted lines): L = 60 to 180 numerical data; purple (solid line) + arrow: CFT predictions (the BE's are added to the CFT formulas when necessary); (i), (j): FSS relative to panels (a) and (e) (best fits use the 5-parameters formula y = a0 +a1x a 2 +a3x a 4 ; dots are numerical data, solid lines best fits and squares the CFT predictions).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"REE's in the model (15) (I). Black dotted lines: L = 100 numerical data; red solid line: CFT predictions (the BE's are added to the CFT formulas when necessary).\",\n \"content\": \"FIG. 2 :\\n2REE's in the model (15) (I). Black dotted lines: L = 100 numerical data; red solid line: CFT predictions (the BE's are added to the CFT formulas when necessary).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"REE's in the model (15) (II). See the caption of figure 2. of the REE's of quantum chains with open boundaries identify the primary operators in the CFT, providing, in principle, a tool for deepening the understanding of the operator content of a CFT.\",\n \"content\": \"FIG. 3 :\\n3REE's in the model (15) (II). See the caption of figure 2. of the REE's of quantum chains with open boundaries identify the primary operators in the CFT, providing, in principle, a tool for deepening the understanding of the operator content of a CFT.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". P Calabrese, J Cardy, B Doyon, J. Phys. A.: Math. Theor. 42500301P. Calabrese, J. Cardy, B. Doyon, J. Phys. A.: Math. Theor. 42, 500301 (2009).","doi":""},{"text":". J Eisert, M Cramer, M B Plenio, Rev. Mod. Phys. 82277J. Eisert, M. Cramer, M. B. Plenio, Rev. Mod. Phys. 82, 277 (2010).","doi":""},{"text":". A A Belavin, A M Polyakov, A B Zamolodchikov, Nuclear Physics. 241333A. A. Belavin, A. M. Polyakov, A. B. Zamolodchikov, Nuclear Physics B241, 333 (1984).","doi":""},{"text":"P Di Francesco, P Mathieu, D Sénéchal, Conformal Field Theory. SpringerP. Di Francesco, P. Mathieu, D. Sénéchal, Conformal Field Theory, Springer (1997).","doi":""},{"text":". C Holzhey, F Larsen, F Wilczek, Nucl. Phys. B. 424443C. Holzhey, F. Larsen, F. Wilczek, Nucl. Phys. B 424, 443 (1994).","doi":""},{"text":". G Vidal, Phys. Rev. Lett. 90227902G. Vidal et al., Phys. Rev. Lett. 90 227902 (2003).","doi":""},{"text":". P Calabrese, J Cardy, J. Stat. Mech. 42504005J. Phys. AP. Calabrese, J. Cardy, J. Stat. Mech. (2004), P06002; J. Phys. A 42, 504005 (2009).","doi":""},{"text":". H W J Blöte, J L Cardy, M M Nightingale, Phys. Rev. Lett. 56742H. W. J. Blöte, J. L. Cardy, M. M. Nightingale, Phys. Rev. Lett. 56 742 (1986);","doi":""},{"text":". I Affleck, Phys. Rev. Lett. 56746I. Affleck, Phys. Rev. Lett. 56 746 (1986).","doi":""},{"text":". F C Alcaraz, M I Berganza, G Sierra, Phys. Rev. Lett. 106201601F. C. Alcaraz, M. I. Berganza, G. Sierra, Phys. Rev. Lett. 106, 201601 (2011).","doi":""},{"text":". M I Berganza, F C Alcaraz, G Sierra, J. Stat. Mech. 1016M. I. Berganza, F. C. Alcaraz, G. Sierra, J. Stat. Mech. (2012), P01016.","doi":""},{"text":". J L Cardy, Nucl. Phys. 275200J. L. Cardy, Nucl. Phys. B275, 200 (1986).","doi":""},{"text":"Topological aspects of low dimensional systems. arXiv:cond-mat/9812110H. Saleur in A. Comtet et al.SpringerH. Saleur in A. Comtet et al. (ed.), Topological aspects of low dimensional systems, Springer (1998); arXiv:cond- mat/9812110.","doi":""},{"text":"J Polchinski, String Theory Vols. I. New YorkCambridge Univ. PressIIJ. Polchinski, String Theory Vols. I, II, Cambridge Univ. Press, New York (1998).","doi":""},{"text":". H.-Q Zhou, Phys. Rev. A. 7450305H.-Q. Zhou et al., Phys. Rev. A 74, 050305 (2006).","doi":""},{"text":". I Affleck, N Laflorencie, E S Sørensen, J. Phys. A: Math. Theor. 42504009I. Affleck, N. Laflorencie, E. S. Sørensen, J. Phys. A: Math. Theor. 42, 504009 (2009).","doi":""},{"text":". J C Xavier, F C Alcaraz, Phys. Rev. B. 8524418J. C. Xavier, F. C. Alcaraz, Phys. Rev. B 85, 024418 (2012);","doi":""},{"text":". J C Xavier, F C Alcaraz, Phys. Rev. B. 8494410J. C. Xavier, F. C. Alcaraz, Phys. Rev. B 84, 094410 (2011).","doi":""},{"text":". N Laflorencie, Phys. Rev. Lett. 96100603N. Laflorencie et al., Phys. Rev. Lett. 96, 100603 (2006).","doi":""},{"text":". P Calabrese, Phys. Rev. Lett. 10495701P. Calabrese et al., Phys. Rev. Lett. 104, 095701 (2010).","doi":""},{"text":". M Dalmonte, E Ercolessi, L Taddia, Phys. Rev. B. 85165112M. Dalmonte, E. Ercolessi, L. Taddia, Phys. Rev. B 85, 165112 (2012).","doi":""},{"text":". D Eloy, J C Xavier, Phys. Rev. B. 8664421D. Eloy, J. C. Xavier, Phys. Rev. B 86, 064421 (2012).","doi":""},{"text":". E Lieb, T Schultz, D Mattis, Ann. Phys. 16407E. Lieb, T. Schultz, D. Mattis, Ann. Phys. 16, 407 (1961).","doi":""},{"text":". I Peschel, J. Phys. A: Math. Gen. 36205I. Peschel, J. Phys. A: Math. Gen. 36 (2003), L205.","doi":""},{"text":". I Affleck, A W W Ludwig, Phys. Rev. Lett. 67161I. Affleck, A. W. W. Ludwig, Phys. Rev. Lett. 67, 161 (1991).","doi":""},{"text":". F Iglói, R Juhász, EPL. 8157003F. Iglói, R. Juhász, EPL 81, 57003 (2008).","doi":""},{"text":". S R White, Phys. Rev. Lett. 692863S. R. White, Phys. Rev. Lett. 69, 2863 (1992);","doi":""},{"text":". Phys. Rev. B. 4810345Phys. Rev. B 48, 10345 (1993).","doi":""},{"text":". J L Cardy, Nucl. Phys. 324581J. L. Cardy, Nucl. Phys. B324, 581 (1989).","doi":""},{"text":". F H L Essler, A M Läuchli, P Calabrese, arXiv:1211.2474F. H. L. Essler, A. M. Läuchli, P. Calabrese, arXiv:1211.2474.","doi":""},{"text":". E Ardonne, G Sierra, J. Phys. A: Math. Theor. 43505402E. Ardonne, G. Sierra, J. Phys. A: Math. Theor. 43 505402 (2010).","doi":""},{"text":". U Bilstein, J. Phys. A: Math. Gen. 334437U. Bilstein, J. Phys. A: Math. Gen. 33, 4437 (2000).","doi":""},{"text":". M Dalmonte, E Ercolessi, L Taddia, Phys. Rev. B. 8485110M. Dalmonte, E. Ercolessi, L. Taddia, Phys. Rev. B 84, 085110 (2011).","doi":""},{"text":". C. Degli Esposti Boschi, F Ortolani, Eur , J. Phys. B. 41503C. Degli Esposti Boschi, F. Ortolani, Eur. J. Phys. B 41, 503 (2004).","doi":""}],"string":"[\n {\n \"text\": \". P Calabrese, J Cardy, B Doyon, J. Phys. A.: Math. Theor. 42500301P. Calabrese, J. Cardy, B. Doyon, J. Phys. A.: Math. Theor. 42, 500301 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Eisert, M Cramer, M B Plenio, Rev. Mod. Phys. 82277J. Eisert, M. Cramer, M. B. Plenio, Rev. Mod. Phys. 82, 277 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A A Belavin, A M Polyakov, A B Zamolodchikov, Nuclear Physics. 241333A. A. Belavin, A. M. Polyakov, A. B. Zamolodchikov, Nuclear Physics B241, 333 (1984).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"P Di Francesco, P Mathieu, D Sénéchal, Conformal Field Theory. SpringerP. Di Francesco, P. Mathieu, D. Sénéchal, Conformal Field Theory, Springer (1997).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Holzhey, F Larsen, F Wilczek, Nucl. Phys. B. 424443C. Holzhey, F. Larsen, F. Wilczek, Nucl. Phys. B 424, 443 (1994).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Vidal, Phys. Rev. Lett. 90227902G. Vidal et al., Phys. Rev. Lett. 90 227902 (2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Calabrese, J Cardy, J. Stat. Mech. 42504005J. Phys. AP. Calabrese, J. Cardy, J. Stat. Mech. (2004), P06002; J. Phys. A 42, 504005 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H W J Blöte, J L Cardy, M M Nightingale, Phys. Rev. Lett. 56742H. W. J. Blöte, J. L. Cardy, M. M. Nightingale, Phys. Rev. Lett. 56 742 (1986);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I Affleck, Phys. Rev. Lett. 56746I. Affleck, Phys. Rev. Lett. 56 746 (1986).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F C Alcaraz, M I Berganza, G Sierra, Phys. Rev. Lett. 106201601F. C. Alcaraz, M. I. Berganza, G. Sierra, Phys. Rev. Lett. 106, 201601 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M I Berganza, F C Alcaraz, G Sierra, J. Stat. Mech. 1016M. I. Berganza, F. C. Alcaraz, G. Sierra, J. Stat. Mech. (2012), P01016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J L Cardy, Nucl. Phys. 275200J. L. Cardy, Nucl. Phys. B275, 200 (1986).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Topological aspects of low dimensional systems. arXiv:cond-mat/9812110H. Saleur in A. Comtet et al.SpringerH. Saleur in A. Comtet et al. (ed.), Topological aspects of low dimensional systems, Springer (1998); arXiv:cond- mat/9812110.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J Polchinski, String Theory Vols. I. New YorkCambridge Univ. PressIIJ. Polchinski, String Theory Vols. I, II, Cambridge Univ. Press, New York (1998).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H.-Q Zhou, Phys. Rev. A. 7450305H.-Q. Zhou et al., Phys. Rev. A 74, 050305 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I Affleck, N Laflorencie, E S Sørensen, J. Phys. A: Math. Theor. 42504009I. Affleck, N. Laflorencie, E. S. Sørensen, J. Phys. A: Math. Theor. 42, 504009 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J C Xavier, F C Alcaraz, Phys. Rev. B. 8524418J. C. Xavier, F. C. Alcaraz, Phys. Rev. B 85, 024418 (2012);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J C Xavier, F C Alcaraz, Phys. Rev. B. 8494410J. C. Xavier, F. C. Alcaraz, Phys. Rev. B 84, 094410 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Laflorencie, Phys. Rev. Lett. 96100603N. Laflorencie et al., Phys. Rev. Lett. 96, 100603 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Calabrese, Phys. Rev. Lett. 10495701P. Calabrese et al., Phys. Rev. Lett. 104, 095701 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Dalmonte, E Ercolessi, L Taddia, Phys. Rev. B. 85165112M. Dalmonte, E. Ercolessi, L. Taddia, Phys. Rev. B 85, 165112 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Eloy, J C Xavier, Phys. Rev. B. 8664421D. Eloy, J. C. Xavier, Phys. Rev. B 86, 064421 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Lieb, T Schultz, D Mattis, Ann. Phys. 16407E. Lieb, T. Schultz, D. Mattis, Ann. Phys. 16, 407 (1961).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I Peschel, J. Phys. A: Math. Gen. 36205I. Peschel, J. Phys. A: Math. Gen. 36 (2003), L205.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I Affleck, A W W Ludwig, Phys. Rev. Lett. 67161I. Affleck, A. W. W. Ludwig, Phys. Rev. Lett. 67, 161 (1991).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Iglói, R Juhász, EPL. 8157003F. Iglói, R. Juhász, EPL 81, 57003 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S R White, Phys. Rev. Lett. 692863S. R. White, Phys. Rev. Lett. 69, 2863 (1992);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Phys. Rev. B. 4810345Phys. Rev. B 48, 10345 (1993).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J L Cardy, Nucl. Phys. 324581J. L. Cardy, Nucl. Phys. B324, 581 (1989).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F H L Essler, A M Läuchli, P Calabrese, arXiv:1211.2474F. H. L. Essler, A. M. Läuchli, P. Calabrese, arXiv:1211.2474.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Ardonne, G Sierra, J. Phys. A: Math. Theor. 43505402E. Ardonne, G. Sierra, J. Phys. A: Math. Theor. 43 505402 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". U Bilstein, J. Phys. A: Math. Gen. 334437U. Bilstein, J. Phys. A: Math. Gen. 33, 4437 (2000).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Dalmonte, E Ercolessi, L Taddia, Phys. Rev. B. 8485110M. Dalmonte, E. Ercolessi, L. Taddia, Phys. Rev. B 84, 085110 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C. Degli Esposti Boschi, F Ortolani, Eur , J. Phys. B. 41503C. Degli Esposti Boschi, F. Ortolani, Eur. J. Phys. B 41, 503 (2004).\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":485,"cells":{"text":{"kind":"string","value":"\nVisual Attention-based Self-supervised Absolute Depth Estimation using Geometric Priors in Autonomous Driving\n\n\nJie Xiang \nYun Wang \nLifeng An \nHaiyang Liu \nZijun Wang \nJian Liu \nVisual Attention-based Self-supervised Absolute Depth Estimation using Geometric Priors in Autonomous Driving\n10.1109/LRA.2022.3210298\nAlthough existing monocular depth estimation methods have made great progress, predicting an accurate absolute depth map from a single image is still challenging due to the limited modeling capacity of networks and the scale ambiguity issue. In this paper, we introduce a fully Visual Attentionbased Depth (VADepth) network, where spatial attention and channel attention are applied to all stages. By continuously extracting the dependencies of features along the spatial and channel dimensions over a long distance, VADepth network can effectively preserve important details and suppress interfering features to better perceive the scene structure for more accurate depth estimates. In addition, we utilize geometric priors to form scale constraints for scale-aware model training. Specifically, we construct a novel scale-aware loss using the distance between the camera and a plane fitted by the ground points corresponding to the pixels of the rectangular area in the bottom middle of the image. Experimental results on the KITTI dataset show that this architecture achieves the state-of-the-art performance and our method can directly output absolute depth without post-processing. Moreover, our experiments on the SeasonDepth dataset also demonstrate the robustness of our model to multiple unseen environments.Index Terms-Deep learning for visual perception, computer vision for transportation, range sensing.\n\nI. INTRODUCTION\n\nM ONOCULAR Depth Estimation (MDE) is a fundamental problem in the field of computer vision, which refers to predicting the corresponding depth map from a single image. Depth Estimation has broad application prospects in autonomous driving, such as 3D object detection [1], scene understanding [2], and obstacle avoidance [3]. Compared to other ways of measuring depth [4], [5], monocular depth estimation has unique advantages in obtaining dense depth maps at low cost. Therefore, MDE has aroused the interest of many researchers. In the past few years, many deep learning-based MDE methods [6]- [23] have emerged and continual improvements [7]- [23] have been made. These methods can be divided into supervised methods [6]- [9] and self-supervised methods [10]- [23]. Unlike supervised methods, self-supervised methods do not need to rely on ground truth depth for training. Furthermore, existing self-supervised MDE methods can be divided into two categories: self-supervised stereo training methods [10]- [12] and monocular training methods [13]- [23]. The former use stereo image pairs while the latter use monocular videos as training data. In contrast to stereo training, monocular training is a more general form of selfsupervised methods and easier to obtain training data.\n\nHowever, existing self-supervised monocular training methods still have difficulties in predicting accurate metric depth. This unsatisfied performance may come from two reasons: the limited modeling power of the model and the limited supervision of loss functions. Many existing self-supervised depth estimation networks [16]- [18], [23], [24] adopt residual networks [25] based on an encoder-decoder architecture [26]. Such a depth network shows a powerful capacity for capturing local information and representing hierarchical abstract features, but is insufficient to distinguish important details from interfering information. Due to these characteristics of the model, it is difficult for these methods to predict accurate depth estimation results on patterned surfaces or tiny structures. Thus, we introduce a fully Visual Attention-based Depth (VADepth) network. Specifically, we build a decoder based on the designed visual attention blocks and use a visual attention network followed by a dual attention module as the encoder. Benefiting from the ability to dynamically handle long-range dependencies brought by combining channel attention and spatial attention at all stages of the encoder-decoder architecture, our model not only continuously preserves important details but also effectively filters out noise.\n\nOn the other hand, most depth networks [13]- [22] trained on monocular videos only output relative depth due to lacking scale constraints. The commonly used scale recovery method called median scaling [13] requires ground truth depth data that is unavailable at test time in many practical applications. Training a model that directly produces absolute depth in metric units can omit the scale recovery operation like median scaling. An intuitive idea to this end is to use object size [27] but it suffers from seeking ubiquitous fixed-size objects and detecting the appearance size of detected objects. A simple but effective approach is to use camera height with respect 1 arXiv:2205.08780v3 [cs.CV] 6 Oct 2022 2 to ground surface because the ground truth camera height in autonomous driving scenarios is usually fixed and known in advance. Furthermore, we also observe that the middle bottom regions of almost all images captured in these scenarios belong to the ground surface. Considering that not all points on the ground surface are coplanar, we propose to detect all coplanar points on the ground surface from the estimated depth, based on the assumption that a small rectangular region in the middle bottom of any image is part of the ground surface. Using the height errors calculated based on detection results, we construct an absolute scale loss. Combining the scale loss with photometric reconstruction loss [24], the visual attention-based depth network is able to output accurate absolute depth map without post-processing.\n\nIn summary, we make the contributions as follows:\n\n• A novel depth network architecture that fully utilizes the long-range perception properties of visual attention to better perceive the scene structure and predict more accurate depth maps. • An absolute scale loss that leverages the geometric priors in autonomous driving scenarios to train a scale-aware depth estimation model. • We conduct extensive experiments to verify the effectiveness of our network structure and loss function. Experimental results [28] show that our method outperforms previous state-of-the-art methods on the KITTI dataset and our model generalizes well on the unseen Season-Depth dataset [29] that contains multiple challenging environments without fine-tuning.\n\n\nII. RELATED WORK\n\n\nA. Monocular Depth Estimation\n\nEarly deep learning-based works [6], [7] on monocular depth estimation mainly focused on supervised methods and dramatically improved the state-of-the-art performance. However, ground truth depth maps for supervised methods are hard to obtain. To avoid the heavy work of collecting ground truth depth, Garg et al. [10] proposed a self-supervised method inspired by view synthesis task [30], which leverages stereo image pairs to construct the reconstruction loss. Compared to stereo image pairs, monocular videos are much easier to obtain. Therefore, SfMLearner [13] leveraged monocular videos to train MDE models. Unlike stereo training approaches [10]- [12], monocular training methods [13], [16] need to simultaneously estimate ego-motion and depth at training time.\n\nRecently, monocular training methods have been extensively studied. Some works [14], [15], [17], [20], [24], [31] explored to design more effective loss functions based on the photometric reconstruction loss. For example, edge-aware depthnormal consistency loss was used in [14], [15]. Godard et al. [24] proposed a per-pixel minimum reprojection loss to solve the occlusion problem, a multi-scale loss for alleviating the local gradient problem, and the auto-masking method to filter out stationary pixels. Based on Mondepth2 [24], the use of feature-metric loss [31], temporal geometric consistency [17] and semantic-depth consistency [20] further improved monocular depth estimation accuracy. Our method also follows the general pipeline of Monodepth2 [24].\n\nBesides, some works [22], [32] proposed more powerful convolutional networks to enhance the accuracy of depth estimation. Beyond ResNet-based U-Net, Packnet-sfm [22] leveraged 3D convolutions to improve representational capacity of the network. HR-Depth [32] introduced more dense skipconnections to better fuse multi-scale feature maps. Because of the improvement of loss functions and network architectures, the performance of self-supervised monocular depth estimation has been improved a lot. However, these methods still face challenges in predicting metrically accurate depth due to the scale ambiguity issue and lacking the ability to distinguish important features from noise.\n\n\nB. Scale Ambiguity\n\nBecause of lacking scale constraints, monocular vision suffers from the scale ambiguity issue. In order to produce absolute depth at test time, some MDE works [13]- [22] leveraged post-processing for scale recovery. The typical postprocessing technique of median scaling proposed in [13] scales the relative depth map by the ratio of the median of the ground truth depth values to the median of the estimated depth values. Median scaling has been adopted by many subsequent works [17]- [22], [24], [31]. Nevertheless, ground truth depth maps required by median scaling are not always available at test time in many cases. As the use of camera height in classical monocular visual odometry [33], DNet [16] leveraged a dense geometrical constraints module to determine the scale factor by the estimated camera height from every ground point.\n\nCompared to post-processing techniques, training a scaleaware model is more attractive due to its simplicity of testing. To learn metrically accurate depth estimation, [34] pretrained depth network on stereo data of one dataset and then finetuned the model on monocular videos of another dataset to preserve the absolute scale, and PackNet-sfm [22] leveraged the camera's velocity and [35] utilized 4-beam Lidar data to construct scale-aware loss. All of these works require additional sensor data for training. In order to avoid using other sensor data, [23] proposed a camera height-based loss function to enforce metrically-scaled depth during training. Similar to [23], we also leverage camera height to construct scale constraints. However, our method directly determines the pixels corresponding to the ground plane based on predicted depth instead of relying on an extra pretrained ground plane segmentation model as in [23]. Thus, we simplify the training procedure and improve depth estimation accuracy by mitigating the effects of the limited accuracy of the ground plane segmentation model and the uneven ground surface.\n\n\nC. Attention Mechanism\n\nAttention mechanism is a dynamic process that diverts attention to important features [36]. The pioneering work of visual attention is the spatial attention network called RAM [37]. After that, more attention mechanisms [38]- [41] were proposed, such as self-attention [42] and channel attention [38]. Nowadays, attention mechanisms have been applied in 3 many visual tasks [37]- [41], [43], [44]. As for monocular depth estimation, the attention-based network has been applied not only to supervised methods [48], [49] but also to selfsupervised methods [19]- [21]. In [19], an attention module was used to better perceive contextual information. FSRE-Depth [20] utilized an attention module to fuse depth features and semantic features. CADepth-Net [21] applied channel attention for structure perception as well as skip-connection. However, these networks [48], [49], [19]- [21] just added several attention modules to the original network, and the body of the depth network is still the usual localized convolutional network. Therefore, we introduce the visual attention mechanism throughout the processing of the network, so that our model can continuously and adaptively extract important features and remove noise interference at all stages for more accurate depth predictions.\n\n\nIII. METHOD\n\nIn this section, we first introduce the basic paradigm for self-supervised monocular depth estimation. Then, we describe the details of our VADepth network. Finally, we introduce a simple method for detecting coplanar points and construct a scale loss based on the planar geometric prior.\n\n\nA. Self-Supervised Monocular Depth Estimation\n\nDuring the self-supervised MDE model training, the model is optimized by minimizing the difference between the target image and the image synthesized by the source image and the predicted target depth map. In the training process using monocular videos, the source frame is chosen from the same monocular image sequence as the target frame. Following [13], the source frame I s is an adjacent frame to the target frame I t , i.e. s = t ± 1. Since the relative pose between two adjacent frames is unknown, we need to estimate not only the depth but also the ego-motion during training. The depth network takes an RGB image I t as input and outputs a depth map D t . Simultaneously, the pose network predicts the relative pose T t→s between I s and I t . Let K denote the known camera intrinsic matrix. Then we can project the target pixel coordinates (u, v) to the source view and obtain the 2D coordinates of the projected depths D t (u, v) in I s as follows:\n(u s , v s ) = KT t→s P t (u, v),(1)and P t (u, v) = D t (u, v)K −1 (u, v, 1) T .(2)\nHere, (u, v) and (u s , v s ) are the projected coordinates in I t and I s of the same 3D points P t (u, v), respectively. For simplify of notation, the conversions between inhomogeneous and homogeneous coordinates are omitted in (1). Because the calculated coordinates (u s , v s ) are not always integer values, we use bilinear interpolation to compute pixel values I s (u s , v s ) as in [45]. Thus, we can synthesize the target image: where · denotes bilinear sampling operation. Following [11], we combine the L1 loss and structural similarity index measure (SSIM) [46] to formulate the photometric error:\nI s→t (u, v) = I s (u s , v s ) ,(3)Encoder Stage 1 3 H W Encoder Stage 2 Encoder Stage 3 4 4 H W C 2 8 8 H W C 5 16 16 H W C DAM VAB 1 e X VAB VAB VAB 2 e X Encoder Stage 4 3 e X VAB 8 32 32 H W C 4 e X 0 D 1 D 2 D 3 D 4 d X 3 d X 2 d X 1 d X 0 d X Image DepthP E(I a , I b ) = α||I a − I b || 1 + (1 − α) 1 − SSIM (I a , I b ) 2 .(4)\nTo avoid the false photometric error of occluded pixels in the source frame, we apply per-pixel minimum photometric loss [24], i.e.\nL ph = min s∈{t−1,t+1} P E(I t , I s→t ).(5)\nBesides, we adopt the auto-masking method [24] to filter out pixels that do not change the appearance between I s and I t .\n\nThe binary mask µ is computed as\nµ = [min s P E(I t , I s→t ) < min s P E(I t , I s )],(6)\nwhere [·] is the Inverse bracket. Like [11], we also use the edge-aware smoothness loss:\nL sm = |∂ u d * t |e −|∂uIt| + |∂ v d * t |e −|∂vIt| ,(7)\nwhere d * t = d t /d t is the mean-normalized inverse depth. Considering the gradient locality of the bilinear sampler, we predict multi-scale depth maps in the decoder and compute the individual losses at each scale following [24]. Thus, the final baseline loss with a hyperparameter λ sm is defined as\nL baseline = 1 S S−1 i=0 µL ph + λ sm L sm ,(8)\nwhere S refers to the number of multi-scale depth maps.\n4 C Mean Max Conv 7×7 . C Conv 3×3 GAP . Upsample Sigmoid Conv 3×3 FC ReLU FC Conv 3×3 Sigmoid Sigmoid C 1 e i X 1 d i X d i X SAM CAM i D\n\nConcatenation\n\n. Element-wise Multiplication \n\n\nB. VADepth Network Architecture\n\nAs shown in Fig. 1, VADepth adopts the improved U-shape encoder-decoder architecture, which takes a single RGB image as input and outputs multi-scale depth maps. Unlike U-net architecture [26], the encoder and decoder in VADepth are not completely symmetrical, and an additional Dual Attention Module (DAM) head (see details in [40]) is used to connect the encoder and the decoder.\n\nFor the encoder, we employ a visual attention network (VAN) [44] to extract multi-scale feature maps X e i , i = 1, 2, 3, 4. A VAN has four stages, where spatial adaptability and channel adaptability are efficiently implemented by the large kernel attention.\n\nThen the internal features of the encoder are fed into the decoder with skip connections and the lowest resolution features are taken as the input of the DAM. The DAM adaptively selects the discriminative features as the initial input of the decoder by learning spatial and channel interdependencies of features in parallel.\n\nAs for the decoder, we design a novel Visual Attention Block (VAB). There are five successive VABs in the decoder. As illustrated in Fig. 2, the i-th VAB has two inputs, the internal features X e i−1 from the encoder and the features X d i from the previous VAB. X e i−a is fed into a Spatial Attention Module (SAM), which emphasizes the important regions and suppresses the unimportant or noisy regions as in [39]. In the SAM, the initial spatial attention maps are formed by max pooling and average pooling along the channel axis. Then the concatenated initial attention maps are passed through a convolutional layer with a sigmoid function to obtain the final spatial attention map. The output of the SAM is the element-wise multiplication of the broadcasted final attention map and X e i−1 . At the same time, X d i is accordingly passed through a convolutional layer and a nearest upsampling layer with a factor of 2 to recover the resolution. Then a Channel Attention Module (CAM) is used to concatenate and fuse the low-level feature from the SAM and the high-level feature from the upsampling operation. For the CAM, as in [38], the concatenated features are processed through the global average pooling layer, fully-connected (FC) layer with ReLU, and FC layer with sigmoid function to generate the channel attention map. Similar to the SAM, the output of the CAM is obtained by element-wise multiplication of the attention map and the original features. After the process of CAM, a convolutional layer is used to output X d i−1 . Finally, a convolutional layer with sigmoid is required to process X d i−1 when necessary, and the sigmoid output σ can be converted to the depth map D i with 1/(aσ + b), which is omitted in Fig. 2 for simplicity. Note that the SAM is removed from VAB when the internal feature from the encoder are unavailable.\n\n\nC. Co-planar Points Detection on Ground Surface\n\nThe depth estimation model obtained by minimizing the baseline loss defined in (8) can only directly output the relative depth. In order to train a depth estimation model that can output absolute depth maps, we need to utilize the geometric information that contains absolute scale. In most self-driving scenarios, the ubiquitous camera height can be considered as a known constant value. Therefore, the prior knowledge about camera height is suitable for constructing scale constraints by computing the camera height errors derived from the estimated depth.\n\nTo estimate the camera height from the predicted depth map, we need to detect the co-planar points on the ground surface. We observe that the middle-lower region of the image is part of the ground surface in almost all self-driving cases. Based on such geometric prior knowledge, we suppose that a small stationary rectangular region in image I t is the ground plane. We use two hyperparameters α u = 0.075 and α v = 0.875 to predefine an H × W binary mask M rect (as shown in Fig. 5) to distinguish whether a pixel is in the rectangular region.\nM rect (u, v) = 1 if |0.5 − u W | < α u and v H > α v , 0 others.(9)\nLet P rect ∈ R Nrect×3 be a matrix for a set of 3D points\n{P t (u, v)|M p (u, v) = 1} in the predefined rectangular region, where N rect = 2α u W (1 − α v )H and P t (u, v)\nis calculated from the estimated depth according to (2). Based on these 3D points, we can fit a plane P l rect that does not pass through the origin by solving the equation: P rect n = 1. We take the Moore-Penrose inverse of P rect to obtain the least-squares solution of the contradictory equations:\nn = P + rect 1.(10)\nAfter plane fitting, we need to detect all 3D points co-planar with the plane P l rect in the target view using a binary mask:\nM p (u, v) = 1 if |P t (u, v)n − 1| < δ, 0 others.(11)\nIn M p , an element whose value is equal to one indicates that the corresponding point is on the plane P l rect . The threshold 5 All models are tested with the resolution of 192 × 640 unless otherwise specified, for the maximum depth of 80m. † means the newer results from github with the resolution of 128 × 416. The best scores for each category are in bold. In the \"Train\" column, we list the supervision for each method with M -Self-supervised monocular supervision, Sem -Semantic supervision, v -velocity supervision. In the \"Scale Factor\" column, \"GT\" refers to determining a per-image scale factor using the ground truth depth for median scaling based scale recovery, \"Cam. Height\" indicates leveraging the known camera height w.r.t the ground surface to recover scaled depth, while \"None\" means test without any post-processing.\n\nδ is set to 0.01, which is used to judge whether the points are coplanar or not.\n\n\nD. Absolute Scale Loss\n\nAfter co-planar points detection on the ground surface, we construct an absolute scale loss term using the camera height. To form a scale constraint, we need to compute the camera height from the estimated depth results:\nH cam (u, v) = P t (u, v)n e ,(12)\nwhere n e = n ||n||1 is the normalized vector of n. Using the estimated heights H cam and the ground truth height h gt , we define the absolute scale loss as follows:\nL as = 1 ||M p || 1 ||M p • (H cam − h gt )|| 1 ,(13)\nwhere • is Hadamard product operator. Combining L as with L baseline , we have the final objective function:\nL = 1 S S−1 i=0 µL ph + λ sm L sm + λ as L as .(14)\nIn our setting, λ sm , λ as and S are set to 0.001, 0.01 and 4, respectively.\n\n\nIV. EXPERIMENTAL RESULTS\n\n\nA. Implementation details\n\nWe use a single Nvidia GeForce RTX 2080 Ti to implement our models in Pytorch. In our setting, we use the VADepth network as our depth network with an input/output resolution of 192 × 640 . For pose estimation, we use a modified ResNet18 [25] to predict a single 6-DoF ego-motion of two adjacent frames following [24]. The pose network adopts the same input resolution as the depth network. Both the encoder of the depth network and the encoder of the pose network are initialized with weights pretrained on ImageNet [47]. The depth network and pose network are jointly trained with Adam optimizer for 20 epochs. The learning rate is set to 5.0 × 10 −5 for the first 15 epochs and then drops by half for the remaining epochs. The source code and models are available at https://github.com/xjixzz/vadepth-net.\n\n\nB. Evaluation on KITTI\n\nKITTI dataset [28] is the commonly used dataset for autonomous driving. We use the Eigen split [6] of KITTI Stereo dataset to evaluate our method with the metrics proposed in [6]. Following [13], we also remove the static frames before training, which results in 39810 monocular triplets for training, 4424 images for validation and 697 images for test. The resolution of these images is approximately 375 × 1242. All images are required to resize to 192 × 640 for training and evaluation. At test time, the depth network only outputs the depth map with the resolution of 192 × 640, which is then resized to the full resolution of the original RGB image for evaluation.\n\nTo evaluate the performance of our depth network, we first train VADepth network only with the baseline loss function. We report the results of applying median scaling at the top of Table I, where we can see that our model achieves the best scores in the metrics of Abs Rel, RMSE log, δ 1 , and δ 2 , and ranks second or third in the remaining metrics. Among these methods, CADepth [21] achieves close performance to VADepth but requires much more parameters, due to the underutilization of visual attention, especially ignoring the channel and spatial interdependencies of low-level features.\n\nBesides, we also combine our absolute scale loss to train our model so that the depth network can directly output metrically accurate depth without post-processing that rely on the ground 6 \n\n\nRGB\n\nOurs Monodepth2 FSRE-Depth CADepth-Net Fig. 3. Quantitative results on the Eigen split of KITTI dataset. Compared to other methods [20], [21], [24], our model not only outputs smoother depth in patterned planar regions, but also preserves better details for small objects. White boxes highlight the difference. Best viewed in color and zoom in. In the Train Column, D refers to ground truth depth supervision and M refers to self-supervised monocular supervision. ↓ means the lower the better while ↑ means the higher the better. Methods marked with ‡ indicate that the corresponding test results are taken from [29]. truth depth maps. As listed at the bottom of Table I, we compare our methods with existing self-supervised absolute depth estimation methods that do not rely on the ground truth depth. Among these methods, our method achieves the best results on all metrics without requiring extra pretrained road segmentation network [23] or velocity supervision [22] at training time, or post-processing [16] at test time.\n\nThe qualitative results are presented in Fig. 3. Compared with [20], [21], [24], our model predicts smoother and more accurate depth for patterned surfaces (e.g. the traffic sign in row 1, the banner in row 3, and the bus in row 5), which validates the effectiveness of our model to suppress unnecessary information. Meanwhile, our model also works well for thin things, such as the thin poles in row 2 and the traffic sign in row 4, which reflects the ability to highlight the important details. Taken together, these results illustrate that incorporating the attention mechanism throughout the whole depth network can help to improve the ability of the model to automatically distinguish important contextual information from redundant information for better depth estimation.\n\n\nC. Generalization performance\n\nSeasonDepth dataset [29] is an MDE dataset that contains the same multi-traverse routes under 12 different environmen-tal conditions. The test set of SeasonDepth contains 17225 RGB images and corresponding ground truth depth maps with the resolution of 768 × 1024. We use the model trained on KITTI to evaluate the generalization ability of our method on the test set of SeasonDepth without fine-tuning. Results in Table II show that our model generalizes better than existing supervised methods and self-supervised monocular training methods, which proves that our model is more capable of learning transferable feature representation.\n\nWe also provide the quantitative comparison in Fig. 4. The inferior results of CADepth [21] and FSRE [20] suggest that applying channel attention in the decoder is not enough to obtain high-level features that are robust to lighting and weather changes. Despite extreme light conditions or challenging weather conditions, the output of our model still correctly reflects the scene structure. This further verifies that robust low-level feature representations extracted from our encoder and SAM are also important for forming robust high-level features.\n\n\nD. Ablation Study\n\nTo verify the effectiveness of each component in our method, we perform an ablation study on the KITTI dataset.\n\nAblations of the VADepth network are reported in Table III, where all models are trained with the loss in (14). We use Monodepth2 [24] as the baseline, which adopts a ResNet18based encoder-decoder without any attention mechanisms. The model of row 2 is composed of a VAN [44] based encoder and a convolutional decoder similar to [24], which performs better than the baseline. Adding either DAM or VAB leads to improvements on all metrics. Combining DAM with SAM or CAM can further improve the performance. Incorporating all modules together results in the best overall performance. We also ablate the absolute scale loss L as using different network architectures. It can be seen from Table IV that adding L as with proper α u and α v leads to the similar performance or even better performance compared to the baseline loss when ground truth depth maps are used for median scaling at test time. However, when testing without median scaling, the model trained without absolute scale loss performs poorly. When trained with absolute scale loss, both network architectures achieve good performance without any post-processing. These results indicate that the scale-aware depth network can be obtained with the camera height-based scale constraints. Besides, we also list the results of selecting different values for α u and α v , which determine the size of the rectangular area defined in (9), in the lower part of Table IV. Selecting a too large or too small rectangle degrades performance. Thus, we empirically select α u and α v as 0.075 and 0.875, respectively. Fig. 5. Quantitative results for ablation study of absolute scale loss on the KITTI dataset. The green box in the RGB image is the rectangular region defined by (9). Using the absolute scale loss, the predicted results at the white solid line are smoother and more accurate.\n\nMoreover, we find that absolute scale loss helps alleviate the impact of lane lines on depth estimation. As shown in Fig. 5, although these networks can not predict smooth depth at the solid white line unless absolute scale loss is used, both VADepth and Monodepth2 [24] trained with the absolute scale loss output more accurate depths at the lane line, which suggests that adopting supervision of camera height can lead to smooth depth estimates of the ground surface, in addition to absolute scaled depth. This further demonstrates that our network can effectively learn the geometric knowledge of autonomous driving scenarios with the proposed loss.\n\n\nV. CONCLUSION\n\nIn this work, we presented a novel deep learning architecture for self-supervised monocular depth estimation and proposed an absolute scale loss to supervise the scale of estimated depth maps. Experimental results show that our method achieves state-of-the-art performance with or without median scaling on the KITTI dataset and generalizes well on the unseen changing environments. In the future, we plan to further improve the generalization capacity and interpretability of the depth estimation network by visualizing the attention map and using the results of semantic segmentation to guide the network to strengthen the responses of different channels to specific semantic information or improve the distinction between important features and noises.\n\nFig. 1 .\n1Network structure of VADepth. The encoder extracts feature maps X e i at different resolutions. The Dual Attention Module (DAM) takes the last level feature map X e 4 as input and emphasizes the important features to produce the initial input of the decoder. The decoder has five successive Visual Attention Blocks (VABs) and outputs multi-scale depth maps for the top four VABs.\n\nFig. 2 .\n2Network details of Visual Attention Block (VAB). A Spatial Attention Module (SAM) and a Channel Attention Module (CAM) are used in the VAB.\n\nFig. 4 .\n4Quantitative results on the test set of SeasonDepth dataset. Our method generates depth maps that are more consistent with the scene structure in challenging environments.\n\n\nManuscript received: May 1, 2022; Revised August 2, 2022; Accepted September 17, 2022. This paper was recommended for publication by Editor Cesar Cadena Lerma upon evaluation of the Associate Editor and Reviewers' comments. This work was supported by the National Key Research and Development Program of China (No. 2021YFB2501403). (Corresponding author: Jie Xiang.) 1 Jie Xiang is with the Institute of Microelectronics, Chinese Academy of Sciences, Beijing 100029, China, and with the School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing 100049, China. (Email: xiangjie@ime.ac.cn). 2 Yun Wang, Lifeng An, Haiyang Liu, Zijun Wang and Jian Liu are with the Institute of Microelectronics, Chinese Academy of Sciences, Beijing 100029, China. (Email: {wangyun, anlifeng, liuhaiyang, wangzijun, liujian}@ime.ac.cn).Digital Object Identifier (DOI): 10.1109/LRA.2022.3210298. \n\n\n\nTABLE I QUANTITATIVE\nIRESULTS ON THE EIGEN SPLIT OF KITTI DATASETMethod \nTrain \nScale Factor \nThe lower the better \nThe higher the better \n#Param. \nAbs Rel Sq Rel RMSE RMSE log \nδ 1 \nδ 2 \nδ 3 \nSfMlearner [13] † \nM \nGT \n0.183 \n1.595 \n6.709 \n0.270 \n0.734 0.902 0.959 \n31.6M \nMonodepth2 [24] \nM \nGT \n0.115 \n0.903 \n4.863 \n0.193 \n0.877 0.959 0.981 \n14.3M \nDNet [16] \nM \nGT \n0.113 \n0.864 \n4.812 \n0.191 \n0.877 0.960 0.981 \n14.3M \nSGDepth [18] \nM+Sem \nGT \n0.113 \n0.835 \n4.693 \n0.191 \n0.879 0.961 0.981 \n14.3M \nPacknet-sfm [22] \nM \nGT \n0.111 \n0.785 \n4.601 \n0.189 \n0.878 0.960 0.982 \n128M \nHR-Depth [32] \nM \nGT \n0.109 \n0.792 \n4.632 \n0.185 \n0.884 0.962 0.983 \n14.6M \nJohnston et al. [19] \nM \nGT \n0.106 \n0.861 \n4.699 \n0.185 \n0.889 0.962 0.982 \n-\nWang et al. [17] \nM \nGT \n0.109 \n0.779 \n4.641 \n0.186 \n0.883 0.962 0.982 \n-\nFSRE-Depth [20] \nM+Sem \nGT \n0.105 \n0.722 \n4.547 \n0.182 \n0.886 0.964 \n0.984 \n25.2M \nCADepth [21] \nM \nGT \n0.105 \n0.769 \n4.535 \n0.181 \n0.892 \n0.964 0.983 \n58.3M \nVADepth (Ours) w/o Las \nM \nGT \n0.104 \n0.774 \n4.552 \n0.181 \n0.892 0.965 \n0.983 \n18.8M \nDNet [16] \nM \nCam. Height \n0.118 \n0.925 \n4.918 \n0.199 \n0.862 0.953 0.979 \n14.3M \nPacknet-sfm [22] \nM+v \nNone \n0.111 \n0.829 \n4.788 \n0.199 \n0.864 0.954 0.980 \n128M \nWagstaff et al. [23] \nM \nNone \n0.123 \n0.996 \n5.253 \n0.213 \n0.840 0.947 0.978 \n-\nMonodepth2 [24] with Las \nM \nNone \n0.112 \n0.875 \n4.905 \n0.199 \n0.863 0.955 0.980 \n14.3M \nVADepth (Ours) with Las \nM \nNone \n0.109 \n0.785 \n4.624 \n0.190 \n0.875 0.960 0.982 \n18.8M \n\n\n\nTABLE II\nIIGENERALIZATION PERFORMANCE ON SEASONDEPTH \n\nSeasonDepth test \nMethod \nTrain \nResolution \nKITTI Eigen test \nAverage \nVariance(10 −2 ) \nRelative Range \nAbsRel ↓ \na 1 ↑ \nAbsRel ↓ \na 1 ↑ \nAbsRel ↓ \na 1 ↓ \nAbsRel ↓ 1 − a 1 ↓ \nEigen et al. [6] ‡ \nD \n-\n0.203 \n0.702 \n1.093 \n0.340 \n0.346 \n0.0170 \n0.206 \n0.0746 \nVNL(ResNext101) [8] ‡ \nD \n-\n0.072 \n0.938 \n0.306 \n0.527 \n0.126 \n0.166 \n0.400 \n0.290 \nBTS(DenseNet161) [9] ‡ \nD \n-\n0.060 \n0.955 \n0.676 \n0.209 \n0.545 \n0.0650 \n0.405 \n0.129 \nSfMLearner [13] ‡ \nM \n128 × 416 \n0.181 \n0.733 \n0.360 \n0.495 \n0.0801 \n0.0628 \n0.269 \n0.182 \nMonodepth2 [24] \nM \n192 × 640 \n0.115 \n0.877 \n0.266 \n0.611 \n0.0410 \n0.0457 \n0.266 \n0.202 \nFSRE-Depth [20] \nM \n192 × 640 \n0.105 \n0.886 \n0.256 \n0.624 \n0.0288 \n0.0283 \n0.227 \n0.158 \nCADepth-Net [21] \nM \n192 × 640 \n0.105 \n0.892 \n0.257 \n0.625 \n0.0447 \n0.0725 \n0.265 \n0.278 \nVADepth (Ours) \nM \n192 × 640 \n0.104 \n0.892 \n0.230 \n0.667 \n0.0158 \n0.0215 \n0.205 \n0.179 \n\n\n\nTABLE III ABLATION\nIIISTUDY OF OUR NETWORK STRUCTURE VAN DAM VAB The lower the better SAM CAM AbsRel SqRel RMSE RMSE log0.112 0.875 4.905 \n0.199 \n0.111 0.876 4.762 \n0.195 \n0.108 0.815 4.686 \n0.191 \n0.110 0.821 4.681 \n0.193 \n0.106 0.789 4.672 \n0.190 \n0.109 0.798 4.631 \n0.192 \n0.109 0.785 4.624 \n0.190 \n\n\n\nTABLE IV ABLATION\nIVSTUDY ON ABSOLUTE SCALE LOSSScale \nLas \nThe lower the better \nδ 1 ↑ \nFactor \nαu \nαv \nAbsRel SqRel RMSE RMSElog \nMonodepth2 \n\nGT \nw/o Las \n0.115 \n0.903 \n4.863 \n0.193 \n0.877 \n0.075 0.875 0.112 \n0.864 \n4.804 \n0.190 \n0.878 \n\nNone \nw/o Las \n0.968 15.094 19.176 \n3.459 \n0.000 \n0.075 0.875 0.112 \n0.875 \n4.905 \n0.199 \n0.863 \nVADepth \n\nGT \nw/o Las \n0.104 \n0.774 \n4.552 \n0.181 \n0.892 \n0.075 0.875 0.105 \n0.757 \n4.501 \n0.180 \n0.891 \n\nNone \n\nw/o Las \n0.966 15.045 19.147 \n3.398 \n0.000 \n0.025 0.950 0.127 \n0.932 \n4.920 \n0.207 \n0.850 \n0.050 0.900 0.110 \n0.870 \n4.719 \n0.191 \n0.876 \n0.050 0.875 0.110 \n0.792 \n4.654 \n0.192 \n0.872 \n0.075 0.900 0.109 \n0.877 \n4.702 \n0.190 \n0.879 \n0.075 0.875 0.109 \n0.785 \n4.624 \n0.190 \n0.875 \n0.075 0.850 0.109 \n0.827 \n4.709 \n0.192 \n0.875 \n0.100 0.875 0.109 \n0.825 \n4.680 \n0.190 \n0.876 \n0.100 0.800 0.109 \n0.818 \n4.674 \n0.192 \n0.875 \n0.200 0.600 0.126 \n0.863 \n4.858 \n0.209 \n0.850 \n\nRGB \nVADepth w/o L as \n\nMonodepth2 \nCADepth-Net \nMonodepth2 with L as \n\nVADepth with L as \n\n\n\nIs pseudolidar needed for monocular 3d object detection. D Park, R Ambrus, V Guizilini, J Li, A Gaidon, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. VisD. Park, R. Ambrus, V. Guizilini, J. Li, and A. Gaidon, \"Is pseudo- lidar needed for monocular 3d object detection?\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., October 2021, pp. 3142-3152.\n\nMgnet: Monocular geometric scene understanding for autonomous driving. M Schön, M Buchholz, K Dietmayer, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. VisM. Schön, M. Buchholz, and K. Dietmayer, \"Mgnet: Monocular geo- metric scene understanding for autonomous driving,\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., 2021, pp. 15 804-15 815.\n\nJ-mod 2: Joint monocular obstacle detection and depth estimation. M Mancini, G Costante, P Valigi, T A Ciarfuglia, IEEE Robot. Autom. Lett. 33M. Mancini, G. Costante, P. Valigi, and T. A. Ciarfuglia, \"J-mod 2: Joint monocular obstacle detection and depth estimation,\" IEEE Robot. Autom. Lett., vol. 3, no. 3, pp. 1490-1497, 2018.\n\nSparsity invariant cnns. J Uhrig, N Schneider, L Schneider, U Franke, T Brox, A Geiger, Proc. Int. Conf. 3D Vis. Int. Conf. 3D VisJ. Uhrig, N. Schneider, L. Schneider, U. Franke, T. Brox, and A. Geiger, \"Sparsity invariant cnns,\" in Proc. Int. Conf. 3D Vis., 2017, pp. 11-20.\n\nMultiple view geometry in computer vision. R Hartley, A Zisserman, Cambridge Univ. PressR. Hartley and A. Zisserman, Multiple view geometry in computer vision. Cambridge Univ. Press, 2003.\n\nDepth map prediction from a single image using a multi-scale deep network. D Eigen, C Puhrsch, R Fergus, Proc. Adv. Neural Inf. Process. Syst. 27D. Eigen, C. Puhrsch, and R. Fergus, \"Depth map prediction from a single image using a multi-scale deep network,\" Proc. Adv. Neural Inf. Process. Syst., vol. 27, pp. 2366-2374, 2014.\n\nPredicting depth, surface normals and semantic labels with a common multi-scale convolutional architecture. D Eigen, R Fergus, Proc. IEEE Int. Conf. Comput. Vis. IEEE Int. Conf. Comput. VisD. Eigen and R. Fergus, \"Predicting depth, surface normals and semantic labels with a common multi-scale convolutional architecture,\" in Proc. IEEE Int. Conf. Comput. Vis., December 2015, pp. 2650-2658.\n\nEnforcing geometric constraints of virtual normal for depth prediction. W Yin, Y Liu, C Shen, Y Yan, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. VisW. Yin, Y. Liu, C. Shen, and Y. Yan, \"Enforcing geometric constraints of virtual normal for depth prediction,\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., 2019, pp. 5684-5693.\n\nFrom big to small: Multi-scale local planar guidance for monocular depth estimation. J H Lee, M.-K Han, D W Ko, I H Suh, arXiv:1907.10326arXiv preprintJ. H. Lee, M.-K. Han, D. W. Ko, and I. H. Suh, \"From big to small: Multi-scale local planar guidance for monocular depth estimation,\" arXiv preprint arXiv:1907.10326, 2019.\n\nUnsupervised cnn for single view depth estimation: Geometry to the rescue. R Garg, V K Bg, G Carneiro, I Reid, Proc. Eur. Conf. Comput. Vis. Eur. Conf. Comput. VisR. Garg, V. K. Bg, G. Carneiro, and I. Reid, \"Unsupervised cnn for single view depth estimation: Geometry to the rescue,\" in Proc. Eur. Conf. Comput. Vis., 2016, pp. 740-756.\n\nUnsupervised monocular depth estimation with left-right consistency. C Godard, O Mac Aodha, G J Brostow, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. IEEE Conf. Comput. Vis. Pattern RecognitC. Godard, O. Mac Aodha, and G. J. Brostow, \"Unsupervised monocular depth estimation with left-right consistency,\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., 2017, pp. 270-279.\n\nExcavating the potential capacity of self-supervised monocular depth estimation. R Peng, R Wang, Y Lai, L Tang, Y Cai, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. Vis15R. Peng, R. Wang, Y. Lai, L. Tang, and Y. Cai, \"Excavating the potential capacity of self-supervised monocular depth estimation,\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., 2021, pp. 15 560-15 569.\n\nUnsupervised learning of depth and ego-motion from video. T Zhou, M Brown, N Snavely, D G Lowe, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. IEEE Conf. Comput. Vis. Pattern RecognitT. Zhou, M. Brown, N. Snavely, and D. G. Lowe, \"Unsupervised learning of depth and ego-motion from video,\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., 2017, pp. 1851-1858.\n\nUnsupervised learning of geometry from videos with edge-aware depth-normal consistency. Z Yang, P Wang, W Xu, L Zhao, R Nevatia, Proc. AAAI Conf. AAAI ConfZ. Yang, P. Wang, W. Xu, L. Zhao, and R. Nevatia, \"Unsupervised learning of geometry from videos with edge-aware depth-normal con- sistency,\" in Proc. AAAI Conf. Artif. Intell., 2018, pp. 7493-7500.\n\nLego: Learning edge with geometry all at once by watching videos. Z Yang, P Wang, Y Wang, W Xu, R Nevatia, Proc. IEEE Conf. Comput. Vis. IEEE Conf. Comput. VisZ. Yang, P. Wang, Y. Wang, W. Xu, and R. Nevatia, \"Lego: Learning edge with geometry all at once by watching videos,\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., 2018, pp. 225-234.\n\nToward hierarchical self-supervised monocular absolute depth estimation for autonomous driving applications. F Xue, G Zhuo, Z Huang, W Fu, Z Wu, M H Ang, Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst. IEEE/RSJ Int. Conf. Intell. Robots SystF. Xue, G. Zhuo, Z. Huang, W. Fu, Z. Wu, and M. H. Ang, \"Toward hierarchical self-supervised monocular absolute depth estimation for autonomous driving applications,\" in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., 2020, pp. 2330-2337.\n\nCan scale-consistent monocular depth be learned in a self-supervised scaleinvariant manner. L Wang, Y Wang, L Wang, Y Zhan, Y Wang, H Lu, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. VisL. Wang, Y. Wang, L. Wang, Y. Zhan, Y. Wang, and H. Lu, \"Can scale-consistent monocular depth be learned in a self-supervised scale- invariant manner?\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., 2021, pp. 12 727-12 736.\n\nSelfsupervised monocular depth estimation: Solving the dynamic object problem by semantic guidance. M Klingner, J.-A Termöhlen, J Mikolajczyk, T Fingscheidt, Proc. Eur. Conf. Comput. Vis. Eur. Conf. Comput. VisM. Klingner, J.-A. Termöhlen, J. Mikolajczyk, and T. Fingscheidt, \"Self- supervised monocular depth estimation: Solving the dynamic object problem by semantic guidance,\" in Proc. Eur. Conf. Comput. Vis., 2020, pp. 582-600.\n\nSelf-supervised monocular trained depth estimation using self-attention and discrete disparity volume. A Johnston, G Carneiro, Proc. nullA. Johnston and G. Carneiro, \"Self-supervised monocular trained depth estimation using self-attention and discrete disparity volume,\" in Proc.\n\nIeee/Cvf, Conf, Comput. Vis. Pattern Recognit. IEEE/CVF Conf. Comput. Vis. Pattern Recognit., 2020, pp. 4756-4765.\n\nFine-grained semantics-aware representation enhancement for self-supervised monocular depth estimation. H Jung, E Park, S Yoo, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. VisH. Jung, E. Park, and S. Yoo, \"Fine-grained semantics-aware represen- tation enhancement for self-supervised monocular depth estimation,\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., 2021, pp. 12 642-12 652.\n\nChannel-wise attention-based network for self-supervised monocular depth estimation. J Yan, H Zhao, P Bu, Y Jin, Proc. Int. Conf. 3D Vis. Int. Conf. 3D VisJ. Yan, H. Zhao, P. Bu, and Y. Jin, \"Channel-wise attention-based network for self-supervised monocular depth estimation,\" in Proc. Int. Conf. 3D Vis., 2021, pp. 464-473.\n\n3d packing for self-supervised monocular depth estimation. V Guizilini, R Ambrus, S Pillai, A Raventos, A Gaidon, Proc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit. IEEE/CVF Conf. Comput. Vis. Pattern RecognitV. Guizilini, R. Ambrus, S. Pillai, A. Raventos, and A. Gaidon, \"3d pack- ing for self-supervised monocular depth estimation,\" in Proc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit., 2020, pp. 2485-2494.\n\nSelf-supervised scale recovery for monocular depth and egomotion estimation. B Wagstaff, J Kelly, IEEE/RSJ Int. Conf. Intell. Robots Syst. B. Wagstaff and J. Kelly, \"Self-supervised scale recovery for monocular depth and egomotion estimation,\" in IEEE/RSJ Int. Conf. Intell. Robots Syst., 2020, pp. 2620-2627.\n\nDigging into self-supervised monocular depth estimation. C Godard, O Mac Aodha, M Firman, G J Brostow, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. VisC. Godard, O. Mac Aodha, M. Firman, and G. J. Brostow, \"Digging into self-supervised monocular depth estimation,\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., 2019, pp. 3828-3838.\n\nDeep residual learning for image recognition. K He, X Zhang, S Ren, J Sun, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. IEEE Conf. Comput. Vis. Pattern RecognitK. He, X. Zhang, S. Ren, and J. Sun, \"Deep residual learning for image recognition,\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., 2016, pp. 770-778.\n\nU-net: Convolutional networks for biomedical image segmentation. O Ronneberger, P Fischer, T Brox, Proc. Int. Conf. Med. Image Comput. Comput.-Assisted Intervention. Int. Conf. Med. Image Comput. Comput.-Assisted InterventionChamSpringer International PublishingO. Ronneberger, P. Fischer, and T. Brox, \"U-net: Convolutional networks for biomedical image segmentation,\" in Proc. Int. Conf. Med. Image Comput. Comput.-Assisted Intervention. Cham: Springer International Publishing, 2015, pp. 234-241.\n\nProbabilistic global scale estimation for monoslam based on generic object detection. E Sucar, J.-B Hayet, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. IEEE Conf. Comput. Vis. Pattern RecognitE. Sucar and J.-B. Hayet, \"Probabilistic global scale estimation for monoslam based on generic object detection,\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit. Workshops, 2017, pp. 48-56.\n\nAre we ready for autonomous driving? the kitti vision benchmark suite. A Geiger, P Lenz, R Urtasun, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. IEEE Conf. Comput. Vis. Pattern RecognitA. Geiger, P. Lenz, and R. Urtasun, \"Are we ready for autonomous driving? the kitti vision benchmark suite,\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., 2012, pp. 3354-3361.\n\nSeasondepth: Cross-season monocular depth prediction dataset and benchmark under multiple environments. H Hu, B Yang, Z Qiao, D Zhao, H Wang, arXiv:2011.04408arXiv preprintH. Hu, B. Yang, Z. Qiao, D. Zhao, and H. Wang, \"Seasondepth: Cross-season monocular depth prediction dataset and benchmark under multiple environments,\" arXiv preprint arXiv:2011.04408, 2020.\n\nDeep3d: Fully automatic 2d-to-3d video conversion with deep convolutional neural networks. J Xie, R Girshick, A Farhadi, Proc. Eur. Conf. Comput. Vis. Eur. Conf. Comput. VisJ. Xie, R. Girshick, and A. Farhadi, \"Deep3d: Fully automatic 2d-to- 3d video conversion with deep convolutional neural networks,\" in Proc. Eur. Conf. Comput. Vis., 2016, pp. 842-857.\n\nFeature-metric loss for selfsupervised learning of depth and egomotion. C Shu, K Yu, Z Duan, K Yang, Proc. Eur. Conf. Comput. Vis. Eur. Conf. Comput. VisC. Shu, K. Yu, Z. Duan, and K. Yang, \"Feature-metric loss for self- supervised learning of depth and egomotion,\" in Proc. Eur. Conf. Comput. Vis., 2020, pp. 572-588.\n\nHr-depth: High resolution self-supervised monocular depth estimation. X Lyu, L Liu, M Wang, X Kong, L Liu, Y Liu, X Chen, Y Yuan, Proc. AAAI Conf. AAAI Conf35X. Lyu, L. Liu, M. Wang, X. Kong, L. Liu, Y. Liu, X. Chen, and Y. Yuan, \"Hr-depth: High resolution self-supervised monocular depth estimation,\" in Proc. AAAI Conf. Artif. Intell., vol. 35, no. 3, 2021, pp. 2294-2301.\n\nMonocular visual odometry scale recovery using geometrical constraint. X Wang, H Zhang, X Yin, M Du, Q Chen, Proc. nullX. Wang, H. Zhang, X. Yin, M. Du, and Q. Chen, \"Monocular visual odometry scale recovery using geometrical constraint,\" in Proc. IEEE Int. Conf. Robot. Autom., 2018, pp. 988-995.\n\nMonocular depth estimation in new environments with absolute scale. T Roussel, L Van Eycken, T Tuytelaars, Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst. IEEE/RSJ Int. Conf. Intell. Robots SystT. Roussel, L. Van Eycken, and T. Tuytelaars, \"Monocular depth estimation in new environments with absolute scale,\" in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., 2019, pp. 1735-1741.\n\nLidartouch: Monocular metric depth estimation with a few-beam lidar. F Bartoccioni, É Zablocki, P Pérez, M Cord, K Alahari, arXiv:2109.03569arXiv preprintF. Bartoccioni,É. Zablocki, P. Pérez, M. Cord, and K. Alahari, \"Lidar- touch: Monocular metric depth estimation with a few-beam lidar,\" arXiv preprint arXiv:2109.03569, 2021.\n\nAttention mechanisms in computer vision: A survey. M.-H Guo, T.-X Xu, J.-J Liu, Z.-N Liu, P.-T Jiang, T.-J Mu, S.-H Zhang, R R Martin, M.-M Cheng, S.-M Hu, Comput. Vis. Media. M.-H. Guo, T.-X. Xu, J.-J. Liu, Z.-N. Liu, P.-T. Jiang, T.-J. Mu, S.- H. Zhang, R. R. Martin, M.-M. Cheng, and S.-M. Hu, \"Attention mechanisms in computer vision: A survey,\" Comput. Vis. Media, Mar 2022.\n\nRecurrent models of visual attention. V Mnih, N Heess, A Graves, Proc. Adv. Neural Inf. Process. Syst. 27V. Mnih, N. Heess, A. Graves et al., \"Recurrent models of visual attention,\" Proc. Adv. Neural Inf. Process. Syst., vol. 27, pp. 1-9, 2014.\n\nSqueeze-and-excitation networks. J Hu, L Shen, G Sun, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. IEEE Conf. Comput. Vis. Pattern RecognitJ. Hu, L. Shen, and G. Sun, \"Squeeze-and-excitation networks,\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., June 2018, pp. 7132-7141.\n\nCbam: Convolutional block attention module. S Woo, J Park, J.-Y. Lee, I S Kweon, Proc. Eur. Conf. Comput. Vis. Eur. Conf. Comput. VisS. Woo, J. Park, J.-Y. Lee, and I. S. Kweon, \"Cbam: Convolutional block attention module,\" in Proc. Eur. Conf. Comput. Vis., September 2018, pp. 3-19.\n\nDual attention network for scene segmentation. J Fu, J Liu, H Tian, Y Li, Y Bao, Z Fang, H Lu, Proc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit. IEEE/CVF Conf. Comput. Vis. Pattern RecognitJ. Fu, J. Liu, H. Tian, Y. Li, Y. Bao, Z. Fang, and H. Lu, \"Dual attention network for scene segmentation,\" in Proc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit., June 2019, pp. 3146-3154.\n\nAn image is worth 16x16 words: Transformers for image recognition at scale. A Dosovitskiy, L Beyer, A Kolesnikov, D Weissenborn, X Zhai, T Unterthiner, M Dehghani, M Minderer, G Heigold, S Gelly, J Uszkoreit, N Houlsby, Proc. Int. Conf. Learn. Represent. Int. Conf. Learn. RepresentA. Dosovitskiy, L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, T. Unterthiner, M. Dehghani, M. Minderer, G. Heigold, S. Gelly, J. Uszkoreit, and N. Houlsby, \"An image is worth 16x16 words: Transformers for image recognition at scale,\" in Proc. Int. Conf. Learn. Represent., 2021.\n\nAttention is all you need. A Vaswani, N Shazeer, N Parmar, J Uszkoreit, L Jones, A N Gomez, L Kaiser, I Polosukhin, Proc. Adv. Neural Inf. Process. Syst. 30A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, \"Attention is all you need,\" Proc. Adv. Neural Inf. Process. Syst., vol. 30, pp. 5998-6008, 2017.\n\nVision transformers for dense prediction. R Ranftl, A Bochkovskiy, V Koltun, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. VisR. Ranftl, A. Bochkovskiy, and V. Koltun, \"Vision transformers for dense prediction,\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., 2021, pp. 12 179-12 188.\n\nVisual attention network. M.-H Guo, C.-Z Lu, Z.-N Liu, M.-M Cheng, S.-M Hu, arXiv:2202.09741arXiv preprintM.-H. Guo, C.-Z. Lu, Z.-N. Liu, M.-M. Cheng, and S.-M. Hu, \"Visual attention network,\" arXiv preprint arXiv:2202.09741, 2022.\n\nSpatial transformer networks. M Jaderberg, K Simonyan, A Zisserman, Proc. Adv. Neural Inf. Process. Syst. 28M. Jaderberg, K. Simonyan, A. Zisserman et al., \"Spatial transformer networks,\" Proc. Adv. Neural Inf. Process. Syst., vol. 28, 2015.\n\nImage quality assessment: from error visibility to structural similarity. Z Wang, A C Bovik, H R Sheikh, E P Simoncelli, IEEE Trans. Image Process. 134Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, \"Image quality assessment: from error visibility to structural similarity,\" IEEE Trans. Image Process., vol. 13, no. 4, pp. 600-612, 2004.\n\nImagenet: A large-scale hierarchical image database. J Deng, W Dong, R Socher, L.-J Li, K Li, L Fei-Fei, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. IEEE Conf. Comput. Vis. Pattern RecognitJ. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei, \"Imagenet: A large-scale hierarchical image database,\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., 2009, pp. 248-255.\n\nAttention-based context aggregation network for monocular depth estimation. Y Chen, H Zhao, Z Hu, J Peng, Int. J. Mach. Learn. Cybern. 126Y. Chen, H. Zhao, Z. Hu, and J. Peng, \"Attention-based context aggre- gation network for monocular depth estimation,\" Int. J. Mach. Learn. Cybern., vol. 12, no. 6, pp. 1583-1596, 2021.\n\nEdgeconv with attention module for monocular depth estimation. M Lee, S Hwang, C Park, S Lee, Proc. IEEE/CVF Winter Conf. Appl. of Comput. Vis. (WACV). IEEE/CVF Winter Conf. Appl. of Comput. Vis. (WACV)M. Lee, S. Hwang, C. Park, and S. Lee, \"Edgeconv with attention module for monocular depth estimation,\" in Proc. IEEE/CVF Winter Conf. Appl. of Comput. Vis. (WACV), January 2022, pp. 2858-2867.\n"},"corpus_id":{"kind":"number","value":248863189,"string":"248,863,189"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2205.08780\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"journals/ral/XiangWALWL22\",\n \"doi\": \"10.1109/lra.2022.3210298\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/2205.08780v3.pdf\"\n ],\n \"pdf_sha\": \"2869090a3a6dd3f51c33bc0aaa6563c6cd5c96fa\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/2205.08780\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Visual Attention-based Self-supervised Absolute Depth Estimation using Geometric Priors in Autonomous Driving"},"authors":{"kind":"list like","value":["Jie Xiang","Yun Wang","Lifeng An","Haiyang Liu","Zijun Wang","Jian Liu"],"string":"[\n \"Jie Xiang\",\n \"Yun Wang\",\n \"Lifeng An\",\n \"Haiyang Liu\",\n \"Zijun Wang\",\n \"Jian Liu\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Although existing monocular depth estimation methods have made great progress, predicting an accurate absolute depth map from a single image is still challenging due to the limited modeling capacity of networks and the scale ambiguity issue. In this paper, we introduce a fully Visual Attentionbased Depth (VADepth) network, where spatial attention and channel attention are applied to all stages. By continuously extracting the dependencies of features along the spatial and channel dimensions over a long distance, VADepth network can effectively preserve important details and suppress interfering features to better perceive the scene structure for more accurate depth estimates. In addition, we utilize geometric priors to form scale constraints for scale-aware model training. Specifically, we construct a novel scale-aware loss using the distance between the camera and a plane fitted by the ground points corresponding to the pixels of the rectangular area in the bottom middle of the image. Experimental results on the KITTI dataset show that this architecture achieves the state-of-the-art performance and our method can directly output absolute depth without post-processing. Moreover, our experiments on the SeasonDepth dataset also demonstrate the robustness of our model to multiple unseen environments.Index Terms-Deep learning for visual perception, computer vision for transportation, range sensing."},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":"M ONOCULAR Depth Estimation (MDE) is a fundamental problem in the field of computer vision, which refers to predicting the corresponding depth map from a single image. Depth Estimation has broad application prospects in autonomous driving, such as 3D object detection [1], scene understanding [2], and obstacle avoidance [3]. Compared to other ways of measuring depth [4], [5], monocular depth estimation has unique advantages in obtaining dense depth maps at low cost. Therefore, MDE has aroused the interest of many researchers. In the past few years, many deep learning-based MDE methods [6]- [23] have emerged and continual improvements [7]- [23] have been made. These methods can be divided into supervised methods [6]- [9] and self-supervised methods [10]- [23]. Unlike supervised methods, self-supervised methods do not need to rely on ground truth depth for training. Furthermore, existing self-supervised MDE methods can be divided into two categories: self-supervised stereo training methods [10]- [12] and monocular training methods [13]- [23]. The former use stereo image pairs while the latter use monocular videos as training data. In contrast to stereo training, monocular training is a more general form of selfsupervised methods and easier to obtain training data.\nHowever, existing self-supervised monocular training methods still have difficulties in predicting accurate metric depth. This unsatisfied performance may come from two reasons: the limited modeling power of the model and the limited supervision of loss functions. Many existing self-supervised depth estimation networks [16]- [18], [23], [24] adopt residual networks [25] based on an encoder-decoder architecture [26]. Such a depth network shows a powerful capacity for capturing local information and representing hierarchical abstract features, but is insufficient to distinguish important details from interfering information. Due to these characteristics of the model, it is difficult for these methods to predict accurate depth estimation results on patterned surfaces or tiny structures. Thus, we introduce a fully Visual Attention-based Depth (VADepth) network. Specifically, we build a decoder based on the designed visual attention blocks and use a visual attention network followed by a dual attention module as the encoder. Benefiting from the ability to dynamically handle long-range dependencies brought by combining channel attention and spatial attention at all stages of the encoder-decoder architecture, our model not only continuously preserves important details but also effectively filters out noise.\nOn the other hand, most depth networks [13]- [22] trained on monocular videos only output relative depth due to lacking scale constraints. The commonly used scale recovery method called median scaling [13] requires ground truth depth data that is unavailable at test time in many practical applications. Training a model that directly produces absolute depth in metric units can omit the scale recovery operation like median scaling. An intuitive idea to this end is to use object size [27] but it suffers from seeking ubiquitous fixed-size objects and detecting the appearance size of detected objects. A simple but effective approach is to use camera height with respect 1 arXiv:2205.08780v3 [cs.CV] 6 Oct 2022 2 to ground surface because the ground truth camera height in autonomous driving scenarios is usually fixed and known in advance. Furthermore, we also observe that the middle bottom regions of almost all images captured in these scenarios belong to the ground surface. Considering that not all points on the ground surface are coplanar, we propose to detect all coplanar points on the ground surface from the estimated depth, based on the assumption that a small rectangular region in the middle bottom of any image is part of the ground surface. Using the height errors calculated based on detection results, we construct an absolute scale loss. Combining the scale loss with photometric reconstruction loss [24], the visual attention-based depth network is able to output accurate absolute depth map without post-processing.\nIn summary, we make the contributions as follows:\n• A novel depth network architecture that fully utilizes the long-range perception properties of visual attention to better perceive the scene structure and predict more accurate depth maps. • An absolute scale loss that leverages the geometric priors in autonomous driving scenarios to train a scale-aware depth estimation model. • We conduct extensive experiments to verify the effectiveness of our network structure and loss function. Experimental results [28] show that our method outperforms previous state-of-the-art methods on the KITTI dataset and our model generalizes well on the unseen Season-Depth dataset [29] that contains multiple challenging environments without fine-tuning."},{"title":"II. RELATED WORK","content":""},{"title":"A. Monocular Depth Estimation","content":"Early deep learning-based works [6], [7] on monocular depth estimation mainly focused on supervised methods and dramatically improved the state-of-the-art performance. However, ground truth depth maps for supervised methods are hard to obtain. To avoid the heavy work of collecting ground truth depth, Garg et al. [10] proposed a self-supervised method inspired by view synthesis task [30], which leverages stereo image pairs to construct the reconstruction loss. Compared to stereo image pairs, monocular videos are much easier to obtain. Therefore, SfMLearner [13] leveraged monocular videos to train MDE models. Unlike stereo training approaches [10]- [12], monocular training methods [13], [16] need to simultaneously estimate ego-motion and depth at training time.\nRecently, monocular training methods have been extensively studied. Some works [14], [15], [17], [20], [24], [31] explored to design more effective loss functions based on the photometric reconstruction loss. For example, edge-aware depthnormal consistency loss was used in [14], [15]. Godard et al. [24] proposed a per-pixel minimum reprojection loss to solve the occlusion problem, a multi-scale loss for alleviating the local gradient problem, and the auto-masking method to filter out stationary pixels. Based on Mondepth2 [24], the use of feature-metric loss [31], temporal geometric consistency [17] and semantic-depth consistency [20] further improved monocular depth estimation accuracy. Our method also follows the general pipeline of Monodepth2 [24].\nBesides, some works [22], [32] proposed more powerful convolutional networks to enhance the accuracy of depth estimation. Beyond ResNet-based U-Net, Packnet-sfm [22] leveraged 3D convolutions to improve representational capacity of the network. HR-Depth [32] introduced more dense skipconnections to better fuse multi-scale feature maps. Because of the improvement of loss functions and network architectures, the performance of self-supervised monocular depth estimation has been improved a lot. However, these methods still face challenges in predicting metrically accurate depth due to the scale ambiguity issue and lacking the ability to distinguish important features from noise."},{"title":"B. Scale Ambiguity","content":"Because of lacking scale constraints, monocular vision suffers from the scale ambiguity issue. In order to produce absolute depth at test time, some MDE works [13]- [22] leveraged post-processing for scale recovery. The typical postprocessing technique of median scaling proposed in [13] scales the relative depth map by the ratio of the median of the ground truth depth values to the median of the estimated depth values. Median scaling has been adopted by many subsequent works [17]- [22], [24], [31]. Nevertheless, ground truth depth maps required by median scaling are not always available at test time in many cases. As the use of camera height in classical monocular visual odometry [33], DNet [16] leveraged a dense geometrical constraints module to determine the scale factor by the estimated camera height from every ground point.\nCompared to post-processing techniques, training a scaleaware model is more attractive due to its simplicity of testing. To learn metrically accurate depth estimation, [34] pretrained depth network on stereo data of one dataset and then finetuned the model on monocular videos of another dataset to preserve the absolute scale, and PackNet-sfm [22] leveraged the camera's velocity and [35] utilized 4-beam Lidar data to construct scale-aware loss. All of these works require additional sensor data for training. In order to avoid using other sensor data, [23] proposed a camera height-based loss function to enforce metrically-scaled depth during training. Similar to [23], we also leverage camera height to construct scale constraints. However, our method directly determines the pixels corresponding to the ground plane based on predicted depth instead of relying on an extra pretrained ground plane segmentation model as in [23]. Thus, we simplify the training procedure and improve depth estimation accuracy by mitigating the effects of the limited accuracy of the ground plane segmentation model and the uneven ground surface."},{"title":"C. Attention Mechanism","content":"Attention mechanism is a dynamic process that diverts attention to important features [36]. The pioneering work of visual attention is the spatial attention network called RAM [37]. After that, more attention mechanisms [38]- [41] were proposed, such as self-attention [42] and channel attention [38]. Nowadays, attention mechanisms have been applied in 3 many visual tasks [37]- [41], [43], [44]. As for monocular depth estimation, the attention-based network has been applied not only to supervised methods [48], [49] but also to selfsupervised methods [19]- [21]. In [19], an attention module was used to better perceive contextual information. FSRE-Depth [20] utilized an attention module to fuse depth features and semantic features. CADepth-Net [21] applied channel attention for structure perception as well as skip-connection. However, these networks [48], [49], [19]- [21] just added several attention modules to the original network, and the body of the depth network is still the usual localized convolutional network. Therefore, we introduce the visual attention mechanism throughout the processing of the network, so that our model can continuously and adaptively extract important features and remove noise interference at all stages for more accurate depth predictions."},{"title":"III. METHOD","content":"In this section, we first introduce the basic paradigm for self-supervised monocular depth estimation. Then, we describe the details of our VADepth network. Finally, we introduce a simple method for detecting coplanar points and construct a scale loss based on the planar geometric prior."},{"title":"A. Self-Supervised Monocular Depth Estimation","content":"During the self-supervised MDE model training, the model is optimized by minimizing the difference between the target image and the image synthesized by the source image and the predicted target depth map. In the training process using monocular videos, the source frame is chosen from the same monocular image sequence as the target frame. Following [13], the source frame I s is an adjacent frame to the target frame I t , i.e. s = t ± 1. Since the relative pose between two adjacent frames is unknown, we need to estimate not only the depth but also the ego-motion during training. The depth network takes an RGB image I t as input and outputs a depth map D t . Simultaneously, the pose network predicts the relative pose T t→s between I s and I t . Let K denote the known camera intrinsic matrix. Then we can project the target pixel coordinates (u, v) to the source view and obtain the 2D coordinates of the projected depths D t (u, v) in I s as follows:\nHere, (u, v) and (u s , v s ) are the projected coordinates in I t and I s of the same 3D points P t (u, v), respectively. For simplify of notation, the conversions between inhomogeneous and homogeneous coordinates are omitted in (1). Because the calculated coordinates (u s , v s ) are not always integer values, we use bilinear interpolation to compute pixel values I s (u s , v s ) as in [45]. Thus, we can synthesize the target image: where · denotes bilinear sampling operation. Following [11], we combine the L1 loss and structural similarity index measure (SSIM) [46] to formulate the photometric error:\nTo avoid the false photometric error of occluded pixels in the source frame, we apply per-pixel minimum photometric loss [24], i.e.\nBesides, we adopt the auto-masking method [24] to filter out pixels that do not change the appearance between I s and I t .\nThe binary mask µ is computed as\nwhere [·] is the Inverse bracket. Like [11], we also use the edge-aware smoothness loss:\nwhere d * t = d t /d t is the mean-normalized inverse depth. Considering the gradient locality of the bilinear sampler, we predict multi-scale depth maps in the decoder and compute the individual losses at each scale following [24]. Thus, the final baseline loss with a hyperparameter λ sm is defined as\nwhere S refers to the number of multi-scale depth maps."},{"title":"Concatenation","content":". Element-wise Multiplication"},{"title":"B. VADepth Network Architecture","content":"As shown in Fig. 1, VADepth adopts the improved U-shape encoder-decoder architecture, which takes a single RGB image as input and outputs multi-scale depth maps. Unlike U-net architecture [26], the encoder and decoder in VADepth are not completely symmetrical, and an additional Dual Attention Module (DAM) head (see details in [40]) is used to connect the encoder and the decoder.\nFor the encoder, we employ a visual attention network (VAN) [44] to extract multi-scale feature maps X e i , i = 1, 2, 3, 4. A VAN has four stages, where spatial adaptability and channel adaptability are efficiently implemented by the large kernel attention.\nThen the internal features of the encoder are fed into the decoder with skip connections and the lowest resolution features are taken as the input of the DAM. The DAM adaptively selects the discriminative features as the initial input of the decoder by learning spatial and channel interdependencies of features in parallel.\nAs for the decoder, we design a novel Visual Attention Block (VAB). There are five successive VABs in the decoder. As illustrated in Fig. 2, the i-th VAB has two inputs, the internal features X e i−1 from the encoder and the features X d i from the previous VAB. X e i−a is fed into a Spatial Attention Module (SAM), which emphasizes the important regions and suppresses the unimportant or noisy regions as in [39]. In the SAM, the initial spatial attention maps are formed by max pooling and average pooling along the channel axis. Then the concatenated initial attention maps are passed through a convolutional layer with a sigmoid function to obtain the final spatial attention map. The output of the SAM is the element-wise multiplication of the broadcasted final attention map and X e i−1 . At the same time, X d i is accordingly passed through a convolutional layer and a nearest upsampling layer with a factor of 2 to recover the resolution. Then a Channel Attention Module (CAM) is used to concatenate and fuse the low-level feature from the SAM and the high-level feature from the upsampling operation. For the CAM, as in [38], the concatenated features are processed through the global average pooling layer, fully-connected (FC) layer with ReLU, and FC layer with sigmoid function to generate the channel attention map. Similar to the SAM, the output of the CAM is obtained by element-wise multiplication of the attention map and the original features. After the process of CAM, a convolutional layer is used to output X d i−1 . Finally, a convolutional layer with sigmoid is required to process X d i−1 when necessary, and the sigmoid output σ can be converted to the depth map D i with 1/(aσ + b), which is omitted in Fig. 2 for simplicity. Note that the SAM is removed from VAB when the internal feature from the encoder are unavailable."},{"title":"C. Co-planar Points Detection on Ground Surface","content":"The depth estimation model obtained by minimizing the baseline loss defined in (8) can only directly output the relative depth. In order to train a depth estimation model that can output absolute depth maps, we need to utilize the geometric information that contains absolute scale. In most self-driving scenarios, the ubiquitous camera height can be considered as a known constant value. Therefore, the prior knowledge about camera height is suitable for constructing scale constraints by computing the camera height errors derived from the estimated depth.\nTo estimate the camera height from the predicted depth map, we need to detect the co-planar points on the ground surface. We observe that the middle-lower region of the image is part of the ground surface in almost all self-driving cases. Based on such geometric prior knowledge, we suppose that a small stationary rectangular region in image I t is the ground plane. We use two hyperparameters α u = 0.075 and α v = 0.875 to predefine an H × W binary mask M rect (as shown in Fig. 5) to distinguish whether a pixel is in the rectangular region.\nLet P rect ∈ R Nrect×3 be a matrix for a set of 3D points\nis calculated from the estimated depth according to (2). Based on these 3D points, we can fit a plane P l rect that does not pass through the origin by solving the equation: P rect n = 1. We take the Moore-Penrose inverse of P rect to obtain the least-squares solution of the contradictory equations:\nAfter plane fitting, we need to detect all 3D points co-planar with the plane P l rect in the target view using a binary mask:\nIn M p , an element whose value is equal to one indicates that the corresponding point is on the plane P l rect . The threshold 5 All models are tested with the resolution of 192 × 640 unless otherwise specified, for the maximum depth of 80m. † means the newer results from github with the resolution of 128 × 416. The best scores for each category are in bold. In the \"Train\" column, we list the supervision for each method with M -Self-supervised monocular supervision, Sem -Semantic supervision, v -velocity supervision. In the \"Scale Factor\" column, \"GT\" refers to determining a per-image scale factor using the ground truth depth for median scaling based scale recovery, \"Cam. Height\" indicates leveraging the known camera height w.r.t the ground surface to recover scaled depth, while \"None\" means test without any post-processing.\nδ is set to 0.01, which is used to judge whether the points are coplanar or not."},{"title":"D. Absolute Scale Loss","content":"After co-planar points detection on the ground surface, we construct an absolute scale loss term using the camera height. To form a scale constraint, we need to compute the camera height from the estimated depth results:\nwhere n e = n ||n||1 is the normalized vector of n. Using the estimated heights H cam and the ground truth height h gt , we define the absolute scale loss as follows:\nwhere • is Hadamard product operator. Combining L as with L baseline , we have the final objective function:\nIn our setting, λ sm , λ as and S are set to 0.001, 0.01 and 4, respectively."},{"title":"IV. EXPERIMENTAL RESULTS","content":""},{"title":"A. Implementation details","content":"We use a single Nvidia GeForce RTX 2080 Ti to implement our models in Pytorch. In our setting, we use the VADepth network as our depth network with an input/output resolution of 192 × 640 . For pose estimation, we use a modified ResNet18 [25] to predict a single 6-DoF ego-motion of two adjacent frames following [24]. The pose network adopts the same input resolution as the depth network. Both the encoder of the depth network and the encoder of the pose network are initialized with weights pretrained on ImageNet [47]. The depth network and pose network are jointly trained with Adam optimizer for 20 epochs. The learning rate is set to 5.0 × 10 −5 for the first 15 epochs and then drops by half for the remaining epochs. The source code and models are available at https://github.com/xjixzz/vadepth-net."},{"title":"B. Evaluation on KITTI","content":"KITTI dataset [28] is the commonly used dataset for autonomous driving. We use the Eigen split [6] of KITTI Stereo dataset to evaluate our method with the metrics proposed in [6]. Following [13], we also remove the static frames before training, which results in 39810 monocular triplets for training, 4424 images for validation and 697 images for test. The resolution of these images is approximately 375 × 1242. All images are required to resize to 192 × 640 for training and evaluation. At test time, the depth network only outputs the depth map with the resolution of 192 × 640, which is then resized to the full resolution of the original RGB image for evaluation.\nTo evaluate the performance of our depth network, we first train VADepth network only with the baseline loss function. We report the results of applying median scaling at the top of Table I, where we can see that our model achieves the best scores in the metrics of Abs Rel, RMSE log, δ 1 , and δ 2 , and ranks second or third in the remaining metrics. Among these methods, CADepth [21] achieves close performance to VADepth but requires much more parameters, due to the underutilization of visual attention, especially ignoring the channel and spatial interdependencies of low-level features.\nBesides, we also combine our absolute scale loss to train our model so that the depth network can directly output metrically accurate depth without post-processing that rely on the ground 6"},{"title":"RGB","content":"Ours Monodepth2 FSRE-Depth CADepth-Net Fig. 3. Quantitative results on the Eigen split of KITTI dataset. Compared to other methods [20], [21], [24], our model not only outputs smoother depth in patterned planar regions, but also preserves better details for small objects. White boxes highlight the difference. Best viewed in color and zoom in. In the Train Column, D refers to ground truth depth supervision and M refers to self-supervised monocular supervision. ↓ means the lower the better while ↑ means the higher the better. Methods marked with ‡ indicate that the corresponding test results are taken from [29]. truth depth maps. As listed at the bottom of Table I, we compare our methods with existing self-supervised absolute depth estimation methods that do not rely on the ground truth depth. Among these methods, our method achieves the best results on all metrics without requiring extra pretrained road segmentation network [23] or velocity supervision [22] at training time, or post-processing [16] at test time.\nThe qualitative results are presented in Fig. 3. Compared with [20], [21], [24], our model predicts smoother and more accurate depth for patterned surfaces (e.g. the traffic sign in row 1, the banner in row 3, and the bus in row 5), which validates the effectiveness of our model to suppress unnecessary information. Meanwhile, our model also works well for thin things, such as the thin poles in row 2 and the traffic sign in row 4, which reflects the ability to highlight the important details. Taken together, these results illustrate that incorporating the attention mechanism throughout the whole depth network can help to improve the ability of the model to automatically distinguish important contextual information from redundant information for better depth estimation."},{"title":"C. Generalization performance","content":"SeasonDepth dataset [29] is an MDE dataset that contains the same multi-traverse routes under 12 different environmen-tal conditions. The test set of SeasonDepth contains 17225 RGB images and corresponding ground truth depth maps with the resolution of 768 × 1024. We use the model trained on KITTI to evaluate the generalization ability of our method on the test set of SeasonDepth without fine-tuning. Results in Table II show that our model generalizes better than existing supervised methods and self-supervised monocular training methods, which proves that our model is more capable of learning transferable feature representation.\nWe also provide the quantitative comparison in Fig. 4. The inferior results of CADepth [21] and FSRE [20] suggest that applying channel attention in the decoder is not enough to obtain high-level features that are robust to lighting and weather changes. Despite extreme light conditions or challenging weather conditions, the output of our model still correctly reflects the scene structure. This further verifies that robust low-level feature representations extracted from our encoder and SAM are also important for forming robust high-level features."},{"title":"D. Ablation Study","content":"To verify the effectiveness of each component in our method, we perform an ablation study on the KITTI dataset.\nAblations of the VADepth network are reported in Table III, where all models are trained with the loss in (14). We use Monodepth2 [24] as the baseline, which adopts a ResNet18based encoder-decoder without any attention mechanisms. The model of row 2 is composed of a VAN [44] based encoder and a convolutional decoder similar to [24], which performs better than the baseline. Adding either DAM or VAB leads to improvements on all metrics. Combining DAM with SAM or CAM can further improve the performance. Incorporating all modules together results in the best overall performance. We also ablate the absolute scale loss L as using different network architectures. It can be seen from Table IV that adding L as with proper α u and α v leads to the similar performance or even better performance compared to the baseline loss when ground truth depth maps are used for median scaling at test time. However, when testing without median scaling, the model trained without absolute scale loss performs poorly. When trained with absolute scale loss, both network architectures achieve good performance without any post-processing. These results indicate that the scale-aware depth network can be obtained with the camera height-based scale constraints. Besides, we also list the results of selecting different values for α u and α v , which determine the size of the rectangular area defined in (9), in the lower part of Table IV. Selecting a too large or too small rectangle degrades performance. Thus, we empirically select α u and α v as 0.075 and 0.875, respectively. Fig. 5. Quantitative results for ablation study of absolute scale loss on the KITTI dataset. The green box in the RGB image is the rectangular region defined by (9). Using the absolute scale loss, the predicted results at the white solid line are smoother and more accurate.\nMoreover, we find that absolute scale loss helps alleviate the impact of lane lines on depth estimation. As shown in Fig. 5, although these networks can not predict smooth depth at the solid white line unless absolute scale loss is used, both VADepth and Monodepth2 [24] trained with the absolute scale loss output more accurate depths at the lane line, which suggests that adopting supervision of camera height can lead to smooth depth estimates of the ground surface, in addition to absolute scaled depth. This further demonstrates that our network can effectively learn the geometric knowledge of autonomous driving scenarios with the proposed loss."},{"title":"V. CONCLUSION","content":"In this work, we presented a novel deep learning architecture for self-supervised monocular depth estimation and proposed an absolute scale loss to supervise the scale of estimated depth maps. Experimental results show that our method achieves state-of-the-art performance with or without median scaling on the KITTI dataset and generalizes well on the unseen changing environments. In the future, we plan to further improve the generalization capacity and interpretability of the depth estimation network by visualizing the attention map and using the results of semantic segmentation to guide the network to strengthen the responses of different channels to specific semantic information or improve the distinction between important features and noises."},{"title":"Fig. 1 .","content":""},{"title":"Fig. 2 .","content":""},{"title":"Fig. 4 .","content":""},{"title":"TABLE I QUANTITATIVE","content":""},{"title":"TABLE II","content":""},{"title":"TABLE III ABLATION","content":""},{"title":"TABLE IV ABLATION","content":""}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"M ONOCULAR Depth Estimation (MDE) is a fundamental problem in the field of computer vision, which refers to predicting the corresponding depth map from a single image. Depth Estimation has broad application prospects in autonomous driving, such as 3D object detection [1], scene understanding [2], and obstacle avoidance [3]. Compared to other ways of measuring depth [4], [5], monocular depth estimation has unique advantages in obtaining dense depth maps at low cost. Therefore, MDE has aroused the interest of many researchers. In the past few years, many deep learning-based MDE methods [6]- [23] have emerged and continual improvements [7]- [23] have been made. These methods can be divided into supervised methods [6]- [9] and self-supervised methods [10]- [23]. Unlike supervised methods, self-supervised methods do not need to rely on ground truth depth for training. Furthermore, existing self-supervised MDE methods can be divided into two categories: self-supervised stereo training methods [10]- [12] and monocular training methods [13]- [23]. The former use stereo image pairs while the latter use monocular videos as training data. In contrast to stereo training, monocular training is a more general form of selfsupervised methods and easier to obtain training data.\\nHowever, existing self-supervised monocular training methods still have difficulties in predicting accurate metric depth. This unsatisfied performance may come from two reasons: the limited modeling power of the model and the limited supervision of loss functions. Many existing self-supervised depth estimation networks [16]- [18], [23], [24] adopt residual networks [25] based on an encoder-decoder architecture [26]. Such a depth network shows a powerful capacity for capturing local information and representing hierarchical abstract features, but is insufficient to distinguish important details from interfering information. Due to these characteristics of the model, it is difficult for these methods to predict accurate depth estimation results on patterned surfaces or tiny structures. Thus, we introduce a fully Visual Attention-based Depth (VADepth) network. Specifically, we build a decoder based on the designed visual attention blocks and use a visual attention network followed by a dual attention module as the encoder. Benefiting from the ability to dynamically handle long-range dependencies brought by combining channel attention and spatial attention at all stages of the encoder-decoder architecture, our model not only continuously preserves important details but also effectively filters out noise.\\nOn the other hand, most depth networks [13]- [22] trained on monocular videos only output relative depth due to lacking scale constraints. The commonly used scale recovery method called median scaling [13] requires ground truth depth data that is unavailable at test time in many practical applications. Training a model that directly produces absolute depth in metric units can omit the scale recovery operation like median scaling. An intuitive idea to this end is to use object size [27] but it suffers from seeking ubiquitous fixed-size objects and detecting the appearance size of detected objects. A simple but effective approach is to use camera height with respect 1 arXiv:2205.08780v3 [cs.CV] 6 Oct 2022 2 to ground surface because the ground truth camera height in autonomous driving scenarios is usually fixed and known in advance. Furthermore, we also observe that the middle bottom regions of almost all images captured in these scenarios belong to the ground surface. Considering that not all points on the ground surface are coplanar, we propose to detect all coplanar points on the ground surface from the estimated depth, based on the assumption that a small rectangular region in the middle bottom of any image is part of the ground surface. Using the height errors calculated based on detection results, we construct an absolute scale loss. Combining the scale loss with photometric reconstruction loss [24], the visual attention-based depth network is able to output accurate absolute depth map without post-processing.\\nIn summary, we make the contributions as follows:\\n• A novel depth network architecture that fully utilizes the long-range perception properties of visual attention to better perceive the scene structure and predict more accurate depth maps. • An absolute scale loss that leverages the geometric priors in autonomous driving scenarios to train a scale-aware depth estimation model. • We conduct extensive experiments to verify the effectiveness of our network structure and loss function. Experimental results [28] show that our method outperforms previous state-of-the-art methods on the KITTI dataset and our model generalizes well on the unseen Season-Depth dataset [29] that contains multiple challenging environments without fine-tuning.\"\n },\n {\n \"title\": \"II. RELATED WORK\",\n \"content\": \"\"\n },\n {\n \"title\": \"A. Monocular Depth Estimation\",\n \"content\": \"Early deep learning-based works [6], [7] on monocular depth estimation mainly focused on supervised methods and dramatically improved the state-of-the-art performance. However, ground truth depth maps for supervised methods are hard to obtain. To avoid the heavy work of collecting ground truth depth, Garg et al. [10] proposed a self-supervised method inspired by view synthesis task [30], which leverages stereo image pairs to construct the reconstruction loss. Compared to stereo image pairs, monocular videos are much easier to obtain. Therefore, SfMLearner [13] leveraged monocular videos to train MDE models. Unlike stereo training approaches [10]- [12], monocular training methods [13], [16] need to simultaneously estimate ego-motion and depth at training time.\\nRecently, monocular training methods have been extensively studied. Some works [14], [15], [17], [20], [24], [31] explored to design more effective loss functions based on the photometric reconstruction loss. For example, edge-aware depthnormal consistency loss was used in [14], [15]. Godard et al. [24] proposed a per-pixel minimum reprojection loss to solve the occlusion problem, a multi-scale loss for alleviating the local gradient problem, and the auto-masking method to filter out stationary pixels. Based on Mondepth2 [24], the use of feature-metric loss [31], temporal geometric consistency [17] and semantic-depth consistency [20] further improved monocular depth estimation accuracy. Our method also follows the general pipeline of Monodepth2 [24].\\nBesides, some works [22], [32] proposed more powerful convolutional networks to enhance the accuracy of depth estimation. Beyond ResNet-based U-Net, Packnet-sfm [22] leveraged 3D convolutions to improve representational capacity of the network. HR-Depth [32] introduced more dense skipconnections to better fuse multi-scale feature maps. Because of the improvement of loss functions and network architectures, the performance of self-supervised monocular depth estimation has been improved a lot. However, these methods still face challenges in predicting metrically accurate depth due to the scale ambiguity issue and lacking the ability to distinguish important features from noise.\"\n },\n {\n \"title\": \"B. Scale Ambiguity\",\n \"content\": \"Because of lacking scale constraints, monocular vision suffers from the scale ambiguity issue. In order to produce absolute depth at test time, some MDE works [13]- [22] leveraged post-processing for scale recovery. The typical postprocessing technique of median scaling proposed in [13] scales the relative depth map by the ratio of the median of the ground truth depth values to the median of the estimated depth values. Median scaling has been adopted by many subsequent works [17]- [22], [24], [31]. Nevertheless, ground truth depth maps required by median scaling are not always available at test time in many cases. As the use of camera height in classical monocular visual odometry [33], DNet [16] leveraged a dense geometrical constraints module to determine the scale factor by the estimated camera height from every ground point.\\nCompared to post-processing techniques, training a scaleaware model is more attractive due to its simplicity of testing. To learn metrically accurate depth estimation, [34] pretrained depth network on stereo data of one dataset and then finetuned the model on monocular videos of another dataset to preserve the absolute scale, and PackNet-sfm [22] leveraged the camera's velocity and [35] utilized 4-beam Lidar data to construct scale-aware loss. All of these works require additional sensor data for training. In order to avoid using other sensor data, [23] proposed a camera height-based loss function to enforce metrically-scaled depth during training. Similar to [23], we also leverage camera height to construct scale constraints. However, our method directly determines the pixels corresponding to the ground plane based on predicted depth instead of relying on an extra pretrained ground plane segmentation model as in [23]. Thus, we simplify the training procedure and improve depth estimation accuracy by mitigating the effects of the limited accuracy of the ground plane segmentation model and the uneven ground surface.\"\n },\n {\n \"title\": \"C. Attention Mechanism\",\n \"content\": \"Attention mechanism is a dynamic process that diverts attention to important features [36]. The pioneering work of visual attention is the spatial attention network called RAM [37]. After that, more attention mechanisms [38]- [41] were proposed, such as self-attention [42] and channel attention [38]. Nowadays, attention mechanisms have been applied in 3 many visual tasks [37]- [41], [43], [44]. As for monocular depth estimation, the attention-based network has been applied not only to supervised methods [48], [49] but also to selfsupervised methods [19]- [21]. In [19], an attention module was used to better perceive contextual information. FSRE-Depth [20] utilized an attention module to fuse depth features and semantic features. CADepth-Net [21] applied channel attention for structure perception as well as skip-connection. However, these networks [48], [49], [19]- [21] just added several attention modules to the original network, and the body of the depth network is still the usual localized convolutional network. Therefore, we introduce the visual attention mechanism throughout the processing of the network, so that our model can continuously and adaptively extract important features and remove noise interference at all stages for more accurate depth predictions.\"\n },\n {\n \"title\": \"III. METHOD\",\n \"content\": \"In this section, we first introduce the basic paradigm for self-supervised monocular depth estimation. Then, we describe the details of our VADepth network. Finally, we introduce a simple method for detecting coplanar points and construct a scale loss based on the planar geometric prior.\"\n },\n {\n \"title\": \"A. Self-Supervised Monocular Depth Estimation\",\n \"content\": \"During the self-supervised MDE model training, the model is optimized by minimizing the difference between the target image and the image synthesized by the source image and the predicted target depth map. In the training process using monocular videos, the source frame is chosen from the same monocular image sequence as the target frame. Following [13], the source frame I s is an adjacent frame to the target frame I t , i.e. s = t ± 1. Since the relative pose between two adjacent frames is unknown, we need to estimate not only the depth but also the ego-motion during training. The depth network takes an RGB image I t as input and outputs a depth map D t . Simultaneously, the pose network predicts the relative pose T t→s between I s and I t . Let K denote the known camera intrinsic matrix. Then we can project the target pixel coordinates (u, v) to the source view and obtain the 2D coordinates of the projected depths D t (u, v) in I s as follows:\\nHere, (u, v) and (u s , v s ) are the projected coordinates in I t and I s of the same 3D points P t (u, v), respectively. For simplify of notation, the conversions between inhomogeneous and homogeneous coordinates are omitted in (1). Because the calculated coordinates (u s , v s ) are not always integer values, we use bilinear interpolation to compute pixel values I s (u s , v s ) as in [45]. Thus, we can synthesize the target image: where · denotes bilinear sampling operation. Following [11], we combine the L1 loss and structural similarity index measure (SSIM) [46] to formulate the photometric error:\\nTo avoid the false photometric error of occluded pixels in the source frame, we apply per-pixel minimum photometric loss [24], i.e.\\nBesides, we adopt the auto-masking method [24] to filter out pixels that do not change the appearance between I s and I t .\\nThe binary mask µ is computed as\\nwhere [·] is the Inverse bracket. Like [11], we also use the edge-aware smoothness loss:\\nwhere d * t = d t /d t is the mean-normalized inverse depth. Considering the gradient locality of the bilinear sampler, we predict multi-scale depth maps in the decoder and compute the individual losses at each scale following [24]. Thus, the final baseline loss with a hyperparameter λ sm is defined as\\nwhere S refers to the number of multi-scale depth maps.\"\n },\n {\n \"title\": \"Concatenation\",\n \"content\": \". Element-wise Multiplication\"\n },\n {\n \"title\": \"B. VADepth Network Architecture\",\n \"content\": \"As shown in Fig. 1, VADepth adopts the improved U-shape encoder-decoder architecture, which takes a single RGB image as input and outputs multi-scale depth maps. Unlike U-net architecture [26], the encoder and decoder in VADepth are not completely symmetrical, and an additional Dual Attention Module (DAM) head (see details in [40]) is used to connect the encoder and the decoder.\\nFor the encoder, we employ a visual attention network (VAN) [44] to extract multi-scale feature maps X e i , i = 1, 2, 3, 4. A VAN has four stages, where spatial adaptability and channel adaptability are efficiently implemented by the large kernel attention.\\nThen the internal features of the encoder are fed into the decoder with skip connections and the lowest resolution features are taken as the input of the DAM. The DAM adaptively selects the discriminative features as the initial input of the decoder by learning spatial and channel interdependencies of features in parallel.\\nAs for the decoder, we design a novel Visual Attention Block (VAB). There are five successive VABs in the decoder. As illustrated in Fig. 2, the i-th VAB has two inputs, the internal features X e i−1 from the encoder and the features X d i from the previous VAB. X e i−a is fed into a Spatial Attention Module (SAM), which emphasizes the important regions and suppresses the unimportant or noisy regions as in [39]. In the SAM, the initial spatial attention maps are formed by max pooling and average pooling along the channel axis. Then the concatenated initial attention maps are passed through a convolutional layer with a sigmoid function to obtain the final spatial attention map. The output of the SAM is the element-wise multiplication of the broadcasted final attention map and X e i−1 . At the same time, X d i is accordingly passed through a convolutional layer and a nearest upsampling layer with a factor of 2 to recover the resolution. Then a Channel Attention Module (CAM) is used to concatenate and fuse the low-level feature from the SAM and the high-level feature from the upsampling operation. For the CAM, as in [38], the concatenated features are processed through the global average pooling layer, fully-connected (FC) layer with ReLU, and FC layer with sigmoid function to generate the channel attention map. Similar to the SAM, the output of the CAM is obtained by element-wise multiplication of the attention map and the original features. After the process of CAM, a convolutional layer is used to output X d i−1 . Finally, a convolutional layer with sigmoid is required to process X d i−1 when necessary, and the sigmoid output σ can be converted to the depth map D i with 1/(aσ + b), which is omitted in Fig. 2 for simplicity. Note that the SAM is removed from VAB when the internal feature from the encoder are unavailable.\"\n },\n {\n \"title\": \"C. Co-planar Points Detection on Ground Surface\",\n \"content\": \"The depth estimation model obtained by minimizing the baseline loss defined in (8) can only directly output the relative depth. In order to train a depth estimation model that can output absolute depth maps, we need to utilize the geometric information that contains absolute scale. In most self-driving scenarios, the ubiquitous camera height can be considered as a known constant value. Therefore, the prior knowledge about camera height is suitable for constructing scale constraints by computing the camera height errors derived from the estimated depth.\\nTo estimate the camera height from the predicted depth map, we need to detect the co-planar points on the ground surface. We observe that the middle-lower region of the image is part of the ground surface in almost all self-driving cases. Based on such geometric prior knowledge, we suppose that a small stationary rectangular region in image I t is the ground plane. We use two hyperparameters α u = 0.075 and α v = 0.875 to predefine an H × W binary mask M rect (as shown in Fig. 5) to distinguish whether a pixel is in the rectangular region.\\nLet P rect ∈ R Nrect×3 be a matrix for a set of 3D points\\nis calculated from the estimated depth according to (2). Based on these 3D points, we can fit a plane P l rect that does not pass through the origin by solving the equation: P rect n = 1. We take the Moore-Penrose inverse of P rect to obtain the least-squares solution of the contradictory equations:\\nAfter plane fitting, we need to detect all 3D points co-planar with the plane P l rect in the target view using a binary mask:\\nIn M p , an element whose value is equal to one indicates that the corresponding point is on the plane P l rect . The threshold 5 All models are tested with the resolution of 192 × 640 unless otherwise specified, for the maximum depth of 80m. † means the newer results from github with the resolution of 128 × 416. The best scores for each category are in bold. In the \\\"Train\\\" column, we list the supervision for each method with M -Self-supervised monocular supervision, Sem -Semantic supervision, v -velocity supervision. In the \\\"Scale Factor\\\" column, \\\"GT\\\" refers to determining a per-image scale factor using the ground truth depth for median scaling based scale recovery, \\\"Cam. Height\\\" indicates leveraging the known camera height w.r.t the ground surface to recover scaled depth, while \\\"None\\\" means test without any post-processing.\\nδ is set to 0.01, which is used to judge whether the points are coplanar or not.\"\n },\n {\n \"title\": \"D. Absolute Scale Loss\",\n \"content\": \"After co-planar points detection on the ground surface, we construct an absolute scale loss term using the camera height. To form a scale constraint, we need to compute the camera height from the estimated depth results:\\nwhere n e = n ||n||1 is the normalized vector of n. Using the estimated heights H cam and the ground truth height h gt , we define the absolute scale loss as follows:\\nwhere • is Hadamard product operator. Combining L as with L baseline , we have the final objective function:\\nIn our setting, λ sm , λ as and S are set to 0.001, 0.01 and 4, respectively.\"\n },\n {\n \"title\": \"IV. EXPERIMENTAL RESULTS\",\n \"content\": \"\"\n },\n {\n \"title\": \"A. Implementation details\",\n \"content\": \"We use a single Nvidia GeForce RTX 2080 Ti to implement our models in Pytorch. In our setting, we use the VADepth network as our depth network with an input/output resolution of 192 × 640 . For pose estimation, we use a modified ResNet18 [25] to predict a single 6-DoF ego-motion of two adjacent frames following [24]. The pose network adopts the same input resolution as the depth network. Both the encoder of the depth network and the encoder of the pose network are initialized with weights pretrained on ImageNet [47]. The depth network and pose network are jointly trained with Adam optimizer for 20 epochs. The learning rate is set to 5.0 × 10 −5 for the first 15 epochs and then drops by half for the remaining epochs. The source code and models are available at https://github.com/xjixzz/vadepth-net.\"\n },\n {\n \"title\": \"B. Evaluation on KITTI\",\n \"content\": \"KITTI dataset [28] is the commonly used dataset for autonomous driving. We use the Eigen split [6] of KITTI Stereo dataset to evaluate our method with the metrics proposed in [6]. Following [13], we also remove the static frames before training, which results in 39810 monocular triplets for training, 4424 images for validation and 697 images for test. The resolution of these images is approximately 375 × 1242. All images are required to resize to 192 × 640 for training and evaluation. At test time, the depth network only outputs the depth map with the resolution of 192 × 640, which is then resized to the full resolution of the original RGB image for evaluation.\\nTo evaluate the performance of our depth network, we first train VADepth network only with the baseline loss function. We report the results of applying median scaling at the top of Table I, where we can see that our model achieves the best scores in the metrics of Abs Rel, RMSE log, δ 1 , and δ 2 , and ranks second or third in the remaining metrics. Among these methods, CADepth [21] achieves close performance to VADepth but requires much more parameters, due to the underutilization of visual attention, especially ignoring the channel and spatial interdependencies of low-level features.\\nBesides, we also combine our absolute scale loss to train our model so that the depth network can directly output metrically accurate depth without post-processing that rely on the ground 6\"\n },\n {\n \"title\": \"RGB\",\n \"content\": \"Ours Monodepth2 FSRE-Depth CADepth-Net Fig. 3. Quantitative results on the Eigen split of KITTI dataset. Compared to other methods [20], [21], [24], our model not only outputs smoother depth in patterned planar regions, but also preserves better details for small objects. White boxes highlight the difference. Best viewed in color and zoom in. In the Train Column, D refers to ground truth depth supervision and M refers to self-supervised monocular supervision. ↓ means the lower the better while ↑ means the higher the better. Methods marked with ‡ indicate that the corresponding test results are taken from [29]. truth depth maps. As listed at the bottom of Table I, we compare our methods with existing self-supervised absolute depth estimation methods that do not rely on the ground truth depth. Among these methods, our method achieves the best results on all metrics without requiring extra pretrained road segmentation network [23] or velocity supervision [22] at training time, or post-processing [16] at test time.\\nThe qualitative results are presented in Fig. 3. Compared with [20], [21], [24], our model predicts smoother and more accurate depth for patterned surfaces (e.g. the traffic sign in row 1, the banner in row 3, and the bus in row 5), which validates the effectiveness of our model to suppress unnecessary information. Meanwhile, our model also works well for thin things, such as the thin poles in row 2 and the traffic sign in row 4, which reflects the ability to highlight the important details. Taken together, these results illustrate that incorporating the attention mechanism throughout the whole depth network can help to improve the ability of the model to automatically distinguish important contextual information from redundant information for better depth estimation.\"\n },\n {\n \"title\": \"C. Generalization performance\",\n \"content\": \"SeasonDepth dataset [29] is an MDE dataset that contains the same multi-traverse routes under 12 different environmen-tal conditions. The test set of SeasonDepth contains 17225 RGB images and corresponding ground truth depth maps with the resolution of 768 × 1024. We use the model trained on KITTI to evaluate the generalization ability of our method on the test set of SeasonDepth without fine-tuning. Results in Table II show that our model generalizes better than existing supervised methods and self-supervised monocular training methods, which proves that our model is more capable of learning transferable feature representation.\\nWe also provide the quantitative comparison in Fig. 4. The inferior results of CADepth [21] and FSRE [20] suggest that applying channel attention in the decoder is not enough to obtain high-level features that are robust to lighting and weather changes. Despite extreme light conditions or challenging weather conditions, the output of our model still correctly reflects the scene structure. This further verifies that robust low-level feature representations extracted from our encoder and SAM are also important for forming robust high-level features.\"\n },\n {\n \"title\": \"D. Ablation Study\",\n \"content\": \"To verify the effectiveness of each component in our method, we perform an ablation study on the KITTI dataset.\\nAblations of the VADepth network are reported in Table III, where all models are trained with the loss in (14). We use Monodepth2 [24] as the baseline, which adopts a ResNet18based encoder-decoder without any attention mechanisms. The model of row 2 is composed of a VAN [44] based encoder and a convolutional decoder similar to [24], which performs better than the baseline. Adding either DAM or VAB leads to improvements on all metrics. Combining DAM with SAM or CAM can further improve the performance. Incorporating all modules together results in the best overall performance. We also ablate the absolute scale loss L as using different network architectures. It can be seen from Table IV that adding L as with proper α u and α v leads to the similar performance or even better performance compared to the baseline loss when ground truth depth maps are used for median scaling at test time. However, when testing without median scaling, the model trained without absolute scale loss performs poorly. When trained with absolute scale loss, both network architectures achieve good performance without any post-processing. These results indicate that the scale-aware depth network can be obtained with the camera height-based scale constraints. Besides, we also list the results of selecting different values for α u and α v , which determine the size of the rectangular area defined in (9), in the lower part of Table IV. Selecting a too large or too small rectangle degrades performance. Thus, we empirically select α u and α v as 0.075 and 0.875, respectively. Fig. 5. Quantitative results for ablation study of absolute scale loss on the KITTI dataset. The green box in the RGB image is the rectangular region defined by (9). Using the absolute scale loss, the predicted results at the white solid line are smoother and more accurate.\\nMoreover, we find that absolute scale loss helps alleviate the impact of lane lines on depth estimation. As shown in Fig. 5, although these networks can not predict smooth depth at the solid white line unless absolute scale loss is used, both VADepth and Monodepth2 [24] trained with the absolute scale loss output more accurate depths at the lane line, which suggests that adopting supervision of camera height can lead to smooth depth estimates of the ground surface, in addition to absolute scaled depth. This further demonstrates that our network can effectively learn the geometric knowledge of autonomous driving scenarios with the proposed loss.\"\n },\n {\n \"title\": \"V. CONCLUSION\",\n \"content\": \"In this work, we presented a novel deep learning architecture for self-supervised monocular depth estimation and proposed an absolute scale loss to supervise the scale of estimated depth maps. Experimental results show that our method achieves state-of-the-art performance with or without median scaling on the KITTI dataset and generalizes well on the unseen changing environments. In the future, we plan to further improve the generalization capacity and interpretability of the depth estimation network by visualizing the attention map and using the results of semantic segmentation to guide the network to strengthen the responses of different channels to specific semantic information or improve the distinction between important features and noises.\"\n },\n {\n \"title\": \"Fig. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"TABLE I QUANTITATIVE\",\n \"content\": \"\"\n },\n {\n \"title\": \"TABLE II\",\n \"content\": \"\"\n },\n {\n \"title\": \"TABLE III ABLATION\",\n \"content\": \"\"\n },\n {\n \"title\": \"TABLE IV ABLATION\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Network structure of VADepth. The encoder extracts feature maps X e i at different resolutions. The Dual Attention Module (DAM) takes the last level feature map X e 4 as input and emphasizes the important features to produce the initial input of the decoder. The decoder has five successive Visual Attention Blocks (VABs) and outputs multi-scale depth maps for the top four VABs.","content":"Fig. 1 .\n1Network structure of VADepth. The encoder extracts feature maps X e i at different resolutions. The Dual Attention Module (DAM) takes the last level feature map X e 4 as input and emphasizes the important features to produce the initial input of the decoder. The decoder has five successive Visual Attention Blocks (VABs) and outputs multi-scale depth maps for the top four VABs."},{"id":"","caption":"Network details of Visual Attention Block (VAB). A Spatial Attention Module (SAM) and a Channel Attention Module (CAM) are used in the VAB.","content":"Fig. 2 .\n2Network details of Visual Attention Block (VAB). A Spatial Attention Module (SAM) and a Channel Attention Module (CAM) are used in the VAB."},{"id":"","caption":"Quantitative results on the test set of SeasonDepth dataset. Our method generates depth maps that are more consistent with the scene structure in challenging environments.","content":"Fig. 4 .\n4Quantitative results on the test set of SeasonDepth dataset. Our method generates depth maps that are more consistent with the scene structure in challenging environments."},{"id":"","caption":"Manuscript received: May 1, 2022; Revised August 2, 2022; Accepted September 17, 2022. This paper was recommended for publication by Editor Cesar Cadena Lerma upon evaluation of the Associate Editor and Reviewers' comments. This work was supported by the National Key Research and Development Program of China (No. 2021YFB2501403). (Corresponding author: Jie Xiang.) 1 Jie Xiang is with the Institute of Microelectronics, Chinese Academy of Sciences, Beijing 100029, China, and with the School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing 100049, China. (Email: xiangjie@ime.ac.cn). 2 Yun Wang, Lifeng An, Haiyang Liu, Zijun Wang and Jian Liu are with the Institute of Microelectronics, Chinese Academy of Sciences, Beijing 100029, China. (Email: {wangyun, anlifeng, liuhaiyang, wangzijun, liujian}@ime.ac.cn).","content":"Manuscript received: May 1, 2022; Revised August 2, 2022; Accepted September 17, 2022. This paper was recommended for publication by Editor Cesar Cadena Lerma upon evaluation of the Associate Editor and Reviewers' comments. This work was supported by the National Key Research and Development Program of China (No. 2021YFB2501403). (Corresponding author: Jie Xiang.) 1 Jie Xiang is with the Institute of Microelectronics, Chinese Academy of Sciences, Beijing 100029, China, and with the School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing 100049, China. (Email: xiangjie@ime.ac.cn). 2 Yun Wang, Lifeng An, Haiyang Liu, Zijun Wang and Jian Liu are with the Institute of Microelectronics, Chinese Academy of Sciences, Beijing 100029, China. (Email: {wangyun, anlifeng, liuhaiyang, wangzijun, liujian}@ime.ac.cn).Digital Object Identifier (DOI): 10.1109/LRA.2022.3210298."},{"id":"","caption":"RESULTS ON THE EIGEN SPLIT OF KITTI DATASET","content":"TABLE I QUANTITATIVE\nIRESULTS ON THE EIGEN SPLIT OF KITTI DATASETMethod \nTrain \nScale Factor \nThe lower the better \nThe higher the better \n#Param. \nAbs Rel Sq Rel RMSE RMSE log \nδ 1 \nδ 2 \nδ 3 \nSfMlearner [13] † \nM \nGT \n0.183 \n1.595 \n6.709 \n0.270 \n0.734 0.902 0.959 \n31.6M \nMonodepth2 [24] \nM \nGT \n0.115 \n0.903 \n4.863 \n0.193 \n0.877 0.959 0.981 \n14.3M \nDNet [16] \nM \nGT \n0.113 \n0.864 \n4.812 \n0.191 \n0.877 0.960 0.981 \n14.3M \nSGDepth [18] \nM+Sem \nGT \n0.113 \n0.835 \n4.693 \n0.191 \n0.879 0.961 0.981 \n14.3M \nPacknet-sfm [22] \nM \nGT \n0.111 \n0.785 \n4.601 \n0.189 \n0.878 0.960 0.982 \n128M \nHR-Depth [32] \nM \nGT \n0.109 \n0.792 \n4.632 \n0.185 \n0.884 0.962 0.983 \n14.6M \nJohnston et al. [19] \nM \nGT \n0.106 \n0.861 \n4.699 \n0.185 \n0.889 0.962 0.982 \n-\nWang et al. [17] \nM \nGT \n0.109 \n0.779 \n4.641 \n0.186 \n0.883 0.962 0.982 \n-\nFSRE-Depth [20] \nM+Sem \nGT \n0.105 \n0.722 \n4.547 \n0.182 \n0.886 0.964 \n0.984 \n25.2M \nCADepth [21] \nM \nGT \n0.105 \n0.769 \n4.535 \n0.181 \n0.892 \n0.964 0.983 \n58.3M \nVADepth (Ours) w/o Las \nM \nGT \n0.104 \n0.774 \n4.552 \n0.181 \n0.892 0.965 \n0.983 \n18.8M \nDNet [16] \nM \nCam. Height \n0.118 \n0.925 \n4.918 \n0.199 \n0.862 0.953 0.979 \n14.3M \nPacknet-sfm [22] \nM+v \nNone \n0.111 \n0.829 \n4.788 \n0.199 \n0.864 0.954 0.980 \n128M \nWagstaff et al. [23] \nM \nNone \n0.123 \n0.996 \n5.253 \n0.213 \n0.840 0.947 0.978 \n-\nMonodepth2 [24] with Las \nM \nNone \n0.112 \n0.875 \n4.905 \n0.199 \n0.863 0.955 0.980 \n14.3M \nVADepth (Ours) with Las \nM \nNone \n0.109 \n0.785 \n4.624 \n0.190 \n0.875 0.960 0.982 \n18.8M"},{"id":"","caption":"STUDY OF OUR NETWORK STRUCTURE VAN DAM VAB The lower the better SAM CAM AbsRel SqRel RMSE RMSE log","content":"TABLE II\nIIGENERALIZATION PERFORMANCE ON SEASONDEPTH \n\nSeasonDepth test \nMethod \nTrain \nResolution \nKITTI Eigen test \nAverage \nVariance(10 −2 ) \nRelative Range \nAbsRel ↓ \na 1 ↑ \nAbsRel ↓ \na 1 ↑ \nAbsRel ↓ \na 1 ↓ \nAbsRel ↓ 1 − a 1 ↓ \nEigen et al. [6] ‡ \nD \n-\n0.203 \n0.702 \n1.093 \n0.340 \n0.346 \n0.0170 \n0.206 \n0.0746 \nVNL(ResNext101) [8] ‡ \nD \n-\n0.072 \n0.938 \n0.306 \n0.527 \n0.126 \n0.166 \n0.400 \n0.290 \nBTS(DenseNet161) [9] ‡ \nD \n-\n0.060 \n0.955 \n0.676 \n0.209 \n0.545 \n0.0650 \n0.405 \n0.129 \nSfMLearner [13] ‡ \nM \n128 × 416 \n0.181 \n0.733 \n0.360 \n0.495 \n0.0801 \n0.0628 \n0.269 \n0.182 \nMonodepth2 [24] \nM \n192 × 640 \n0.115 \n0.877 \n0.266 \n0.611 \n0.0410 \n0.0457 \n0.266 \n0.202 \nFSRE-Depth [20] \nM \n192 × 640 \n0.105 \n0.886 \n0.256 \n0.624 \n0.0288 \n0.0283 \n0.227 \n0.158 \nCADepth-Net [21] \nM \n192 × 640 \n0.105 \n0.892 \n0.257 \n0.625 \n0.0447 \n0.0725 \n0.265 \n0.278 \nVADepth (Ours) \nM \n192 × 640 \n0.104 \n0.892 \n0.230 \n0.667 \n0.0158 \n0.0215 \n0.205 \n0.179"},{"id":"","caption":"STUDY ON ABSOLUTE SCALE LOSS","content":"TABLE III ABLATION\nIIISTUDY OF OUR NETWORK STRUCTURE VAN DAM VAB The lower the better SAM CAM AbsRel SqRel RMSE RMSE log0.112 0.875 4.905 \n0.199 \n0.111 0.876 4.762 \n0.195 \n0.108 0.815 4.686 \n0.191 \n0.110 0.821 4.681 \n0.193 \n0.106 0.789 4.672 \n0.190 \n0.109 0.798 4.631 \n0.192 \n0.109 0.785 4.624 \n0.190"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Network structure of VADepth. The encoder extracts feature maps X e i at different resolutions. The Dual Attention Module (DAM) takes the last level feature map X e 4 as input and emphasizes the important features to produce the initial input of the decoder. The decoder has five successive Visual Attention Blocks (VABs) and outputs multi-scale depth maps for the top four VABs.\",\n \"content\": \"Fig. 1 .\\n1Network structure of VADepth. The encoder extracts feature maps X e i at different resolutions. The Dual Attention Module (DAM) takes the last level feature map X e 4 as input and emphasizes the important features to produce the initial input of the decoder. The decoder has five successive Visual Attention Blocks (VABs) and outputs multi-scale depth maps for the top four VABs.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Network details of Visual Attention Block (VAB). A Spatial Attention Module (SAM) and a Channel Attention Module (CAM) are used in the VAB.\",\n \"content\": \"Fig. 2 .\\n2Network details of Visual Attention Block (VAB). A Spatial Attention Module (SAM) and a Channel Attention Module (CAM) are used in the VAB.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Quantitative results on the test set of SeasonDepth dataset. Our method generates depth maps that are more consistent with the scene structure in challenging environments.\",\n \"content\": \"Fig. 4 .\\n4Quantitative results on the test set of SeasonDepth dataset. Our method generates depth maps that are more consistent with the scene structure in challenging environments.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Manuscript received: May 1, 2022; Revised August 2, 2022; Accepted September 17, 2022. This paper was recommended for publication by Editor Cesar Cadena Lerma upon evaluation of the Associate Editor and Reviewers' comments. This work was supported by the National Key Research and Development Program of China (No. 2021YFB2501403). (Corresponding author: Jie Xiang.) 1 Jie Xiang is with the Institute of Microelectronics, Chinese Academy of Sciences, Beijing 100029, China, and with the School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing 100049, China. (Email: xiangjie@ime.ac.cn). 2 Yun Wang, Lifeng An, Haiyang Liu, Zijun Wang and Jian Liu are with the Institute of Microelectronics, Chinese Academy of Sciences, Beijing 100029, China. (Email: {wangyun, anlifeng, liuhaiyang, wangzijun, liujian}@ime.ac.cn).\",\n \"content\": \"Manuscript received: May 1, 2022; Revised August 2, 2022; Accepted September 17, 2022. This paper was recommended for publication by Editor Cesar Cadena Lerma upon evaluation of the Associate Editor and Reviewers' comments. This work was supported by the National Key Research and Development Program of China (No. 2021YFB2501403). (Corresponding author: Jie Xiang.) 1 Jie Xiang is with the Institute of Microelectronics, Chinese Academy of Sciences, Beijing 100029, China, and with the School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing 100049, China. (Email: xiangjie@ime.ac.cn). 2 Yun Wang, Lifeng An, Haiyang Liu, Zijun Wang and Jian Liu are with the Institute of Microelectronics, Chinese Academy of Sciences, Beijing 100029, China. (Email: {wangyun, anlifeng, liuhaiyang, wangzijun, liujian}@ime.ac.cn).Digital Object Identifier (DOI): 10.1109/LRA.2022.3210298.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"RESULTS ON THE EIGEN SPLIT OF KITTI DATASET\",\n \"content\": \"TABLE I QUANTITATIVE\\nIRESULTS ON THE EIGEN SPLIT OF KITTI DATASETMethod \\nTrain \\nScale Factor \\nThe lower the better \\nThe higher the better \\n#Param. \\nAbs Rel Sq Rel RMSE RMSE log \\nδ 1 \\nδ 2 \\nδ 3 \\nSfMlearner [13] † \\nM \\nGT \\n0.183 \\n1.595 \\n6.709 \\n0.270 \\n0.734 0.902 0.959 \\n31.6M \\nMonodepth2 [24] \\nM \\nGT \\n0.115 \\n0.903 \\n4.863 \\n0.193 \\n0.877 0.959 0.981 \\n14.3M \\nDNet [16] \\nM \\nGT \\n0.113 \\n0.864 \\n4.812 \\n0.191 \\n0.877 0.960 0.981 \\n14.3M \\nSGDepth [18] \\nM+Sem \\nGT \\n0.113 \\n0.835 \\n4.693 \\n0.191 \\n0.879 0.961 0.981 \\n14.3M \\nPacknet-sfm [22] \\nM \\nGT \\n0.111 \\n0.785 \\n4.601 \\n0.189 \\n0.878 0.960 0.982 \\n128M \\nHR-Depth [32] \\nM \\nGT \\n0.109 \\n0.792 \\n4.632 \\n0.185 \\n0.884 0.962 0.983 \\n14.6M \\nJohnston et al. [19] \\nM \\nGT \\n0.106 \\n0.861 \\n4.699 \\n0.185 \\n0.889 0.962 0.982 \\n-\\nWang et al. [17] \\nM \\nGT \\n0.109 \\n0.779 \\n4.641 \\n0.186 \\n0.883 0.962 0.982 \\n-\\nFSRE-Depth [20] \\nM+Sem \\nGT \\n0.105 \\n0.722 \\n4.547 \\n0.182 \\n0.886 0.964 \\n0.984 \\n25.2M \\nCADepth [21] \\nM \\nGT \\n0.105 \\n0.769 \\n4.535 \\n0.181 \\n0.892 \\n0.964 0.983 \\n58.3M \\nVADepth (Ours) w/o Las \\nM \\nGT \\n0.104 \\n0.774 \\n4.552 \\n0.181 \\n0.892 0.965 \\n0.983 \\n18.8M \\nDNet [16] \\nM \\nCam. Height \\n0.118 \\n0.925 \\n4.918 \\n0.199 \\n0.862 0.953 0.979 \\n14.3M \\nPacknet-sfm [22] \\nM+v \\nNone \\n0.111 \\n0.829 \\n4.788 \\n0.199 \\n0.864 0.954 0.980 \\n128M \\nWagstaff et al. [23] \\nM \\nNone \\n0.123 \\n0.996 \\n5.253 \\n0.213 \\n0.840 0.947 0.978 \\n-\\nMonodepth2 [24] with Las \\nM \\nNone \\n0.112 \\n0.875 \\n4.905 \\n0.199 \\n0.863 0.955 0.980 \\n14.3M \\nVADepth (Ours) with Las \\nM \\nNone \\n0.109 \\n0.785 \\n4.624 \\n0.190 \\n0.875 0.960 0.982 \\n18.8M\"\n },\n {\n \"id\": \"\",\n \"caption\": \"STUDY OF OUR NETWORK STRUCTURE VAN DAM VAB The lower the better SAM CAM AbsRel SqRel RMSE RMSE log\",\n \"content\": \"TABLE II\\nIIGENERALIZATION PERFORMANCE ON SEASONDEPTH \\n\\nSeasonDepth test \\nMethod \\nTrain \\nResolution \\nKITTI Eigen test \\nAverage \\nVariance(10 −2 ) \\nRelative Range \\nAbsRel ↓ \\na 1 ↑ \\nAbsRel ↓ \\na 1 ↑ \\nAbsRel ↓ \\na 1 ↓ \\nAbsRel ↓ 1 − a 1 ↓ \\nEigen et al. [6] ‡ \\nD \\n-\\n0.203 \\n0.702 \\n1.093 \\n0.340 \\n0.346 \\n0.0170 \\n0.206 \\n0.0746 \\nVNL(ResNext101) [8] ‡ \\nD \\n-\\n0.072 \\n0.938 \\n0.306 \\n0.527 \\n0.126 \\n0.166 \\n0.400 \\n0.290 \\nBTS(DenseNet161) [9] ‡ \\nD \\n-\\n0.060 \\n0.955 \\n0.676 \\n0.209 \\n0.545 \\n0.0650 \\n0.405 \\n0.129 \\nSfMLearner [13] ‡ \\nM \\n128 × 416 \\n0.181 \\n0.733 \\n0.360 \\n0.495 \\n0.0801 \\n0.0628 \\n0.269 \\n0.182 \\nMonodepth2 [24] \\nM \\n192 × 640 \\n0.115 \\n0.877 \\n0.266 \\n0.611 \\n0.0410 \\n0.0457 \\n0.266 \\n0.202 \\nFSRE-Depth [20] \\nM \\n192 × 640 \\n0.105 \\n0.886 \\n0.256 \\n0.624 \\n0.0288 \\n0.0283 \\n0.227 \\n0.158 \\nCADepth-Net [21] \\nM \\n192 × 640 \\n0.105 \\n0.892 \\n0.257 \\n0.625 \\n0.0447 \\n0.0725 \\n0.265 \\n0.278 \\nVADepth (Ours) \\nM \\n192 × 640 \\n0.104 \\n0.892 \\n0.230 \\n0.667 \\n0.0158 \\n0.0215 \\n0.205 \\n0.179\"\n },\n {\n \"id\": \"\",\n \"caption\": \"STUDY ON ABSOLUTE SCALE LOSS\",\n \"content\": \"TABLE III ABLATION\\nIIISTUDY OF OUR NETWORK STRUCTURE VAN DAM VAB The lower the better SAM CAM AbsRel SqRel RMSE RMSE log0.112 0.875 4.905 \\n0.199 \\n0.111 0.876 4.762 \\n0.195 \\n0.108 0.815 4.686 \\n0.191 \\n0.110 0.821 4.681 \\n0.193 \\n0.106 0.789 4.672 \\n0.190 \\n0.109 0.798 4.631 \\n0.192 \\n0.109 0.785 4.624 \\n0.190\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Is pseudolidar needed for monocular 3d object detection. D Park, R Ambrus, V Guizilini, J Li, A Gaidon, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. VisD. Park, R. Ambrus, V. Guizilini, J. Li, and A. Gaidon, \"Is pseudo- lidar needed for monocular 3d object detection?\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., October 2021, pp. 3142-3152.","doi":""},{"text":"Mgnet: Monocular geometric scene understanding for autonomous driving. M Schön, M Buchholz, K Dietmayer, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. VisM. Schön, M. Buchholz, and K. Dietmayer, \"Mgnet: Monocular geo- metric scene understanding for autonomous driving,\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., 2021, pp. 15 804-15 815.","doi":""},{"text":"J-mod 2: Joint monocular obstacle detection and depth estimation. M Mancini, G Costante, P Valigi, T A Ciarfuglia, IEEE Robot. Autom. Lett. 33M. Mancini, G. Costante, P. Valigi, and T. A. Ciarfuglia, \"J-mod 2: Joint monocular obstacle detection and depth estimation,\" IEEE Robot. Autom. Lett., vol. 3, no. 3, pp. 1490-1497, 2018.","doi":""},{"text":"Sparsity invariant cnns. J Uhrig, N Schneider, L Schneider, U Franke, T Brox, A Geiger, Proc. Int. Conf. 3D Vis. Int. Conf. 3D VisJ. Uhrig, N. Schneider, L. Schneider, U. Franke, T. Brox, and A. Geiger, \"Sparsity invariant cnns,\" in Proc. Int. Conf. 3D Vis., 2017, pp. 11-20.","doi":""},{"text":"Multiple view geometry in computer vision. R Hartley, A Zisserman, Cambridge Univ. PressR. Hartley and A. Zisserman, Multiple view geometry in computer vision. Cambridge Univ. Press, 2003.","doi":""},{"text":"Depth map prediction from a single image using a multi-scale deep network. D Eigen, C Puhrsch, R Fergus, Proc. Adv. Neural Inf. Process. Syst. 27D. Eigen, C. Puhrsch, and R. Fergus, \"Depth map prediction from a single image using a multi-scale deep network,\" Proc. Adv. Neural Inf. Process. Syst., vol. 27, pp. 2366-2374, 2014.","doi":""},{"text":"Predicting depth, surface normals and semantic labels with a common multi-scale convolutional architecture. D Eigen, R Fergus, Proc. IEEE Int. Conf. Comput. Vis. IEEE Int. Conf. Comput. VisD. Eigen and R. Fergus, \"Predicting depth, surface normals and semantic labels with a common multi-scale convolutional architecture,\" in Proc. IEEE Int. Conf. Comput. Vis., December 2015, pp. 2650-2658.","doi":""},{"text":"Enforcing geometric constraints of virtual normal for depth prediction. W Yin, Y Liu, C Shen, Y Yan, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. VisW. Yin, Y. Liu, C. Shen, and Y. Yan, \"Enforcing geometric constraints of virtual normal for depth prediction,\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., 2019, pp. 5684-5693.","doi":""},{"text":"From big to small: Multi-scale local planar guidance for monocular depth estimation. J H Lee, M.-K Han, D W Ko, I H Suh, arXiv:1907.10326arXiv preprintJ. H. Lee, M.-K. Han, D. W. Ko, and I. H. Suh, \"From big to small: Multi-scale local planar guidance for monocular depth estimation,\" arXiv preprint arXiv:1907.10326, 2019.","doi":"arXiv:1907.10326"},{"text":"Unsupervised cnn for single view depth estimation: Geometry to the rescue. R Garg, V K Bg, G Carneiro, I Reid, Proc. Eur. Conf. Comput. Vis. Eur. Conf. Comput. VisR. Garg, V. K. Bg, G. Carneiro, and I. Reid, \"Unsupervised cnn for single view depth estimation: Geometry to the rescue,\" in Proc. Eur. Conf. Comput. Vis., 2016, pp. 740-756.","doi":""},{"text":"Unsupervised monocular depth estimation with left-right consistency. C Godard, O Mac Aodha, G J Brostow, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. IEEE Conf. Comput. Vis. Pattern RecognitC. Godard, O. Mac Aodha, and G. J. Brostow, \"Unsupervised monocular depth estimation with left-right consistency,\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., 2017, pp. 270-279.","doi":""},{"text":"Excavating the potential capacity of self-supervised monocular depth estimation. R Peng, R Wang, Y Lai, L Tang, Y Cai, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. Vis15R. Peng, R. Wang, Y. Lai, L. Tang, and Y. Cai, \"Excavating the potential capacity of self-supervised monocular depth estimation,\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., 2021, pp. 15 560-15 569.","doi":""},{"text":"Unsupervised learning of depth and ego-motion from video. T Zhou, M Brown, N Snavely, D G Lowe, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. IEEE Conf. Comput. Vis. Pattern RecognitT. Zhou, M. Brown, N. Snavely, and D. G. Lowe, \"Unsupervised learning of depth and ego-motion from video,\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., 2017, pp. 1851-1858.","doi":""},{"text":"Unsupervised learning of geometry from videos with edge-aware depth-normal consistency. Z Yang, P Wang, W Xu, L Zhao, R Nevatia, Proc. AAAI Conf. AAAI ConfZ. Yang, P. Wang, W. Xu, L. Zhao, and R. Nevatia, \"Unsupervised learning of geometry from videos with edge-aware depth-normal con- sistency,\" in Proc. AAAI Conf. Artif. Intell., 2018, pp. 7493-7500.","doi":""},{"text":"Lego: Learning edge with geometry all at once by watching videos. Z Yang, P Wang, Y Wang, W Xu, R Nevatia, Proc. IEEE Conf. Comput. Vis. IEEE Conf. Comput. VisZ. Yang, P. Wang, Y. Wang, W. Xu, and R. Nevatia, \"Lego: Learning edge with geometry all at once by watching videos,\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., 2018, pp. 225-234.","doi":""},{"text":"Toward hierarchical self-supervised monocular absolute depth estimation for autonomous driving applications. F Xue, G Zhuo, Z Huang, W Fu, Z Wu, M H Ang, Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst. IEEE/RSJ Int. Conf. Intell. Robots SystF. Xue, G. Zhuo, Z. Huang, W. Fu, Z. Wu, and M. H. Ang, \"Toward hierarchical self-supervised monocular absolute depth estimation for autonomous driving applications,\" in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., 2020, pp. 2330-2337.","doi":""},{"text":"Can scale-consistent monocular depth be learned in a self-supervised scaleinvariant manner. L Wang, Y Wang, L Wang, Y Zhan, Y Wang, H Lu, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. VisL. Wang, Y. Wang, L. Wang, Y. Zhan, Y. Wang, and H. Lu, \"Can scale-consistent monocular depth be learned in a self-supervised scale- invariant manner?\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., 2021, pp. 12 727-12 736.","doi":""},{"text":"Selfsupervised monocular depth estimation: Solving the dynamic object problem by semantic guidance. M Klingner, J.-A Termöhlen, J Mikolajczyk, T Fingscheidt, Proc. Eur. Conf. Comput. Vis. Eur. Conf. Comput. VisM. Klingner, J.-A. Termöhlen, J. Mikolajczyk, and T. Fingscheidt, \"Self- supervised monocular depth estimation: Solving the dynamic object problem by semantic guidance,\" in Proc. Eur. Conf. Comput. Vis., 2020, pp. 582-600.","doi":""},{"text":"Self-supervised monocular trained depth estimation using self-attention and discrete disparity volume. A Johnston, G Carneiro, Proc. nullA. Johnston and G. Carneiro, \"Self-supervised monocular trained depth estimation using self-attention and discrete disparity volume,\" in Proc.","doi":""},{"text":"Ieee/Cvf, Conf, Comput. Vis. Pattern Recognit. IEEE/CVF Conf. Comput. Vis. Pattern Recognit., 2020, pp. 4756-4765.","doi":""},{"text":"Fine-grained semantics-aware representation enhancement for self-supervised monocular depth estimation. H Jung, E Park, S Yoo, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. VisH. Jung, E. Park, and S. Yoo, \"Fine-grained semantics-aware represen- tation enhancement for self-supervised monocular depth estimation,\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., 2021, pp. 12 642-12 652.","doi":""},{"text":"Channel-wise attention-based network for self-supervised monocular depth estimation. J Yan, H Zhao, P Bu, Y Jin, Proc. Int. Conf. 3D Vis. Int. Conf. 3D VisJ. Yan, H. Zhao, P. Bu, and Y. Jin, \"Channel-wise attention-based network for self-supervised monocular depth estimation,\" in Proc. Int. Conf. 3D Vis., 2021, pp. 464-473.","doi":""},{"text":"3d packing for self-supervised monocular depth estimation. V Guizilini, R Ambrus, S Pillai, A Raventos, A Gaidon, Proc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit. IEEE/CVF Conf. Comput. Vis. Pattern RecognitV. Guizilini, R. Ambrus, S. Pillai, A. Raventos, and A. Gaidon, \"3d pack- ing for self-supervised monocular depth estimation,\" in Proc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit., 2020, pp. 2485-2494.","doi":""},{"text":"Self-supervised scale recovery for monocular depth and egomotion estimation. B Wagstaff, J Kelly, IEEE/RSJ Int. Conf. Intell. Robots Syst. B. Wagstaff and J. Kelly, \"Self-supervised scale recovery for monocular depth and egomotion estimation,\" in IEEE/RSJ Int. Conf. Intell. Robots Syst., 2020, pp. 2620-2627.","doi":""},{"text":"Digging into self-supervised monocular depth estimation. C Godard, O Mac Aodha, M Firman, G J Brostow, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. VisC. Godard, O. Mac Aodha, M. Firman, and G. J. Brostow, \"Digging into self-supervised monocular depth estimation,\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., 2019, pp. 3828-3838.","doi":""},{"text":"Deep residual learning for image recognition. K He, X Zhang, S Ren, J Sun, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. IEEE Conf. Comput. Vis. Pattern RecognitK. He, X. Zhang, S. Ren, and J. Sun, \"Deep residual learning for image recognition,\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., 2016, pp. 770-778.","doi":""},{"text":"U-net: Convolutional networks for biomedical image segmentation. O Ronneberger, P Fischer, T Brox, Proc. Int. Conf. Med. Image Comput. Comput.-Assisted Intervention. Int. Conf. Med. Image Comput. Comput.-Assisted InterventionChamSpringer International PublishingO. Ronneberger, P. Fischer, and T. Brox, \"U-net: Convolutional networks for biomedical image segmentation,\" in Proc. Int. Conf. Med. Image Comput. Comput.-Assisted Intervention. Cham: Springer International Publishing, 2015, pp. 234-241.","doi":""},{"text":"Probabilistic global scale estimation for monoslam based on generic object detection. E Sucar, J.-B Hayet, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. IEEE Conf. Comput. Vis. Pattern RecognitE. Sucar and J.-B. Hayet, \"Probabilistic global scale estimation for monoslam based on generic object detection,\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit. Workshops, 2017, pp. 48-56.","doi":""},{"text":"Are we ready for autonomous driving? the kitti vision benchmark suite. A Geiger, P Lenz, R Urtasun, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. IEEE Conf. Comput. Vis. Pattern RecognitA. Geiger, P. Lenz, and R. Urtasun, \"Are we ready for autonomous driving? the kitti vision benchmark suite,\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., 2012, pp. 3354-3361.","doi":""},{"text":"Seasondepth: Cross-season monocular depth prediction dataset and benchmark under multiple environments. H Hu, B Yang, Z Qiao, D Zhao, H Wang, arXiv:2011.04408arXiv preprintH. Hu, B. Yang, Z. Qiao, D. Zhao, and H. Wang, \"Seasondepth: Cross-season monocular depth prediction dataset and benchmark under multiple environments,\" arXiv preprint arXiv:2011.04408, 2020.","doi":"arXiv:2011.04408"},{"text":"Deep3d: Fully automatic 2d-to-3d video conversion with deep convolutional neural networks. J Xie, R Girshick, A Farhadi, Proc. Eur. Conf. Comput. Vis. Eur. Conf. Comput. VisJ. Xie, R. Girshick, and A. Farhadi, \"Deep3d: Fully automatic 2d-to- 3d video conversion with deep convolutional neural networks,\" in Proc. Eur. Conf. Comput. Vis., 2016, pp. 842-857.","doi":""},{"text":"Feature-metric loss for selfsupervised learning of depth and egomotion. C Shu, K Yu, Z Duan, K Yang, Proc. Eur. Conf. Comput. Vis. Eur. Conf. Comput. VisC. Shu, K. Yu, Z. Duan, and K. Yang, \"Feature-metric loss for self- supervised learning of depth and egomotion,\" in Proc. Eur. Conf. Comput. Vis., 2020, pp. 572-588.","doi":""},{"text":"Hr-depth: High resolution self-supervised monocular depth estimation. X Lyu, L Liu, M Wang, X Kong, L Liu, Y Liu, X Chen, Y Yuan, Proc. AAAI Conf. AAAI Conf35X. Lyu, L. Liu, M. Wang, X. Kong, L. Liu, Y. Liu, X. Chen, and Y. Yuan, \"Hr-depth: High resolution self-supervised monocular depth estimation,\" in Proc. AAAI Conf. Artif. Intell., vol. 35, no. 3, 2021, pp. 2294-2301.","doi":""},{"text":"Monocular visual odometry scale recovery using geometrical constraint. X Wang, H Zhang, X Yin, M Du, Q Chen, Proc. nullX. Wang, H. Zhang, X. Yin, M. Du, and Q. Chen, \"Monocular visual odometry scale recovery using geometrical constraint,\" in Proc. IEEE Int. Conf. Robot. Autom., 2018, pp. 988-995.","doi":""},{"text":"Monocular depth estimation in new environments with absolute scale. T Roussel, L Van Eycken, T Tuytelaars, Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst. IEEE/RSJ Int. Conf. Intell. Robots SystT. Roussel, L. Van Eycken, and T. Tuytelaars, \"Monocular depth estimation in new environments with absolute scale,\" in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., 2019, pp. 1735-1741.","doi":""},{"text":"Lidartouch: Monocular metric depth estimation with a few-beam lidar. F Bartoccioni, É Zablocki, P Pérez, M Cord, K Alahari, arXiv:2109.03569arXiv preprintF. Bartoccioni,É. Zablocki, P. Pérez, M. Cord, and K. Alahari, \"Lidar- touch: Monocular metric depth estimation with a few-beam lidar,\" arXiv preprint arXiv:2109.03569, 2021.","doi":"arXiv:2109.03569"},{"text":"Attention mechanisms in computer vision: A survey. M.-H Guo, T.-X Xu, J.-J Liu, Z.-N Liu, P.-T Jiang, T.-J Mu, S.-H Zhang, R R Martin, M.-M Cheng, S.-M Hu, Comput. Vis. Media. M.-H. Guo, T.-X. Xu, J.-J. Liu, Z.-N. Liu, P.-T. Jiang, T.-J. Mu, S.- H. Zhang, R. R. Martin, M.-M. Cheng, and S.-M. Hu, \"Attention mechanisms in computer vision: A survey,\" Comput. Vis. Media, Mar 2022.","doi":""},{"text":"Recurrent models of visual attention. V Mnih, N Heess, A Graves, Proc. Adv. Neural Inf. Process. Syst. 27V. Mnih, N. Heess, A. Graves et al., \"Recurrent models of visual attention,\" Proc. Adv. Neural Inf. Process. Syst., vol. 27, pp. 1-9, 2014.","doi":""},{"text":"Squeeze-and-excitation networks. J Hu, L Shen, G Sun, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. IEEE Conf. Comput. Vis. Pattern RecognitJ. Hu, L. Shen, and G. Sun, \"Squeeze-and-excitation networks,\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., June 2018, pp. 7132-7141.","doi":""},{"text":"Cbam: Convolutional block attention module. S Woo, J Park, J.-Y. Lee, I S Kweon, Proc. Eur. Conf. Comput. Vis. Eur. Conf. Comput. VisS. Woo, J. Park, J.-Y. Lee, and I. S. Kweon, \"Cbam: Convolutional block attention module,\" in Proc. Eur. Conf. Comput. Vis., September 2018, pp. 3-19.","doi":""},{"text":"Dual attention network for scene segmentation. J Fu, J Liu, H Tian, Y Li, Y Bao, Z Fang, H Lu, Proc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit. IEEE/CVF Conf. Comput. Vis. Pattern RecognitJ. Fu, J. Liu, H. Tian, Y. Li, Y. Bao, Z. Fang, and H. Lu, \"Dual attention network for scene segmentation,\" in Proc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit., June 2019, pp. 3146-3154.","doi":""},{"text":"An image is worth 16x16 words: Transformers for image recognition at scale. A Dosovitskiy, L Beyer, A Kolesnikov, D Weissenborn, X Zhai, T Unterthiner, M Dehghani, M Minderer, G Heigold, S Gelly, J Uszkoreit, N Houlsby, Proc. Int. Conf. Learn. Represent. Int. Conf. Learn. RepresentA. Dosovitskiy, L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, T. Unterthiner, M. Dehghani, M. Minderer, G. Heigold, S. Gelly, J. Uszkoreit, and N. Houlsby, \"An image is worth 16x16 words: Transformers for image recognition at scale,\" in Proc. Int. Conf. Learn. Represent., 2021.","doi":""},{"text":"Attention is all you need. A Vaswani, N Shazeer, N Parmar, J Uszkoreit, L Jones, A N Gomez, L Kaiser, I Polosukhin, Proc. Adv. Neural Inf. Process. Syst. 30A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, \"Attention is all you need,\" Proc. Adv. Neural Inf. Process. Syst., vol. 30, pp. 5998-6008, 2017.","doi":""},{"text":"Vision transformers for dense prediction. R Ranftl, A Bochkovskiy, V Koltun, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. VisR. Ranftl, A. Bochkovskiy, and V. Koltun, \"Vision transformers for dense prediction,\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., 2021, pp. 12 179-12 188.","doi":""},{"text":"Visual attention network. M.-H Guo, C.-Z Lu, Z.-N Liu, M.-M Cheng, S.-M Hu, arXiv:2202.09741arXiv preprintM.-H. Guo, C.-Z. Lu, Z.-N. Liu, M.-M. Cheng, and S.-M. Hu, \"Visual attention network,\" arXiv preprint arXiv:2202.09741, 2022.","doi":"arXiv:2202.09741"},{"text":"Spatial transformer networks. M Jaderberg, K Simonyan, A Zisserman, Proc. Adv. Neural Inf. Process. Syst. 28M. Jaderberg, K. Simonyan, A. Zisserman et al., \"Spatial transformer networks,\" Proc. Adv. Neural Inf. Process. Syst., vol. 28, 2015.","doi":""},{"text":"Image quality assessment: from error visibility to structural similarity. Z Wang, A C Bovik, H R Sheikh, E P Simoncelli, IEEE Trans. Image Process. 134Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, \"Image quality assessment: from error visibility to structural similarity,\" IEEE Trans. Image Process., vol. 13, no. 4, pp. 600-612, 2004.","doi":""},{"text":"Imagenet: A large-scale hierarchical image database. J Deng, W Dong, R Socher, L.-J Li, K Li, L Fei-Fei, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. IEEE Conf. Comput. Vis. Pattern RecognitJ. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei, \"Imagenet: A large-scale hierarchical image database,\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., 2009, pp. 248-255.","doi":""},{"text":"Attention-based context aggregation network for monocular depth estimation. Y Chen, H Zhao, Z Hu, J Peng, Int. J. Mach. Learn. Cybern. 126Y. Chen, H. Zhao, Z. Hu, and J. Peng, \"Attention-based context aggre- gation network for monocular depth estimation,\" Int. J. Mach. Learn. Cybern., vol. 12, no. 6, pp. 1583-1596, 2021.","doi":""},{"text":"Edgeconv with attention module for monocular depth estimation. M Lee, S Hwang, C Park, S Lee, Proc. IEEE/CVF Winter Conf. Appl. of Comput. Vis. (WACV). IEEE/CVF Winter Conf. Appl. of Comput. Vis. (WACV)M. Lee, S. Hwang, C. Park, and S. Lee, \"Edgeconv with attention module for monocular depth estimation,\" in Proc. IEEE/CVF Winter Conf. Appl. of Comput. Vis. (WACV), January 2022, pp. 2858-2867.","doi":""}],"string":"[\n {\n \"text\": \"Is pseudolidar needed for monocular 3d object detection. D Park, R Ambrus, V Guizilini, J Li, A Gaidon, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. VisD. Park, R. Ambrus, V. Guizilini, J. Li, and A. Gaidon, \\\"Is pseudo- lidar needed for monocular 3d object detection?\\\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., October 2021, pp. 3142-3152.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Mgnet: Monocular geometric scene understanding for autonomous driving. M Schön, M Buchholz, K Dietmayer, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. VisM. Schön, M. Buchholz, and K. Dietmayer, \\\"Mgnet: Monocular geo- metric scene understanding for autonomous driving,\\\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., 2021, pp. 15 804-15 815.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J-mod 2: Joint monocular obstacle detection and depth estimation. M Mancini, G Costante, P Valigi, T A Ciarfuglia, IEEE Robot. Autom. Lett. 33M. Mancini, G. Costante, P. Valigi, and T. A. Ciarfuglia, \\\"J-mod 2: Joint monocular obstacle detection and depth estimation,\\\" IEEE Robot. Autom. Lett., vol. 3, no. 3, pp. 1490-1497, 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Sparsity invariant cnns. J Uhrig, N Schneider, L Schneider, U Franke, T Brox, A Geiger, Proc. Int. Conf. 3D Vis. Int. Conf. 3D VisJ. Uhrig, N. Schneider, L. Schneider, U. Franke, T. Brox, and A. Geiger, \\\"Sparsity invariant cnns,\\\" in Proc. Int. Conf. 3D Vis., 2017, pp. 11-20.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Multiple view geometry in computer vision. R Hartley, A Zisserman, Cambridge Univ. PressR. Hartley and A. Zisserman, Multiple view geometry in computer vision. Cambridge Univ. Press, 2003.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Depth map prediction from a single image using a multi-scale deep network. D Eigen, C Puhrsch, R Fergus, Proc. Adv. Neural Inf. Process. Syst. 27D. Eigen, C. Puhrsch, and R. Fergus, \\\"Depth map prediction from a single image using a multi-scale deep network,\\\" Proc. Adv. Neural Inf. Process. Syst., vol. 27, pp. 2366-2374, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Predicting depth, surface normals and semantic labels with a common multi-scale convolutional architecture. D Eigen, R Fergus, Proc. IEEE Int. Conf. Comput. Vis. IEEE Int. Conf. Comput. VisD. Eigen and R. Fergus, \\\"Predicting depth, surface normals and semantic labels with a common multi-scale convolutional architecture,\\\" in Proc. IEEE Int. Conf. Comput. Vis., December 2015, pp. 2650-2658.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Enforcing geometric constraints of virtual normal for depth prediction. W Yin, Y Liu, C Shen, Y Yan, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. VisW. Yin, Y. Liu, C. Shen, and Y. Yan, \\\"Enforcing geometric constraints of virtual normal for depth prediction,\\\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., 2019, pp. 5684-5693.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"From big to small: Multi-scale local planar guidance for monocular depth estimation. J H Lee, M.-K Han, D W Ko, I H Suh, arXiv:1907.10326arXiv preprintJ. H. Lee, M.-K. Han, D. W. Ko, and I. H. Suh, \\\"From big to small: Multi-scale local planar guidance for monocular depth estimation,\\\" arXiv preprint arXiv:1907.10326, 2019.\",\n \"doi\": \"arXiv:1907.10326\"\n },\n {\n \"text\": \"Unsupervised cnn for single view depth estimation: Geometry to the rescue. R Garg, V K Bg, G Carneiro, I Reid, Proc. Eur. Conf. Comput. Vis. Eur. Conf. Comput. VisR. Garg, V. K. Bg, G. Carneiro, and I. Reid, \\\"Unsupervised cnn for single view depth estimation: Geometry to the rescue,\\\" in Proc. Eur. Conf. Comput. Vis., 2016, pp. 740-756.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Unsupervised monocular depth estimation with left-right consistency. C Godard, O Mac Aodha, G J Brostow, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. IEEE Conf. Comput. Vis. Pattern RecognitC. Godard, O. Mac Aodha, and G. J. Brostow, \\\"Unsupervised monocular depth estimation with left-right consistency,\\\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., 2017, pp. 270-279.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Excavating the potential capacity of self-supervised monocular depth estimation. R Peng, R Wang, Y Lai, L Tang, Y Cai, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. Vis15R. Peng, R. Wang, Y. Lai, L. Tang, and Y. Cai, \\\"Excavating the potential capacity of self-supervised monocular depth estimation,\\\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., 2021, pp. 15 560-15 569.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Unsupervised learning of depth and ego-motion from video. T Zhou, M Brown, N Snavely, D G Lowe, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. IEEE Conf. Comput. Vis. Pattern RecognitT. Zhou, M. Brown, N. Snavely, and D. G. Lowe, \\\"Unsupervised learning of depth and ego-motion from video,\\\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., 2017, pp. 1851-1858.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Unsupervised learning of geometry from videos with edge-aware depth-normal consistency. Z Yang, P Wang, W Xu, L Zhao, R Nevatia, Proc. AAAI Conf. AAAI ConfZ. Yang, P. Wang, W. Xu, L. Zhao, and R. Nevatia, \\\"Unsupervised learning of geometry from videos with edge-aware depth-normal con- sistency,\\\" in Proc. AAAI Conf. Artif. Intell., 2018, pp. 7493-7500.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Lego: Learning edge with geometry all at once by watching videos. Z Yang, P Wang, Y Wang, W Xu, R Nevatia, Proc. IEEE Conf. Comput. Vis. IEEE Conf. Comput. VisZ. Yang, P. Wang, Y. Wang, W. Xu, and R. Nevatia, \\\"Lego: Learning edge with geometry all at once by watching videos,\\\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., 2018, pp. 225-234.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Toward hierarchical self-supervised monocular absolute depth estimation for autonomous driving applications. F Xue, G Zhuo, Z Huang, W Fu, Z Wu, M H Ang, Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst. IEEE/RSJ Int. Conf. Intell. Robots SystF. Xue, G. Zhuo, Z. Huang, W. Fu, Z. Wu, and M. H. Ang, \\\"Toward hierarchical self-supervised monocular absolute depth estimation for autonomous driving applications,\\\" in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., 2020, pp. 2330-2337.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Can scale-consistent monocular depth be learned in a self-supervised scaleinvariant manner. L Wang, Y Wang, L Wang, Y Zhan, Y Wang, H Lu, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. VisL. Wang, Y. Wang, L. Wang, Y. Zhan, Y. Wang, and H. Lu, \\\"Can scale-consistent monocular depth be learned in a self-supervised scale- invariant manner?\\\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., 2021, pp. 12 727-12 736.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Selfsupervised monocular depth estimation: Solving the dynamic object problem by semantic guidance. M Klingner, J.-A Termöhlen, J Mikolajczyk, T Fingscheidt, Proc. Eur. Conf. Comput. Vis. Eur. Conf. Comput. VisM. Klingner, J.-A. Termöhlen, J. Mikolajczyk, and T. Fingscheidt, \\\"Self- supervised monocular depth estimation: Solving the dynamic object problem by semantic guidance,\\\" in Proc. Eur. Conf. Comput. Vis., 2020, pp. 582-600.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Self-supervised monocular trained depth estimation using self-attention and discrete disparity volume. A Johnston, G Carneiro, Proc. nullA. Johnston and G. Carneiro, \\\"Self-supervised monocular trained depth estimation using self-attention and discrete disparity volume,\\\" in Proc.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Ieee/Cvf, Conf, Comput. Vis. Pattern Recognit. IEEE/CVF Conf. Comput. Vis. Pattern Recognit., 2020, pp. 4756-4765.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fine-grained semantics-aware representation enhancement for self-supervised monocular depth estimation. H Jung, E Park, S Yoo, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. VisH. Jung, E. Park, and S. Yoo, \\\"Fine-grained semantics-aware represen- tation enhancement for self-supervised monocular depth estimation,\\\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., 2021, pp. 12 642-12 652.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Channel-wise attention-based network for self-supervised monocular depth estimation. J Yan, H Zhao, P Bu, Y Jin, Proc. Int. Conf. 3D Vis. Int. Conf. 3D VisJ. Yan, H. Zhao, P. Bu, and Y. Jin, \\\"Channel-wise attention-based network for self-supervised monocular depth estimation,\\\" in Proc. Int. Conf. 3D Vis., 2021, pp. 464-473.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"3d packing for self-supervised monocular depth estimation. V Guizilini, R Ambrus, S Pillai, A Raventos, A Gaidon, Proc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit. IEEE/CVF Conf. Comput. Vis. Pattern RecognitV. Guizilini, R. Ambrus, S. Pillai, A. Raventos, and A. Gaidon, \\\"3d pack- ing for self-supervised monocular depth estimation,\\\" in Proc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit., 2020, pp. 2485-2494.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Self-supervised scale recovery for monocular depth and egomotion estimation. B Wagstaff, J Kelly, IEEE/RSJ Int. Conf. Intell. Robots Syst. B. Wagstaff and J. Kelly, \\\"Self-supervised scale recovery for monocular depth and egomotion estimation,\\\" in IEEE/RSJ Int. Conf. Intell. Robots Syst., 2020, pp. 2620-2627.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Digging into self-supervised monocular depth estimation. C Godard, O Mac Aodha, M Firman, G J Brostow, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. VisC. Godard, O. Mac Aodha, M. Firman, and G. J. Brostow, \\\"Digging into self-supervised monocular depth estimation,\\\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., 2019, pp. 3828-3838.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Deep residual learning for image recognition. K He, X Zhang, S Ren, J Sun, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. IEEE Conf. Comput. Vis. Pattern RecognitK. He, X. Zhang, S. Ren, and J. Sun, \\\"Deep residual learning for image recognition,\\\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., 2016, pp. 770-778.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"U-net: Convolutional networks for biomedical image segmentation. O Ronneberger, P Fischer, T Brox, Proc. Int. Conf. Med. Image Comput. Comput.-Assisted Intervention. Int. Conf. Med. Image Comput. Comput.-Assisted InterventionChamSpringer International PublishingO. Ronneberger, P. Fischer, and T. Brox, \\\"U-net: Convolutional networks for biomedical image segmentation,\\\" in Proc. Int. Conf. Med. Image Comput. Comput.-Assisted Intervention. Cham: Springer International Publishing, 2015, pp. 234-241.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Probabilistic global scale estimation for monoslam based on generic object detection. E Sucar, J.-B Hayet, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. IEEE Conf. Comput. Vis. Pattern RecognitE. Sucar and J.-B. Hayet, \\\"Probabilistic global scale estimation for monoslam based on generic object detection,\\\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit. Workshops, 2017, pp. 48-56.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Are we ready for autonomous driving? the kitti vision benchmark suite. A Geiger, P Lenz, R Urtasun, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. IEEE Conf. Comput. Vis. Pattern RecognitA. Geiger, P. Lenz, and R. Urtasun, \\\"Are we ready for autonomous driving? the kitti vision benchmark suite,\\\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., 2012, pp. 3354-3361.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Seasondepth: Cross-season monocular depth prediction dataset and benchmark under multiple environments. H Hu, B Yang, Z Qiao, D Zhao, H Wang, arXiv:2011.04408arXiv preprintH. Hu, B. Yang, Z. Qiao, D. Zhao, and H. Wang, \\\"Seasondepth: Cross-season monocular depth prediction dataset and benchmark under multiple environments,\\\" arXiv preprint arXiv:2011.04408, 2020.\",\n \"doi\": \"arXiv:2011.04408\"\n },\n {\n \"text\": \"Deep3d: Fully automatic 2d-to-3d video conversion with deep convolutional neural networks. J Xie, R Girshick, A Farhadi, Proc. Eur. Conf. Comput. Vis. Eur. Conf. Comput. VisJ. Xie, R. Girshick, and A. Farhadi, \\\"Deep3d: Fully automatic 2d-to- 3d video conversion with deep convolutional neural networks,\\\" in Proc. Eur. Conf. Comput. Vis., 2016, pp. 842-857.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Feature-metric loss for selfsupervised learning of depth and egomotion. C Shu, K Yu, Z Duan, K Yang, Proc. Eur. Conf. Comput. Vis. Eur. Conf. Comput. VisC. Shu, K. Yu, Z. Duan, and K. Yang, \\\"Feature-metric loss for self- supervised learning of depth and egomotion,\\\" in Proc. Eur. Conf. Comput. Vis., 2020, pp. 572-588.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hr-depth: High resolution self-supervised monocular depth estimation. X Lyu, L Liu, M Wang, X Kong, L Liu, Y Liu, X Chen, Y Yuan, Proc. AAAI Conf. AAAI Conf35X. Lyu, L. Liu, M. Wang, X. Kong, L. Liu, Y. Liu, X. Chen, and Y. Yuan, \\\"Hr-depth: High resolution self-supervised monocular depth estimation,\\\" in Proc. AAAI Conf. Artif. Intell., vol. 35, no. 3, 2021, pp. 2294-2301.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Monocular visual odometry scale recovery using geometrical constraint. X Wang, H Zhang, X Yin, M Du, Q Chen, Proc. nullX. Wang, H. Zhang, X. Yin, M. Du, and Q. Chen, \\\"Monocular visual odometry scale recovery using geometrical constraint,\\\" in Proc. IEEE Int. Conf. Robot. Autom., 2018, pp. 988-995.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Monocular depth estimation in new environments with absolute scale. T Roussel, L Van Eycken, T Tuytelaars, Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst. IEEE/RSJ Int. Conf. Intell. Robots SystT. Roussel, L. Van Eycken, and T. Tuytelaars, \\\"Monocular depth estimation in new environments with absolute scale,\\\" in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., 2019, pp. 1735-1741.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Lidartouch: Monocular metric depth estimation with a few-beam lidar. F Bartoccioni, É Zablocki, P Pérez, M Cord, K Alahari, arXiv:2109.03569arXiv preprintF. Bartoccioni,É. Zablocki, P. Pérez, M. Cord, and K. Alahari, \\\"Lidar- touch: Monocular metric depth estimation with a few-beam lidar,\\\" arXiv preprint arXiv:2109.03569, 2021.\",\n \"doi\": \"arXiv:2109.03569\"\n },\n {\n \"text\": \"Attention mechanisms in computer vision: A survey. M.-H Guo, T.-X Xu, J.-J Liu, Z.-N Liu, P.-T Jiang, T.-J Mu, S.-H Zhang, R R Martin, M.-M Cheng, S.-M Hu, Comput. Vis. Media. M.-H. Guo, T.-X. Xu, J.-J. Liu, Z.-N. Liu, P.-T. Jiang, T.-J. Mu, S.- H. Zhang, R. R. Martin, M.-M. Cheng, and S.-M. Hu, \\\"Attention mechanisms in computer vision: A survey,\\\" Comput. Vis. Media, Mar 2022.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Recurrent models of visual attention. V Mnih, N Heess, A Graves, Proc. Adv. Neural Inf. Process. Syst. 27V. Mnih, N. Heess, A. Graves et al., \\\"Recurrent models of visual attention,\\\" Proc. Adv. Neural Inf. Process. Syst., vol. 27, pp. 1-9, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Squeeze-and-excitation networks. J Hu, L Shen, G Sun, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. IEEE Conf. Comput. Vis. Pattern RecognitJ. Hu, L. Shen, and G. Sun, \\\"Squeeze-and-excitation networks,\\\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., June 2018, pp. 7132-7141.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cbam: Convolutional block attention module. S Woo, J Park, J.-Y. Lee, I S Kweon, Proc. Eur. Conf. Comput. Vis. Eur. Conf. Comput. VisS. Woo, J. Park, J.-Y. Lee, and I. S. Kweon, \\\"Cbam: Convolutional block attention module,\\\" in Proc. Eur. Conf. Comput. Vis., September 2018, pp. 3-19.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dual attention network for scene segmentation. J Fu, J Liu, H Tian, Y Li, Y Bao, Z Fang, H Lu, Proc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit. IEEE/CVF Conf. Comput. Vis. Pattern RecognitJ. Fu, J. Liu, H. Tian, Y. Li, Y. Bao, Z. Fang, and H. Lu, \\\"Dual attention network for scene segmentation,\\\" in Proc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit., June 2019, pp. 3146-3154.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An image is worth 16x16 words: Transformers for image recognition at scale. A Dosovitskiy, L Beyer, A Kolesnikov, D Weissenborn, X Zhai, T Unterthiner, M Dehghani, M Minderer, G Heigold, S Gelly, J Uszkoreit, N Houlsby, Proc. Int. Conf. Learn. Represent. Int. Conf. Learn. RepresentA. Dosovitskiy, L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, T. Unterthiner, M. Dehghani, M. Minderer, G. Heigold, S. Gelly, J. Uszkoreit, and N. Houlsby, \\\"An image is worth 16x16 words: Transformers for image recognition at scale,\\\" in Proc. Int. Conf. Learn. Represent., 2021.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Attention is all you need. A Vaswani, N Shazeer, N Parmar, J Uszkoreit, L Jones, A N Gomez, L Kaiser, I Polosukhin, Proc. Adv. Neural Inf. Process. Syst. 30A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, \\\"Attention is all you need,\\\" Proc. Adv. Neural Inf. Process. Syst., vol. 30, pp. 5998-6008, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Vision transformers for dense prediction. R Ranftl, A Bochkovskiy, V Koltun, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. VisR. Ranftl, A. Bochkovskiy, and V. Koltun, \\\"Vision transformers for dense prediction,\\\" in Proc. IEEE/CVF Int. Conf. Comput. Vis., 2021, pp. 12 179-12 188.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Visual attention network. M.-H Guo, C.-Z Lu, Z.-N Liu, M.-M Cheng, S.-M Hu, arXiv:2202.09741arXiv preprintM.-H. Guo, C.-Z. Lu, Z.-N. Liu, M.-M. Cheng, and S.-M. Hu, \\\"Visual attention network,\\\" arXiv preprint arXiv:2202.09741, 2022.\",\n \"doi\": \"arXiv:2202.09741\"\n },\n {\n \"text\": \"Spatial transformer networks. M Jaderberg, K Simonyan, A Zisserman, Proc. Adv. Neural Inf. Process. Syst. 28M. Jaderberg, K. Simonyan, A. Zisserman et al., \\\"Spatial transformer networks,\\\" Proc. Adv. Neural Inf. Process. Syst., vol. 28, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Image quality assessment: from error visibility to structural similarity. Z Wang, A C Bovik, H R Sheikh, E P Simoncelli, IEEE Trans. Image Process. 134Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, \\\"Image quality assessment: from error visibility to structural similarity,\\\" IEEE Trans. Image Process., vol. 13, no. 4, pp. 600-612, 2004.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Imagenet: A large-scale hierarchical image database. J Deng, W Dong, R Socher, L.-J Li, K Li, L Fei-Fei, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. IEEE Conf. Comput. Vis. Pattern RecognitJ. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei, \\\"Imagenet: A large-scale hierarchical image database,\\\" in Proc. IEEE Conf. Comput. Vis. Pattern Recognit., 2009, pp. 248-255.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Attention-based context aggregation network for monocular depth estimation. Y Chen, H Zhao, Z Hu, J Peng, Int. J. Mach. Learn. Cybern. 126Y. Chen, H. Zhao, Z. Hu, and J. Peng, \\\"Attention-based context aggre- gation network for monocular depth estimation,\\\" Int. J. Mach. Learn. Cybern., vol. 12, no. 6, pp. 1583-1596, 2021.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Edgeconv with attention module for monocular depth estimation. M Lee, S Hwang, C Park, S Lee, Proc. IEEE/CVF Winter Conf. Appl. of Comput. Vis. (WACV). IEEE/CVF Winter Conf. Appl. of Comput. Vis. (WACV)M. Lee, S. Hwang, C. Park, and S. Lee, \\\"Edgeconv with attention module for monocular depth estimation,\\\" in Proc. IEEE/CVF Winter Conf. Appl. of Comput. Vis. (WACV), January 2022, pp. 2858-2867.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":486,"cells":{"text":{"kind":"string","value":"\nEnhancing synchronizability of weighted dynamical networks using betweenness centrality\n\n\nMahdi Jalili \nSchool of Computer and Communication Sciences\nLaboratory of Nonlinear Systems\nEcole Polytechnique Fédérale de Lausanne\n1015LausanneCHSwitzerland\n\nAli Ajdari Rad \nSchool of Computer and Communication Sciences\nLaboratory of Nonlinear Systems\nEcole Polytechnique Fédérale de Lausanne\n1015LausanneCHSwitzerland\n\nMartin Hasler \nSchool of Computer and Communication Sciences\nLaboratory of Nonlinear Systems\nEcole Polytechnique Fédérale de Lausanne\n1015LausanneCHSwitzerland\n\nEnhancing synchronizability of weighted dynamical networks using betweenness centrality\n1\nBy considering the eigenratio of the Laplacian of the connection graph as synchronizability measure, we propose a procedure for weighting dynamical networks to enhance their synchronizability. The method is based on node and edge betweenness centrality measures and is tested on artificially constructed scale-free, Watts-Strogatz and random networks as well as on some real-world graphs. It is also numerically shown that the same procedure could be used to enhance the phase synchronizability of networks of nonidentical oscillators.PACS NUMBERS05.45.Xt (Synchronization; coupled oscillators), 89.75.-k (Complex systems).\n\nsuch complex networks on the one hand and dynamics located on their nodes on the other hand has attracted a great deal of attention [3][4][5]. Much of this interest is motivated by the fact that many real-world networks share some common structural properties such as small-world [6] and/or scale-free [7] attributes.\n\nIn recent years an avalanche of studies on the most conspicuous form of collective behavior, synchronization, has appeared [8]. Fundamental assumption of the most of the works in this field is that the individual dynamical systems are diffusively coupled with uniform strength over unweighted networks, but most relevant dynamical networks are inherently weighted and directed such as brain networks [9], ecological systems [10], traffic load of a road [11], social networks [12], metabolic networks [13] and technological networks [1,14]. Thus, a natural question arises, namely: \"Given a network, how one can assign proper connection weights to enhance its synchronizability?\" This may provide us with insights into the behavior of real-world complex dynamical networks and guide us in designing large artificial networks. In technological networks with desirable synchronizability, assigning the appropriate interaction weights between dynamical units is important [14]. It has been recently shown that networks with properly assigned weights can be distinctly more synchronizable than unweighted networks [15].\n\nIn this paper, we give an algorithm for assigning connection weights to enhance the synchronizability of dynamical networks. Starting with a connected undirected and unweighted network, and by considering its local and global structural properties such as degree, node and edge betweenness centralities, we end up with an asymmetric weighted network with enhanced synchronizability. Compared to the other methods in this context [16][17][18][19][20], we give evidence that the proposed method leads to higher synchronizability for a class of scale-free, Watts-Strogatz and random networks as well as for many real-world complex networks. We also show that the same weighting procedure enhances the phase synchronizability of coupled nonidentical dynamical systems.\n\n\nII. WEIGHTING ALGORITHM\n\nLet us start with a dynamical network of N linearly coupled identical systems with the following equations of the motion The variational equations of manifold synchronized solution ( )\n( ) ( ) 1 , 1, , N i i i j j j g i N σ = = − = ∑ x F x H x & K ,(1)( ) ( ), i t t i = ∀ x s can be diagonalized into N blocks of the form ( ) ( ) ( ) i i i i D s t σλ = − F H & ζ ζ ζ , where i\nλ are the eigenvalues of G , ordered as 0 = λ 1 ≤ λ 2 ≤ … ≤ λ N and λ 1 is associated with the synchronized manifold. In the sequel, we suppose that the connection graph is connected, which implies the strict inequality 0 = λ 1 < λ 2 .\n\nThe largest Lyapunov exponent of the above variational equation, Λ(σλ i ), called masterstability-function [21], gives a sufficient condition for the local stability of the synchronization manifold: if the synchronization manifold is locally stable we must have Λ(σλ i ) < 0 , i = 2,…,N.\n\nFor a number of systems such as x-coupled Rössler oscillators, the master-stability-function is negative only within a bounded interval (ν 1 , ν 2 ) [21]. Requiring all coupling strengths lie within such an interval, i.e. ν 1 < σλ 2 ≤ … ≤ σλ N < ν 2 , leads to the following condition for the local stability of the synchronization manifold: λ N /λ 2 < ν 2 /ν 1 . The left-hand side of the inequality depends solely on the structure of the graph, while the right-hand side depends on the dynamics of the individual systems and on the coupling configuration. There is a number of interpretations for synchronizability of dynamical networks [15]. We adopt here the following interpretation: the larger the range of parameters of the individual dynamical systems that allows for synchronization, the better the synchronizability of the network. This relates the synchronizability to the eigenratio λ N /λ 2 , and concludes that the smaller the eigenratio λ N /λ 2 of a network, the better its synchronizability [22].\n\nIn general, there are two possible ways to enhance the synchronizability of dynamical networks: rewiring of the links [23] and/or assigning proper weights for the existing links [15].\n\nFor many applications it is not possible to change the network topology and the only option to enhance the synchronizability is weighting the links. In Networks with good synchronizability, couplings between the nodes are neither necessarily uniform nor symmetric. A very first attempt to assign the proper connection weights for enhancing the synchronizability was proposed in [18,19], where the coupling on the right hand side of the equation (1) was taken as\n( ) ( ) / i i i j j N k β σ ∈ −\n∑ H x x , with k i the degree of node i and N i the set of neighbors of node i. The value of β = 1 was found to be optimal in [19] for synchronizability. Further enhancement of the synchronizability was achieved by scaling the weight of each edge by its load [17], where the coupling takes the form as ( )\n( ) / i i ij ij i j j N j N β β σ ρ ρ ∈ ∈ − ∑ ∑ H x x ,\nwhere ρ ij is the load of the edge e ij between the i-th and j-th node. The load (also known as edge betweenness centrality) of e ij is defined as\n( ) ( ) / ij pq ij pq p q e ρ ≠ = Γ Γ ∑ ,\nwhere Γ pq is the number of shortest paths from the p-th to the q-th node and Γ pq (e ij ) is the number of these paths making use of e ij . The optimal condition β = 1 was found for synchronizability [17] that itself performs better than the optimal case of [19]. In this way, not only the local structural information but also the effects of network structure at a global level are taken into account. Very recently, by considering the concepts of gradient networks, another weighting algorithm as ( )\n( ) / i i j j i j j N j N k k β β σ ∈ ∈ − ∑ ∑\nH x x has been proposed [20].\n\nHere we show that further enhancement of synchronizability can be achieved by considering not only connection loads but also node betweenness centralities. Node betweenness centrality i C is a centrality measure of the i-th node in a graph, which counts the number of shortest paths making use of that node (except shortest paths between the i -th node and other nodes) [24]. More precisely,\n( ) ( ) / i p q p q p i q C i ≠ ≠ = Γ Γ ∑ ,\nwhere Γ pq is the number of shortest paths from the p-th to the q-th node and Γ pq (i) is the number of these shortest paths making use of the ith node. In the weighting procedure we propose, the weight of each edge will be a function of its load and the betweenness centrality of the tail node. The resulting weighted network becomes directed; the links go from the head nodes to the tail nodes. The weight of an edge should essentially be proportional to the betweenness centrality of the edge as proposed in [17]. The dependence of the weight to the betweenness centrality of the tail node is also straight forward.\n\nNodes with high values of betweenness centrality can be regarded as hub nodes, i.e. many shortest paths make use of them. Therefore, it is reasonable to increase the weight of the links ending to these nodes. More precisely, the network equations read\n( ) ( ) ( ) ( ) ; 1, 2,..., i i i i j i j i j j N j i j j N C i N C α α σ ε ρ ε ρ ∈ ∈ = − + − = + ∑ ∑ x F x H x x & ,(2)\nwhere α is a real tunable parameter, and ε is a small positive value to make (ε + C j α ) > 0 (some nodes may have betweenness centrality equal to zero). Here, we take ε = 1.\n\nBy this construction of connection weights, the diagonal elements of G are always normalized to one, thus preventing the coupling to be arbitrary large or small. Although G becomes asymmetric for any value of α, it can be written as G = D l WD r , where W is a zero row-\nsum matrix with off-diagonal elements W ij = -ρ ij , ( ) 1 1 diag 1/ , ,1/ N l j N j j N j N D ρ ρ ∈ ∈ = ∑ ∑ K and ( ) ( ) ( ) 1 diag , , r N D C C α α ε ε = + + K\n. It can easily be shown that the eigenvalues of G, λ i (i = 1,…,N), are the same as the eigenvalues of D r 1/2 D l 1/2 WD l 1/2 D r 1/2 , i.e. real and non-negative with smallest eigenvalue as λ 1 = 0. For this case, Gerschgorin circle theorem [25] guarantees that 0 <\nλ 2 ≤ … ≤ λ N ≤ 2.\nAnother important term concerning our procedure is the effect of different values of α. It is worth mentioning that the case with α = 0 corresponds to the optimal situation proposed in [17], which itself has the optimal case of [18,19] as a special case. For large absolute values of α, it may happen that the resulting weighted network is approximately disconnected [17] and thus λ 2 is close to zero; therefore, we limit ourselves to α with small absolute values.\n\n\nIII. NUMERICAL SIMULATIONS\n\n\nA. Performance in artificially constructed networks\n\nBy sweeping α and calculating the values of λ N /λ 2 we can study the synchronizability profile of dynamical networks with different topological properties such as scale-free, Watts-Strogatz and random networks. We will construct scale-free networks using an algorithm proposed in [17], which itself is a generalization of the preferential attachment growing procedure λ N /λ 2 is rapidly decreasing, i.e. synchronizability is enhanced. Note that α = 0 recovers the optimal condition of [17], which itself has the optimal situation of [19] as a special case. For m = 2, the situation is somewhat different; for small values of B, λ N /λ 2 is decreased by increasing α.\n\nHowever, for larger values of B (less heterogeneity in the degree distribution), there is a local minimum in α ~ 0.5, and then by increasing α, the eigenratio is also increased and by further increasing α over a value around 1, the eigenratio starts decreasing. Although α = 1 is not the exact optimal point, to avoid the network from being disconnected (high values of α may lead the network to be disconnected), we consider α = 1 for weighting the scale-free networks in order to enhance their synchronizability. Considering node betweenness centrality in addition to edge load makes the resulting weighted network more homogeneous and thus enhances its synchronizability.\n\n\nFIG. 1.\n\nFor comparison, we have also applied our weighting procedure to Watts-Strogatz networks that exhibit more homogeneity in the network structure than scale-free networks. We consider Fig. 2b and Fig. 2d for N = 1000 and N = 2000, respectively. As it is seen, there is a clear optimum in α ~ 1. Also, the effect of the Watts-Strogatz phenomenon on the synchronizability of the network is clearly seen from these graphs,\n\ni.e. the synchronizability of the network is greatly enhanced by introducing some rewirings.\n\n\nFIG. 2.\n\nHere we also report the behavior of the synchronizability of the weighted networks with α = 1 as a function of network size N. Fig. 3a shows the results for scale-free networks with different values of B and random networks, all the cases with m = 1. As expected, the propensity of synchronization for weighted scale-free with lower values of B is better regardless of N [16].\n\nAlso, weighted scale-free networks exhibit better synchronizability than random networks. For m = 2 the profile of λ N /λ 2 as a function of N is depicted in Fig. 3b, where again weighted scale-free networks show better synchronizability than Watts-Strogatz and random networks. It also illustrates the fact that synchronizability of the weighted networks is almost independent of network size. Indeed, the average degree seems to be the only important factor affecting the synchronizability of the weighted networks. \n\n\nB. Performance in real-world networks\n\nAlthough we showed that applying (2) for weighting the edges of dynamical networks greatly enhances the synchronizability of the network, i.e. reduces the eigenratio λ N /λ 2 , many realworld networks cannot be simply modeled by these scale-free, Watts-Strogatz or random network models. Real-world networks may possess a number of complex topological properties, which can make it difficult to construct a model that mimics all of them. Thus, we consider some available real-world undirected networks 1 and apply the proposed weighting algorithm (2) to study the behavior of λ N /λ 2 in the resulting weighted networks. We consider some real-world networks including protein structure network [26], network of power grid [6], US airport network 2 [27], Email network [28], protein-protein interaction network [29], yeast protein interaction network [30], Internet on the level of autonomous system [31], and the network of coauthorship [32]. Table I summarizes the results. For all networks, our proposed weighting algorithm is the best. Let us remark that since for the algorithm of [20] there is no optimal value of β, thus for each network we adopt the least λ N /λ 2 among the cases with β = 1, β = 2, and β = 3. For the methods of [18] and [17] we used the optimal condition β = 1 and for our proposed method we used α = 1. Since the method of [18] is a simple scaling based on the degree of the nodes, it is incapable of capturing all the useful information, and thus, its performance is always worse than that of the one proposed by [17] that indeed considered the edge betweenness centrality as well as scaling based on the nodes. The method proposed in [20] uses the degree of nodes in a different way, and its performance is not always better than [17]. Indeed, it can be seen that by increase of the heterogeneity of the network, i.e. increase of the standard deviation of the node-degrees, the algorithm of [20] performs better than [17]. Our proposed algorithm is an intelligent extension of [17]; to consider the heterogeneity of the network, it considers the node betweenness centrality in addition to the edge betweenness centrality. Thus, it always outperforms to [17]. However, for networks with high levels of heterogeneity, [20] gives results better than [17] and closed to our results. \n\n\nC. Enhancing phase synchronizability in coupled nonidentical dynamical systems\n\nThe rational behind taking the eigenratio λ N /λ 2 as a synchronizability measure of dynamical networks is the master-stability-function [21]. Since the theory of the master-stability-function was developed for local stability of synchronization of identical dynamical systems, it can not be directly applied to coupled nonidentical systems. However, coupled nonidentical dynamical systems may exhibit some weaker types of synchronization such as phase synchronization [33].\n\nHere, we study the collective behavior of nonidentical Rössler oscillators [34] on scale-free, Watts-Strogatz and random networks to study how much the proposed weighting algorithm can improve the degree of phase synchronization. The dynamics of motion is governed by (2) [17]. Here i ω is the natural frequency of the i-th system that is randomly chosen from a Gaussian distribution with mean value ω mean = 1 and standard deviation Δω = 0.03.\ni i i i i i i i i i b c a b c a ω ω = ⎡− − + + − ⎤ ⎣ ⎦ F x , and ( ) a = H x\nTo study the phase synchronization among coupled oscillators one can monitor the order parameter [8,33] ( ) ( ) results in better phase synchronizability than the case with α = 0 (dashed lines). This improvement is well pronounced for the case with m = 1 (Fig.4a). These results confirm that the master-stability-function gives also valid information for nonidentical oscillators. Note that synchronizability is improved if the σ range for which Ф ≈ 1 increases.\n\n\nFIG. 4.\n\n\nIV. CONCLUSIONS\n\nBy considering the eigenratio λ N /λ 2 of the Laplacian of the connection as the synchronizability measure, we proposed a procedure for assigning proper connection weights to enhance the synchronizability of dynamical networks. To form the weights, we used the information of the node and the edge betweenness centrality measures. The algorithm was tested on artificially constructed networks such as scale-free, Watts-Strogatz and random networks as well as on some real-world networks. This method was also powerful in enhancing the phase synchronizability in networks of nonidentical dynamical systems. size N, average node-degree <k>, and standard deviation of node-degrees std(k), respectively.\n\n\nACKNOWLEDGMENTS\n\nFifth to eighth columns: the eigenratio λ N /λ 2 using different weighting algorithms.\n\nReal-world networks N <k> std(k) λ N /λ 2 [18] λ N /λ 2 [17] λ N /λ 2 [20] λ N /λ 2 using (2) \n\nFig. 1\n1introduced in[7]. Namely, starting with a network of m + 1 all-to-all connected nodes, at each step a new node is added with m links that are connected to node i with probability k i is the degree of the node and B a tunable real parameter controlling the heterogeneity of the network[17]. Watts-Strogatz networks with average degree <k> = 2m are constructed based on the Watts-Strogatz algorithm[6] with probability of rewiring P. Since we are also interested in studying networks with small mean degree (e.g. <k> = 2), and construction of connected Watts-Strogatz networks with m = 1 for large N is difficult, we study the λ N /λ 2 for the largest connected component of such networks. We also consider a class of connected random networks with predefined average degree, where in order to build the network with N nodes and exactly mN edges, i.e. <k> = 2m, first [mN/2] of possible N(N-1)/2 edges are selected randomly, which results in Q connected components. Then, these connected components are randomly connected through (shows the logarithm of the eigenratio λ N /λ 2 in the parameter space (α,B) for scalefree networks of different size and topological properties. For the case with m = 1 by increasing α,\n\n\nWatts-Strogatz networks with m = 1 and m = 2. For the cases with m = 1, we consider the largest connected component of the network, where its average degree is about 2.23 ± 0.1.Fig. 2a (Fig. 2c) shows the logarithm of λ N /λ 2 as a function of α and P for largest connected component ofWatts-Strogatz networks with m = 1 and N = 1000 (N = 2000). The eigenratio profile for Watts-Strogatz networks with m = 2 (<k> = 4) is shown in\n\nFIG. 3 .\n3FIG. 3.\n\n\n, but with nonidentical individual dynamical systems, where the dynamics of the i-th node with\n\n\nof the j-th oscillator, and t ⋅ makes time averaging. The values Ф ≈ 1 indicate that the systems are phase synchronized. Behavior of Ф as a function of the general coupling strength σ is shown in Fig. 4a for m = 1 and Fig. 4b for m = 2 using different networks topologies with N = 1500. For all of the cases, the weighting approach (2) with α = 1 (solid lines)\n\nFIG. 3 .\n3(color online) Eigenratio λ N /λ 2 (in logarithmic scale) as a function of the network size N for different classes of networks (scale-free (SF), Watts-Strogatz (SW), and random) with α = 1 and with a) m = 1, b) m = 2. Data refers to averages over 20 realizations.\n\nTABLE I .\nI\n\nTABLE I .\nIEnhancing the synchronizability of some real-world networks with different algorithms. First column: the name of the networks. Second, third, and fourths columns: network\nAll of these networks are downloadable from Internet in the web site provided by the authors of the original works; interested readers may refer to the cited work.\nThe original version of the US airport network is a weighted network, but here we have considered only the unweighted version.\n\n. S H Strogatz, Nature. 410268S. H. Strogatz, Nature 410 (2001) 268.\n\nM Newman, A.-L Barabasi, D J Watts, The structure and dynamics of networks. Princeton University PressM. Newman, A.-L. Barabasi, and D. J. Watts, The structure and dynamics of networks, Princeton University Press, 2006.\n\n. S Boccaletti, V Latora, Y Moreno, M Chavez, D U Hwang, Physics Reports. 424175S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D. U. Hwang, Physics Reports 424 (2006) 175.\n\n. M E J Newman, SIAM Review. 45167M. E. J. Newman, SIAM Review 45 (2003) 167.\n\nG V Osipov, J Kurths, C Zhou, Synchronization in Oscillatory Networks. SpringerG. V. Osipov, J. Kurths, and C. Zhou, Synchronization in Oscillatory Networks, Springer, 2007.\n\n. D J Watts, S H Strogatz, Nature. 393440D. J. Watts and S. H. Strogatz, Nature 393 (1998) 440.\n\n. A.-L Barabasi, R Albert, Science. 2865009A.-L. Barabasi and R. Albert, Science 286 (1999) 5009.\n\nSynchronization: a universal concept in nonlinear sciences. A Pikovsky, M Rosenblum, J Kurths, Cambridge University PressA. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: a universal concept in nonlinear sciences, Cambridge University Press, 2003.\n\n. G Buzsaki, C Geisler, D A Henze, X.-J Wang, Trends in Neurosciences. 27186G. Buzsaki, C. Geisler, D. A. Henze, and X.-J. Wang, Trends in Neurosciences 27 (2004) 186.\n\n. G A Polis, Nature. 395744G. A. Polis, Nature 395 (1998) 744.\n\n. V Latora, M Marchiori, Physical Review Letters. 87V. Latora and M. Marchiori, Physical Review Letters 87 (2001) 198701.\n\n. J J Ramasco, S N Dorogovtsev, R Pastor-Satorras, Physical Review E. 7036106J. J. Ramasco, S. N. Dorogovtsev, and R. Pastor-Satorras, Physical Review E 70 (2004) 036106.\n\n. E Fischer, U Sauer, Nature Genetics. 37636E. Fischer and U. Sauer, Nature Genetics 37 (2005) 636.\n\n. G Korniss, M A Novotny, H Guclu, Z Toroczkai, P A Rikvold, Science. 299677G. Korniss, M. A. Novotny, H. Guclu, Z. Toroczkai, and P. A. Rikvold, Science 299 (2003) 677.\n\n. M Jalili, A Rad, M Hasler, International Journal of Circuit Theory and Applications. 35611M. Jalili, A. Ajdari Rad, and M. Hasler, International Journal of Circuit Theory and Applications 35 (2007) 611.\n\n. M Chavez, D.-U Hwang, A Amann, S Boccaletti, Chaos. 1615106M. Chavez, D.-U. Hwang, A. Amann, and S. Boccaletti, Chaos 16 (2006) 015106.\n\n. M Chavez, D.-U Hwang, A Amann, H G E Hentschel, S Boccaletti, Physical Review Letters. 94218701M. Chavez, D.-U. Hwang, A. Amann, H. G. E. Hentschel, and S. Boccaletti, Physical Review Letters 94 (2005) 218701.\n\n. A E Motter, C Zhou, J Kurths, Physical Review E. 7116116A. E. Motter, C. Zhou, and J. Kurths, Physical Review E 71 (2005) 016116.\n\n. A E Motter, C S Zhou, J Kurths, Europhysics Letters. 69334A. E. Motter, C. S. Zhou, and J. Kurths, Europhysics Letters 69 (2005) 334.\n\n. X Wang, Y.-C Lai, C H Lai, Physical Review E. 7556205X. Wang, Y.-C. Lai, and C. H. Lai, Physical Review E 75 (2007) 056205.\n\n. L M Pecora, T L Carroll, Physical Review Letters. 802109L. M. Pecora and T. L. Carroll, Physical Review Letters 80 (1998) 2109.\n\nWe have indeed showed that, in general, different interpretations of synchronizability, i.e. the eigenratio, cost of synchronization, time to synchronize and phase order parameter. do not lie in the same line. However, if the networks are weighted using the algorithm proposed in this paper, they do (not yet publishedWe have indeed showed that, in general, different interpretations of synchronizability, i.e. the eigenratio, cost of synchronization, time to synchronize and phase order parameter, do not lie in the same line. However, if the networks are weighted using the algorithm proposed in this paper, they do (not yet published).\n\n. A Rad, M Jalili, M Hasler, Chaos. A. Ajdari Rad, M. Jalili, and M. Hasler, Chaos (2008)\n\n. L C Freeman, Siociometry. 4035L. C. Freeman, Siociometry 40 (1977) 35.\n\n. S A Gerschgorin, Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk. 7749S. A. Gerschgorin, Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk 7 (1931) 749.\n\n. R Milo, S Itzkovitz, N Kashtan, R Levitt, S Shen-Orr, I Ayzenshtat, M Sheffer, U Alon, Science. 3031538R. Milo, S. Itzkovitz, N. Kashtan, R. Levitt, S. Shen-Orr, I. Ayzenshtat, M. Sheffer, and U. Alon, Science 303 (2004) 1538.\n\n. V Colizza, A Barrat, M Barthelemy, A Vespignani, Proceedings of the National Academy of Science of the United States of America. 103V. Colizza, A. Barrat, M. Barthelemy, and A. Vespignani, Proceedings of the National Academy of Science of the United States of America 103 (2006) 2015.\n\n. R Guimera, L Danon, A Diaz-Guilera, F Giralt, A Arenas, Physical review E. 6865103R. Guimera, L. Danon, A. Diaz-Guilera, F. Giralt, and A. Arenas, Physical review E 68 (2003) 065103\n\nT Ito, T Chiba, R Ozawa, M Yoshida, M Hattori, Y Sakaki, Proceedings of the National Academy of Science of the United States of America. the National Academy of Science of the United States of America984569T. Ito, T. Chiba, R. Ozawa, M. Yoshida, M. Hattori, and Y. Sakaki, Proceedings of the National Academy of Science of the United States of America 98 (2001) 4569.\n\n. H Jeong, S P Mason, A.-L Barabási, Z N Oltvai, Nature. 41141H. Jeong, S. P. Mason, A.-L. Barabási, and Z. N. Oltvai, Nature 411 (2001) 41.\n\n. V Krishnamurthy, M Faloutsos, M Chrobak, L Lao, J.-H Cui, A G Percus, Lecture Notes in Computer Science. 3462V. Krishnamurthy, M. Faloutsos, M. Chrobak, L. Lao, J.-H. Cui, and A. G. Percus, Lecture Notes in Computer Science 3462 (2005) 328-341.\n\n. M E J Newman, Proceedings of the National Academy of Science of the United States of America. 101M. E. J. Newman, Proceedings of the National Academy of Science of the United States of America 101 (2004) 5200-5205.\n\n. S Boccaletti, J Kurths, G Osipov, D L Valladares, C S Zhou, Physics Reports. 3661S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares, and C. S. Zhou, Physics Reports 366 (2002) 1.\n\n. O E Rössler, Physics Letters A. 57397O. E. Rössler, Physics Letters A 57 (1976) 397.\n\nLogarithm of the eigenratio λ N /λ 2 as a function of (α,B) for scale-free networks with a) N = 1000, m = 1, b) N = 1000, m = 2, c) N = 2000, m = 1, and d) N =. FIG. 1. (color online. m = 2. The graphs refer to averaging over 20 realizations of the networksFIG. 1. (color online) Logarithm of the eigenratio λ N /λ 2 as a function of (α,B) for scale-free networks with a) N = 1000, m = 1, b) N = 1000, m = 2, c) N = 2000, m = 1, and d) N = 2000, m = 2. The graphs refer to averaging over 20 realizations of the networks.\n\nLogarithm of the eigenratio λ N /λ 2 as a function of (α,P) for Watts-Strogatz networks with a) N = 1000, m = 1 (the largest component), b) N = 1000, m = 2, c) N = 2000, m = 1 (the largest component), and d) N = 2000, m = 2. The graphs refer to averaging over 20 2 and B = 5, Watts-Strogatz (SW) networks with m = 2 and P = 0.5. The network. FIG. 2. (color online. size in all case is N = 1500FIG. 2. (color online) Logarithm of the eigenratio λ N /λ 2 as a function of (α,P) for Watts-Strogatz networks with a) N = 1000, m = 1 (the largest component), b) N = 1000, m = 2, c) N = 2000, m = 1 (the largest component), and d) N = 2000, m = 2. The graphs refer to averaging over 20 2 and B = 5, Watts-Strogatz (SW) networks with m = 2 and P = 0.5. The network size in all case is N = 1500.\n"},"corpus_id":{"kind":"number","value":19797201,"string":"19,797,201"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"0812.4720\",\n \"mag\": 2079897686,\n \"acl\": null,\n \"pubmed\": \"18764018\",\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1103/physreve.78.016105\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/0812.4720v1.pdf\"\n ],\n \"pdf_sha\": \"6a6e20664c2fa7838851898b5dd455167668802b\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/0812.4720\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Enhancing synchronizability of weighted dynamical networks using betweenness centrality"},"authors":{"kind":"list like","value":["Mahdi Jalili \nSchool of Computer and Communication Sciences\nLaboratory of Nonlinear Systems\nEcole Polytechnique Fédérale de Lausanne\n1015LausanneCHSwitzerland","Ali Ajdari Rad \nSchool of Computer and Communication Sciences\nLaboratory of Nonlinear Systems\nEcole Polytechnique Fédérale de Lausanne\n1015LausanneCHSwitzerland","Martin Hasler \nSchool of Computer and Communication Sciences\nLaboratory of Nonlinear Systems\nEcole Polytechnique Fédérale de Lausanne\n1015LausanneCHSwitzerland"],"string":"[\n \"Mahdi Jalili \\nSchool of Computer and Communication Sciences\\nLaboratory of Nonlinear Systems\\nEcole Polytechnique Fédérale de Lausanne\\n1015LausanneCHSwitzerland\",\n \"Ali Ajdari Rad \\nSchool of Computer and Communication Sciences\\nLaboratory of Nonlinear Systems\\nEcole Polytechnique Fédérale de Lausanne\\n1015LausanneCHSwitzerland\",\n \"Martin Hasler \\nSchool of Computer and Communication Sciences\\nLaboratory of Nonlinear Systems\\nEcole Polytechnique Fédérale de Lausanne\\n1015LausanneCHSwitzerland\"\n]","isConversational":false},"abstract":{"kind":"string","value":"By considering the eigenratio of the Laplacian of the connection graph as synchronizability measure, we propose a procedure for weighting dynamical networks to enhance their synchronizability. The method is based on node and edge betweenness centrality measures and is tested on artificially constructed scale-free, Watts-Strogatz and random networks as well as on some real-world graphs. It is also numerically shown that the same procedure could be used to enhance the phase synchronizability of networks of nonidentical oscillators.PACS NUMBERS05.45.Xt (Synchronization; coupled oscillators), 89.75.-k (Complex systems)."},"sections":{"kind":"list like","value":[{"title":"II. WEIGHTING ALGORITHM","content":"Let us start with a dynamical network of N linearly coupled identical systems with the following equations of the motion The variational equations of manifold synchronized solution ( )\nλ are the eigenvalues of G , ordered as 0 = λ 1 ≤ λ 2 ≤ … ≤ λ N and λ 1 is associated with the synchronized manifold. In the sequel, we suppose that the connection graph is connected, which implies the strict inequality 0 = λ 1 < λ 2 .\nThe largest Lyapunov exponent of the above variational equation, Λ(σλ i ), called masterstability-function [21], gives a sufficient condition for the local stability of the synchronization manifold: if the synchronization manifold is locally stable we must have Λ(σλ i ) < 0 , i = 2,…,N.\nFor a number of systems such as x-coupled Rössler oscillators, the master-stability-function is negative only within a bounded interval (ν 1 , ν 2 ) [21]. Requiring all coupling strengths lie within such an interval, i.e. ν 1 < σλ 2 ≤ … ≤ σλ N < ν 2 , leads to the following condition for the local stability of the synchronization manifold: λ N /λ 2 < ν 2 /ν 1 . The left-hand side of the inequality depends solely on the structure of the graph, while the right-hand side depends on the dynamics of the individual systems and on the coupling configuration. There is a number of interpretations for synchronizability of dynamical networks [15]. We adopt here the following interpretation: the larger the range of parameters of the individual dynamical systems that allows for synchronization, the better the synchronizability of the network. This relates the synchronizability to the eigenratio λ N /λ 2 , and concludes that the smaller the eigenratio λ N /λ 2 of a network, the better its synchronizability [22].\nIn general, there are two possible ways to enhance the synchronizability of dynamical networks: rewiring of the links [23] and/or assigning proper weights for the existing links [15].\nFor many applications it is not possible to change the network topology and the only option to enhance the synchronizability is weighting the links. In Networks with good synchronizability, couplings between the nodes are neither necessarily uniform nor symmetric. A very first attempt to assign the proper connection weights for enhancing the synchronizability was proposed in [18,19], where the coupling on the right hand side of the equation (1) was taken as\n∑ H x x , with k i the degree of node i and N i the set of neighbors of node i. The value of β = 1 was found to be optimal in [19] for synchronizability. Further enhancement of the synchronizability was achieved by scaling the weight of each edge by its load [17], where the coupling takes the form as ( )\nwhere ρ ij is the load of the edge e ij between the i-th and j-th node. The load (also known as edge betweenness centrality) of e ij is defined as\nwhere Γ pq is the number of shortest paths from the p-th to the q-th node and Γ pq (e ij ) is the number of these paths making use of e ij . The optimal condition β = 1 was found for synchronizability [17] that itself performs better than the optimal case of [19]. In this way, not only the local structural information but also the effects of network structure at a global level are taken into account. Very recently, by considering the concepts of gradient networks, another weighting algorithm as ( )\nH x x has been proposed [20].\nHere we show that further enhancement of synchronizability can be achieved by considering not only connection loads but also node betweenness centralities. Node betweenness centrality i C is a centrality measure of the i-th node in a graph, which counts the number of shortest paths making use of that node (except shortest paths between the i -th node and other nodes) [24]. More precisely,\nwhere Γ pq is the number of shortest paths from the p-th to the q-th node and Γ pq (i) is the number of these shortest paths making use of the ith node. In the weighting procedure we propose, the weight of each edge will be a function of its load and the betweenness centrality of the tail node. The resulting weighted network becomes directed; the links go from the head nodes to the tail nodes. The weight of an edge should essentially be proportional to the betweenness centrality of the edge as proposed in [17]. The dependence of the weight to the betweenness centrality of the tail node is also straight forward.\nNodes with high values of betweenness centrality can be regarded as hub nodes, i.e. many shortest paths make use of them. Therefore, it is reasonable to increase the weight of the links ending to these nodes. More precisely, the network equations read\nwhere α is a real tunable parameter, and ε is a small positive value to make (ε + C j α ) > 0 (some nodes may have betweenness centrality equal to zero). Here, we take ε = 1.\nBy this construction of connection weights, the diagonal elements of G are always normalized to one, thus preventing the coupling to be arbitrary large or small. Although G becomes asymmetric for any value of α, it can be written as G = D l WD r , where W is a zero row-\n. It can easily be shown that the eigenvalues of G, λ i (i = 1,…,N), are the same as the eigenvalues of D r 1/2 D l 1/2 WD l 1/2 D r 1/2 , i.e. real and non-negative with smallest eigenvalue as λ 1 = 0. For this case, Gerschgorin circle theorem [25] guarantees that 0 <\nAnother important term concerning our procedure is the effect of different values of α. It is worth mentioning that the case with α = 0 corresponds to the optimal situation proposed in [17], which itself has the optimal case of [18,19] as a special case. For large absolute values of α, it may happen that the resulting weighted network is approximately disconnected [17] and thus λ 2 is close to zero; therefore, we limit ourselves to α with small absolute values."},{"title":"III. NUMERICAL SIMULATIONS","content":""},{"title":"A. Performance in artificially constructed networks","content":"By sweeping α and calculating the values of λ N /λ 2 we can study the synchronizability profile of dynamical networks with different topological properties such as scale-free, Watts-Strogatz and random networks. We will construct scale-free networks using an algorithm proposed in [17], which itself is a generalization of the preferential attachment growing procedure λ N /λ 2 is rapidly decreasing, i.e. synchronizability is enhanced. Note that α = 0 recovers the optimal condition of [17], which itself has the optimal situation of [19] as a special case. For m = 2, the situation is somewhat different; for small values of B, λ N /λ 2 is decreased by increasing α.\nHowever, for larger values of B (less heterogeneity in the degree distribution), there is a local minimum in α ~ 0.5, and then by increasing α, the eigenratio is also increased and by further increasing α over a value around 1, the eigenratio starts decreasing. Although α = 1 is not the exact optimal point, to avoid the network from being disconnected (high values of α may lead the network to be disconnected), we consider α = 1 for weighting the scale-free networks in order to enhance their synchronizability. Considering node betweenness centrality in addition to edge load makes the resulting weighted network more homogeneous and thus enhances its synchronizability."},{"title":"FIG. 1.","content":"For comparison, we have also applied our weighting procedure to Watts-Strogatz networks that exhibit more homogeneity in the network structure than scale-free networks. We consider Fig. 2b and Fig. 2d for N = 1000 and N = 2000, respectively. As it is seen, there is a clear optimum in α ~ 1. Also, the effect of the Watts-Strogatz phenomenon on the synchronizability of the network is clearly seen from these graphs,\ni.e. the synchronizability of the network is greatly enhanced by introducing some rewirings."},{"title":"FIG. 2.","content":"Here we also report the behavior of the synchronizability of the weighted networks with α = 1 as a function of network size N. Fig. 3a shows the results for scale-free networks with different values of B and random networks, all the cases with m = 1. As expected, the propensity of synchronization for weighted scale-free with lower values of B is better regardless of N [16].\nAlso, weighted scale-free networks exhibit better synchronizability than random networks. For m = 2 the profile of λ N /λ 2 as a function of N is depicted in Fig. 3b, where again weighted scale-free networks show better synchronizability than Watts-Strogatz and random networks. It also illustrates the fact that synchronizability of the weighted networks is almost independent of network size. Indeed, the average degree seems to be the only important factor affecting the synchronizability of the weighted networks."},{"title":"B. Performance in real-world networks","content":"Although we showed that applying (2) for weighting the edges of dynamical networks greatly enhances the synchronizability of the network, i.e. reduces the eigenratio λ N /λ 2 , many realworld networks cannot be simply modeled by these scale-free, Watts-Strogatz or random network models. Real-world networks may possess a number of complex topological properties, which can make it difficult to construct a model that mimics all of them. Thus, we consider some available real-world undirected networks 1 and apply the proposed weighting algorithm (2) to study the behavior of λ N /λ 2 in the resulting weighted networks. We consider some real-world networks including protein structure network [26], network of power grid [6], US airport network 2 [27], Email network [28], protein-protein interaction network [29], yeast protein interaction network [30], Internet on the level of autonomous system [31], and the network of coauthorship [32]. Table I summarizes the results. For all networks, our proposed weighting algorithm is the best. Let us remark that since for the algorithm of [20] there is no optimal value of β, thus for each network we adopt the least λ N /λ 2 among the cases with β = 1, β = 2, and β = 3. For the methods of [18] and [17] we used the optimal condition β = 1 and for our proposed method we used α = 1. Since the method of [18] is a simple scaling based on the degree of the nodes, it is incapable of capturing all the useful information, and thus, its performance is always worse than that of the one proposed by [17] that indeed considered the edge betweenness centrality as well as scaling based on the nodes. The method proposed in [20] uses the degree of nodes in a different way, and its performance is not always better than [17]. Indeed, it can be seen that by increase of the heterogeneity of the network, i.e. increase of the standard deviation of the node-degrees, the algorithm of [20] performs better than [17]. Our proposed algorithm is an intelligent extension of [17]; to consider the heterogeneity of the network, it considers the node betweenness centrality in addition to the edge betweenness centrality. Thus, it always outperforms to [17]. However, for networks with high levels of heterogeneity, [20] gives results better than [17] and closed to our results."},{"title":"C. Enhancing phase synchronizability in coupled nonidentical dynamical systems","content":"The rational behind taking the eigenratio λ N /λ 2 as a synchronizability measure of dynamical networks is the master-stability-function [21]. Since the theory of the master-stability-function was developed for local stability of synchronization of identical dynamical systems, it can not be directly applied to coupled nonidentical systems. However, coupled nonidentical dynamical systems may exhibit some weaker types of synchronization such as phase synchronization [33].\nHere, we study the collective behavior of nonidentical Rössler oscillators [34] on scale-free, Watts-Strogatz and random networks to study how much the proposed weighting algorithm can improve the degree of phase synchronization. The dynamics of motion is governed by (2) [17]. Here i ω is the natural frequency of the i-th system that is randomly chosen from a Gaussian distribution with mean value ω mean = 1 and standard deviation Δω = 0.03.\nTo study the phase synchronization among coupled oscillators one can monitor the order parameter [8,33] ( ) ( ) results in better phase synchronizability than the case with α = 0 (dashed lines). This improvement is well pronounced for the case with m = 1 (Fig.4a). These results confirm that the master-stability-function gives also valid information for nonidentical oscillators. Note that synchronizability is improved if the σ range for which Ф ≈ 1 increases."},{"title":"FIG. 4.","content":""},{"title":"IV. CONCLUSIONS","content":"By considering the eigenratio λ N /λ 2 of the Laplacian of the connection as the synchronizability measure, we proposed a procedure for assigning proper connection weights to enhance the synchronizability of dynamical networks. To form the weights, we used the information of the node and the edge betweenness centrality measures. The algorithm was tested on artificially constructed networks such as scale-free, Watts-Strogatz and random networks as well as on some real-world networks. This method was also powerful in enhancing the phase synchronizability in networks of nonidentical dynamical systems. size N, average node-degree <k>, and standard deviation of node-degrees std(k), respectively."},{"title":"ACKNOWLEDGMENTS","content":"Fifth to eighth columns: the eigenratio λ N /λ 2 using different weighting algorithms.\nReal-world networks N <k> std(k) λ N /λ 2 [18] λ N /λ 2 [17] λ N /λ 2 [20] λ N /λ 2 using (2)"},{"title":"Fig. 1","content":""},{"title":"FIG. 3 .","content":""},{"title":"FIG. 3 .","content":""},{"title":"TABLE I .","content":""},{"title":"TABLE I .","content":""}],"string":"[\n {\n \"title\": \"II. WEIGHTING ALGORITHM\",\n \"content\": \"Let us start with a dynamical network of N linearly coupled identical systems with the following equations of the motion The variational equations of manifold synchronized solution ( )\\nλ are the eigenvalues of G , ordered as 0 = λ 1 ≤ λ 2 ≤ … ≤ λ N and λ 1 is associated with the synchronized manifold. In the sequel, we suppose that the connection graph is connected, which implies the strict inequality 0 = λ 1 < λ 2 .\\nThe largest Lyapunov exponent of the above variational equation, Λ(σλ i ), called masterstability-function [21], gives a sufficient condition for the local stability of the synchronization manifold: if the synchronization manifold is locally stable we must have Λ(σλ i ) < 0 , i = 2,…,N.\\nFor a number of systems such as x-coupled Rössler oscillators, the master-stability-function is negative only within a bounded interval (ν 1 , ν 2 ) [21]. Requiring all coupling strengths lie within such an interval, i.e. ν 1 < σλ 2 ≤ … ≤ σλ N < ν 2 , leads to the following condition for the local stability of the synchronization manifold: λ N /λ 2 < ν 2 /ν 1 . The left-hand side of the inequality depends solely on the structure of the graph, while the right-hand side depends on the dynamics of the individual systems and on the coupling configuration. There is a number of interpretations for synchronizability of dynamical networks [15]. We adopt here the following interpretation: the larger the range of parameters of the individual dynamical systems that allows for synchronization, the better the synchronizability of the network. This relates the synchronizability to the eigenratio λ N /λ 2 , and concludes that the smaller the eigenratio λ N /λ 2 of a network, the better its synchronizability [22].\\nIn general, there are two possible ways to enhance the synchronizability of dynamical networks: rewiring of the links [23] and/or assigning proper weights for the existing links [15].\\nFor many applications it is not possible to change the network topology and the only option to enhance the synchronizability is weighting the links. In Networks with good synchronizability, couplings between the nodes are neither necessarily uniform nor symmetric. A very first attempt to assign the proper connection weights for enhancing the synchronizability was proposed in [18,19], where the coupling on the right hand side of the equation (1) was taken as\\n∑ H x x , with k i the degree of node i and N i the set of neighbors of node i. The value of β = 1 was found to be optimal in [19] for synchronizability. Further enhancement of the synchronizability was achieved by scaling the weight of each edge by its load [17], where the coupling takes the form as ( )\\nwhere ρ ij is the load of the edge e ij between the i-th and j-th node. The load (also known as edge betweenness centrality) of e ij is defined as\\nwhere Γ pq is the number of shortest paths from the p-th to the q-th node and Γ pq (e ij ) is the number of these paths making use of e ij . The optimal condition β = 1 was found for synchronizability [17] that itself performs better than the optimal case of [19]. In this way, not only the local structural information but also the effects of network structure at a global level are taken into account. Very recently, by considering the concepts of gradient networks, another weighting algorithm as ( )\\nH x x has been proposed [20].\\nHere we show that further enhancement of synchronizability can be achieved by considering not only connection loads but also node betweenness centralities. Node betweenness centrality i C is a centrality measure of the i-th node in a graph, which counts the number of shortest paths making use of that node (except shortest paths between the i -th node and other nodes) [24]. More precisely,\\nwhere Γ pq is the number of shortest paths from the p-th to the q-th node and Γ pq (i) is the number of these shortest paths making use of the ith node. In the weighting procedure we propose, the weight of each edge will be a function of its load and the betweenness centrality of the tail node. The resulting weighted network becomes directed; the links go from the head nodes to the tail nodes. The weight of an edge should essentially be proportional to the betweenness centrality of the edge as proposed in [17]. The dependence of the weight to the betweenness centrality of the tail node is also straight forward.\\nNodes with high values of betweenness centrality can be regarded as hub nodes, i.e. many shortest paths make use of them. Therefore, it is reasonable to increase the weight of the links ending to these nodes. More precisely, the network equations read\\nwhere α is a real tunable parameter, and ε is a small positive value to make (ε + C j α ) > 0 (some nodes may have betweenness centrality equal to zero). Here, we take ε = 1.\\nBy this construction of connection weights, the diagonal elements of G are always normalized to one, thus preventing the coupling to be arbitrary large or small. Although G becomes asymmetric for any value of α, it can be written as G = D l WD r , where W is a zero row-\\n. It can easily be shown that the eigenvalues of G, λ i (i = 1,…,N), are the same as the eigenvalues of D r 1/2 D l 1/2 WD l 1/2 D r 1/2 , i.e. real and non-negative with smallest eigenvalue as λ 1 = 0. For this case, Gerschgorin circle theorem [25] guarantees that 0 <\\nAnother important term concerning our procedure is the effect of different values of α. It is worth mentioning that the case with α = 0 corresponds to the optimal situation proposed in [17], which itself has the optimal case of [18,19] as a special case. For large absolute values of α, it may happen that the resulting weighted network is approximately disconnected [17] and thus λ 2 is close to zero; therefore, we limit ourselves to α with small absolute values.\"\n },\n {\n \"title\": \"III. NUMERICAL SIMULATIONS\",\n \"content\": \"\"\n },\n {\n \"title\": \"A. Performance in artificially constructed networks\",\n \"content\": \"By sweeping α and calculating the values of λ N /λ 2 we can study the synchronizability profile of dynamical networks with different topological properties such as scale-free, Watts-Strogatz and random networks. We will construct scale-free networks using an algorithm proposed in [17], which itself is a generalization of the preferential attachment growing procedure λ N /λ 2 is rapidly decreasing, i.e. synchronizability is enhanced. Note that α = 0 recovers the optimal condition of [17], which itself has the optimal situation of [19] as a special case. For m = 2, the situation is somewhat different; for small values of B, λ N /λ 2 is decreased by increasing α.\\nHowever, for larger values of B (less heterogeneity in the degree distribution), there is a local minimum in α ~ 0.5, and then by increasing α, the eigenratio is also increased and by further increasing α over a value around 1, the eigenratio starts decreasing. Although α = 1 is not the exact optimal point, to avoid the network from being disconnected (high values of α may lead the network to be disconnected), we consider α = 1 for weighting the scale-free networks in order to enhance their synchronizability. Considering node betweenness centrality in addition to edge load makes the resulting weighted network more homogeneous and thus enhances its synchronizability.\"\n },\n {\n \"title\": \"FIG. 1.\",\n \"content\": \"For comparison, we have also applied our weighting procedure to Watts-Strogatz networks that exhibit more homogeneity in the network structure than scale-free networks. We consider Fig. 2b and Fig. 2d for N = 1000 and N = 2000, respectively. As it is seen, there is a clear optimum in α ~ 1. Also, the effect of the Watts-Strogatz phenomenon on the synchronizability of the network is clearly seen from these graphs,\\ni.e. the synchronizability of the network is greatly enhanced by introducing some rewirings.\"\n },\n {\n \"title\": \"FIG. 2.\",\n \"content\": \"Here we also report the behavior of the synchronizability of the weighted networks with α = 1 as a function of network size N. Fig. 3a shows the results for scale-free networks with different values of B and random networks, all the cases with m = 1. As expected, the propensity of synchronization for weighted scale-free with lower values of B is better regardless of N [16].\\nAlso, weighted scale-free networks exhibit better synchronizability than random networks. For m = 2 the profile of λ N /λ 2 as a function of N is depicted in Fig. 3b, where again weighted scale-free networks show better synchronizability than Watts-Strogatz and random networks. It also illustrates the fact that synchronizability of the weighted networks is almost independent of network size. Indeed, the average degree seems to be the only important factor affecting the synchronizability of the weighted networks.\"\n },\n {\n \"title\": \"B. Performance in real-world networks\",\n \"content\": \"Although we showed that applying (2) for weighting the edges of dynamical networks greatly enhances the synchronizability of the network, i.e. reduces the eigenratio λ N /λ 2 , many realworld networks cannot be simply modeled by these scale-free, Watts-Strogatz or random network models. Real-world networks may possess a number of complex topological properties, which can make it difficult to construct a model that mimics all of them. Thus, we consider some available real-world undirected networks 1 and apply the proposed weighting algorithm (2) to study the behavior of λ N /λ 2 in the resulting weighted networks. We consider some real-world networks including protein structure network [26], network of power grid [6], US airport network 2 [27], Email network [28], protein-protein interaction network [29], yeast protein interaction network [30], Internet on the level of autonomous system [31], and the network of coauthorship [32]. Table I summarizes the results. For all networks, our proposed weighting algorithm is the best. Let us remark that since for the algorithm of [20] there is no optimal value of β, thus for each network we adopt the least λ N /λ 2 among the cases with β = 1, β = 2, and β = 3. For the methods of [18] and [17] we used the optimal condition β = 1 and for our proposed method we used α = 1. Since the method of [18] is a simple scaling based on the degree of the nodes, it is incapable of capturing all the useful information, and thus, its performance is always worse than that of the one proposed by [17] that indeed considered the edge betweenness centrality as well as scaling based on the nodes. The method proposed in [20] uses the degree of nodes in a different way, and its performance is not always better than [17]. Indeed, it can be seen that by increase of the heterogeneity of the network, i.e. increase of the standard deviation of the node-degrees, the algorithm of [20] performs better than [17]. Our proposed algorithm is an intelligent extension of [17]; to consider the heterogeneity of the network, it considers the node betweenness centrality in addition to the edge betweenness centrality. Thus, it always outperforms to [17]. However, for networks with high levels of heterogeneity, [20] gives results better than [17] and closed to our results.\"\n },\n {\n \"title\": \"C. Enhancing phase synchronizability in coupled nonidentical dynamical systems\",\n \"content\": \"The rational behind taking the eigenratio λ N /λ 2 as a synchronizability measure of dynamical networks is the master-stability-function [21]. Since the theory of the master-stability-function was developed for local stability of synchronization of identical dynamical systems, it can not be directly applied to coupled nonidentical systems. However, coupled nonidentical dynamical systems may exhibit some weaker types of synchronization such as phase synchronization [33].\\nHere, we study the collective behavior of nonidentical Rössler oscillators [34] on scale-free, Watts-Strogatz and random networks to study how much the proposed weighting algorithm can improve the degree of phase synchronization. The dynamics of motion is governed by (2) [17]. Here i ω is the natural frequency of the i-th system that is randomly chosen from a Gaussian distribution with mean value ω mean = 1 and standard deviation Δω = 0.03.\\nTo study the phase synchronization among coupled oscillators one can monitor the order parameter [8,33] ( ) ( ) results in better phase synchronizability than the case with α = 0 (dashed lines). This improvement is well pronounced for the case with m = 1 (Fig.4a). These results confirm that the master-stability-function gives also valid information for nonidentical oscillators. Note that synchronizability is improved if the σ range for which Ф ≈ 1 increases.\"\n },\n {\n \"title\": \"FIG. 4.\",\n \"content\": \"\"\n },\n {\n \"title\": \"IV. CONCLUSIONS\",\n \"content\": \"By considering the eigenratio λ N /λ 2 of the Laplacian of the connection as the synchronizability measure, we proposed a procedure for assigning proper connection weights to enhance the synchronizability of dynamical networks. To form the weights, we used the information of the node and the edge betweenness centrality measures. The algorithm was tested on artificially constructed networks such as scale-free, Watts-Strogatz and random networks as well as on some real-world networks. This method was also powerful in enhancing the phase synchronizability in networks of nonidentical dynamical systems. size N, average node-degree <k>, and standard deviation of node-degrees std(k), respectively.\"\n },\n {\n \"title\": \"ACKNOWLEDGMENTS\",\n \"content\": \"Fifth to eighth columns: the eigenratio λ N /λ 2 using different weighting algorithms.\\nReal-world networks N <k> std(k) λ N /λ 2 [18] λ N /λ 2 [17] λ N /λ 2 [20] λ N /λ 2 using (2)\"\n },\n {\n \"title\": \"Fig. 1\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"TABLE I .\",\n \"content\": \"\"\n },\n {\n \"title\": \"TABLE I .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"introduced in[7]. Namely, starting with a network of m + 1 all-to-all connected nodes, at each step a new node is added with m links that are connected to node i with probability k i is the degree of the node and B a tunable real parameter controlling the heterogeneity of the network[17]. Watts-Strogatz networks with average degree <k> = 2m are constructed based on the Watts-Strogatz algorithm[6] with probability of rewiring P. Since we are also interested in studying networks with small mean degree (e.g. <k> = 2), and construction of connected Watts-Strogatz networks with m = 1 for large N is difficult, we study the λ N /λ 2 for the largest connected component of such networks. We also consider a class of connected random networks with predefined average degree, where in order to build the network with N nodes and exactly mN edges, i.e. <k> = 2m, first [mN/2] of possible N(N-1)/2 edges are selected randomly, which results in Q connected components. Then, these connected components are randomly connected through (shows the logarithm of the eigenratio λ N /λ 2 in the parameter space (α,B) for scalefree networks of different size and topological properties. For the case with m = 1 by increasing α,","content":"Fig. 1\n1introduced in[7]. Namely, starting with a network of m + 1 all-to-all connected nodes, at each step a new node is added with m links that are connected to node i with probability k i is the degree of the node and B a tunable real parameter controlling the heterogeneity of the network[17]. Watts-Strogatz networks with average degree <k> = 2m are constructed based on the Watts-Strogatz algorithm[6] with probability of rewiring P. Since we are also interested in studying networks with small mean degree (e.g. <k> = 2), and construction of connected Watts-Strogatz networks with m = 1 for large N is difficult, we study the λ N /λ 2 for the largest connected component of such networks. We also consider a class of connected random networks with predefined average degree, where in order to build the network with N nodes and exactly mN edges, i.e. <k> = 2m, first [mN/2] of possible N(N-1)/2 edges are selected randomly, which results in Q connected components. Then, these connected components are randomly connected through (shows the logarithm of the eigenratio λ N /λ 2 in the parameter space (α,B) for scalefree networks of different size and topological properties. For the case with m = 1 by increasing α,"},{"id":"","caption":"Watts-Strogatz networks with m = 1 and m = 2. For the cases with m = 1, we consider the largest connected component of the network, where its average degree is about 2.23 ± 0.1.Fig. 2a (Fig. 2c) shows the logarithm of λ N /λ 2 as a function of α and P for largest connected component ofWatts-Strogatz networks with m = 1 and N = 1000 (N = 2000). The eigenratio profile for Watts-Strogatz networks with m = 2 (<k> = 4) is shown in","content":"Watts-Strogatz networks with m = 1 and m = 2. For the cases with m = 1, we consider the largest connected component of the network, where its average degree is about 2.23 ± 0.1.Fig. 2a (Fig. 2c) shows the logarithm of λ N /λ 2 as a function of α and P for largest connected component ofWatts-Strogatz networks with m = 1 and N = 1000 (N = 2000). The eigenratio profile for Watts-Strogatz networks with m = 2 (<k> = 4) is shown in"},{"id":"","caption":"FIG. 3.","content":"FIG. 3 .\n3FIG. 3."},{"id":"","caption":", but with nonidentical individual dynamical systems, where the dynamics of the i-th node with","content":", but with nonidentical individual dynamical systems, where the dynamics of the i-th node with"},{"id":"","caption":"of the j-th oscillator, and t ⋅ makes time averaging. The values Ф ≈ 1 indicate that the systems are phase synchronized. Behavior of Ф as a function of the general coupling strength σ is shown in Fig. 4a for m = 1 and Fig. 4b for m = 2 using different networks topologies with N = 1500. For all of the cases, the weighting approach (2) with α = 1 (solid lines)","content":"of the j-th oscillator, and t ⋅ makes time averaging. The values Ф ≈ 1 indicate that the systems are phase synchronized. Behavior of Ф as a function of the general coupling strength σ is shown in Fig. 4a for m = 1 and Fig. 4b for m = 2 using different networks topologies with N = 1500. For all of the cases, the weighting approach (2) with α = 1 (solid lines)"},{"id":"","caption":"(color online) Eigenratio λ N /λ 2 (in logarithmic scale) as a function of the network size N for different classes of networks (scale-free (SF), Watts-Strogatz (SW), and random) with α = 1 and with a) m = 1, b) m = 2. Data refers to averages over 20 realizations.","content":"FIG. 3 .\n3(color online) Eigenratio λ N /λ 2 (in logarithmic scale) as a function of the network size N for different classes of networks (scale-free (SF), Watts-Strogatz (SW), and random) with α = 1 and with a) m = 1, b) m = 2. Data refers to averages over 20 realizations."},{"id":"","caption":"Enhancing the synchronizability of some real-world networks with different algorithms. First column: the name of the networks. Second, third, and fourths columns: network","content":"TABLE I .\nI"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"introduced in[7]. Namely, starting with a network of m + 1 all-to-all connected nodes, at each step a new node is added with m links that are connected to node i with probability k i is the degree of the node and B a tunable real parameter controlling the heterogeneity of the network[17]. Watts-Strogatz networks with average degree <k> = 2m are constructed based on the Watts-Strogatz algorithm[6] with probability of rewiring P. Since we are also interested in studying networks with small mean degree (e.g. <k> = 2), and construction of connected Watts-Strogatz networks with m = 1 for large N is difficult, we study the λ N /λ 2 for the largest connected component of such networks. We also consider a class of connected random networks with predefined average degree, where in order to build the network with N nodes and exactly mN edges, i.e. <k> = 2m, first [mN/2] of possible N(N-1)/2 edges are selected randomly, which results in Q connected components. Then, these connected components are randomly connected through (shows the logarithm of the eigenratio λ N /λ 2 in the parameter space (α,B) for scalefree networks of different size and topological properties. For the case with m = 1 by increasing α,\",\n \"content\": \"Fig. 1\\n1introduced in[7]. Namely, starting with a network of m + 1 all-to-all connected nodes, at each step a new node is added with m links that are connected to node i with probability k i is the degree of the node and B a tunable real parameter controlling the heterogeneity of the network[17]. Watts-Strogatz networks with average degree <k> = 2m are constructed based on the Watts-Strogatz algorithm[6] with probability of rewiring P. Since we are also interested in studying networks with small mean degree (e.g. <k> = 2), and construction of connected Watts-Strogatz networks with m = 1 for large N is difficult, we study the λ N /λ 2 for the largest connected component of such networks. We also consider a class of connected random networks with predefined average degree, where in order to build the network with N nodes and exactly mN edges, i.e. <k> = 2m, first [mN/2] of possible N(N-1)/2 edges are selected randomly, which results in Q connected components. Then, these connected components are randomly connected through (shows the logarithm of the eigenratio λ N /λ 2 in the parameter space (α,B) for scalefree networks of different size and topological properties. For the case with m = 1 by increasing α,\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Watts-Strogatz networks with m = 1 and m = 2. For the cases with m = 1, we consider the largest connected component of the network, where its average degree is about 2.23 ± 0.1.Fig. 2a (Fig. 2c) shows the logarithm of λ N /λ 2 as a function of α and P for largest connected component ofWatts-Strogatz networks with m = 1 and N = 1000 (N = 2000). The eigenratio profile for Watts-Strogatz networks with m = 2 (<k> = 4) is shown in\",\n \"content\": \"Watts-Strogatz networks with m = 1 and m = 2. For the cases with m = 1, we consider the largest connected component of the network, where its average degree is about 2.23 ± 0.1.Fig. 2a (Fig. 2c) shows the logarithm of λ N /λ 2 as a function of α and P for largest connected component ofWatts-Strogatz networks with m = 1 and N = 1000 (N = 2000). The eigenratio profile for Watts-Strogatz networks with m = 2 (<k> = 4) is shown in\"\n },\n {\n \"id\": \"\",\n \"caption\": \"FIG. 3.\",\n \"content\": \"FIG. 3 .\\n3FIG. 3.\"\n },\n {\n \"id\": \"\",\n \"caption\": \", but with nonidentical individual dynamical systems, where the dynamics of the i-th node with\",\n \"content\": \", but with nonidentical individual dynamical systems, where the dynamics of the i-th node with\"\n },\n {\n \"id\": \"\",\n \"caption\": \"of the j-th oscillator, and t ⋅ makes time averaging. The values Ф ≈ 1 indicate that the systems are phase synchronized. Behavior of Ф as a function of the general coupling strength σ is shown in Fig. 4a for m = 1 and Fig. 4b for m = 2 using different networks topologies with N = 1500. For all of the cases, the weighting approach (2) with α = 1 (solid lines)\",\n \"content\": \"of the j-th oscillator, and t ⋅ makes time averaging. The values Ф ≈ 1 indicate that the systems are phase synchronized. Behavior of Ф as a function of the general coupling strength σ is shown in Fig. 4a for m = 1 and Fig. 4b for m = 2 using different networks topologies with N = 1500. For all of the cases, the weighting approach (2) with α = 1 (solid lines)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(color online) Eigenratio λ N /λ 2 (in logarithmic scale) as a function of the network size N for different classes of networks (scale-free (SF), Watts-Strogatz (SW), and random) with α = 1 and with a) m = 1, b) m = 2. Data refers to averages over 20 realizations.\",\n \"content\": \"FIG. 3 .\\n3(color online) Eigenratio λ N /λ 2 (in logarithmic scale) as a function of the network size N for different classes of networks (scale-free (SF), Watts-Strogatz (SW), and random) with α = 1 and with a) m = 1, b) m = 2. Data refers to averages over 20 realizations.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Enhancing the synchronizability of some real-world networks with different algorithms. First column: the name of the networks. Second, third, and fourths columns: network\",\n \"content\": \"TABLE I .\\nI\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". S H Strogatz, Nature. 410268S. H. Strogatz, Nature 410 (2001) 268.","doi":""},{"text":"M Newman, A.-L Barabasi, D J Watts, The structure and dynamics of networks. Princeton University PressM. Newman, A.-L. Barabasi, and D. J. Watts, The structure and dynamics of networks, Princeton University Press, 2006.","doi":""},{"text":". S Boccaletti, V Latora, Y Moreno, M Chavez, D U Hwang, Physics Reports. 424175S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D. U. Hwang, Physics Reports 424 (2006) 175.","doi":""},{"text":". M E J Newman, SIAM Review. 45167M. E. J. Newman, SIAM Review 45 (2003) 167.","doi":""},{"text":"G V Osipov, J Kurths, C Zhou, Synchronization in Oscillatory Networks. SpringerG. V. Osipov, J. Kurths, and C. Zhou, Synchronization in Oscillatory Networks, Springer, 2007.","doi":""},{"text":". D J Watts, S H Strogatz, Nature. 393440D. J. Watts and S. H. Strogatz, Nature 393 (1998) 440.","doi":""},{"text":". A.-L Barabasi, R Albert, Science. 2865009A.-L. Barabasi and R. Albert, Science 286 (1999) 5009.","doi":""},{"text":"Synchronization: a universal concept in nonlinear sciences. A Pikovsky, M Rosenblum, J Kurths, Cambridge University PressA. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: a universal concept in nonlinear sciences, Cambridge University Press, 2003.","doi":""},{"text":". G Buzsaki, C Geisler, D A Henze, X.-J Wang, Trends in Neurosciences. 27186G. Buzsaki, C. Geisler, D. A. Henze, and X.-J. Wang, Trends in Neurosciences 27 (2004) 186.","doi":""},{"text":". G A Polis, Nature. 395744G. A. Polis, Nature 395 (1998) 744.","doi":""},{"text":". V Latora, M Marchiori, Physical Review Letters. 87V. Latora and M. Marchiori, Physical Review Letters 87 (2001) 198701.","doi":""},{"text":". J J Ramasco, S N Dorogovtsev, R Pastor-Satorras, Physical Review E. 7036106J. J. Ramasco, S. N. Dorogovtsev, and R. Pastor-Satorras, Physical Review E 70 (2004) 036106.","doi":""},{"text":". E Fischer, U Sauer, Nature Genetics. 37636E. Fischer and U. Sauer, Nature Genetics 37 (2005) 636.","doi":""},{"text":". G Korniss, M A Novotny, H Guclu, Z Toroczkai, P A Rikvold, Science. 299677G. Korniss, M. A. Novotny, H. Guclu, Z. Toroczkai, and P. A. Rikvold, Science 299 (2003) 677.","doi":""},{"text":". M Jalili, A Rad, M Hasler, International Journal of Circuit Theory and Applications. 35611M. Jalili, A. Ajdari Rad, and M. Hasler, International Journal of Circuit Theory and Applications 35 (2007) 611.","doi":""},{"text":". M Chavez, D.-U Hwang, A Amann, S Boccaletti, Chaos. 1615106M. Chavez, D.-U. Hwang, A. Amann, and S. Boccaletti, Chaos 16 (2006) 015106.","doi":""},{"text":". M Chavez, D.-U Hwang, A Amann, H G E Hentschel, S Boccaletti, Physical Review Letters. 94218701M. Chavez, D.-U. Hwang, A. Amann, H. G. E. Hentschel, and S. Boccaletti, Physical Review Letters 94 (2005) 218701.","doi":""},{"text":". A E Motter, C Zhou, J Kurths, Physical Review E. 7116116A. E. Motter, C. Zhou, and J. Kurths, Physical Review E 71 (2005) 016116.","doi":""},{"text":". A E Motter, C S Zhou, J Kurths, Europhysics Letters. 69334A. E. Motter, C. S. Zhou, and J. Kurths, Europhysics Letters 69 (2005) 334.","doi":""},{"text":". X Wang, Y.-C Lai, C H Lai, Physical Review E. 7556205X. Wang, Y.-C. Lai, and C. H. Lai, Physical Review E 75 (2007) 056205.","doi":""},{"text":". L M Pecora, T L Carroll, Physical Review Letters. 802109L. M. Pecora and T. L. Carroll, Physical Review Letters 80 (1998) 2109.","doi":""},{"text":"We have indeed showed that, in general, different interpretations of synchronizability, i.e. the eigenratio, cost of synchronization, time to synchronize and phase order parameter. do not lie in the same line. However, if the networks are weighted using the algorithm proposed in this paper, they do (not yet publishedWe have indeed showed that, in general, different interpretations of synchronizability, i.e. the eigenratio, cost of synchronization, time to synchronize and phase order parameter, do not lie in the same line. However, if the networks are weighted using the algorithm proposed in this paper, they do (not yet published).","doi":""},{"text":". A Rad, M Jalili, M Hasler, Chaos. A. Ajdari Rad, M. Jalili, and M. Hasler, Chaos (2008)","doi":""},{"text":". L C Freeman, Siociometry. 4035L. C. Freeman, Siociometry 40 (1977) 35.","doi":""},{"text":". S A Gerschgorin, Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk. 7749S. A. Gerschgorin, Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk 7 (1931) 749.","doi":""},{"text":". R Milo, S Itzkovitz, N Kashtan, R Levitt, S Shen-Orr, I Ayzenshtat, M Sheffer, U Alon, Science. 3031538R. Milo, S. Itzkovitz, N. Kashtan, R. Levitt, S. Shen-Orr, I. Ayzenshtat, M. Sheffer, and U. Alon, Science 303 (2004) 1538.","doi":""},{"text":". V Colizza, A Barrat, M Barthelemy, A Vespignani, Proceedings of the National Academy of Science of the United States of America. 103V. Colizza, A. Barrat, M. Barthelemy, and A. Vespignani, Proceedings of the National Academy of Science of the United States of America 103 (2006) 2015.","doi":""},{"text":". R Guimera, L Danon, A Diaz-Guilera, F Giralt, A Arenas, Physical review E. 6865103R. Guimera, L. Danon, A. Diaz-Guilera, F. Giralt, and A. Arenas, Physical review E 68 (2003) 065103","doi":""},{"text":"T Ito, T Chiba, R Ozawa, M Yoshida, M Hattori, Y Sakaki, Proceedings of the National Academy of Science of the United States of America. the National Academy of Science of the United States of America984569T. Ito, T. Chiba, R. Ozawa, M. Yoshida, M. Hattori, and Y. Sakaki, Proceedings of the National Academy of Science of the United States of America 98 (2001) 4569.","doi":""},{"text":". H Jeong, S P Mason, A.-L Barabási, Z N Oltvai, Nature. 41141H. Jeong, S. P. Mason, A.-L. Barabási, and Z. N. Oltvai, Nature 411 (2001) 41.","doi":""},{"text":". V Krishnamurthy, M Faloutsos, M Chrobak, L Lao, J.-H Cui, A G Percus, Lecture Notes in Computer Science. 3462V. Krishnamurthy, M. Faloutsos, M. Chrobak, L. Lao, J.-H. Cui, and A. G. Percus, Lecture Notes in Computer Science 3462 (2005) 328-341.","doi":""},{"text":". M E J Newman, Proceedings of the National Academy of Science of the United States of America. 101M. E. J. Newman, Proceedings of the National Academy of Science of the United States of America 101 (2004) 5200-5205.","doi":""},{"text":". S Boccaletti, J Kurths, G Osipov, D L Valladares, C S Zhou, Physics Reports. 3661S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares, and C. S. Zhou, Physics Reports 366 (2002) 1.","doi":""},{"text":". O E Rössler, Physics Letters A. 57397O. E. Rössler, Physics Letters A 57 (1976) 397.","doi":""},{"text":"Logarithm of the eigenratio λ N /λ 2 as a function of (α,B) for scale-free networks with a) N = 1000, m = 1, b) N = 1000, m = 2, c) N = 2000, m = 1, and d) N =. FIG. 1. (color online. m = 2. The graphs refer to averaging over 20 realizations of the networksFIG. 1. (color online) Logarithm of the eigenratio λ N /λ 2 as a function of (α,B) for scale-free networks with a) N = 1000, m = 1, b) N = 1000, m = 2, c) N = 2000, m = 1, and d) N = 2000, m = 2. The graphs refer to averaging over 20 realizations of the networks.","doi":""},{"text":"Logarithm of the eigenratio λ N /λ 2 as a function of (α,P) for Watts-Strogatz networks with a) N = 1000, m = 1 (the largest component), b) N = 1000, m = 2, c) N = 2000, m = 1 (the largest component), and d) N = 2000, m = 2. The graphs refer to averaging over 20 2 and B = 5, Watts-Strogatz (SW) networks with m = 2 and P = 0.5. The network. FIG. 2. (color online. size in all case is N = 1500FIG. 2. (color online) Logarithm of the eigenratio λ N /λ 2 as a function of (α,P) for Watts-Strogatz networks with a) N = 1000, m = 1 (the largest component), b) N = 1000, m = 2, c) N = 2000, m = 1 (the largest component), and d) N = 2000, m = 2. The graphs refer to averaging over 20 2 and B = 5, Watts-Strogatz (SW) networks with m = 2 and P = 0.5. The network size in all case is N = 1500.","doi":""}],"string":"[\n {\n \"text\": \". S H Strogatz, Nature. 410268S. H. Strogatz, Nature 410 (2001) 268.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"M Newman, A.-L Barabasi, D J Watts, The structure and dynamics of networks. Princeton University PressM. Newman, A.-L. Barabasi, and D. J. Watts, The structure and dynamics of networks, Princeton University Press, 2006.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Boccaletti, V Latora, Y Moreno, M Chavez, D U Hwang, Physics Reports. 424175S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D. U. Hwang, Physics Reports 424 (2006) 175.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M E J Newman, SIAM Review. 45167M. E. J. Newman, SIAM Review 45 (2003) 167.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"G V Osipov, J Kurths, C Zhou, Synchronization in Oscillatory Networks. SpringerG. V. Osipov, J. Kurths, and C. Zhou, Synchronization in Oscillatory Networks, Springer, 2007.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D J Watts, S H Strogatz, Nature. 393440D. J. Watts and S. H. Strogatz, Nature 393 (1998) 440.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A.-L Barabasi, R Albert, Science. 2865009A.-L. Barabasi and R. Albert, Science 286 (1999) 5009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Synchronization: a universal concept in nonlinear sciences. A Pikovsky, M Rosenblum, J Kurths, Cambridge University PressA. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: a universal concept in nonlinear sciences, Cambridge University Press, 2003.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Buzsaki, C Geisler, D A Henze, X.-J Wang, Trends in Neurosciences. 27186G. Buzsaki, C. Geisler, D. A. Henze, and X.-J. Wang, Trends in Neurosciences 27 (2004) 186.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G A Polis, Nature. 395744G. A. Polis, Nature 395 (1998) 744.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Latora, M Marchiori, Physical Review Letters. 87V. Latora and M. Marchiori, Physical Review Letters 87 (2001) 198701.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J J Ramasco, S N Dorogovtsev, R Pastor-Satorras, Physical Review E. 7036106J. J. Ramasco, S. N. Dorogovtsev, and R. Pastor-Satorras, Physical Review E 70 (2004) 036106.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Fischer, U Sauer, Nature Genetics. 37636E. Fischer and U. Sauer, Nature Genetics 37 (2005) 636.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Korniss, M A Novotny, H Guclu, Z Toroczkai, P A Rikvold, Science. 299677G. Korniss, M. A. Novotny, H. Guclu, Z. Toroczkai, and P. A. Rikvold, Science 299 (2003) 677.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Jalili, A Rad, M Hasler, International Journal of Circuit Theory and Applications. 35611M. Jalili, A. Ajdari Rad, and M. Hasler, International Journal of Circuit Theory and Applications 35 (2007) 611.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Chavez, D.-U Hwang, A Amann, S Boccaletti, Chaos. 1615106M. Chavez, D.-U. Hwang, A. Amann, and S. Boccaletti, Chaos 16 (2006) 015106.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Chavez, D.-U Hwang, A Amann, H G E Hentschel, S Boccaletti, Physical Review Letters. 94218701M. Chavez, D.-U. Hwang, A. Amann, H. G. E. Hentschel, and S. Boccaletti, Physical Review Letters 94 (2005) 218701.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A E Motter, C Zhou, J Kurths, Physical Review E. 7116116A. E. Motter, C. Zhou, and J. Kurths, Physical Review E 71 (2005) 016116.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A E Motter, C S Zhou, J Kurths, Europhysics Letters. 69334A. E. Motter, C. S. Zhou, and J. Kurths, Europhysics Letters 69 (2005) 334.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". X Wang, Y.-C Lai, C H Lai, Physical Review E. 7556205X. Wang, Y.-C. Lai, and C. H. Lai, Physical Review E 75 (2007) 056205.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L M Pecora, T L Carroll, Physical Review Letters. 802109L. M. Pecora and T. L. Carroll, Physical Review Letters 80 (1998) 2109.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"We have indeed showed that, in general, different interpretations of synchronizability, i.e. the eigenratio, cost of synchronization, time to synchronize and phase order parameter. do not lie in the same line. However, if the networks are weighted using the algorithm proposed in this paper, they do (not yet publishedWe have indeed showed that, in general, different interpretations of synchronizability, i.e. the eigenratio, cost of synchronization, time to synchronize and phase order parameter, do not lie in the same line. However, if the networks are weighted using the algorithm proposed in this paper, they do (not yet published).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Rad, M Jalili, M Hasler, Chaos. A. Ajdari Rad, M. Jalili, and M. Hasler, Chaos (2008)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L C Freeman, Siociometry. 4035L. C. Freeman, Siociometry 40 (1977) 35.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S A Gerschgorin, Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk. 7749S. A. Gerschgorin, Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk 7 (1931) 749.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Milo, S Itzkovitz, N Kashtan, R Levitt, S Shen-Orr, I Ayzenshtat, M Sheffer, U Alon, Science. 3031538R. Milo, S. Itzkovitz, N. Kashtan, R. Levitt, S. Shen-Orr, I. Ayzenshtat, M. Sheffer, and U. Alon, Science 303 (2004) 1538.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Colizza, A Barrat, M Barthelemy, A Vespignani, Proceedings of the National Academy of Science of the United States of America. 103V. Colizza, A. Barrat, M. Barthelemy, and A. Vespignani, Proceedings of the National Academy of Science of the United States of America 103 (2006) 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Guimera, L Danon, A Diaz-Guilera, F Giralt, A Arenas, Physical review E. 6865103R. Guimera, L. Danon, A. Diaz-Guilera, F. Giralt, and A. Arenas, Physical review E 68 (2003) 065103\",\n \"doi\": \"\"\n },\n {\n \"text\": \"T Ito, T Chiba, R Ozawa, M Yoshida, M Hattori, Y Sakaki, Proceedings of the National Academy of Science of the United States of America. the National Academy of Science of the United States of America984569T. Ito, T. Chiba, R. Ozawa, M. Yoshida, M. Hattori, and Y. Sakaki, Proceedings of the National Academy of Science of the United States of America 98 (2001) 4569.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Jeong, S P Mason, A.-L Barabási, Z N Oltvai, Nature. 41141H. Jeong, S. P. Mason, A.-L. Barabási, and Z. N. Oltvai, Nature 411 (2001) 41.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Krishnamurthy, M Faloutsos, M Chrobak, L Lao, J.-H Cui, A G Percus, Lecture Notes in Computer Science. 3462V. Krishnamurthy, M. Faloutsos, M. Chrobak, L. Lao, J.-H. Cui, and A. G. Percus, Lecture Notes in Computer Science 3462 (2005) 328-341.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M E J Newman, Proceedings of the National Academy of Science of the United States of America. 101M. E. J. Newman, Proceedings of the National Academy of Science of the United States of America 101 (2004) 5200-5205.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Boccaletti, J Kurths, G Osipov, D L Valladares, C S Zhou, Physics Reports. 3661S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares, and C. S. Zhou, Physics Reports 366 (2002) 1.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". O E Rössler, Physics Letters A. 57397O. E. Rössler, Physics Letters A 57 (1976) 397.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Logarithm of the eigenratio λ N /λ 2 as a function of (α,B) for scale-free networks with a) N = 1000, m = 1, b) N = 1000, m = 2, c) N = 2000, m = 1, and d) N =. FIG. 1. (color online. m = 2. The graphs refer to averaging over 20 realizations of the networksFIG. 1. (color online) Logarithm of the eigenratio λ N /λ 2 as a function of (α,B) for scale-free networks with a) N = 1000, m = 1, b) N = 1000, m = 2, c) N = 2000, m = 1, and d) N = 2000, m = 2. The graphs refer to averaging over 20 realizations of the networks.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Logarithm of the eigenratio λ N /λ 2 as a function of (α,P) for Watts-Strogatz networks with a) N = 1000, m = 1 (the largest component), b) N = 1000, m = 2, c) N = 2000, m = 1 (the largest component), and d) N = 2000, m = 2. The graphs refer to averaging over 20 2 and B = 5, Watts-Strogatz (SW) networks with m = 2 and P = 0.5. The network. FIG. 2. (color online. size in all case is N = 1500FIG. 2. (color online) Logarithm of the eigenratio λ N /λ 2 as a function of (α,P) for Watts-Strogatz networks with a) N = 1000, m = 1 (the largest component), b) N = 1000, m = 2, c) N = 2000, m = 1 (the largest component), and d) N = 2000, m = 2. The graphs refer to averaging over 20 2 and B = 5, Watts-Strogatz (SW) networks with m = 2 and P = 0.5. The network size in all case is N = 1500.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":487,"cells":{"text":{"kind":"string","value":"\nSigned Distance Laplacian Matrices for Signed Graphs\n8 Oct 2020\n\nRoshni T Roy roshnitroy@gmail.com \nDepartment of Mathematics\nCentral University of Kerala\nKasaragod -671316, Ker-alaIndia\n\nDepartment of Mathematics\nCentral University of Kerala\nKasaragod -671316, Ker-alaIndia\n\nShahul Hameed \nDepartment of Mathematics\nK M M Government Women's College\nKannur -670004KeralaIndia\n\nThomas Zaslavsky \nDepartment of Mathematical Sciences\nBinghamton University (SUNY)\n13902-6000BinghamtonNYU.S.A\n\nSigned Distance Laplacian Matrices for Signed Graphs\n8 Oct 2020BC1FD37B1782D0AD2EB5D50FBBD03DF5arXiv:2010.04204v1[math.CO]Signed graphsSigned distance matrixSigned distance laplacian matrixSigned distance laplacian spectrum Mathematics Subject Classification (2010): Primary 05C12Secondary 05C2205C5005C75\nA signed graph is a graph whose edges are labeled either positive or negative.Corresponding to the two signed distance matrices defined for signed graphs, we define two signed distance laplacian matrices.We characterize balance in signed graphs using these matrices and find signed distance laplacian spectra of some classes of unbalanced signed graphs.\n\nIntroduction\n\nThroughout this article, unless otherwise mentioned, by a graph we mean a finite, connected, simple graph.For any terms which are not mentioned here, the reader may refer to [4].\n\nA signed graph Σ = (G, σ) is an underlying graph G = (V, E) with a signature function σ : E → {1, −1} .Two types of signed distances matrices are introduced by Hameed et al. [3].In this paper, we define two signed distance laplacian matrices for signed graphs and characterize balance in signed graphs using these matrices.\n\nThe signed distance laplacian spectra of some classes of unbalanced signed graphs are also studied.\n\nFirst we recall the definition of signed distances and corresponding signed distance matrices defined in [3].Given a signed graph Σ = (G, σ) , the sign of a path P in Σ is defined as σ(P ) = e∈E(P ) σ(e) .The shortest path between two given vertices u and v is denoted by P (u,v) and the collection of all shortest paths P (u,v) by P (u,v) ; and d(u, v) denotes the usual distance between u and v .Definition 1.1 (Signed distance matrices [3]).Auxiliary signs are defined as: Signed distances are:\n(S1) σ max (u, v) = −1 if all shortest uv -paths(d1) d max (u, v) = σ max (u, v)d(u, v) = max{σ(P (u,v) ) : P (u,v) ∈ P (u,v) }d(u, v). (d2) d min (u, v) = σ min (u, v)d(u, v) = min{σ(P (u,v) ) : P (u,v) ∈ P (u,v) }d(u, v).\nAnd the signed distance matrices are: The distance laplacian matrix of a graph was introduced and studied by Mustapha et al. in [1].We introduce two signed distance laplacian matrices for signed graphs as follows.\n(D1) D max (Σ) = (d max (u, v)) n×n . (D2) D min (Σ) = (d min (u, v)) n×n .\nThe transmission T r(v) of a vertex v is defined to be the sum of the distances from v to all other vertices in G .That is, T r(v) = u∈V (G) d(v, u) .The transmission matrix T r(G) for a graph is the diagonal matrix with diagonal entries T r(v i ) .\n\nDefinition 1.4.Now we define two signed distance laplacian matrices for signed graphs as\n(L1) L max (Σ) = T r(G) − D max (Σ). (L2) L min (Σ) = T r(G) − D min (Σ). When Σ is compatible, L max (Σ) = L min (Σ) = L ± (Σ) .\n\nBalanced signed graphs\n\nTo study balance in signed graphs using signed distance laplacian matrices, it is necessary to study the laplacian matrix of weighted signed graphs.Let Σ = (G, σ) be a signed graph and let w be a positive weight function defined on the edges of Σ .We denote a weighted signed graph by (Σ, w) .For a weighted signed graph (Σ, w) , w(Σ) is the product of all the weights given to the edges of Σ .We use the notation u ∼ v when the vertices u and v are adjacent and similar notation for the incidence of an edge on a vertex.Definition 2.1.Let (Σ, w) be a weighted signed graph.Its adjacency matrix A(Σ, w) = (a ij ) n is defined as the square matrix of order n = |V (G)| where\na ij = σ(v i v j )w(v i v j ) if v i ∼ v j 0 otherwise.\nDefinition 2.2.For a weighted signed graph (Σ, w) , its weighted laplacian matrix is defined as L(Σ, w) = D(Σ, w) − A(Σ, w) where the diagonal matrix D(Σ, w) is diag e:v i ∼e w(e) .The matrix D(Σ, w) is the weighted degree matrix of (Σ, w).\n\nTo discuss oriented incidence matrix, we orient the edges (in an arbitrarily but fixed way) of the weighted signed graph.For an oriented edge e j = − − → v i v k we take v i as the tail of that edge and v k as its head and we write t( e j ) = v i and h( e j ) = v k .\n\nWe choose weights w from the set of positive real numbers and take √ w as the positive square root of w .\n\nDefinition 2.3.Given a weighted signed graph (Σ, w) , its (oriented) weighted incidence matrix is defined as H(Σ, w) = (η v i e j ) where\nη v i e j = σ(e j ) w(e j ) if t( e j ) = v i , − w(e j ) if h( e j ) = v i , 0 otherwise. Let H T (Σ, w) = (η ′ e i v j\n) be the transpose of the weighted incidence matrix H(Σ, w) .Thus, η ′ e i v j = η v j e i .\n\nTheorem 2.4.For a weighted signed graph (Σ, w) , L(Σ, w) = H(Σ, w)H T (Σ, w).\n\nProof.Let v 1 , v 2 , . . ., v n and e 1 , e 2 , . . ., e m be the vertices and edges in Σ , respectively.The (i, j) th entry of HH T is\nm k=1 η v i e k η ′ e k v j . For i = j , η v i e k η ′ e k v j = η 2 v i e k = 0 if and only if e k is incident to v i . Then η v i e k η ′ e k v j = (± w(e k )) 2 = w(e k ) .\nThus, the diagonal entry in HH T is e:v i ∼e w(e).\n\nFor i = j , η v i e k η ′ e k v j = 0 if and only if e k is an edge joining v i and v j .Then\nη v i e k η ′ e k v j = −σ(e k )w(e k ) .\nIn both cases, the (i, j) th entry of HH T coincides with the (i, j) th entry of L(Σ, w) , hence the proof.\n\nLemma 2.5.For a weighted signed tree (Σ, w) , det L(Σ, w) = 0.\n\nProof.A tree on n vertices has n − 1 edges.Thus, H(Σ, w) is a matrix of order n × (n − 1) and hence L(Σ, w) = H(Σ, w)H T (Σ, w) has rank less than n .This implies det L(Σ, w) = 0.\n\nLemma 2.6.Let (Σ, w) be a weighted signed graph where the underlying graph is\na cycle C n of order n . Then det L(Σ, w) = 2w(C n )(1 − σ(C n )) .\nProof.Let the cycle be Proof.Define the orientation of edges so that for i < j the edge e i,j has tail v i and head v j .Let C = v 1 e 1 v 2 e 2 • • • v p e p v 1 be the unique cycle, and label the vertices so that each edge e ij not in C has v i nearer to C than v j ; in other words, the vertex labels increase when moving away from C .Then the incidence matrix H(Σ, w) has the following form: Lemma 2.8.Let (Σ, w) be a signed graph whose underlying graph is a 1 -forest.\nC n = v 1 e 1 v 2 e 2 v 3 e 3 • • • v n−1 e n−1 v n e n v 1 . The weighted inci- dence matrix H(Σ, w) is σ(e= (σ(C n ) − 1) w(C n ). Now, det H T (Σ, w) = det H(Σ, w), hence, det L(Σ, w) = (σ(C n ) − 1) w(C n ).(σ(C n ) − 1) w(C n ) = 2w(C n )(1 − σ(C n )). H(C,Then det L(Σ, w) = w(Σ) ψ 2(1 − σ(C ψ ))\n, where the product runs over all component 1 -trees ψ having unique cycle C ψ .\n\nProof.By suitable reordering of vertices and edges we can make the laplacian L(Σ, w) into a block diagonal matrix, where the blocks correspond to the components of the 1 -forest;.Thus, det L(Σ, w) is the product of the laplacian determinants of the components.The components are 1 -trees.By using Lemma 2.7 we get the expression for the determinant.\n\nA signed graph is said to be contrabalanced if it contains no positive cycles [5].\n\nSigned Distance Laplacian Matrices for Signed Graphs Lemma 2.9.Let L(Σ, w) be a weighted signed graph having n vertices and Ψ be a spanning subgraph of (Σ, w) having exactly n edges.Then, det L(Ψ, w) = 0 if and only if Ψ is a contrabalanced 1 -forest.\n\nProof.A spanning subgraph Ψ of n edges that is not a 1 -forest must have a component that has fewer edges than vertices, that is, it is a tree T , and det L(T, w) = 0 by Lemma 2.5.Since det L(Ψ, w) is the product of the laplacians of the components of (Ψ, w) , det L(Ψ, w) = 0 if Ψ is not a 1 -forest.Now, assuming (Ψ, w) is a 1 -forest, by Lemma 2.8, det L(Ψ, w) = 0 if and only if Ψ contains no positive cycles, which means that Ψ is a contrabalanced 1 -forest.\n\nLet c(Ψ) denote the number of components of a signed graph Ψ .\n\nTheorem 2.10.Let (Σ, w) be a signed graph; then\ndet L(Σ, w) = Ψ 4 c(Ψ) w(Ψ),\nwhere the summation runs over all contrabalanced spanning 1 -forests Ψ of G .\n\nProof.Since L(Σ, w) = H(Σ, w)H T (Σ, w) , by the Binet-Cauchy theorem [2] we get det L(Σ, w)\n= J det H(J) det H T (J) = J det L(J, w),\nwhere J is a spanning subgraph of G with exactly n edges.\n\nThus by Lemma 2.9, det L(Σ, w) = Ψ det L(Ψ) where the summation runs over all contrabalanced spanning 1 -forests Ψ of G. Since every cycle C ψ in Ψ is negative, the factor 1 − σ(C ψ ) = 2 .That gives the formula of the theorem.\n\nRecall that we assume the signed graph Σ is connected.\n\nTheorem 2.11.A weighted signed graph (Σ, w) is balanced if and only if the de-Proof.In the formula of Theorem 2.10 every term is positive, so the determinant is 0 if and only if there is no contrabalanced 1 -forest in Σ .Since Σ is connected, that implies it has no negative circle, that is, it is balanced.\n\nWe recall the characterization theorem for balance in signed graphs using signed distances proved by Hameed et al. in [3].Hence, by Theorem 2.12 we get the required characterization.\n\n\nSigned distance laplacian spectrum\n\nNow, we move to the signed distance laplacian spectrum of some types of signed graph.Thus, L ± (Σ) is similiar to L(G) , which implies that L ± (Σ) is cospectral with L(G) .\n\nConversely, suppose L max (Σ) = L min (Σ) = L ± (Σ) and L ± is cospectral with L(G) .Thus, det L ± (Σ) = det L(G) = 0 and hence, by Theorem 2.13, Σ is balanced.\n\n\n\nare negative, and +1 otherwise.(S2) σ min (u, v) = +1 if all shortest uv -paths are positive, and −1 otherwise.\n\n\nDefinition 1 . 2 (\n12\n[3]).Two vertices u and v in a signed graph Σ are said to bedistance-compatible (briefly, compatible) if d min (u, v) = d max (u, v) .And Σ is said to be (distance-)compatible if every two vertices are compatible.Then D max (Σ) = D min (Σ) = D ± (Σ).The two complete signed graphs from the distance matrices D max and D min is defined as follows.Definition 1.3 ([3]).The associated signed complete graph K D max (Σ) with respect to D max (Σ) is obtained by joining the non-adjacent vertices of Σ with edges having signs σ(uv) = σ max (uv).The associated signed complete graph K D min (Σ) with respect to D min (Σ) is obtained by joining the non-adjacent vertices of Σ with edges having signs σ(uv) = σ min (uv).\n\n\n.Since M 1 , 1\n11\n1 ) w(e 1 ) 0 . . .0 − w(e n ) − w(e 1 ) σ(e 2 ) w(e 2 ) . . .Expanding along the first row to find the determinant we get det H(Σ, w) = σ(e 1 ) w(e 1 )M 1,1 + (−1) n w(e n )M 1e 1 ) σ(e 2 ) w(e 2 ) . . .σ(e n−1 ) w(e n−1 ) and M 1,n are determinants of triangular matrices, det H(Σ, w) = σ(e 1 ) w(e 1 )σ(e 2 ) w(e 2 ) • • • σ(e n ) w(e n ) + (−1) n w(e n )(− w(e 1 )) • • • (− w(e n−1 ))\n\n\nLemma 2 . 7 .\n27\nLet (Σ, w) be a signed graph, where the underlying graph is a unicyclic graph of order n with unique cycle C .Then det L(Σ, w) = 2w(Σ)(1 − σ(C)).\n\n\n,\n\nwhich is an upper-triangular block matrix whose first diagonal block is H(C, w) and whose other diagonal elements correspond to the heads of edges not in C .Hence,det H(Σ, w) = det H(C, w) • n k=p+1 (− w(e)) = (σ(C) − 1)(−1) n−p w(Σ).Thus, det L(Σ, w) = det H(Σ, w) det H T (Σ, w) = 2w(Σ)(1 − σ(C)).\n\n\nTheorem 2 .\n2\n12 ([3]).For a signed graph Σ the following statements are equivalent:(i) Σ is balanced.(ii)The associated signed complete graph K D max (Σ) is balanced.(iii)The associated signed complete graph K D min (Σ) is balanced.(iv) D max (Σ) = D min (Σ) and the associated signed complete graph K D ± (Σ) is balanced.Now we are ready to characterize balance in signed graphs using signed distance laplacian matrices.Theorem 2.13.The following properties of a signed graph Σ are equivalent.(i) Σ is balanced.(ii) The max-signed distance laplacian determinant det L max (Σ) = 0. (iii) The min-signed distance laplacian determinant det L min (Σ) = 0. (iv) L max (Σ) = L min (Σ) and det L ± (Σ) = 0. Proof.Corresponding to the associated signed complete graph K D max (Σ) , we define a weighted signed complete graph (K D max (Σ), w) where w(e) = d max (u, v) and (K D min (Σ), w) , where w(e) = d min (u, v) for an edge e = uv .Then L(K D max (Σ), w) = L max (Σ) and L(K D min (Σ), w) = L min (Σ) .Thus, by Theorem 2.11, det L max (Σ) = 0 if and only if K D max (Σ) is balanced, and det L min (Σ) = 0 if and only if K D min (Σ) is balanced.\n\n\nTheorem 3 . 1 .\n31\nA signed graph Σ is balanced if and only if L max (Σ) = L min (Σ) = L ± (Σ) and L ± (Σ) is cospectral with L(G) .Proof.Suppose Σ = (G, σ) is balanced.Then Σ can be switched to an all positive signed graph Σ ζ = (G, +) .Now, D ± (Σ) exists[3] and is similar toD ± (Σ ζ ) , that is, D ± (Σ ζ ) = SD ± (Σ)S −1 [3].So, SL ± (Σ)S −1 = S(T r(G) − D ± (Σ))S −1 = T r(G) − D ± (Σ ζ ) = L ± (Σ ζ ).\n\nAcknowledgementThe first author would like to acknowledge her gratitude to Department of Science and Technology, Govt. of India for the financial support under INSPIRE Fellowship scheme Reg No: IF180462.The second author would like to acknowledge her gratitude to Science and Engineering Research Board (SERB), Govt. of India, for the financial support under the scheme Mathematical Research Impact Centric Support (MATRICS), vide order no.: File No. MTR/2017/000689.A signed graph Σ is t -transmission regular if T r(v) = u∈V (G) d(v, u) = t for all v ∈ V (G).Odd cycles C 2k+1 are k(k + 1) -transmission regular and even cyclesFor the odd unbalanced cycle C − n , where n = 2k + 1 , there is a unique shortest path between any two vertices.Thus,The signed distance spectrum of an odd unbalanced cycle is given in[3].Thus, we get, the signed distance laplacian spectrum of C − n as an immidiate corollary.\nTwo Laplacians for the distance matrix of a graph. M Aouchiche, P Hansen, Linear Algebra Appl. 4392013\n\nComprehensive Introduction to Linear Algebra. J G Broida, S G Williamson, 1989Addison WesleyRedwood City, Cal\n\nSigned distance in signed graphs. S Hameed, K Shijin, P Soorya, T Germina K A, Zaslavsky, Linear Algebra Appl. 6082021\n\nF Harary, Graph Theory. Reading, Mass.Addison Wesley1969\n\nBiased graphs. I. Bias, balance, and gains. T Zaslavsky, J. Combinatorial Theory Ser. B. 471989\n"},"corpus_id":{"kind":"number","value":222272048,"string":"222,272,048"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2010.04204\",\n \"mag\": 3092419419,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1080/03081087.2022.2158165\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/2010.04204v1.pdf\"\n ],\n \"pdf_sha\": \"4f42f693b90293d8b352f226594aced3e52683f1\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"https://arxiv.org/pdf/2010.04204\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Signed Distance Laplacian Matrices for Signed Graphs"},"authors":{"kind":"list like","value":["Roshni T Roy roshnitroy@gmail.com \nDepartment of Mathematics\nCentral University of Kerala\nKasaragod -671316, Ker-alaIndia\n\nDepartment of Mathematics\nCentral University of Kerala\nKasaragod -671316, Ker-alaIndia","Shahul Hameed \nDepartment of Mathematics\nK M M Government Women's College\nKannur -670004KeralaIndia","Thomas Zaslavsky \nDepartment of Mathematical Sciences\nBinghamton University (SUNY)\n13902-6000BinghamtonNYU.S.A"],"string":"[\n \"Roshni T Roy roshnitroy@gmail.com \\nDepartment of Mathematics\\nCentral University of Kerala\\nKasaragod -671316, Ker-alaIndia\\n\\nDepartment of Mathematics\\nCentral University of Kerala\\nKasaragod -671316, Ker-alaIndia\",\n \"Shahul Hameed \\nDepartment of Mathematics\\nK M M Government Women's College\\nKannur -670004KeralaIndia\",\n \"Thomas Zaslavsky \\nDepartment of Mathematical Sciences\\nBinghamton University (SUNY)\\n13902-6000BinghamtonNYU.S.A\"\n]","isConversational":false},"abstract":{"kind":"string","value":"A signed graph is a graph whose edges are labeled either positive or negative.Corresponding to the two signed distance matrices defined for signed graphs, we define two signed distance laplacian matrices.We characterize balance in signed graphs using these matrices and find signed distance laplacian spectra of some classes of unbalanced signed graphs."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Throughout this article, unless otherwise mentioned, by a graph we mean a finite, connected, simple graph.For any terms which are not mentioned here, the reader may refer to [4].\nA signed graph Σ = (G, σ) is an underlying graph G = (V, E) with a signature function σ : E → {1, −1} .Two types of signed distances matrices are introduced by Hameed et al. [3].In this paper, we define two signed distance laplacian matrices for signed graphs and characterize balance in signed graphs using these matrices.\nThe signed distance laplacian spectra of some classes of unbalanced signed graphs are also studied.\nFirst we recall the definition of signed distances and corresponding signed distance matrices defined in [3].Given a signed graph Σ = (G, σ) , the sign of a path P in Σ is defined as σ(P ) = e∈E(P ) σ(e) .The shortest path between two given vertices u and v is denoted by P (u,v) and the collection of all shortest paths P (u,v) by P (u,v) ; and d(u, v) denotes the usual distance between u and v .Definition 1.1 (Signed distance matrices [3]).Auxiliary signs are defined as: Signed distances are:\nAnd the signed distance matrices are: The distance laplacian matrix of a graph was introduced and studied by Mustapha et al. in [1].We introduce two signed distance laplacian matrices for signed graphs as follows.\nThe transmission T r(v) of a vertex v is defined to be the sum of the distances from v to all other vertices in G .That is, T r(v) = u∈V (G) d(v, u) .The transmission matrix T r(G) for a graph is the diagonal matrix with diagonal entries T r(v i ) .\nDefinition 1.4.Now we define two signed distance laplacian matrices for signed graphs as"},{"title":"Balanced signed graphs","content":"To study balance in signed graphs using signed distance laplacian matrices, it is necessary to study the laplacian matrix of weighted signed graphs.Let Σ = (G, σ) be a signed graph and let w be a positive weight function defined on the edges of Σ .We denote a weighted signed graph by (Σ, w) .For a weighted signed graph (Σ, w) , w(Σ) is the product of all the weights given to the edges of Σ .We use the notation u ∼ v when the vertices u and v are adjacent and similar notation for the incidence of an edge on a vertex.Definition 2.1.Let (Σ, w) be a weighted signed graph.Its adjacency matrix A(Σ, w) = (a ij ) n is defined as the square matrix of order n = |V (G)| where\nDefinition 2.2.For a weighted signed graph (Σ, w) , its weighted laplacian matrix is defined as L(Σ, w) = D(Σ, w) − A(Σ, w) where the diagonal matrix D(Σ, w) is diag e:v i ∼e w(e) .The matrix D(Σ, w) is the weighted degree matrix of (Σ, w).\nTo discuss oriented incidence matrix, we orient the edges (in an arbitrarily but fixed way) of the weighted signed graph.For an oriented edge e j = − − → v i v k we take v i as the tail of that edge and v k as its head and we write t( e j ) = v i and h( e j ) = v k .\nWe choose weights w from the set of positive real numbers and take √ w as the positive square root of w .\nDefinition 2.3.Given a weighted signed graph (Σ, w) , its (oriented) weighted incidence matrix is defined as H(Σ, w) = (η v i e j ) where\n) be the transpose of the weighted incidence matrix H(Σ, w) .Thus, η ′ e i v j = η v j e i .\nTheorem 2.4.For a weighted signed graph (Σ, w) , L(Σ, w) = H(Σ, w)H T (Σ, w).\nProof.Let v 1 , v 2 , . . ., v n and e 1 , e 2 , . . ., e m be the vertices and edges in Σ , respectively.The (i, j) th entry of HH T is\nThus, the diagonal entry in HH T is e:v i ∼e w(e).\nFor i = j , η v i e k η ′ e k v j = 0 if and only if e k is an edge joining v i and v j .Then\nIn both cases, the (i, j) th entry of HH T coincides with the (i, j) th entry of L(Σ, w) , hence the proof.\nLemma 2.5.For a weighted signed tree (Σ, w) , det L(Σ, w) = 0.\nProof.A tree on n vertices has n − 1 edges.Thus, H(Σ, w) is a matrix of order n × (n − 1) and hence L(Σ, w) = H(Σ, w)H T (Σ, w) has rank less than n .This implies det L(Σ, w) = 0.\nLemma 2.6.Let (Σ, w) be a weighted signed graph where the underlying graph is\nProof.Let the cycle be Proof.Define the orientation of edges so that for i < j the edge e i,j has tail v i and head v j .Let C = v 1 e 1 v 2 e 2 • • • v p e p v 1 be the unique cycle, and label the vertices so that each edge e ij not in C has v i nearer to C than v j ; in other words, the vertex labels increase when moving away from C .Then the incidence matrix H(Σ, w) has the following form: Lemma 2.8.Let (Σ, w) be a signed graph whose underlying graph is a 1 -forest.\n, where the product runs over all component 1 -trees ψ having unique cycle C ψ .\nProof.By suitable reordering of vertices and edges we can make the laplacian L(Σ, w) into a block diagonal matrix, where the blocks correspond to the components of the 1 -forest;.Thus, det L(Σ, w) is the product of the laplacian determinants of the components.The components are 1 -trees.By using Lemma 2.7 we get the expression for the determinant.\nA signed graph is said to be contrabalanced if it contains no positive cycles [5].\nSigned Distance Laplacian Matrices for Signed Graphs Lemma 2.9.Let L(Σ, w) be a weighted signed graph having n vertices and Ψ be a spanning subgraph of (Σ, w) having exactly n edges.Then, det L(Ψ, w) = 0 if and only if Ψ is a contrabalanced 1 -forest.\nProof.A spanning subgraph Ψ of n edges that is not a 1 -forest must have a component that has fewer edges than vertices, that is, it is a tree T , and det L(T, w) = 0 by Lemma 2.5.Since det L(Ψ, w) is the product of the laplacians of the components of (Ψ, w) , det L(Ψ, w) = 0 if Ψ is not a 1 -forest.Now, assuming (Ψ, w) is a 1 -forest, by Lemma 2.8, det L(Ψ, w) = 0 if and only if Ψ contains no positive cycles, which means that Ψ is a contrabalanced 1 -forest.\nLet c(Ψ) denote the number of components of a signed graph Ψ .\nTheorem 2.10.Let (Σ, w) be a signed graph; then\nwhere the summation runs over all contrabalanced spanning 1 -forests Ψ of G .\nProof.Since L(Σ, w) = H(Σ, w)H T (Σ, w) , by the Binet-Cauchy theorem [2] we get det L(Σ, w)\nwhere J is a spanning subgraph of G with exactly n edges.\nThus by Lemma 2.9, det L(Σ, w) = Ψ det L(Ψ) where the summation runs over all contrabalanced spanning 1 -forests Ψ of G. Since every cycle C ψ in Ψ is negative, the factor 1 − σ(C ψ ) = 2 .That gives the formula of the theorem.\nRecall that we assume the signed graph Σ is connected.\nTheorem 2.11.A weighted signed graph (Σ, w) is balanced if and only if the de-Proof.In the formula of Theorem 2.10 every term is positive, so the determinant is 0 if and only if there is no contrabalanced 1 -forest in Σ .Since Σ is connected, that implies it has no negative circle, that is, it is balanced.\nWe recall the characterization theorem for balance in signed graphs using signed distances proved by Hameed et al. in [3].Hence, by Theorem 2.12 we get the required characterization."},{"title":"Signed distance laplacian spectrum","content":"Now, we move to the signed distance laplacian spectrum of some types of signed graph.Thus, L ± (Σ) is similiar to L(G) , which implies that L ± (Σ) is cospectral with L(G) .\nConversely, suppose L max (Σ) = L min (Σ) = L ± (Σ) and L ± is cospectral with L(G) .Thus, det L ± (Σ) = det L(G) = 0 and hence, by Theorem 2.13, Σ is balanced.\nare negative, and +1 otherwise.(S2) σ min (u, v) = +1 if all shortest uv -paths are positive, and −1 otherwise."},{"title":"Definition 1 . 2 (","content":"[3]).Two vertices u and v in a signed graph Σ are said to bedistance-compatible (briefly, compatible) if d min (u, v) = d max (u, v) .And Σ is said to be (distance-)compatible if every two vertices are compatible.Then D max (Σ) = D min (Σ) = D ± (Σ).The two complete signed graphs from the distance matrices D max and D min is defined as follows.Definition 1.3 ([3]).The associated signed complete graph K D max (Σ) with respect to D max (Σ) is obtained by joining the non-adjacent vertices of Σ with edges having signs σ(uv) = σ max (uv).The associated signed complete graph K D min (Σ) with respect to D min (Σ) is obtained by joining the non-adjacent vertices of Σ with edges having signs σ(uv) = σ min (uv)."},{"title":".Since M 1 , 1","content":"1 ) w(e 1 ) 0 . . .0 − w(e n ) − w(e 1 ) σ(e 2 ) w(e 2 ) . . .Expanding along the first row to find the determinant we get det H(Σ, w) = σ(e 1 ) w(e 1 )M 1,1 + (−1) n w(e n )M 1e 1 ) σ(e 2 ) w(e 2 ) . . .σ(e n−1 ) w(e n−1 ) and M 1,n are determinants of triangular matrices, det H(Σ, w) = σ(e 1 ) w(e 1 )σ(e 2 ) w(e 2 ) • • • σ(e n ) w(e n ) + (−1) n w(e n )(− w(e 1 )) • • • (− w(e n−1 ))"},{"title":"Lemma 2 . 7 .","content":"Let (Σ, w) be a signed graph, where the underlying graph is a unicyclic graph of order n with unique cycle C .Then det L(Σ, w) = 2w(Σ)(1 − σ(C))."},{"title":",","content":"which is an upper-triangular block matrix whose first diagonal block is H(C, w) and whose other diagonal elements correspond to the heads of edges not in C .Hence,det H(Σ, w) = det H(C, w) • n k=p+1 (− w(e)) = (σ(C) − 1)(−1) n−p w(Σ).Thus, det L(Σ, w) = det H(Σ, w) det H T (Σ, w) = 2w(Σ)(1 − σ(C))."},{"title":"Theorem 2 .","content":"12 ([3]).For a signed graph Σ the following statements are equivalent:(i) Σ is balanced.(ii)The associated signed complete graph K D max (Σ) is balanced.(iii)The associated signed complete graph K D min (Σ) is balanced.(iv) D max (Σ) = D min (Σ) and the associated signed complete graph K D ± (Σ) is balanced.Now we are ready to characterize balance in signed graphs using signed distance laplacian matrices.Theorem 2.13.The following properties of a signed graph Σ are equivalent.(i) Σ is balanced.(ii) The max-signed distance laplacian determinant det L max (Σ) = 0. (iii) The min-signed distance laplacian determinant det L min (Σ) = 0. (iv) L max (Σ) = L min (Σ) and det L ± (Σ) = 0. Proof.Corresponding to the associated signed complete graph K D max (Σ) , we define a weighted signed complete graph (K D max (Σ), w) where w(e) = d max (u, v) and (K D min (Σ), w) , where w(e) = d min (u, v) for an edge e = uv .Then L(K D max (Σ), w) = L max (Σ) and L(K D min (Σ), w) = L min (Σ) .Thus, by Theorem 2.11, det L max (Σ) = 0 if and only if K D max (Σ) is balanced, and det L min (Σ) = 0 if and only if K D min (Σ) is balanced."},{"title":"Theorem 3 . 1 .","content":"A signed graph Σ is balanced if and only if L max (Σ) = L min (Σ) = L ± (Σ) and L ± (Σ) is cospectral with L(G) .Proof.Suppose Σ = (G, σ) is balanced.Then Σ can be switched to an all positive signed graph Σ ζ = (G, +) .Now, D ± (Σ) exists[3] and is similar toD ± (Σ ζ ) , that is, D ± (Σ ζ ) = SD ± (Σ)S −1 [3].So, SL ± (Σ)S −1 = S(T r(G) − D ± (Σ))S −1 = T r(G) − D ± (Σ ζ ) = L ± (Σ ζ )."}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Throughout this article, unless otherwise mentioned, by a graph we mean a finite, connected, simple graph.For any terms which are not mentioned here, the reader may refer to [4].\\nA signed graph Σ = (G, σ) is an underlying graph G = (V, E) with a signature function σ : E → {1, −1} .Two types of signed distances matrices are introduced by Hameed et al. [3].In this paper, we define two signed distance laplacian matrices for signed graphs and characterize balance in signed graphs using these matrices.\\nThe signed distance laplacian spectra of some classes of unbalanced signed graphs are also studied.\\nFirst we recall the definition of signed distances and corresponding signed distance matrices defined in [3].Given a signed graph Σ = (G, σ) , the sign of a path P in Σ is defined as σ(P ) = e∈E(P ) σ(e) .The shortest path between two given vertices u and v is denoted by P (u,v) and the collection of all shortest paths P (u,v) by P (u,v) ; and d(u, v) denotes the usual distance between u and v .Definition 1.1 (Signed distance matrices [3]).Auxiliary signs are defined as: Signed distances are:\\nAnd the signed distance matrices are: The distance laplacian matrix of a graph was introduced and studied by Mustapha et al. in [1].We introduce two signed distance laplacian matrices for signed graphs as follows.\\nThe transmission T r(v) of a vertex v is defined to be the sum of the distances from v to all other vertices in G .That is, T r(v) = u∈V (G) d(v, u) .The transmission matrix T r(G) for a graph is the diagonal matrix with diagonal entries T r(v i ) .\\nDefinition 1.4.Now we define two signed distance laplacian matrices for signed graphs as\"\n },\n {\n \"title\": \"Balanced signed graphs\",\n \"content\": \"To study balance in signed graphs using signed distance laplacian matrices, it is necessary to study the laplacian matrix of weighted signed graphs.Let Σ = (G, σ) be a signed graph and let w be a positive weight function defined on the edges of Σ .We denote a weighted signed graph by (Σ, w) .For a weighted signed graph (Σ, w) , w(Σ) is the product of all the weights given to the edges of Σ .We use the notation u ∼ v when the vertices u and v are adjacent and similar notation for the incidence of an edge on a vertex.Definition 2.1.Let (Σ, w) be a weighted signed graph.Its adjacency matrix A(Σ, w) = (a ij ) n is defined as the square matrix of order n = |V (G)| where\\nDefinition 2.2.For a weighted signed graph (Σ, w) , its weighted laplacian matrix is defined as L(Σ, w) = D(Σ, w) − A(Σ, w) where the diagonal matrix D(Σ, w) is diag e:v i ∼e w(e) .The matrix D(Σ, w) is the weighted degree matrix of (Σ, w).\\nTo discuss oriented incidence matrix, we orient the edges (in an arbitrarily but fixed way) of the weighted signed graph.For an oriented edge e j = − − → v i v k we take v i as the tail of that edge and v k as its head and we write t( e j ) = v i and h( e j ) = v k .\\nWe choose weights w from the set of positive real numbers and take √ w as the positive square root of w .\\nDefinition 2.3.Given a weighted signed graph (Σ, w) , its (oriented) weighted incidence matrix is defined as H(Σ, w) = (η v i e j ) where\\n) be the transpose of the weighted incidence matrix H(Σ, w) .Thus, η ′ e i v j = η v j e i .\\nTheorem 2.4.For a weighted signed graph (Σ, w) , L(Σ, w) = H(Σ, w)H T (Σ, w).\\nProof.Let v 1 , v 2 , . . ., v n and e 1 , e 2 , . . ., e m be the vertices and edges in Σ , respectively.The (i, j) th entry of HH T is\\nThus, the diagonal entry in HH T is e:v i ∼e w(e).\\nFor i = j , η v i e k η ′ e k v j = 0 if and only if e k is an edge joining v i and v j .Then\\nIn both cases, the (i, j) th entry of HH T coincides with the (i, j) th entry of L(Σ, w) , hence the proof.\\nLemma 2.5.For a weighted signed tree (Σ, w) , det L(Σ, w) = 0.\\nProof.A tree on n vertices has n − 1 edges.Thus, H(Σ, w) is a matrix of order n × (n − 1) and hence L(Σ, w) = H(Σ, w)H T (Σ, w) has rank less than n .This implies det L(Σ, w) = 0.\\nLemma 2.6.Let (Σ, w) be a weighted signed graph where the underlying graph is\\nProof.Let the cycle be Proof.Define the orientation of edges so that for i < j the edge e i,j has tail v i and head v j .Let C = v 1 e 1 v 2 e 2 • • • v p e p v 1 be the unique cycle, and label the vertices so that each edge e ij not in C has v i nearer to C than v j ; in other words, the vertex labels increase when moving away from C .Then the incidence matrix H(Σ, w) has the following form: Lemma 2.8.Let (Σ, w) be a signed graph whose underlying graph is a 1 -forest.\\n, where the product runs over all component 1 -trees ψ having unique cycle C ψ .\\nProof.By suitable reordering of vertices and edges we can make the laplacian L(Σ, w) into a block diagonal matrix, where the blocks correspond to the components of the 1 -forest;.Thus, det L(Σ, w) is the product of the laplacian determinants of the components.The components are 1 -trees.By using Lemma 2.7 we get the expression for the determinant.\\nA signed graph is said to be contrabalanced if it contains no positive cycles [5].\\nSigned Distance Laplacian Matrices for Signed Graphs Lemma 2.9.Let L(Σ, w) be a weighted signed graph having n vertices and Ψ be a spanning subgraph of (Σ, w) having exactly n edges.Then, det L(Ψ, w) = 0 if and only if Ψ is a contrabalanced 1 -forest.\\nProof.A spanning subgraph Ψ of n edges that is not a 1 -forest must have a component that has fewer edges than vertices, that is, it is a tree T , and det L(T, w) = 0 by Lemma 2.5.Since det L(Ψ, w) is the product of the laplacians of the components of (Ψ, w) , det L(Ψ, w) = 0 if Ψ is not a 1 -forest.Now, assuming (Ψ, w) is a 1 -forest, by Lemma 2.8, det L(Ψ, w) = 0 if and only if Ψ contains no positive cycles, which means that Ψ is a contrabalanced 1 -forest.\\nLet c(Ψ) denote the number of components of a signed graph Ψ .\\nTheorem 2.10.Let (Σ, w) be a signed graph; then\\nwhere the summation runs over all contrabalanced spanning 1 -forests Ψ of G .\\nProof.Since L(Σ, w) = H(Σ, w)H T (Σ, w) , by the Binet-Cauchy theorem [2] we get det L(Σ, w)\\nwhere J is a spanning subgraph of G with exactly n edges.\\nThus by Lemma 2.9, det L(Σ, w) = Ψ det L(Ψ) where the summation runs over all contrabalanced spanning 1 -forests Ψ of G. Since every cycle C ψ in Ψ is negative, the factor 1 − σ(C ψ ) = 2 .That gives the formula of the theorem.\\nRecall that we assume the signed graph Σ is connected.\\nTheorem 2.11.A weighted signed graph (Σ, w) is balanced if and only if the de-Proof.In the formula of Theorem 2.10 every term is positive, so the determinant is 0 if and only if there is no contrabalanced 1 -forest in Σ .Since Σ is connected, that implies it has no negative circle, that is, it is balanced.\\nWe recall the characterization theorem for balance in signed graphs using signed distances proved by Hameed et al. in [3].Hence, by Theorem 2.12 we get the required characterization.\"\n },\n {\n \"title\": \"Signed distance laplacian spectrum\",\n \"content\": \"Now, we move to the signed distance laplacian spectrum of some types of signed graph.Thus, L ± (Σ) is similiar to L(G) , which implies that L ± (Σ) is cospectral with L(G) .\\nConversely, suppose L max (Σ) = L min (Σ) = L ± (Σ) and L ± is cospectral with L(G) .Thus, det L ± (Σ) = det L(G) = 0 and hence, by Theorem 2.13, Σ is balanced.\\nare negative, and +1 otherwise.(S2) σ min (u, v) = +1 if all shortest uv -paths are positive, and −1 otherwise.\"\n },\n {\n \"title\": \"Definition 1 . 2 (\",\n \"content\": \"[3]).Two vertices u and v in a signed graph Σ are said to bedistance-compatible (briefly, compatible) if d min (u, v) = d max (u, v) .And Σ is said to be (distance-)compatible if every two vertices are compatible.Then D max (Σ) = D min (Σ) = D ± (Σ).The two complete signed graphs from the distance matrices D max and D min is defined as follows.Definition 1.3 ([3]).The associated signed complete graph K D max (Σ) with respect to D max (Σ) is obtained by joining the non-adjacent vertices of Σ with edges having signs σ(uv) = σ max (uv).The associated signed complete graph K D min (Σ) with respect to D min (Σ) is obtained by joining the non-adjacent vertices of Σ with edges having signs σ(uv) = σ min (uv).\"\n },\n {\n \"title\": \".Since M 1 , 1\",\n \"content\": \"1 ) w(e 1 ) 0 . . .0 − w(e n ) − w(e 1 ) σ(e 2 ) w(e 2 ) . . .Expanding along the first row to find the determinant we get det H(Σ, w) = σ(e 1 ) w(e 1 )M 1,1 + (−1) n w(e n )M 1e 1 ) σ(e 2 ) w(e 2 ) . . .σ(e n−1 ) w(e n−1 ) and M 1,n are determinants of triangular matrices, det H(Σ, w) = σ(e 1 ) w(e 1 )σ(e 2 ) w(e 2 ) • • • σ(e n ) w(e n ) + (−1) n w(e n )(− w(e 1 )) • • • (− w(e n−1 ))\"\n },\n {\n \"title\": \"Lemma 2 . 7 .\",\n \"content\": \"Let (Σ, w) be a signed graph, where the underlying graph is a unicyclic graph of order n with unique cycle C .Then det L(Σ, w) = 2w(Σ)(1 − σ(C)).\"\n },\n {\n \"title\": \",\",\n \"content\": \"which is an upper-triangular block matrix whose first diagonal block is H(C, w) and whose other diagonal elements correspond to the heads of edges not in C .Hence,det H(Σ, w) = det H(C, w) • n k=p+1 (− w(e)) = (σ(C) − 1)(−1) n−p w(Σ).Thus, det L(Σ, w) = det H(Σ, w) det H T (Σ, w) = 2w(Σ)(1 − σ(C)).\"\n },\n {\n \"title\": \"Theorem 2 .\",\n \"content\": \"12 ([3]).For a signed graph Σ the following statements are equivalent:(i) Σ is balanced.(ii)The associated signed complete graph K D max (Σ) is balanced.(iii)The associated signed complete graph K D min (Σ) is balanced.(iv) D max (Σ) = D min (Σ) and the associated signed complete graph K D ± (Σ) is balanced.Now we are ready to characterize balance in signed graphs using signed distance laplacian matrices.Theorem 2.13.The following properties of a signed graph Σ are equivalent.(i) Σ is balanced.(ii) The max-signed distance laplacian determinant det L max (Σ) = 0. (iii) The min-signed distance laplacian determinant det L min (Σ) = 0. (iv) L max (Σ) = L min (Σ) and det L ± (Σ) = 0. Proof.Corresponding to the associated signed complete graph K D max (Σ) , we define a weighted signed complete graph (K D max (Σ), w) where w(e) = d max (u, v) and (K D min (Σ), w) , where w(e) = d min (u, v) for an edge e = uv .Then L(K D max (Σ), w) = L max (Σ) and L(K D min (Σ), w) = L min (Σ) .Thus, by Theorem 2.11, det L max (Σ) = 0 if and only if K D max (Σ) is balanced, and det L min (Σ) = 0 if and only if K D min (Σ) is balanced.\"\n },\n {\n \"title\": \"Theorem 3 . 1 .\",\n \"content\": \"A signed graph Σ is balanced if and only if L max (Σ) = L min (Σ) = L ± (Σ) and L ± (Σ) is cospectral with L(G) .Proof.Suppose Σ = (G, σ) is balanced.Then Σ can be switched to an all positive signed graph Σ ζ = (G, +) .Now, D ± (Σ) exists[3] and is similar toD ± (Σ ζ ) , that is, D ± (Σ ζ ) = SD ± (Σ)S −1 [3].So, SL ± (Σ)S −1 = S(T r(G) − D ± (Σ))S −1 = T r(G) − D ± (Σ ζ ) = L ± (Σ ζ ).\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"are negative, and +1 otherwise.(S2) σ min (u, v) = +1 if all shortest uv -paths are positive, and −1 otherwise.","content":"are negative, and +1 otherwise.(S2) σ min (u, v) = +1 if all shortest uv -paths are positive, and −1 otherwise."},{"id":"","caption":"[3]).Two vertices u and v in a signed graph Σ are said to bedistance-compatible (briefly, compatible) if d min (u, v) = d max (u, v) .And Σ is said to be (distance-)compatible if every two vertices are compatible.Then D max (Σ) = D min (Σ) = D ± (Σ).The two complete signed graphs from the distance matrices D max and D min is defined as follows.Definition 1.3 ([3]).The associated signed complete graph K D max (Σ) with respect to D max (Σ) is obtained by joining the non-adjacent vertices of Σ with edges having signs σ(uv) = σ max (uv).The associated signed complete graph K D min (Σ) with respect to D min (Σ) is obtained by joining the non-adjacent vertices of Σ with edges having signs σ(uv) = σ min (uv).","content":"Definition 1 . 2 (\n12\n[3]).Two vertices u and v in a signed graph Σ are said to bedistance-compatible (briefly, compatible) if d min (u, v) = d max (u, v) .And Σ is said to be (distance-)compatible if every two vertices are compatible.Then D max (Σ) = D min (Σ) = D ± (Σ).The two complete signed graphs from the distance matrices D max and D min is defined as follows.Definition 1.3 ([3]).The associated signed complete graph K D max (Σ) with respect to D max (Σ) is obtained by joining the non-adjacent vertices of Σ with edges having signs σ(uv) = σ max (uv).The associated signed complete graph K D min (Σ) with respect to D min (Σ) is obtained by joining the non-adjacent vertices of Σ with edges having signs σ(uv) = σ min (uv)."},{"id":"","caption":"1 ) w(e 1 ) 0 . . .0 − w(e n ) − w(e 1 ) σ(e 2 ) w(e 2 ) . . .Expanding along the first row to find the determinant we get det H(Σ, w) = σ(e 1 ) w(e 1 )M 1,1 + (−1) n w(e n )M 1e 1 ) σ(e 2 ) w(e 2 ) . . .σ(e n−1 ) w(e n−1 ) and M 1,n are determinants of triangular matrices, det H(Σ, w) = σ(e 1 ) w(e 1 )σ(e 2 ) w(e 2 ) • • • σ(e n ) w(e n ) + (−1) n w(e n )(− w(e 1 )) • • • (− w(e n−1 ))","content":".Since M 1 , 1\n11\n1 ) w(e 1 ) 0 . . .0 − w(e n ) − w(e 1 ) σ(e 2 ) w(e 2 ) . . .Expanding along the first row to find the determinant we get det H(Σ, w) = σ(e 1 ) w(e 1 )M 1,1 + (−1) n w(e n )M 1e 1 ) σ(e 2 ) w(e 2 ) . . .σ(e n−1 ) w(e n−1 ) and M 1,n are determinants of triangular matrices, det H(Σ, w) = σ(e 1 ) w(e 1 )σ(e 2 ) w(e 2 ) • • • σ(e n ) w(e n ) + (−1) n w(e n )(− w(e 1 )) • • • (− w(e n−1 ))"},{"id":"","caption":"Let (Σ, w) be a signed graph, where the underlying graph is a unicyclic graph of order n with unique cycle C .Then det L(Σ, w) = 2w(Σ)(1 − σ(C)).","content":"Lemma 2 . 7 .\n27\nLet (Σ, w) be a signed graph, where the underlying graph is a unicyclic graph of order n with unique cycle C .Then det L(Σ, w) = 2w(Σ)(1 − σ(C))."},{"id":"","caption":"which is an upper-triangular block matrix whose first diagonal block is H(C, w) and whose other diagonal elements correspond to the heads of edges not in C .Hence,det H(Σ, w) = det H(C, w) • n k=p+1 (− w(e)) = (σ(C) − 1)(−1) n−p w(Σ).Thus, det L(Σ, w) = det H(Σ, w) det H T (Σ, w) = 2w(Σ)(1 − σ(C)).","content":",\n\nwhich is an upper-triangular block matrix whose first diagonal block is H(C, w) and whose other diagonal elements correspond to the heads of edges not in C .Hence,det H(Σ, w) = det H(C, w) • n k=p+1 (− w(e)) = (σ(C) − 1)(−1) n−p w(Σ).Thus, det L(Σ, w) = det H(Σ, w) det H T (Σ, w) = 2w(Σ)(1 − σ(C))."},{"id":"","caption":"12 ([3]).For a signed graph Σ the following statements are equivalent:(i) Σ is balanced.(ii)The associated signed complete graph K D max (Σ) is balanced.(iii)The associated signed complete graph K D min (Σ) is balanced.(iv) D max (Σ) = D min (Σ) and the associated signed complete graph K D ± (Σ) is balanced.Now we are ready to characterize balance in signed graphs using signed distance laplacian matrices.Theorem 2.13.The following properties of a signed graph Σ are equivalent.(i) Σ is balanced.(ii) The max-signed distance laplacian determinant det L max (Σ) = 0. (iii) The min-signed distance laplacian determinant det L min (Σ) = 0. (iv) L max (Σ) = L min (Σ) and det L ± (Σ) = 0. Proof.Corresponding to the associated signed complete graph K D max (Σ) , we define a weighted signed complete graph (K D max (Σ), w) where w(e) = d max (u, v) and (K D min (Σ), w) , where w(e) = d min (u, v) for an edge e = uv .Then L(K D max (Σ), w) = L max (Σ) and L(K D min (Σ), w) = L min (Σ) .Thus, by Theorem 2.11, det L max (Σ) = 0 if and only if K D max (Σ) is balanced, and det L min (Σ) = 0 if and only if K D min (Σ) is balanced.","content":"Theorem 2 .\n2\n12 ([3]).For a signed graph Σ the following statements are equivalent:(i) Σ is balanced.(ii)The associated signed complete graph K D max (Σ) is balanced.(iii)The associated signed complete graph K D min (Σ) is balanced.(iv) D max (Σ) = D min (Σ) and the associated signed complete graph K D ± (Σ) is balanced.Now we are ready to characterize balance in signed graphs using signed distance laplacian matrices.Theorem 2.13.The following properties of a signed graph Σ are equivalent.(i) Σ is balanced.(ii) The max-signed distance laplacian determinant det L max (Σ) = 0. (iii) The min-signed distance laplacian determinant det L min (Σ) = 0. (iv) L max (Σ) = L min (Σ) and det L ± (Σ) = 0. Proof.Corresponding to the associated signed complete graph K D max (Σ) , we define a weighted signed complete graph (K D max (Σ), w) where w(e) = d max (u, v) and (K D min (Σ), w) , where w(e) = d min (u, v) for an edge e = uv .Then L(K D max (Σ), w) = L max (Σ) and L(K D min (Σ), w) = L min (Σ) .Thus, by Theorem 2.11, det L max (Σ) = 0 if and only if K D max (Σ) is balanced, and det L min (Σ) = 0 if and only if K D min (Σ) is balanced."},{"id":"","caption":"A signed graph Σ is balanced if and only if L max (Σ) = L min (Σ) = L ± (Σ) and L ± (Σ) is cospectral with L(G) .Proof.Suppose Σ = (G, σ) is balanced.Then Σ can be switched to an all positive signed graph Σ ζ = (G, +) .Now, D ± (Σ) exists[3] and is similar toD ± (Σ ζ ) , that is, D ± (Σ ζ ) = SD ± (Σ)S −1 [3].So, SL ± (Σ)S −1 = S(T r(G) − D ± (Σ))S −1 = T r(G) − D ± (Σ ζ ) = L ± (Σ ζ ).","content":"Theorem 3 . 1 .\n31\nA signed graph Σ is balanced if and only if L max (Σ) = L min (Σ) = L ± (Σ) and L ± (Σ) is cospectral with L(G) .Proof.Suppose Σ = (G, σ) is balanced.Then Σ can be switched to an all positive signed graph Σ ζ = (G, +) .Now, D ± (Σ) exists[3] and is similar toD ± (Σ ζ ) , that is, D ± (Σ ζ ) = SD ± (Σ)S −1 [3].So, SL ± (Σ)S −1 = S(T r(G) − D ± (Σ))S −1 = T r(G) − D ± (Σ ζ ) = L ± (Σ ζ )."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"are negative, and +1 otherwise.(S2) σ min (u, v) = +1 if all shortest uv -paths are positive, and −1 otherwise.\",\n \"content\": \"are negative, and +1 otherwise.(S2) σ min (u, v) = +1 if all shortest uv -paths are positive, and −1 otherwise.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"[3]).Two vertices u and v in a signed graph Σ are said to bedistance-compatible (briefly, compatible) if d min (u, v) = d max (u, v) .And Σ is said to be (distance-)compatible if every two vertices are compatible.Then D max (Σ) = D min (Σ) = D ± (Σ).The two complete signed graphs from the distance matrices D max and D min is defined as follows.Definition 1.3 ([3]).The associated signed complete graph K D max (Σ) with respect to D max (Σ) is obtained by joining the non-adjacent vertices of Σ with edges having signs σ(uv) = σ max (uv).The associated signed complete graph K D min (Σ) with respect to D min (Σ) is obtained by joining the non-adjacent vertices of Σ with edges having signs σ(uv) = σ min (uv).\",\n \"content\": \"Definition 1 . 2 (\\n12\\n[3]).Two vertices u and v in a signed graph Σ are said to bedistance-compatible (briefly, compatible) if d min (u, v) = d max (u, v) .And Σ is said to be (distance-)compatible if every two vertices are compatible.Then D max (Σ) = D min (Σ) = D ± (Σ).The two complete signed graphs from the distance matrices D max and D min is defined as follows.Definition 1.3 ([3]).The associated signed complete graph K D max (Σ) with respect to D max (Σ) is obtained by joining the non-adjacent vertices of Σ with edges having signs σ(uv) = σ max (uv).The associated signed complete graph K D min (Σ) with respect to D min (Σ) is obtained by joining the non-adjacent vertices of Σ with edges having signs σ(uv) = σ min (uv).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"1 ) w(e 1 ) 0 . . .0 − w(e n ) − w(e 1 ) σ(e 2 ) w(e 2 ) . . .Expanding along the first row to find the determinant we get det H(Σ, w) = σ(e 1 ) w(e 1 )M 1,1 + (−1) n w(e n )M 1e 1 ) σ(e 2 ) w(e 2 ) . . .σ(e n−1 ) w(e n−1 ) and M 1,n are determinants of triangular matrices, det H(Σ, w) = σ(e 1 ) w(e 1 )σ(e 2 ) w(e 2 ) • • • σ(e n ) w(e n ) + (−1) n w(e n )(− w(e 1 )) • • • (− w(e n−1 ))\",\n \"content\": \".Since M 1 , 1\\n11\\n1 ) w(e 1 ) 0 . . .0 − w(e n ) − w(e 1 ) σ(e 2 ) w(e 2 ) . . .Expanding along the first row to find the determinant we get det H(Σ, w) = σ(e 1 ) w(e 1 )M 1,1 + (−1) n w(e n )M 1e 1 ) σ(e 2 ) w(e 2 ) . . .σ(e n−1 ) w(e n−1 ) and M 1,n are determinants of triangular matrices, det H(Σ, w) = σ(e 1 ) w(e 1 )σ(e 2 ) w(e 2 ) • • • σ(e n ) w(e n ) + (−1) n w(e n )(− w(e 1 )) • • • (− w(e n−1 ))\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let (Σ, w) be a signed graph, where the underlying graph is a unicyclic graph of order n with unique cycle C .Then det L(Σ, w) = 2w(Σ)(1 − σ(C)).\",\n \"content\": \"Lemma 2 . 7 .\\n27\\nLet (Σ, w) be a signed graph, where the underlying graph is a unicyclic graph of order n with unique cycle C .Then det L(Σ, w) = 2w(Σ)(1 − σ(C)).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"which is an upper-triangular block matrix whose first diagonal block is H(C, w) and whose other diagonal elements correspond to the heads of edges not in C .Hence,det H(Σ, w) = det H(C, w) • n k=p+1 (− w(e)) = (σ(C) − 1)(−1) n−p w(Σ).Thus, det L(Σ, w) = det H(Σ, w) det H T (Σ, w) = 2w(Σ)(1 − σ(C)).\",\n \"content\": \",\\n\\nwhich is an upper-triangular block matrix whose first diagonal block is H(C, w) and whose other diagonal elements correspond to the heads of edges not in C .Hence,det H(Σ, w) = det H(C, w) • n k=p+1 (− w(e)) = (σ(C) − 1)(−1) n−p w(Σ).Thus, det L(Σ, w) = det H(Σ, w) det H T (Σ, w) = 2w(Σ)(1 − σ(C)).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"12 ([3]).For a signed graph Σ the following statements are equivalent:(i) Σ is balanced.(ii)The associated signed complete graph K D max (Σ) is balanced.(iii)The associated signed complete graph K D min (Σ) is balanced.(iv) D max (Σ) = D min (Σ) and the associated signed complete graph K D ± (Σ) is balanced.Now we are ready to characterize balance in signed graphs using signed distance laplacian matrices.Theorem 2.13.The following properties of a signed graph Σ are equivalent.(i) Σ is balanced.(ii) The max-signed distance laplacian determinant det L max (Σ) = 0. (iii) The min-signed distance laplacian determinant det L min (Σ) = 0. (iv) L max (Σ) = L min (Σ) and det L ± (Σ) = 0. Proof.Corresponding to the associated signed complete graph K D max (Σ) , we define a weighted signed complete graph (K D max (Σ), w) where w(e) = d max (u, v) and (K D min (Σ), w) , where w(e) = d min (u, v) for an edge e = uv .Then L(K D max (Σ), w) = L max (Σ) and L(K D min (Σ), w) = L min (Σ) .Thus, by Theorem 2.11, det L max (Σ) = 0 if and only if K D max (Σ) is balanced, and det L min (Σ) = 0 if and only if K D min (Σ) is balanced.\",\n \"content\": \"Theorem 2 .\\n2\\n12 ([3]).For a signed graph Σ the following statements are equivalent:(i) Σ is balanced.(ii)The associated signed complete graph K D max (Σ) is balanced.(iii)The associated signed complete graph K D min (Σ) is balanced.(iv) D max (Σ) = D min (Σ) and the associated signed complete graph K D ± (Σ) is balanced.Now we are ready to characterize balance in signed graphs using signed distance laplacian matrices.Theorem 2.13.The following properties of a signed graph Σ are equivalent.(i) Σ is balanced.(ii) The max-signed distance laplacian determinant det L max (Σ) = 0. (iii) The min-signed distance laplacian determinant det L min (Σ) = 0. (iv) L max (Σ) = L min (Σ) and det L ± (Σ) = 0. Proof.Corresponding to the associated signed complete graph K D max (Σ) , we define a weighted signed complete graph (K D max (Σ), w) where w(e) = d max (u, v) and (K D min (Σ), w) , where w(e) = d min (u, v) for an edge e = uv .Then L(K D max (Σ), w) = L max (Σ) and L(K D min (Σ), w) = L min (Σ) .Thus, by Theorem 2.11, det L max (Σ) = 0 if and only if K D max (Σ) is balanced, and det L min (Σ) = 0 if and only if K D min (Σ) is balanced.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"A signed graph Σ is balanced if and only if L max (Σ) = L min (Σ) = L ± (Σ) and L ± (Σ) is cospectral with L(G) .Proof.Suppose Σ = (G, σ) is balanced.Then Σ can be switched to an all positive signed graph Σ ζ = (G, +) .Now, D ± (Σ) exists[3] and is similar toD ± (Σ ζ ) , that is, D ± (Σ ζ ) = SD ± (Σ)S −1 [3].So, SL ± (Σ)S −1 = S(T r(G) − D ± (Σ))S −1 = T r(G) − D ± (Σ ζ ) = L ± (Σ ζ ).\",\n \"content\": \"Theorem 3 . 1 .\\n31\\nA signed graph Σ is balanced if and only if L max (Σ) = L min (Σ) = L ± (Σ) and L ± (Σ) is cospectral with L(G) .Proof.Suppose Σ = (G, σ) is balanced.Then Σ can be switched to an all positive signed graph Σ ζ = (G, +) .Now, D ± (Σ) exists[3] and is similar toD ± (Σ ζ ) , that is, D ± (Σ ζ ) = SD ± (Σ)S −1 [3].So, SL ± (Σ)S −1 = S(T r(G) − D ± (Σ))S −1 = T r(G) − D ± (Σ ζ ) = L ± (Σ ζ ).\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Two Laplacians for the distance matrix of a graph. M Aouchiche, P Hansen, Linear Algebra Appl. 4392013","doi":""},{"text":"Comprehensive Introduction to Linear Algebra. J G Broida, S G Williamson, 1989Addison WesleyRedwood City, Cal","doi":""},{"text":"Signed distance in signed graphs. S Hameed, K Shijin, P Soorya, T Germina K A, Zaslavsky, Linear Algebra Appl. 6082021","doi":""},{"text":"F Harary, Graph Theory. Reading, Mass.Addison Wesley1969","doi":""},{"text":"Biased graphs. I. Bias, balance, and gains. T Zaslavsky, J. Combinatorial Theory Ser. B. 471989","doi":""}],"string":"[\n {\n \"text\": \"Two Laplacians for the distance matrix of a graph. M Aouchiche, P Hansen, Linear Algebra Appl. 4392013\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Comprehensive Introduction to Linear Algebra. J G Broida, S G Williamson, 1989Addison WesleyRedwood City, Cal\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Signed distance in signed graphs. S Hameed, K Shijin, P Soorya, T Germina K A, Zaslavsky, Linear Algebra Appl. 6082021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"F Harary, Graph Theory. Reading, Mass.Addison Wesley1969\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Biased graphs. I. Bias, balance, and gains. T Zaslavsky, J. Combinatorial Theory Ser. B. 471989\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":488,"cells":{"text":{"kind":"string","value":"\nNONVANISHING OF HYPERELLIPTIC ZETA FUNCTIONS OVER FINITE FIELDS\n6 Oct 2021\n\nJordan S Ellenberg \nWanlin Li \nMark Shusterman \nNONVANISHING OF HYPERELLIPTIC ZETA FUNCTIONS OVER FINITE FIELDS\n6 Oct 2021\nFixing t ∈ R and a finite field F q of odd characteristic, we give an explicit upper bound on the proportion of genus g hyperelliptic curves over F q whose zeta function vanishes at 1 2 + it. Our upper bound is independent of g and tends to 0 as q grows.\n\nINTRODUCTION\n\nLet p be an odd prime, set q = p k for some positive integer k, and denote by F q the finite field with q elements. To (the smooth completion of) any hyperelliptic curve C over F q one associates a zeta function Z C (s). Weil has shown that Z C (s) = 0 implies that s = 1 2 + it for some t ∈ R. It is widely believed that for any fixed s = 1 2 + it, the 'vast majority' of (hyperelliptic) curves do not have s as a zero of their zeta function. For example, it follows from the work [6] of Chavdarov (and its improvement by Kowalski [13]) that for any fixed (large enough) g, the proportion of genus g hyperelliptic zeta functions vanishing at s tends to 0 as q → ∞.\n\nHere we are concerned with the growing g regime. Namely, for fixed q (and s), we give an upper bound on (1.1) h q,s := sup\ng {C ∈ H g (F q ) : Z C (s) = 0} H g (F q )\nwhere H g (F q ) is the family of genus g hyperelliptic curves over F q . Our bound is better once q is large, as given by our main result. Theorem 1.2 (Theorem 3.2). Fix a prime p, a real number t, and set s = 1 2 + it. Then as k → ∞ we have (1.3) h p k ,s ≪ p −k/276 .\n\nIn particular, h p k ,s tends to 0 as k tends to ∞.\n\nThis complements (but does not quite match) lower bounds on h q,s obtained by Li in [15].\n\nRestricting q to powers of a fixed prime p is not always necessary. In case s = 1 2 , one can show (see [19]) using transcendental number theory (six exponentials theorem [14, Chapter 2, Section 1]) that there are only finitely many p for which p −s is algebraic, so h q,s = 0 for any q not divisible by these p (as Z C (s) is a rational function in q −s ). Hence, it suffices to work with one characteristic at a time, as we do in the theorem above. For s = 1 2 , since the upper bound in Corollary 2.6 holds for any ℓ when q is a perfect square, we can conclude lim q→∞ h q,s = 0 ranging over q which is an even power of a prime. 1 Additional motivation for Theorem 1.2 comes from the ability to write Z C (s) as a rational function in q −s , with the numerator being a quadratic Dirichlet L-function. Interpreted in this language of Dirichlet characters, Theorem 1.2 improves (for all sufficiently large q) upon [5, Corollary 2.1] of Bui and Florea (they give a lower bound of more than 94.27% nonvanishing at s = 1 2 ). Regarding the analogous vanishing problem for quadratic Dirichlet L-functions over Z, we refer to the work [22] of Soundararajan and references therein 1 .\n\nAs we explain in the last section, our theorem can be rephrased as an upper bound for the number of quadratic twists of a constant abelian variety which have positive rank. Motivated by [5,Corollary 2.2] and the analogous results over Z of Conrey, Ghosh and Gonek from [7], we bound the multiplicity of the zeros of Z C , and obtain further information on nonvanishing at s = 1 2 . Theorem 1.5. Let C be a hyperelliptic curve of genus at least 2 over F q and S be the set of Weierstrass points of C. The Frobenius acts on S by permuting the 2g + 2 Weierstrass points via some permutation π. Suppose that either • g is even and π is a (2g + 2)-cycle; or • π is the product of two disjoint cycles of odd length. Then:\n\n(1) The point s = 1 2 is not a zero of Z C . (2) All zeros of Z C are of multiplicity at most 2. Moreover, if π is the product of two disjoint cycles of coprime lengths, all zeros of Z C are simple.\n\nIn the language of Dirichlet characters, this implies in particular the nonvanishing (at the central point) in the case of prime conductor of degree not divisible by 4 and therefore gives an explicit set of size on order X/ log X of Dirichlet characters of conductor at most X which have L-functions nonvanishing at the critical point. See the statement below.\nCorollary 1.6. Let χ be a quadratic character over F q (x) with conductor f ∈ F q [x]. If f is irreducible and 4 ∤ deg f , then L(1/2, χ) = 0.\nIn particular, the number of quadratic characters with irreducible conductor of size at most X whose L-function does not vanish at s = 1/2 is ≫ X/ log X as X → ∞. This result improves on [ In fact, there is nothing special about hyperelliptic curves in Theorem 1.5. A similar \"genus-theory\" argument allows us to handle the case of cyclic ℓ-covers of P 1 for an odd prime ℓ. Theorem 1.7. Let ℓ be an odd prime different from the characteristic of F q . Let C be a smooth projective curve over F q which admits a degree ℓ map f : C → P 1 F q such that f is Galois over F q with Galois group isomorphic to Z/ℓZ. Let S ⊂ P 1 (F q ) be the set of branch points of f . Let π be the permutation induced by the Frobenius action on S, and suppose that π is the composition of disjoint cycles of orders k 1 , k 2 , . . . , k r , all prime to ℓ.\n\n(1) Suppose the k i are mutually coprime and r ≤ 3. Then every zero of Z C has degree ℓ − 1.\n\n(2) Define κ i to be k i if k i is odd and k i /2 if k i is even. Suppose that either • q is congruent to 1 modulo ℓ and r = 2, with both cycles of odd length; or • There is no i such that q κ i is congruent to 1 modulo ℓ. Then the point s = 1 2 is not a zero of Z C . We remark that this theorem, like Theorem 1.5 above, can be used to produce a set of size on order X/ log(X) a of order-ℓ Dirichlet characters of conductor at most X whose zeta functions are non-vanishing at s = 1/2, for some power a ∈ (0, 1]. See Corollary 1.8 for example. This lower bound improves, for ℓ = 3, upon Corollary 1.3 of recent work by David-Florea-Lalin [8] which gives a lower bound of the form X 1−ǫ (for any ǫ > 0). Corollary 1.8. Let ℓ be an odd prime different from the characteristic of F q . Let d be the order of (q mod ℓ) in (Z/ℓZ) * . Let N(X) be the number of primitive degree ℓ Dirichlet characters\nχ f : (F q [t]/ f ) * → C * with conductor f satisfying q deg f ≤ X and L(1/2, χ f ) = 0.\nWe show the following:\n• if d = 1, then N(X) ≫ X log X for X → ∞; • if d is even, then N(X) ≫ X (log X) 1− ℓ−1 2d\nfor X → ∞. In particular, since d | ℓ − 1, we have N(X) ≫ X (log X) 1/2 for X → ∞. We do not get a lower bound of this quality in the case where q has odd order modulo ℓ.\n\nThe main idea that connects all the theorems in this paper is the study of L-functions modulo ℓ. The value of an L-function over F q (x) at a complex number s can be expressed as a polynomial P(T) ∈ Z[T] evaluated at T = q −s . So if we want to prove that P(T) is nonvanishing, it suffices to prove that P(T) is nonvanishing modulo ℓ for some prime ℓ. For Theorem 1.2, we will show that, for suitably chosen ℓ, the vanishing mod ℓ of the L-function is related to the dimension of a certain Frobenius eigenspace in the ℓ-torsion of a hyperelliptic Jacobian over F q ; the average size of this eigenspace can then be controlled by a point count on a moduli space over a finite field, which is a modest generalization and explication of the arguments in [10] and [16] respectively. For Theorem 1.5, on the other hand, we argue that under the given condition on Weierstrass points the L-function of χ f is nonvanishing mod 2 at s = 1/2. For the similar Theorem 1.7, the ℓ is again the order of the Dirichlet character in question. author was partially supported by NSF grant DMS-1700885 and by a fellowship from the Simons Foundation. We thank the referees for the valuable feedback and comments.\n\n\nMAIN THEOREM AND PROOF\n\n2.1. Setup and Notations. Throughout the paper, F q is a finite field of odd characteristic p. Let Q n,q be the set of squarefree polynomials over F q of degree n. For each f ∈ Q n,q , write J f for the Jacobian of the hyperelliptic curve\ny 2 = f (x) and P f (x) ∈ Z[x]\nfor the characteristic polynomial of geometric Frobenius acting on the ℓ-adic Tate module of J f . Let ℓ be a prime not equal to the characteristic of F q and let a be an element of (Z/ℓZ) * . The elements R of J f [ℓ](F q ) which satisfy\nFrob q ·R = aR.\nform a finite-dimensional vector space over Z/ℓZ and we denote by m a ( f ) the number of nonzero elements of this vector space. Note that m 1 ( f ) is just the number of F q -rational nontrivial ℓ-torsion points of J f . Let Q a,ℓ n,q be the set of squarefree polynomials f over F q of degree n such that m a ( f ) is greater than 0.\n\nLet α be a q-Weil number of weight 1 with minimal polynomial g α (x) ∈ Z[x]. Namely, it is an algebraic integer whose absolute values under all complex embeddings equal √ q. Let Q α n,q be the subset of Q n,q defined by { f ∈ Q n,q | P f (α −1 ) = 0}. With notation introduced as above, if g α (a) = 0 mod ℓ, then |Q α n,q | ≤ |Q a,ℓ n,q |. 2.2. Rational points on twisted Hurwitz spaces over finite fields. Our main tool will be the following result about the average size of the subspace of Jac(C)[ℓ](F q ) on which Frobenius acts by some specified scalar a, as C ranges over hyperelliptic curves over F q . More precisely, we study the variation as we range over y 2 = f (x) with f ranging over squarefree polynomials in F q [x]; this amounts to the same, since each isomorphism class of hyperelliptic curves is represented in this form the same number of times (assuming, of course, that the isomorphism classes are weighted inversely to the number of automorphisms they possess.) Proposition 2.1. Let a ∈ (Z/ℓZ) × . With notation as in Section 2.1, there exist constants C ℓ , N ℓ , Q ℓ only depending on ℓ such that\n∑ f ∈Q n,q m a ( f ) |Q n,q | − 1 ≤ C ℓ q −1/2 for all n ≥ N ℓ and q ≥ Q ℓ .\nRemark 2.2. While this paper was in proof, we learned that Proposition 2.1 follows from the proof of Theorem 1.1 of [16], which is in fact more general, and the arguments used are essentially the same as those here. We have left the proof of Proposition 2.1 in the present paper because the form in which we present the proof here is conducive to proving the explicit bounds for the stable range obtained in Proposition 2.7 below. It would be interesting to address the questions of explicit bounds for the stable range in the more general situations considered by [16], where the group-theoretic part of the proof of Proposition 2.7 would presumably be more complicated.\n\nWhen a = 1 and n is odd, Proposition 2.1 is essentially Theorem 8.8 of [10], and indeed the proof here is a modification of the proof of that theorem.\n\nThe reader may note that [10, Thm 8.8] requires not only that q is not a multiple of ℓ but that q is not congruent to 1 modulo ℓ. We face no such restriction here. That's because [10,Thm 8.8] computes arbitrary moments of the Cohen-Lenstra distribution, whereas we are only studying the analogue of the average size of the ℓ-part of the class group. In the language of [10, Thm 8.8], we are only considering the case A = Z/ℓZ. The difference is as follows. In the proof, we will end up estimating the number of F q -points on a moduli space over F p , and the result will depend on that space having just one geometrically irreducible component defined over F q . In the more general setting treated in [10,Thm 8.8], that space has many geometric components, all but one of which have fields of definition containing µ ℓ ; so when q is congruent to 1 mod ℓ there are multiple F q -rational components. In the case treated here, the moduli space in question is geometrically irreducible, so this issue does not arise.\n\nProof. We begin by observing that ∑ f ∈Q n,q m a ( f ) can be interpreted as the number of F qrational points of a certain moduli space.\n\nTo this end we briefly recall the setup of [10, Section 7]. Let k be a field, let G be a finite group with trivial center, denote by e the identity element of G, and let c be a conjugacy-closed subset of G\\e. By a tame G-cover of P 1 with monodromy type c we mean a triple (X, f , φ) where • X is a smooth proper geometrically connected curve X/k; • f : X → P 1 is a tamely ramified finite cover;\n\n• The image of tame inertia at each branch point of f excepting ∞ lies in c;\n\n• f is Galois with group G; that is, Aut( f ) acts transitively on the geometric fibers of f and φ is an automorphism from G to Aut( f ). Here by an isomorphism between two covers f : X → P 1 and f ′ : X ′ → P 1 we mean a morphism ψ :\nX → X ′ with f ′ • ψ = f , not a pair (ψ, ι) with ι a nontrivial automorphism\nof P 1 and f ′ • ψ = ι • f . In other words, our P 1 is \"labeled\". Then, as in [10, Section 7] (more or less immediate from a theorem of Romagny and Wewers [20]), there is a scheme Hn c G,n over Z[1/|G|] whose k-points (as long as k has characteristic prime to |G|) are in bijection with the isomorphism classes of tame G-covers of P 1 which have n branch points on A 1 with monodromy type c. (We do not specify whether or how the cover is branched at ∞.) 2 In fact ([20, Theorem 2.1]), for a scheme S, the set Hn c G,n (S) corresponds to isomorphism classes of tame G-covers over S, suitably defined; we will not need to spell out that definition here. Once the n branch points are chosen on A 1 there are finitely many choices for f and φ. Thus the dimension of Hn c G,n equals to n.\n\nFrom now on, we suppose that k is F q , that G is the dihedral group Z/ℓZ ⋊ Z/2Z, and that c is the conjugacy class of an involution in G. We will now explain the relationship between the space of G-covers and the ℓ-torsion in the Jacobian of hyperelliptic curves.\n\nThe key point is that, for any algebraic curve C, the set of surjections Jac(C)[ℓ] → (Z/ℓZ) k is naturally identified with the set ofétale (Z/ℓZ) k covers of C. For details, see Section 3.9 of [17].\n\nIf f : X → P 1 is a G-cover, the product structure of G allows us to factor f as\nX g → C h → P 1\nwhere h is a hyperelliptic cover and g is a Galois cover with group Z/ℓZ; that is, g is endowed with an isomorphism φ : Z/ℓZ → Aut(g). What's more, the fact that the monodromy in f is of type c implies that g is anétale cover, at least away from the points of C over ∞ ∈ P 1 . What happens over ∞ is slightly more delicate. The double cover h is branched at n points on A 1 , but the total number of branch points of h must be even as C is a smooth proper hyperelliptic curve. Thus, if n is odd, h is branched at ∞. The monodromy around ∞ in the cover X → P 1 is thus an element of G projecting to the nontrivial element of Z/2Z. Such an element must be an involution, and it follows that g is unramified at ∞. If n is even, on the other hand, it is possible for g to be ramified. We thus wish to restrict our attention to those G-covers X → C which are unramified over ∞. These are parametrized by a closed and open subscheme of Hn c G,n (indeed, it is the second term in the disjoint union in the paragraph following (7.3.1) of [10]). Let X n be this subscheme of Hn c G,n when n is even, and Hn c G,n when n is odd. In both cases, dim X n = n. We have explained how every point of X n (k) gives rise to a triple (g, φ, h)/k up to isomorphism, and in fact it is not hard to check that the converse holds as well. (This is essentially the last paragraph of the proof of [10,Proposition 8.7].)\n\nIf a is an element of (Z/ℓZ) * , we denote by a the automorphism of X n which sends (g, φ, h) to (g, aφ, h). We then write X a n for the twist of X n by the homomorphism \nGal(F q /F q ) → Aut(X n ) which∑ f ∈Q n,q m a ( f ) = (q − 1)|X a n (F q )| Proof. A point of X a n (F q ) is a point of X n (F q ) such that Frob q ·x = a · x. In other words, it is a triple (g, φ, h)/F q such that Frob q ·(g, φ, h) is isomorphic to (g, aφ, h).\nThe fact that the isomorphism class of h is fixed by Frobenius implies that the branch locus of h is an\nF q -rational divisor. Let f (x) ∈ F q [x]\nbe the unique monic squarefree polynomial which vanishes precisely at the branch locus of h. Then C is isomorphic (over F q ) to the smooth completion of the hyperelliptic curve defined by y 2 = f (x).\n\nFixing such an h, and thus such a C, we now consider the set of points of X a n (F q ) lying over this h. First of all, the choices of (g, φ) such that (g, φ, h) ∈ X a n (F q ) = X n (F q ) for a specified h are in bijection with the ℓ 2g(C) − 1 surjections from J(C)[ℓ](F q ) to Z/ℓZ. Two such surjections s, s ′ are isomorphic (that is, are parametrized by the same point of X a n (F q )) if and only if s = ±s ′ . The action of Frobenius on the set of surjections sends s to a −1 Frob q s; so s descends to a point of X a n (F q ) if and only if Frob q ·s = ±as. We conclude that the number of points of X a n (F q ) lying over h is (1/2)(m a ( f ) + m −a ( f )).\n\nNow if f in Q n,q is not monic then f = ǫF for some ǫ ∈ F * q and some monic F. The curve C f is isomorphic to C F if ǫ is a quadratic residue and to the nontrivial quadratic twist of C F otherwise. In the former case, m a ( f ) = m a (F), and in the latter, m a ( f ) = m −a (F). In particular, the quantity (1/2)(m a ( f ) + m −a ( f )) is the same for all q − 1 nonzero multiples of F. We conclude that\n∑ f ∈Q n,q (1/2)(m a ( f ) + m −a ( f )) = (q − 1)|X a n (F q )| Moreover, taking ǫ to be a non-residue in F * q , ∑ f ∈Q n,q m a ( f ) = ∑ f ∈Q n,q m a (ǫ f ) = ∑ f ∈Q n,q m −a ( f ) from which we obtain ∑ f ∈Q n,q m a ( f ) = (q − 1)|X a n (F q )| as desired.\nWe now argue exactly as in the proof of [10, Theorem 8.8].\n\nSince |Q n,q | = (q − 1)(q n − q n−1 ), it suffices to prove that (2.4) |q −n |X a n (F q )| − 1| ≤ C ℓ q −1/2 for some C ℓ depending only on ℓ and for all n > N ℓ , q > Q ℓ . Via the Grothendieck-Lefschetz trace formula, we have\n(2.5) |X a n (F q )| = ∑ i (−1) i Tr(Frob q |H i c,ét ((X a n ) F q ; Q λ ).\nwhere λ is a prime greater than max{2ℓ, q, n}. Note that theétale cohomology is that of the base change of X a n to F q , where it becomes isomorphic to the untwisted space X n ; in particular, the choice of a affects the action of Frobenius on theétale cohomology, but not theétale Betti numbers, bounds on which are the main engine of the argument.\n\nWe begin by computing the main term:\nTr(Frob q |H 2n c,ét ((X a n ) F q ; Q λ ) = q n .\nThis follows immediately from the fact that (X a n ) F q ∼ = (X n ) F q is irreducible. When n is odd, this is shown in the proof of [10, Theorem 8.8] as a consequence of a big monodromy theorem of J.K. Yu. (This is actually the only place where we need n to be large, and indeed n = 3 would be enough.) When n is even, we argue as follows. The map from X n to the configuration space Conf n A 1 sending a G-cover to its branch locus is a finite cover ([10, Section 2.2]), and irreducibility of X n is equivalent to the monodromy group of this cover acting transitively on the fiber. It suffices to check that this holds on a closed subvariety of the base. So write Z for the subvariety of Conf n A 1 consisting of those configurations containing some specified point p 0 ∈ P 1 (F q ), and let Y be the preimage of Z in X n . An automorphism of P 1 taking p 0 to ∞ now identifies Y with Hn c G,n−1 , which we know to be irreducible since n − 1 is odd. This implies that X n is irreducible.\n\nWe now turn to the error term. The moduli space X n is a closed and open subscheme of Hn c G,n , so its Betti numbers are bounded by those of Hn c G,n ; by [10, (7.8.1)] we have dim H 2n−i c,ét ((X a n ) F q ; Q λ ) ≤ K ℓ (B ℓ ) i where K ℓ , B ℓ are constants depending only on ℓ.\n\nUsing the Deligne bound [9], the eigenvalue of Frobenius on H i c,ét ((X a n ) F q ; Q λ ) is bounded in absolute value by q i/2 ; so the absolute value of the contribution of all i < 2n to (2.5) is bounded above by the sum of a geometric series which converges for all q > B 2 ℓ . In particular, as in [10, Section 1.8], this contribution is at most\nK ℓ B ℓ q −1/2 (1 − B ℓ q −1/2 ) −1 q n .\nSo if we take Q ℓ = 4B 2 ℓ and q > Q ℓ , we may take C ℓ = 2K ℓ B ℓ and conclude |X a\nn (F q ) − q n | = | 2n−1 ∑ i=0 (−1) i Tr(Frob q |H i c,ét ((X a n ) F q ; Q λ )| < C ℓ q n−1/2\nwhich proves (2.4) and thus the desired result.\n\nProposition 2.1 allows us to bound the proportion of hyperelliptic curves whoseétale cohomology has a Frobenius eigenvalue congruent to a mod ℓ. Recall from Section 2.1 that Q a,ℓ n,q is the set of squarefree polynomials over F q of degree n such that m a ( f ) is greater than 0. Corollary 2.6. There are constants C ′ ℓ , Q ℓ , N ℓ such that for any a ∈ (Z/ℓZ) × we have |Q a,ℓ n,q | |Q n,q |\n≤ 1 ℓ − 1 + C ′ ℓ q −1/2 for all n ≥ N ℓ , q ≥ Q ℓ .\nProof. Write δ for the quantity |Q n,q | −1 |Q a,ℓ n,q | to be bounded. Since m a ( f ) is the number of nonzero elements of a vector space over Z/ℓZ, it is at least ℓ − 1 if it is greater than 0. In particular,\n|Q n,q | −1 ∑ f ∈Q n,q m a ( f ) ≥ |Q n,q | −1 (ℓ − 1)|Q a,ℓ n,q | = (ℓ − 1)δ By Proposition 2.1, we now have (ℓ − 1)δ < 1 + C ℓ q −1/2\nfor all sufficiently large n, q, which yields the desired result by taking C ′ ℓ = C ℓ /(ℓ − 1). So far, we have used the results of [10] as they appear in that paper. However, for the present application, it is useful to compute explicit values C ℓ , Q ℓ for which Corollary 2.6 holds. We do so by going back to the main body of [10] and working out explicit bounds for quantities that are given in [10, (7.8.1)] only as unspecified constants. Proposition 2.7. Corollary 2.6 holds with C ′ ℓ = 2(ℓ − 1) −1 (4ℓ) 138 and Q ℓ = 4 · (4ℓ) 156 .\n\nProof. By the proof of Corollary 2.6, we may take C ′ ℓ to be C ℓ /(ℓ − 1), where C ℓ is the constant appearing in the statement of Proposition 2.1. Moreover, we may take C ℓ to be 2K ℓ B ℓ and Q ℓ to be 4B 2 ℓ , where B ℓ , K ℓ are the constants appearing in the proof of Proposition 2.1 controlling the exponential growth of the Betti numbers of the relevant Hurwitz space. We now explain how to bound B ℓ explicitly.\n\nIn [10, 7.8.1], the bound dim H ié t ((X a n ) F q ; Q λ ) ≤ K ℓ (B ℓ ) i arises from two facts. First, there is a stability theorem [10, 6.2], which tells us in this context that A, B, and D are constants we shall specify. Second, there is an absolute bound [10, 2.5] which tells us that dim H ié t ((X a n ) F q ; Q λ ) ≤ (4ℓ) n . These two facts together imply that dim H ié t ((X a n ) F q ; Q λ ) ≤ (4ℓ) Ai+B+D so we may take B ℓ = (4ℓ) A and K ℓ = (4ℓ) B+D . It remains to compute A, B, and D.\n(2.8) dim H ié t ((X a n ) F q ; Q λ ) = dim H ié t ((X a n+D ) F q ; Q λ ) for all n > Ai + B, where\nThe key object of computation is the ring R defined in [10, §3]. This ring is defined for any finite group G and any conjugacy-closed subset of G; we will consider here just the case relevant to us, which is that where G is the dihedral group of order 2ℓ and c is the class of involutions in G. The set of n-tuples of involutions (τ 1 , . . . , τ n ) ∈ G n carries a natural action of the n-strand braid group; the ring R is a graded Q-algebra whose degree-n part is spanned by the set of orbits of that action, which set we denote Σ n . The multiplication in R is given by concatenation of n-tuples.\n\nThe key fact about R is that it contains a central element U with the property that R[U] and R/UR both have finite degree (that is, they are supported in only finitely many grades.) In the dihedral case, R and U are particularly easy to describe. For any n, there is map from Σ n to G sending (τ 1 , . . . , τ n ) to the product τ 1 . . . , τ n , which is called the boundary monodromy. Each n-tuple in Σ n also has a monodromy group; namely, the group generated by τ 1 , . . . , τ n . The possible monodromy groups are just the order-2 subgroups of G and G itself. It is not hard to check that, for all n ≥ 4, the elements of Σ n are determined by their boundary monodromy and their monodromy group; to be precise, Σ n consists of ℓ orbits consisting of the single element (τ, τ, . . . , τ) as τ ranges over the ℓ involutions, and ℓ more orbits, each of which consists of all n-tuples with monodromy group G and boundary monodromy g, as g ranges over the index-2 cyclic subgroup of G (when n is even) or its nontrivial coset (when n is odd.) In particular, dim R n = 2ℓ for all n ≥ 4. We may take U to be the degree-2 central operator\nU = ∑ τ∈c (τ, τ)\nand check that U induces an isomorphism from R n to R n+2 for all n ≥ 4. In particular, deg R[U] and deg R/UR are both at most 4, where by the degree of a graded ring we mean the highest grade represented in its support.\n\nThis combinatorial information about the dihedral group is what goes into the computation of constants in [10]. The constant D in [10, 6.1] is just the degree of U, which is 2. The stability result in [10, 6.1] is derived from a general theorem [10, 4.2] about R-modules. The R-module M governing the H i of Hurwitz space, to which we apply [10, 4.2] is the one called M i in [10, 6.1], So (using the constants appearing in those theorems) stability begins when n = max(h 0 , h 1 ) + A 0 , where h j is the quantity denoted deg H j (K(M i ) in [10, 6.1]. In turn, as asserted in the first paragraph of the proof of [10,\n6.1], we have deg H j (K(M i )) ≤ A 2 + A 0 (3i + j)\nSo we find that (2.8) holds for all n ≤ A 2 + A 0 (3i + 1)\n+ A 0 = 3A 0 i + (2A 0 + A 2 ).\nIn other words, we may take A = 3A 0 and B = 2A 0 + A 2 . Finally, the values of A 0 and A 2 are given in [10, 4.5.3]. They are defined in terms of\nA(R) = max(deg R[U], deg R/UR), which for us is 4. Now A 0 = 6A(R) + deg U = 26 and A 2 = A(R) + deg U = 6.\nThus, A = 78 and B = 58. Since D = 2, we conclude that we may take B ℓ = (4ℓ) 78 and K ℓ = (4ℓ) 60 . So we have Q ℓ = 4 · (4ℓ) 156 and C ′ ℓ = 2(ℓ − 1) −1 (4ℓ) 138 , as claimed.\n\n\nAPPLICATION TO NONVANISHING OF L-FUNCTIONS\n\nWe can use the above reasoning to bound the number of quadratic L-functions over function fields which vanish at a specified point on the critical line. For the rest of this section we fix an odd prime p and consider only fields of characteristic p. We note that, if χ f is a quadratic character of F q (x), then L(s, χ f ) can vanish only at a point s such that q s is a q-Weil number of weight 1. We first recall the following lemma relating the vanishing of the L-function of a quadratic character in terms of the Frobenius eigenvalues of a hyperelliptic curve; This is immediate from the description of P as the numerator of the zeta function of C, and the connection of the latter to L(s, χ f ) (see, for instance, [21, Section 2]).\n\n\nTheorem 3.2.\n\nFor any squarefree polynomial f ∈ Q n,q , let L(s, χ f ) be the Dirichlet L-function associated to the quadratic character χ f as was defined in Section 2.1. Then for any s = 0,\nlim sup n→∞ |{ f ∈ Q n,q | L(s, χ f ) = 0}| |Q n,q | ≪ q −1/276\nwhere the limit is taken over all powers q of a fixed odd prime number p.\n\nProof. Fix an odd prime number p, and let q be a power of p. By Lemma 3.1, L(s, χ f ) = 0 is equivalent to P(q −s ) = 0 where P(x) ∈ Z[x] is the characteristic polynomial of Frobenius acting on the Jacobian of the hyperelliptic curve defined by y 2 = f (x). Thus, the set { f ∈ Q n,q | L(s, χ f ) = 0} is the same as Q q s n,q .\n\nBy Chebotarev's density theorem, we can (for large enough q) find a prime ℓ = 1 4 q 4 1/276 (1 + o(1)) mod which g p s , the minimal polynomial of p s , splits completely. Let a ∈ Z/lZ such that g p s (a) = 0 mod ℓ. If q = p t , then any f with L(χ f , s) = 0 has m a t ( f ) > 0. So |Q q s n,q | |Q n,q | ≤ |Q a t ,ℓ n,q | |Q n,q | and now we can apply Corollary 2.6 to conclude using the second equation of 2.1 that, for all sufficiently large t, we have\nlim sup n→∞ |Q q s n,q | |Q n,q | ≤ 1 ℓ − 1 + C ′ ℓ q −1/2 .\nThe required bound follows from Proposition 2.7.\n\nResults on the vanishing of quadratic L-functions over function fields can be used to study the rank distribution of quadratic twist families of constant abelian varieties. In the following corollary, we show that as the constant field grows (so the characteristic is not changing), the probability for a quadratic twist of a constant abelian variety to have positive rank goes to 0. In the elliptic curve case, this agrees with the general \"Minimalist Conjecture\" philosophy, which holds that positive ranks should be a density-0 phenomenon except when forced by parity considerations from the functional equation (in this setting the functional equation never forces positive rank, and the rank is always even.) We now prove Theorem 1.5, which makes use of the mod 2 Galois representations on J(C) rather than the representations modulo odd primes.\n\nProof of Theorem 1.5. Let x 1 , ..., x 2g+2 be the set of Weierstrass points of C. The 2-torsion subgroup J(C) [2] is spanned by the degree-0 2-torsion divisors x i − x j . That is, the group of divisors of the form ∑ a i x i with ∑ a i = 0 surjects onto J(C) [2]. Note also that x 1 + . . . + x 2g+2 − (2g + 2)x 1 is a principal divisor and thus x 1 + . . . + x 2g+2 is 0 in J(C) [2]. See [11,Secion 4] for detailed discussion. This identifies J(C) [2] with an explicit subquotient of F 2g+2 2 ; namely, J(C) [2] is the quotient of the subspace (a 1 , . . . , a 2g+2 ) : ∑ a i = 0 by the 1-dimensional subspace spanned by (1, . . . , 1). This identification is equivariant for the Frobenius action on both sides, so it allows us to describe the mod 2 Galois representation afforded by J(C) in terms of the permutation π which Frobenius induces on x 1 , . . . , x 2g+2 . To be precise, the action of S 2g+2 on J(C) [2] is a representation ρ : S 2g+2 → Sp 2g (Z/2Z), and the action of Frobenius on J(C) [2] is given by ρ(π).\n\nThe conditions on π given in Theorem 1.5 are equivalent to the condition that π 2 is a product of two disjoint odd cycles. Thus, the action of π 2 in its permutation representation F 2g+2 2 has eigenvalues given by µ k and µ 2g+2−k for some odd 1 ≤ k ≤ 2g + 1; passing to the subquotient J(C) [2] removes two eigenspaces of ρ(π 2 ) with the eigenvalue 1. So the eigenvalues of Frob 2 on J(C) [2] are the multiset µ ′ k ∪ µ ′ 2g+2−k where µ ′ n denotes the nontrivial nth roots of unity. We see in particular that ρ(π 2 ) does not have 1 as an eigenvalue. But if the zeta function Z C had a zero at 1/2, then √ q would be a Frobenius eigenvalue on C, which would mean that q was an eigenvalue of Frob 2 ; we have shown that Frob 2 has no eigenvalue congruent to 1 mod 2, which rules this out. This proves (1). What's more, the multiset µ ′ k ∪ µ ′ 2g+2−k contains any eigenvalue at most twice, and if (k, 2g + 2 − k) = 1, no eigenvalue appears more than once. This proves (2) (or rather, it proves (2) for the zeta function of C/F q 2 , from which (2) is immediate.)\n\nProof of Corollary 1.6. By assumption, f is an irreducible polynomial over F q . So when deg f = n is even, Frobenius acts on the set of Weierstrass points of C f : y 2 = f (x) as a n-cycle. If deg f = n is odd, then Frobenius acts on the set of Weierstrass points of C f as a disjoint union of a n-cycle and a 1-cycle. In either case, when 4 ∤ deg f we can apply Theorem 1.5 to conclude that Z C f does not vanish at 1/2 and so behaves L(s, χ f ).\n\nFor any X = q 2g+2 , the set of irreducible polynomials of odd degree at most 2g + 1 gives quadratic characters with bounded conductor whose L-function does not vanish at the central point s = 1/2. By the prime number theorem for function fields, the number of irreducible polynomials in F q [x] of degree at most n is ≫ q n /n. This gives the lower bound in statement.\n\nThe proof of Theorem 1.7 is very similar to that of Theorem 1.5, but we treat it separately in order to make the hyperelliptic case above more readable.\n\nProof of Theorem 1.7. Let x 1 , . . . , x m be the ramification points of the (Z/ℓZ)-cover of P 1 in S, where m = k 1 + · · · + k r . The Jacobian J(C) of C carries an action of Z[(Z/ℓZ)]; write λ ∈ Z[(Z/ℓZ)] for ζ ℓ − 1, where ζ ℓ is a generator of (Z/ℓZ). A Riemann-Hurwitz computation shows that the genus of C is (m − 2)(ℓ − 1)/2, so the Tate module T ℓ J(C) is a free Z ℓ [ζ ℓ ]-module of rank m − 2, and J(C) [λ] has dimension m − 2.\n\nThe λ-torsion subgroup of J(C) is spanned by the degree-0 λ-torsion divisors x i − x j . That is, the group of divisors of the form ∑ a i x i with ∑ a i = 0 surjects onto J(C) [λ]. This surjection is not an isomorphism; there is a 1-dimensional kernel, which we can describe as follows. Over F q , the curve C has an affine model of the form y ℓ = f (x) with f a rational function with no zeroes or poles at ∞. Then the principal divisor associated to y is ∑ a i x i where a i = ord x i f . We have now expressed J(C) [λ] as an explicit subquotient of F m ℓ . This identification is equivariant for the Frobenius action on both sides, so it allows us to describe the mod ℓ Galois representation afforded by J(C) in terms of the permutation π which Frobenius induces on x 1 , . . . , x m .\n\nThe action of π splits x 1 , . . . , x m into cycles of length k 1 , . . . , k r , which by hypothesis are prime to ℓ, and which must be multiples of d, where d is the order of q in F * ℓ . So the eigenvalues of π in its action on F m ℓ are the union (as multisets) r j=1 µ k j . Now the composition factors of F m ℓ as a representation of the cyclic group π are J(C) [λ], F ℓ div(y), and the π-trivial one-dimensional representation onto which F m ℓ maps by summing coordinates. The action of π on the latter factor is trivial, while π acts on F ℓ div(y) as multiplication by q. If the zeta function Z C had a zero at 1/2, then √ q would be a Frobenius eigenvalue of C, which would mean that some eigenvalue µ of the action of π on J(C)[λ] satisfied µ 2 = q.\n\nIn case q is congruent to 1 modulo ℓ (i.e., d = 1) the eigenvalues of π in its action on J(C) [λ] are the multiset r j=1 µ ′ k j together with r − 2 copies of 1, where µ ′ n denotes the nontrivial nth roots of unity. The hypotheses r = 2 and k j odd now guarantee that the eigenvalues of π on J(C) [λ] contain no copies of either 1 or −1, completing the proof in this case.\n\nIf d > 1, we note that our condition on q κ j can be satisfied only when d is even. (If d is odd, then κ j is always a multiple of d, so q κ j = 1.) When d is even, our condition in fact says precisely that each k i is a multiple of d but not of 2d. The two square roots of q inF * ℓ both have order 2d, and thus neither can appear among the eigenvalues of Frobenius on J(C) [λ].\n\nWe now turn to the first assertion of the theorem. We note that the coprimality of the k i implies that d = 1, or in other words that q is conguent to 1 modulo ℓ. The fact that J(C) carries an action of Z/ℓZ defined over F q with trivial invariant subspace implies that the Frobenius eigenvalues on J(C) all appear with multiplicity a multiple of ℓ − 1, and that a root of multiplicity k(ℓ − 1) reduces to a root of multiplicity k in the action of Frobenius on J(C) [λ]. So we just need to show that the action of Frobenius on J(C) [λ] has no repeated eigenvalues. The coprimality of the k i guarantees that the union r j=1 µ ′ k j is disjoint; the remaining eigenvalues are r − 2 copies of 1, so since r ≤ 3 we are done.\n\nProof of Corollary 1.8. We first consider the case d = 1 (i.e., q is congruent to 1 modulo ℓ).\n\nLet f 1 , f 2 be two distinct monic irreducible polynomials in F q [t] of degrees being odd and prime to ℓ. Let d 1 = deg f 1 and d 2 = deg f 2 . Let e ∈ {1, . . . , ℓ − 1} be such that ℓ | d 1 + ed 2 . The smooth projective curve C with affine model y ℓ = f 1 f e 2 admits a cyclic degree ℓ map to P 1 F q where the set of branch points are roots of f = f 1 f 2 in the affine line (the condition on d 1 + ed 2 guarantees there is no branching at ∞.) By Theorem 1.7(2), the zeta function Z C does not vanish at s = 1/2. Thus, all degree ℓ Dirichlet characters with conductor f have L-functions nonvanishing at the central point. Evidently, the number of such pairs f 1 , f 2 with d 1 + d 2 ≤ n is at least the number of irreducible polynomials of degree n − 3; the number of characters with non-vanishing L-functions is thus bounded below by a constant multiple of q n /n = X/(log X). This concludes the proof in the d = 1 case.\n\nWe now turn to the case where q ≡ 1 mod ℓ. We recall that d is the order of q mod ℓ in (Z/ℓZ) * . Let Σ d,2d be the set of monic squarefree polynomials such that all of their irreducible factors have degree divisible by d but not divisible by 2d. For any f ∈ Σ d,2d there exists a (Z/ℓZ) field extension K/F q (x) with conductor f . The field K is the function field of a curve C/F q which admits a cyclic degree ℓ map to P 1 F q whose set of branch points are roots of f . By Theorem 1.7(2), the zeta function Z C does not vanish at s = 1/2. Thus, all degree ℓ Dirichlet characters with conductor f have their L-functions do not vanish at the central point. The number of such characters is (ℓ − 1) ω( f ) where ω( f ) denotes the number of irreducible factors of f .\n\nSo it remains to count the number of degree ℓ Dirichlet characters whose conductor is in Σ d,2d , which by the above discussion is given by\n∑ f ∈Σ d,2d ,| f |≤X (ℓ − 1) ω( f ) .\nTo estimate this sum, we start by considering the Dirichlet series (1 + (ℓ − 1)|P| −s ). (1 + |P| −s ) d so that H(s) = (Z d (s)) 2 /Z 2d (s). Since Z d (s) and Z 2d (s) each have a simple pole at s = 1, so does H(s).\nG(s) = ∑ f ∈Σ d,2d (ℓ − 1) ω( f ) | f | −s where | f | = q deg\nWe conclude that G(s) 2d has a pole at s = 1 with order ℓ − 1 and is absolutely convergent for ℜ(s) > 1. By [12,Theorem 3.1], the sum of coefficients of G(s) has the following asymptotic relation\n∑ f ∈Σ d,2d ,| f |≤X (ℓ − 1) ω( f ) ∼ C · X(log X) ℓ−1 2d −1\nwhere the constant C = |(lim s→1 G(s) 2d (s − 1) ℓ−1 ) 1 2d | Γ((ℓ − 1)/(2d)) .\n\nAnd it gives the desired result.\n\nCorollary 1. 4 (\n4Corollary 3.3). Let A be a constant abelian variety defined overF q (x). For each f ∈ F q m [x], denote by A f the quadratic twist of A ⊗ F q m (x) by f . Let R n,m be the set { f ∈ F q m [x],squarefree, of deg n : A f has positive rank}. Then, lim m→∞ lim sup n→∞ |R n,m | q m(n+1) = 0.\n\nLemma 3. 1 .\n1Let f be a monic squarefree polynomial in F q [x] and χ f be the quadratic character with conductor f . Let C be the hyperelliptic curve defined by y 2 = f (x) and let P ∈ Z[x] be the characteristic polynomial of geometric Frobenius acting on the Jacobian of C. Then for any s = 0, L(s, χ f ) = 0 if and only if P(q s ) = 0.\n\nCorollary 3. 3 .\n3Let A be an abelian variety defined over a finite field F q of odd characteristic. For each f ∈ Q n,q m , denote by A f the quadratic twist of A × F q F q m (x) by f . Let R n,m be the set { f ∈ Q n,q m : A f has positive rank}. Then lim m→∞ lim sup n→∞ |R n,m | |Q n,q m | = 0. Proof. Let P(x) be the characteristic polynomial of Frobenius acting on the Tate module of A and let q −s be one of its roots. Then rank A f > 0 is equivalent to L(s, χ f ) = 0. (See [15, Proposition 4.6] for a similar statement with the same proof.) Thus, the statement is a direct application of Theorem 3.2.\n\n\nf . Then G(s) has a Euler product expansion G(s) = ∏ P∈Σ d,2d , irrd.\n\n( 1 +\n1|P| −s ) 2d , we see that G(s) 2d /H(s) ℓ−1 is holomorphic at s = 1. Now define Z d (s) = ∏ d|deg P, P irrd.\n\n\n3, Corollary 2.6] of Andrade and Keating and on [2, Corollary 2.8] of Andrade, Bae, and Jung, which give a proportion on order (log X) −2 , and goes beyond the methods of [1] by Andrade and Baluyot. For the analogous problem over Z, we refer to the recent work [4] of Baluyot and Pratt.\n\n\nsends Frob q to a. (Reference: [18, §4.5].) Lemma 2.3. With notation as in Section 2.1,\nResults in[5] and[22] were stated at point s = 1/2 but the methods can be extended to prove the statement for any point on the critical line.\nThe somewhat artificial special treatment of ∞ in this definition, as in[10], stems from the need to compare with topology, where branched covers of the disc are technically easier to handle than branched covers of the sphere.\nAcknowledgments. The authors are grateful to Chantal David, Zeev Rudnick, Alexandra Florea, and Emmanuel Kowalski for helpful comments and suggestions. The first\nSmall zeros of Dirichlet L-functions of quadratic characters of prime modulus. J Andrade, S Baluyot, arXiv preprintJ. Andrade and S. Baluyot, Small zeros of Dirichlet L-functions of quadratic characters of prime modulus, (2018), arXiv preprint.\n\nAverage values of L-series for real characters in function fields. J C Andrade, S Bae, H Jung, 47. MR 3567720Res. Math. Sci. 3Paper No. 38J. C. Andrade, S. Bae, and H. Jung, Average values of L-series for real characters in function fields, Res. Math. Sci. 3 (2016), Paper No. 38, 47. MR 3567720\n\nMean value theorems for L-functions over prime polynomials for the rational function field. J C Andrade, J P Keating, MR 3150889Acta Arith. 1614J. C. Andrade and J. P. Keating, Mean value theorems for L-functions over prime polynomials for the rational function field, Acta Arith. 161 (2013), no. 4, 371-385. MR 3150889\n\nDirichlet L-functions of quadratic characters of prime conductor at the central point. S Baluyot, K Pratt, arXiv preprintS. Baluyot and K. Pratt, Dirichlet L-functions of quadratic characters of prime conductor at the central point, (2018), arXiv preprint.\n\nZeros of quadratic Dirichlet L-functions in the hyperelliptic ensemble. H M Bui, A Florea, 8013-8045. MR 3852456Trans. Amer. Math. Soc. 37011H. M. Bui and A. Florea, Zeros of quadratic Dirichlet L-functions in the hyperelliptic ensemble, Trans. Amer. Math. Soc. 370 (2018), no. 11, 8013-8045. MR 3852456\n\nThe generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy. N Chavdarov, MR 1440067Duke Math. J. 871N. Chavdarov, The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy, Duke Math. J. 87 (1997), no. 1, 151-180. MR 1440067\n\nSimple zeros of the Riemann zeta-function. J B Conrey, A Ghosh, S M Gonek, MR 1616809Proc. London Math. Soc. 3J. B. Conrey, A. Ghosh, and S. M. Gonek, Simple zeros of the Riemann zeta-function, Proc. London Math. Soc. (3) 76 (1998), no. 3, 497-522. MR 1616809\n\nThe mean values of cubic L-functions over function fields. C David, A Florea, M Lalin, arXiv preprintC. David, A. Florea, and M. Lalin, The mean values of cubic L-functions over function fields, (2019), arXiv preprint, https://arxiv.org/abs/1901.00817.\n\n. Pierre Deligne, La Weil, MR 601520Inst. HautesÉtudes Sci. Publ. Math. II52Pierre Deligne, La conjecture de Weil. II, Inst. HautesÉtudes Sci. Publ. Math. (1980), no. 52, 137-252. MR 601520\n\nHomological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields. J S Ellenberg, A Venkatesh, C Westerland, MR 3488737Ann. of Math. 2J. S. Ellenberg, A. Venkatesh, and C. Westerland, Homological stability for Hurwitz spaces and the Cohen- Lenstra conjecture over function fields, Ann. of Math. (2) 183 (2016), no. 3, 729-786. MR 3488737\n\nHanoi lectures on the arithmetic of hyperelliptic curves. B H Gross, MR 3058664Acta Math. Vietnam. 374B. H. Gross, Hanoi lectures on the arithmetic of hyperelliptic curves, Acta Math. Vietnam. 37 (2012), no. 4, 579-588. MR 3058664\n\nA remark on the Wiener-Ikehara Tauberian theorem. Ryo Kato, MR 3410120Comment. Math. Univ. St. Pauli. 641Ryo Kato, A remark on the Wiener-Ikehara Tauberian theorem, Comment. Math. Univ. St. Pauli 64 (2015), no. 1, 47-58. MR 3410120\n\nThe large sieve, monodromy and zeta functions of curves. E Kowalski, MR 2289204J. Reine Angew. Math. 601E. Kowalski, The large sieve, monodromy and zeta functions of curves, J. Reine Angew. Math. 601 (2006), 29-69. MR 2289204\n\nIntroduction to transcendental numbers, addison-wesley pub. S Lang, S. Lang, Introduction to transcendental numbers, addison-wesley pub. co., (1966).\n\nVanishing of hyperelliptic L-functions at the central point. W Li, MR 3825462J. Number Theory. 191W. Li, Vanishing of hyperelliptic L-functions at the central point, J. Number Theory 191 (2018), 85-103. MR 3825462\n\nCohen-lenstra heuristics forétale group schemes and symplectic pairings. Michael Lipnowski, Jacob Tsimerman, Compositio Mathematica. 1554Michael Lipnowski and Jacob Tsimerman, Cohen-lenstra heuristics forétale group schemes and symplectic pairings, Compositio Mathematica 155 (2019), no. 4, 758-775.\n\nAbelian varieties. J S Milne, v2.00J. S. Milne, Abelian varieties (v2.00), 2008, Available at www.jmilne.org/math/.\n\nRational points on varieties. B Poonen, MR 3729254Graduate Studies in Mathematics. 186American Mathematical SocietyB. Poonen, Rational points on varieties, Graduate Studies in Mathematics, vol. 186, American Mathemat- ical Society, Providence, RI, 2017. MR 3729254\n\nAlgebraic exponential values. A Ray, A. Ray, Algebraic exponential values, https://mathoverflow.net/questions/314098/algebraic- exponential-values.\n\nHurwitz spaces. M Romagny, S Wewers, MR 2316356Groupes de Galois arithmétiques et différentiels. ParisSoc. Math. France13M. Romagny and S. Wewers, Hurwitz spaces, Groupes de Galois arithmétiques et différentiels, Sémin. Congr., vol. 13, Soc. Math. France, Paris, 2006, pp. 313-341. MR 2316356\n\nTraces of high powers of the frobenius class in the hyperelliptic ensemble. Z Rudnick, Acta Arith. 143Z. Rudnick, Traces of high powers of the frobenius class in the hyperelliptic ensemble, Acta Arith. 143 (2010), 81-99.\n\nNonvanishing of quadratic Dirichlet L-functions at s = 1 2. K Soundararajan, MR 1804529Ann. of Math. 2K. Soundararajan, Nonvanishing of quadratic Dirichlet L-functions at s = 1 2 , Ann. of Math. (2) 152 (2000), no. 2, 447-488. MR 1804529\n"},"corpus_id":{"kind":"number","value":119735177,"string":"119,735,177"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1901.08202\",\n \"mag\": 3079316661,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.2140/ant.2020.14.1895\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1901.08202v3.pdf\"\n ],\n \"pdf_sha\": \"eef679288cacab49eaa1110f697171735c9613c0\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1901.08202\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"NONVANISHING OF HYPERELLIPTIC ZETA FUNCTIONS OVER FINITE FIELDS"},"authors":{"kind":"list like","value":["Jordan S Ellenberg","Wanlin Li","Mark Shusterman"],"string":"[\n \"Jordan S Ellenberg\",\n \"Wanlin Li\",\n \"Mark Shusterman\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Fixing t ∈ R and a finite field F q of odd characteristic, we give an explicit upper bound on the proportion of genus g hyperelliptic curves over F q whose zeta function vanishes at 1 2 + it. Our upper bound is independent of g and tends to 0 as q grows."},"sections":{"kind":"list like","value":[{"title":"INTRODUCTION","content":"Let p be an odd prime, set q = p k for some positive integer k, and denote by F q the finite field with q elements. To (the smooth completion of) any hyperelliptic curve C over F q one associates a zeta function Z C (s). Weil has shown that Z C (s) = 0 implies that s = 1 2 + it for some t ∈ R. It is widely believed that for any fixed s = 1 2 + it, the 'vast majority' of (hyperelliptic) curves do not have s as a zero of their zeta function. For example, it follows from the work [6] of Chavdarov (and its improvement by Kowalski [13]) that for any fixed (large enough) g, the proportion of genus g hyperelliptic zeta functions vanishing at s tends to 0 as q → ∞.\nHere we are concerned with the growing g regime. Namely, for fixed q (and s), we give an upper bound on (1.1) h q,s := sup\nwhere H g (F q ) is the family of genus g hyperelliptic curves over F q . Our bound is better once q is large, as given by our main result. Theorem 1.2 (Theorem 3.2). Fix a prime p, a real number t, and set s = 1 2 + it. Then as k → ∞ we have (1.3) h p k ,s ≪ p −k/276 .\nIn particular, h p k ,s tends to 0 as k tends to ∞.\nThis complements (but does not quite match) lower bounds on h q,s obtained by Li in [15].\nRestricting q to powers of a fixed prime p is not always necessary. In case s = 1 2 , one can show (see [19]) using transcendental number theory (six exponentials theorem [14, Chapter 2, Section 1]) that there are only finitely many p for which p −s is algebraic, so h q,s = 0 for any q not divisible by these p (as Z C (s) is a rational function in q −s ). Hence, it suffices to work with one characteristic at a time, as we do in the theorem above. For s = 1 2 , since the upper bound in Corollary 2.6 holds for any ℓ when q is a perfect square, we can conclude lim q→∞ h q,s = 0 ranging over q which is an even power of a prime. 1 Additional motivation for Theorem 1.2 comes from the ability to write Z C (s) as a rational function in q −s , with the numerator being a quadratic Dirichlet L-function. Interpreted in this language of Dirichlet characters, Theorem 1.2 improves (for all sufficiently large q) upon [5, Corollary 2.1] of Bui and Florea (they give a lower bound of more than 94.27% nonvanishing at s = 1 2 ). Regarding the analogous vanishing problem for quadratic Dirichlet L-functions over Z, we refer to the work [22] of Soundararajan and references therein 1 .\nAs we explain in the last section, our theorem can be rephrased as an upper bound for the number of quadratic twists of a constant abelian variety which have positive rank. Motivated by [5,Corollary 2.2] and the analogous results over Z of Conrey, Ghosh and Gonek from [7], we bound the multiplicity of the zeros of Z C , and obtain further information on nonvanishing at s = 1 2 . Theorem 1.5. Let C be a hyperelliptic curve of genus at least 2 over F q and S be the set of Weierstrass points of C. The Frobenius acts on S by permuting the 2g + 2 Weierstrass points via some permutation π. Suppose that either • g is even and π is a (2g + 2)-cycle; or • π is the product of two disjoint cycles of odd length. Then:\n(1) The point s = 1 2 is not a zero of Z C . (2) All zeros of Z C are of multiplicity at most 2. Moreover, if π is the product of two disjoint cycles of coprime lengths, all zeros of Z C are simple.\nIn the language of Dirichlet characters, this implies in particular the nonvanishing (at the central point) in the case of prime conductor of degree not divisible by 4 and therefore gives an explicit set of size on order X/ log X of Dirichlet characters of conductor at most X which have L-functions nonvanishing at the critical point. See the statement below.\nIn particular, the number of quadratic characters with irreducible conductor of size at most X whose L-function does not vanish at s = 1/2 is ≫ X/ log X as X → ∞. This result improves on [ In fact, there is nothing special about hyperelliptic curves in Theorem 1.5. A similar \"genus-theory\" argument allows us to handle the case of cyclic ℓ-covers of P 1 for an odd prime ℓ. Theorem 1.7. Let ℓ be an odd prime different from the characteristic of F q . Let C be a smooth projective curve over F q which admits a degree ℓ map f : C → P 1 F q such that f is Galois over F q with Galois group isomorphic to Z/ℓZ. Let S ⊂ P 1 (F q ) be the set of branch points of f . Let π be the permutation induced by the Frobenius action on S, and suppose that π is the composition of disjoint cycles of orders k 1 , k 2 , . . . , k r , all prime to ℓ.\n(1) Suppose the k i are mutually coprime and r ≤ 3. Then every zero of Z C has degree ℓ − 1.\n(2) Define κ i to be k i if k i is odd and k i /2 if k i is even. Suppose that either • q is congruent to 1 modulo ℓ and r = 2, with both cycles of odd length; or • There is no i such that q κ i is congruent to 1 modulo ℓ. Then the point s = 1 2 is not a zero of Z C . We remark that this theorem, like Theorem 1.5 above, can be used to produce a set of size on order X/ log(X) a of order-ℓ Dirichlet characters of conductor at most X whose zeta functions are non-vanishing at s = 1/2, for some power a ∈ (0, 1]. See Corollary 1.8 for example. This lower bound improves, for ℓ = 3, upon Corollary 1.3 of recent work by David-Florea-Lalin [8] which gives a lower bound of the form X 1−ǫ (for any ǫ > 0). Corollary 1.8. Let ℓ be an odd prime different from the characteristic of F q . Let d be the order of (q mod ℓ) in (Z/ℓZ) * . Let N(X) be the number of primitive degree ℓ Dirichlet characters\nWe show the following:\nfor X → ∞. In particular, since d | ℓ − 1, we have N(X) ≫ X (log X) 1/2 for X → ∞. We do not get a lower bound of this quality in the case where q has odd order modulo ℓ.\nThe main idea that connects all the theorems in this paper is the study of L-functions modulo ℓ. The value of an L-function over F q (x) at a complex number s can be expressed as a polynomial P(T) ∈ Z[T] evaluated at T = q −s . So if we want to prove that P(T) is nonvanishing, it suffices to prove that P(T) is nonvanishing modulo ℓ for some prime ℓ. For Theorem 1.2, we will show that, for suitably chosen ℓ, the vanishing mod ℓ of the L-function is related to the dimension of a certain Frobenius eigenspace in the ℓ-torsion of a hyperelliptic Jacobian over F q ; the average size of this eigenspace can then be controlled by a point count on a moduli space over a finite field, which is a modest generalization and explication of the arguments in [10] and [16] respectively. For Theorem 1.5, on the other hand, we argue that under the given condition on Weierstrass points the L-function of χ f is nonvanishing mod 2 at s = 1/2. For the similar Theorem 1.7, the ℓ is again the order of the Dirichlet character in question. author was partially supported by NSF grant DMS-1700885 and by a fellowship from the Simons Foundation. We thank the referees for the valuable feedback and comments."},{"title":"MAIN THEOREM AND PROOF","content":"2.1. Setup and Notations. Throughout the paper, F q is a finite field of odd characteristic p. Let Q n,q be the set of squarefree polynomials over F q of degree n. For each f ∈ Q n,q , write J f for the Jacobian of the hyperelliptic curve\nfor the characteristic polynomial of geometric Frobenius acting on the ℓ-adic Tate module of J f . Let ℓ be a prime not equal to the characteristic of F q and let a be an element of (Z/ℓZ) * . The elements R of J f [ℓ](F q ) which satisfy\nform a finite-dimensional vector space over Z/ℓZ and we denote by m a ( f ) the number of nonzero elements of this vector space. Note that m 1 ( f ) is just the number of F q -rational nontrivial ℓ-torsion points of J f . Let Q a,ℓ n,q be the set of squarefree polynomials f over F q of degree n such that m a ( f ) is greater than 0.\nLet α be a q-Weil number of weight 1 with minimal polynomial g α (x) ∈ Z[x]. Namely, it is an algebraic integer whose absolute values under all complex embeddings equal √ q. Let Q α n,q be the subset of Q n,q defined by { f ∈ Q n,q | P f (α −1 ) = 0}. With notation introduced as above, if g α (a) = 0 mod ℓ, then |Q α n,q | ≤ |Q a,ℓ n,q |. 2.2. Rational points on twisted Hurwitz spaces over finite fields. Our main tool will be the following result about the average size of the subspace of Jac(C)[ℓ](F q ) on which Frobenius acts by some specified scalar a, as C ranges over hyperelliptic curves over F q . More precisely, we study the variation as we range over y 2 = f (x) with f ranging over squarefree polynomials in F q [x]; this amounts to the same, since each isomorphism class of hyperelliptic curves is represented in this form the same number of times (assuming, of course, that the isomorphism classes are weighted inversely to the number of automorphisms they possess.) Proposition 2.1. Let a ∈ (Z/ℓZ) × . With notation as in Section 2.1, there exist constants C ℓ , N ℓ , Q ℓ only depending on ℓ such that\nRemark 2.2. While this paper was in proof, we learned that Proposition 2.1 follows from the proof of Theorem 1.1 of [16], which is in fact more general, and the arguments used are essentially the same as those here. We have left the proof of Proposition 2.1 in the present paper because the form in which we present the proof here is conducive to proving the explicit bounds for the stable range obtained in Proposition 2.7 below. It would be interesting to address the questions of explicit bounds for the stable range in the more general situations considered by [16], where the group-theoretic part of the proof of Proposition 2.7 would presumably be more complicated.\nWhen a = 1 and n is odd, Proposition 2.1 is essentially Theorem 8.8 of [10], and indeed the proof here is a modification of the proof of that theorem.\nThe reader may note that [10, Thm 8.8] requires not only that q is not a multiple of ℓ but that q is not congruent to 1 modulo ℓ. We face no such restriction here. That's because [10,Thm 8.8] computes arbitrary moments of the Cohen-Lenstra distribution, whereas we are only studying the analogue of the average size of the ℓ-part of the class group. In the language of [10, Thm 8.8], we are only considering the case A = Z/ℓZ. The difference is as follows. In the proof, we will end up estimating the number of F q -points on a moduli space over F p , and the result will depend on that space having just one geometrically irreducible component defined over F q . In the more general setting treated in [10,Thm 8.8], that space has many geometric components, all but one of which have fields of definition containing µ ℓ ; so when q is congruent to 1 mod ℓ there are multiple F q -rational components. In the case treated here, the moduli space in question is geometrically irreducible, so this issue does not arise.\nProof. We begin by observing that ∑ f ∈Q n,q m a ( f ) can be interpreted as the number of F qrational points of a certain moduli space.\nTo this end we briefly recall the setup of [10, Section 7]. Let k be a field, let G be a finite group with trivial center, denote by e the identity element of G, and let c be a conjugacy-closed subset of G\\e. By a tame G-cover of P 1 with monodromy type c we mean a triple (X, f , φ) where • X is a smooth proper geometrically connected curve X/k; • f : X → P 1 is a tamely ramified finite cover;\n• The image of tame inertia at each branch point of f excepting ∞ lies in c;\n• f is Galois with group G; that is, Aut( f ) acts transitively on the geometric fibers of f and φ is an automorphism from G to Aut( f ). Here by an isomorphism between two covers f : X → P 1 and f ′ : X ′ → P 1 we mean a morphism ψ :\nof P 1 and f ′ • ψ = ι • f . In other words, our P 1 is \"labeled\". Then, as in [10, Section 7] (more or less immediate from a theorem of Romagny and Wewers [20]), there is a scheme Hn c G,n over Z[1/|G|] whose k-points (as long as k has characteristic prime to |G|) are in bijection with the isomorphism classes of tame G-covers of P 1 which have n branch points on A 1 with monodromy type c. (We do not specify whether or how the cover is branched at ∞.) 2 In fact ([20, Theorem 2.1]), for a scheme S, the set Hn c G,n (S) corresponds to isomorphism classes of tame G-covers over S, suitably defined; we will not need to spell out that definition here. Once the n branch points are chosen on A 1 there are finitely many choices for f and φ. Thus the dimension of Hn c G,n equals to n.\nFrom now on, we suppose that k is F q , that G is the dihedral group Z/ℓZ ⋊ Z/2Z, and that c is the conjugacy class of an involution in G. We will now explain the relationship between the space of G-covers and the ℓ-torsion in the Jacobian of hyperelliptic curves.\nThe key point is that, for any algebraic curve C, the set of surjections Jac(C)[ℓ] → (Z/ℓZ) k is naturally identified with the set ofétale (Z/ℓZ) k covers of C. For details, see Section 3.9 of [17].\nIf f : X → P 1 is a G-cover, the product structure of G allows us to factor f as\nwhere h is a hyperelliptic cover and g is a Galois cover with group Z/ℓZ; that is, g is endowed with an isomorphism φ : Z/ℓZ → Aut(g). What's more, the fact that the monodromy in f is of type c implies that g is anétale cover, at least away from the points of C over ∞ ∈ P 1 . What happens over ∞ is slightly more delicate. The double cover h is branched at n points on A 1 , but the total number of branch points of h must be even as C is a smooth proper hyperelliptic curve. Thus, if n is odd, h is branched at ∞. The monodromy around ∞ in the cover X → P 1 is thus an element of G projecting to the nontrivial element of Z/2Z. Such an element must be an involution, and it follows that g is unramified at ∞. If n is even, on the other hand, it is possible for g to be ramified. We thus wish to restrict our attention to those G-covers X → C which are unramified over ∞. These are parametrized by a closed and open subscheme of Hn c G,n (indeed, it is the second term in the disjoint union in the paragraph following (7.3.1) of [10]). Let X n be this subscheme of Hn c G,n when n is even, and Hn c G,n when n is odd. In both cases, dim X n = n. We have explained how every point of X n (k) gives rise to a triple (g, φ, h)/k up to isomorphism, and in fact it is not hard to check that the converse holds as well. (This is essentially the last paragraph of the proof of [10,Proposition 8.7].)\nIf a is an element of (Z/ℓZ) * , we denote by a the automorphism of X n which sends (g, φ, h) to (g, aφ, h). We then write X a n for the twist of X n by the homomorphism\nThe fact that the isomorphism class of h is fixed by Frobenius implies that the branch locus of h is an\nbe the unique monic squarefree polynomial which vanishes precisely at the branch locus of h. Then C is isomorphic (over F q ) to the smooth completion of the hyperelliptic curve defined by y 2 = f (x).\nFixing such an h, and thus such a C, we now consider the set of points of X a n (F q ) lying over this h. First of all, the choices of (g, φ) such that (g, φ, h) ∈ X a n (F q ) = X n (F q ) for a specified h are in bijection with the ℓ 2g(C) − 1 surjections from J(C)[ℓ](F q ) to Z/ℓZ. Two such surjections s, s ′ are isomorphic (that is, are parametrized by the same point of X a n (F q )) if and only if s = ±s ′ . The action of Frobenius on the set of surjections sends s to a −1 Frob q s; so s descends to a point of X a n (F q ) if and only if Frob q ·s = ±as. We conclude that the number of points of X a n (F q ) lying over h is (1/2)(m a ( f ) + m −a ( f )).\nNow if f in Q n,q is not monic then f = ǫF for some ǫ ∈ F * q and some monic F. The curve C f is isomorphic to C F if ǫ is a quadratic residue and to the nontrivial quadratic twist of C F otherwise. In the former case, m a ( f ) = m a (F), and in the latter, m a ( f ) = m −a (F). In particular, the quantity (1/2)(m a ( f ) + m −a ( f )) is the same for all q − 1 nonzero multiples of F. We conclude that\nWe now argue exactly as in the proof of [10, Theorem 8.8].\nSince |Q n,q | = (q − 1)(q n − q n−1 ), it suffices to prove that (2.4) |q −n |X a n (F q )| − 1| ≤ C ℓ q −1/2 for some C ℓ depending only on ℓ and for all n > N ℓ , q > Q ℓ . Via the Grothendieck-Lefschetz trace formula, we have\nwhere λ is a prime greater than max{2ℓ, q, n}. Note that theétale cohomology is that of the base change of X a n to F q , where it becomes isomorphic to the untwisted space X n ; in particular, the choice of a affects the action of Frobenius on theétale cohomology, but not theétale Betti numbers, bounds on which are the main engine of the argument.\nWe begin by computing the main term:\nThis follows immediately from the fact that (X a n ) F q ∼ = (X n ) F q is irreducible. When n is odd, this is shown in the proof of [10, Theorem 8.8] as a consequence of a big monodromy theorem of J.K. Yu. (This is actually the only place where we need n to be large, and indeed n = 3 would be enough.) When n is even, we argue as follows. The map from X n to the configuration space Conf n A 1 sending a G-cover to its branch locus is a finite cover ([10, Section 2.2]), and irreducibility of X n is equivalent to the monodromy group of this cover acting transitively on the fiber. It suffices to check that this holds on a closed subvariety of the base. So write Z for the subvariety of Conf n A 1 consisting of those configurations containing some specified point p 0 ∈ P 1 (F q ), and let Y be the preimage of Z in X n . An automorphism of P 1 taking p 0 to ∞ now identifies Y with Hn c G,n−1 , which we know to be irreducible since n − 1 is odd. This implies that X n is irreducible.\nWe now turn to the error term. The moduli space X n is a closed and open subscheme of Hn c G,n , so its Betti numbers are bounded by those of Hn c G,n ; by [10, (7.8.1)] we have dim H 2n−i c,ét ((X a n ) F q ; Q λ ) ≤ K ℓ (B ℓ ) i where K ℓ , B ℓ are constants depending only on ℓ.\nUsing the Deligne bound [9], the eigenvalue of Frobenius on H i c,ét ((X a n ) F q ; Q λ ) is bounded in absolute value by q i/2 ; so the absolute value of the contribution of all i < 2n to (2.5) is bounded above by the sum of a geometric series which converges for all q > B 2 ℓ . In particular, as in [10, Section 1.8], this contribution is at most\nSo if we take Q ℓ = 4B 2 ℓ and q > Q ℓ , we may take C ℓ = 2K ℓ B ℓ and conclude |X a\nwhich proves (2.4) and thus the desired result.\nProposition 2.1 allows us to bound the proportion of hyperelliptic curves whoseétale cohomology has a Frobenius eigenvalue congruent to a mod ℓ. Recall from Section 2.1 that Q a,ℓ n,q is the set of squarefree polynomials over F q of degree n such that m a ( f ) is greater than 0. Corollary 2.6. There are constants C ′ ℓ , Q ℓ , N ℓ such that for any a ∈ (Z/ℓZ) × we have |Q a,ℓ n,q | |Q n,q |\nProof. Write δ for the quantity |Q n,q | −1 |Q a,ℓ n,q | to be bounded. Since m a ( f ) is the number of nonzero elements of a vector space over Z/ℓZ, it is at least ℓ − 1 if it is greater than 0. In particular,\nfor all sufficiently large n, q, which yields the desired result by taking C ′ ℓ = C ℓ /(ℓ − 1). So far, we have used the results of [10] as they appear in that paper. However, for the present application, it is useful to compute explicit values C ℓ , Q ℓ for which Corollary 2.6 holds. We do so by going back to the main body of [10] and working out explicit bounds for quantities that are given in [10, (7.8.1)] only as unspecified constants. Proposition 2.7. Corollary 2.6 holds with C ′ ℓ = 2(ℓ − 1) −1 (4ℓ) 138 and Q ℓ = 4 · (4ℓ) 156 .\nProof. By the proof of Corollary 2.6, we may take C ′ ℓ to be C ℓ /(ℓ − 1), where C ℓ is the constant appearing in the statement of Proposition 2.1. Moreover, we may take C ℓ to be 2K ℓ B ℓ and Q ℓ to be 4B 2 ℓ , where B ℓ , K ℓ are the constants appearing in the proof of Proposition 2.1 controlling the exponential growth of the Betti numbers of the relevant Hurwitz space. We now explain how to bound B ℓ explicitly.\nIn [10, 7.8.1], the bound dim H ié t ((X a n ) F q ; Q λ ) ≤ K ℓ (B ℓ ) i arises from two facts. First, there is a stability theorem [10, 6.2], which tells us in this context that A, B, and D are constants we shall specify. Second, there is an absolute bound [10, 2.5] which tells us that dim H ié t ((X a n ) F q ; Q λ ) ≤ (4ℓ) n . These two facts together imply that dim H ié t ((X a n ) F q ; Q λ ) ≤ (4ℓ) Ai+B+D so we may take B ℓ = (4ℓ) A and K ℓ = (4ℓ) B+D . It remains to compute A, B, and D.\nThe key object of computation is the ring R defined in [10, §3]. This ring is defined for any finite group G and any conjugacy-closed subset of G; we will consider here just the case relevant to us, which is that where G is the dihedral group of order 2ℓ and c is the class of involutions in G. The set of n-tuples of involutions (τ 1 , . . . , τ n ) ∈ G n carries a natural action of the n-strand braid group; the ring R is a graded Q-algebra whose degree-n part is spanned by the set of orbits of that action, which set we denote Σ n . The multiplication in R is given by concatenation of n-tuples.\nThe key fact about R is that it contains a central element U with the property that R[U] and R/UR both have finite degree (that is, they are supported in only finitely many grades.) In the dihedral case, R and U are particularly easy to describe. For any n, there is map from Σ n to G sending (τ 1 , . . . , τ n ) to the product τ 1 . . . , τ n , which is called the boundary monodromy. Each n-tuple in Σ n also has a monodromy group; namely, the group generated by τ 1 , . . . , τ n . The possible monodromy groups are just the order-2 subgroups of G and G itself. It is not hard to check that, for all n ≥ 4, the elements of Σ n are determined by their boundary monodromy and their monodromy group; to be precise, Σ n consists of ℓ orbits consisting of the single element (τ, τ, . . . , τ) as τ ranges over the ℓ involutions, and ℓ more orbits, each of which consists of all n-tuples with monodromy group G and boundary monodromy g, as g ranges over the index-2 cyclic subgroup of G (when n is even) or its nontrivial coset (when n is odd.) In particular, dim R n = 2ℓ for all n ≥ 4. We may take U to be the degree-2 central operator\nand check that U induces an isomorphism from R n to R n+2 for all n ≥ 4. In particular, deg R[U] and deg R/UR are both at most 4, where by the degree of a graded ring we mean the highest grade represented in its support.\nThis combinatorial information about the dihedral group is what goes into the computation of constants in [10]. The constant D in [10, 6.1] is just the degree of U, which is 2. The stability result in [10, 6.1] is derived from a general theorem [10, 4.2] about R-modules. The R-module M governing the H i of Hurwitz space, to which we apply [10, 4.2] is the one called M i in [10, 6.1], So (using the constants appearing in those theorems) stability begins when n = max(h 0 , h 1 ) + A 0 , where h j is the quantity denoted deg H j (K(M i ) in [10, 6.1]. In turn, as asserted in the first paragraph of the proof of [10,\nSo we find that (2.8) holds for all n ≤ A 2 + A 0 (3i + 1)\nIn other words, we may take A = 3A 0 and B = 2A 0 + A 2 . Finally, the values of A 0 and A 2 are given in [10, 4.5.3]. They are defined in terms of\nThus, A = 78 and B = 58. Since D = 2, we conclude that we may take B ℓ = (4ℓ) 78 and K ℓ = (4ℓ) 60 . So we have Q ℓ = 4 · (4ℓ) 156 and C ′ ℓ = 2(ℓ − 1) −1 (4ℓ) 138 , as claimed."},{"title":"APPLICATION TO NONVANISHING OF L-FUNCTIONS","content":"We can use the above reasoning to bound the number of quadratic L-functions over function fields which vanish at a specified point on the critical line. For the rest of this section we fix an odd prime p and consider only fields of characteristic p. We note that, if χ f is a quadratic character of F q (x), then L(s, χ f ) can vanish only at a point s such that q s is a q-Weil number of weight 1. We first recall the following lemma relating the vanishing of the L-function of a quadratic character in terms of the Frobenius eigenvalues of a hyperelliptic curve; This is immediate from the description of P as the numerator of the zeta function of C, and the connection of the latter to L(s, χ f ) (see, for instance, [21, Section 2])."},{"title":"Theorem 3.2.","content":"For any squarefree polynomial f ∈ Q n,q , let L(s, χ f ) be the Dirichlet L-function associated to the quadratic character χ f as was defined in Section 2.1. Then for any s = 0,\nwhere the limit is taken over all powers q of a fixed odd prime number p.\nProof. Fix an odd prime number p, and let q be a power of p. By Lemma 3.1, L(s, χ f ) = 0 is equivalent to P(q −s ) = 0 where P(x) ∈ Z[x] is the characteristic polynomial of Frobenius acting on the Jacobian of the hyperelliptic curve defined by y 2 = f (x). Thus, the set { f ∈ Q n,q | L(s, χ f ) = 0} is the same as Q q s n,q .\nBy Chebotarev's density theorem, we can (for large enough q) find a prime ℓ = 1 4 q 4 1/276 (1 + o(1)) mod which g p s , the minimal polynomial of p s , splits completely. Let a ∈ Z/lZ such that g p s (a) = 0 mod ℓ. If q = p t , then any f with L(χ f , s) = 0 has m a t ( f ) > 0. So |Q q s n,q | |Q n,q | ≤ |Q a t ,ℓ n,q | |Q n,q | and now we can apply Corollary 2.6 to conclude using the second equation of 2.1 that, for all sufficiently large t, we have\nThe required bound follows from Proposition 2.7.\nResults on the vanishing of quadratic L-functions over function fields can be used to study the rank distribution of quadratic twist families of constant abelian varieties. In the following corollary, we show that as the constant field grows (so the characteristic is not changing), the probability for a quadratic twist of a constant abelian variety to have positive rank goes to 0. In the elliptic curve case, this agrees with the general \"Minimalist Conjecture\" philosophy, which holds that positive ranks should be a density-0 phenomenon except when forced by parity considerations from the functional equation (in this setting the functional equation never forces positive rank, and the rank is always even.) We now prove Theorem 1.5, which makes use of the mod 2 Galois representations on J(C) rather than the representations modulo odd primes.\nProof of Theorem 1.5. Let x 1 , ..., x 2g+2 be the set of Weierstrass points of C. The 2-torsion subgroup J(C) [2] is spanned by the degree-0 2-torsion divisors x i − x j . That is, the group of divisors of the form ∑ a i x i with ∑ a i = 0 surjects onto J(C) [2]. Note also that x 1 + . . . + x 2g+2 − (2g + 2)x 1 is a principal divisor and thus x 1 + . . . + x 2g+2 is 0 in J(C) [2]. See [11,Secion 4] for detailed discussion. This identifies J(C) [2] with an explicit subquotient of F 2g+2 2 ; namely, J(C) [2] is the quotient of the subspace (a 1 , . . . , a 2g+2 ) : ∑ a i = 0 by the 1-dimensional subspace spanned by (1, . . . , 1). This identification is equivariant for the Frobenius action on both sides, so it allows us to describe the mod 2 Galois representation afforded by J(C) in terms of the permutation π which Frobenius induces on x 1 , . . . , x 2g+2 . To be precise, the action of S 2g+2 on J(C) [2] is a representation ρ : S 2g+2 → Sp 2g (Z/2Z), and the action of Frobenius on J(C) [2] is given by ρ(π).\nThe conditions on π given in Theorem 1.5 are equivalent to the condition that π 2 is a product of two disjoint odd cycles. Thus, the action of π 2 in its permutation representation F 2g+2 2 has eigenvalues given by µ k and µ 2g+2−k for some odd 1 ≤ k ≤ 2g + 1; passing to the subquotient J(C) [2] removes two eigenspaces of ρ(π 2 ) with the eigenvalue 1. So the eigenvalues of Frob 2 on J(C) [2] are the multiset µ ′ k ∪ µ ′ 2g+2−k where µ ′ n denotes the nontrivial nth roots of unity. We see in particular that ρ(π 2 ) does not have 1 as an eigenvalue. But if the zeta function Z C had a zero at 1/2, then √ q would be a Frobenius eigenvalue on C, which would mean that q was an eigenvalue of Frob 2 ; we have shown that Frob 2 has no eigenvalue congruent to 1 mod 2, which rules this out. This proves (1). What's more, the multiset µ ′ k ∪ µ ′ 2g+2−k contains any eigenvalue at most twice, and if (k, 2g + 2 − k) = 1, no eigenvalue appears more than once. This proves (2) (or rather, it proves (2) for the zeta function of C/F q 2 , from which (2) is immediate.)\nProof of Corollary 1.6. By assumption, f is an irreducible polynomial over F q . So when deg f = n is even, Frobenius acts on the set of Weierstrass points of C f : y 2 = f (x) as a n-cycle. If deg f = n is odd, then Frobenius acts on the set of Weierstrass points of C f as a disjoint union of a n-cycle and a 1-cycle. In either case, when 4 ∤ deg f we can apply Theorem 1.5 to conclude that Z C f does not vanish at 1/2 and so behaves L(s, χ f ).\nFor any X = q 2g+2 , the set of irreducible polynomials of odd degree at most 2g + 1 gives quadratic characters with bounded conductor whose L-function does not vanish at the central point s = 1/2. By the prime number theorem for function fields, the number of irreducible polynomials in F q [x] of degree at most n is ≫ q n /n. This gives the lower bound in statement.\nThe proof of Theorem 1.7 is very similar to that of Theorem 1.5, but we treat it separately in order to make the hyperelliptic case above more readable.\nProof of Theorem 1.7. Let x 1 , . . . , x m be the ramification points of the (Z/ℓZ)-cover of P 1 in S, where m = k 1 + · · · + k r . The Jacobian J(C) of C carries an action of Z[(Z/ℓZ)]; write λ ∈ Z[(Z/ℓZ)] for ζ ℓ − 1, where ζ ℓ is a generator of (Z/ℓZ). A Riemann-Hurwitz computation shows that the genus of C is (m − 2)(ℓ − 1)/2, so the Tate module T ℓ J(C) is a free Z ℓ [ζ ℓ ]-module of rank m − 2, and J(C) [λ] has dimension m − 2.\nThe λ-torsion subgroup of J(C) is spanned by the degree-0 λ-torsion divisors x i − x j . That is, the group of divisors of the form ∑ a i x i with ∑ a i = 0 surjects onto J(C) [λ]. This surjection is not an isomorphism; there is a 1-dimensional kernel, which we can describe as follows. Over F q , the curve C has an affine model of the form y ℓ = f (x) with f a rational function with no zeroes or poles at ∞. Then the principal divisor associated to y is ∑ a i x i where a i = ord x i f . We have now expressed J(C) [λ] as an explicit subquotient of F m ℓ . This identification is equivariant for the Frobenius action on both sides, so it allows us to describe the mod ℓ Galois representation afforded by J(C) in terms of the permutation π which Frobenius induces on x 1 , . . . , x m .\nThe action of π splits x 1 , . . . , x m into cycles of length k 1 , . . . , k r , which by hypothesis are prime to ℓ, and which must be multiples of d, where d is the order of q in F * ℓ . So the eigenvalues of π in its action on F m ℓ are the union (as multisets) r j=1 µ k j . Now the composition factors of F m ℓ as a representation of the cyclic group π are J(C) [λ], F ℓ div(y), and the π-trivial one-dimensional representation onto which F m ℓ maps by summing coordinates. The action of π on the latter factor is trivial, while π acts on F ℓ div(y) as multiplication by q. If the zeta function Z C had a zero at 1/2, then √ q would be a Frobenius eigenvalue of C, which would mean that some eigenvalue µ of the action of π on J(C)[λ] satisfied µ 2 = q.\nIn case q is congruent to 1 modulo ℓ (i.e., d = 1) the eigenvalues of π in its action on J(C) [λ] are the multiset r j=1 µ ′ k j together with r − 2 copies of 1, where µ ′ n denotes the nontrivial nth roots of unity. The hypotheses r = 2 and k j odd now guarantee that the eigenvalues of π on J(C) [λ] contain no copies of either 1 or −1, completing the proof in this case.\nIf d > 1, we note that our condition on q κ j can be satisfied only when d is even. (If d is odd, then κ j is always a multiple of d, so q κ j = 1.) When d is even, our condition in fact says precisely that each k i is a multiple of d but not of 2d. The two square roots of q inF * ℓ both have order 2d, and thus neither can appear among the eigenvalues of Frobenius on J(C) [λ].\nWe now turn to the first assertion of the theorem. We note that the coprimality of the k i implies that d = 1, or in other words that q is conguent to 1 modulo ℓ. The fact that J(C) carries an action of Z/ℓZ defined over F q with trivial invariant subspace implies that the Frobenius eigenvalues on J(C) all appear with multiplicity a multiple of ℓ − 1, and that a root of multiplicity k(ℓ − 1) reduces to a root of multiplicity k in the action of Frobenius on J(C) [λ]. So we just need to show that the action of Frobenius on J(C) [λ] has no repeated eigenvalues. The coprimality of the k i guarantees that the union r j=1 µ ′ k j is disjoint; the remaining eigenvalues are r − 2 copies of 1, so since r ≤ 3 we are done.\nProof of Corollary 1.8. We first consider the case d = 1 (i.e., q is congruent to 1 modulo ℓ).\nLet f 1 , f 2 be two distinct monic irreducible polynomials in F q [t] of degrees being odd and prime to ℓ. Let d 1 = deg f 1 and d 2 = deg f 2 . Let e ∈ {1, . . . , ℓ − 1} be such that ℓ | d 1 + ed 2 . The smooth projective curve C with affine model y ℓ = f 1 f e 2 admits a cyclic degree ℓ map to P 1 F q where the set of branch points are roots of f = f 1 f 2 in the affine line (the condition on d 1 + ed 2 guarantees there is no branching at ∞.) By Theorem 1.7(2), the zeta function Z C does not vanish at s = 1/2. Thus, all degree ℓ Dirichlet characters with conductor f have L-functions nonvanishing at the central point. Evidently, the number of such pairs f 1 , f 2 with d 1 + d 2 ≤ n is at least the number of irreducible polynomials of degree n − 3; the number of characters with non-vanishing L-functions is thus bounded below by a constant multiple of q n /n = X/(log X). This concludes the proof in the d = 1 case.\nWe now turn to the case where q ≡ 1 mod ℓ. We recall that d is the order of q mod ℓ in (Z/ℓZ) * . Let Σ d,2d be the set of monic squarefree polynomials such that all of their irreducible factors have degree divisible by d but not divisible by 2d. For any f ∈ Σ d,2d there exists a (Z/ℓZ) field extension K/F q (x) with conductor f . The field K is the function field of a curve C/F q which admits a cyclic degree ℓ map to P 1 F q whose set of branch points are roots of f . By Theorem 1.7(2), the zeta function Z C does not vanish at s = 1/2. Thus, all degree ℓ Dirichlet characters with conductor f have their L-functions do not vanish at the central point. The number of such characters is (ℓ − 1) ω( f ) where ω( f ) denotes the number of irreducible factors of f .\nSo it remains to count the number of degree ℓ Dirichlet characters whose conductor is in Σ d,2d , which by the above discussion is given by\nTo estimate this sum, we start by considering the Dirichlet series (1 + (ℓ − 1)|P| −s ). (1 + |P| −s ) d so that H(s) = (Z d (s)) 2 /Z 2d (s). Since Z d (s) and Z 2d (s) each have a simple pole at s = 1, so does H(s).\nWe conclude that G(s) 2d has a pole at s = 1 with order ℓ − 1 and is absolutely convergent for ℜ(s) > 1. By [12,Theorem 3.1], the sum of coefficients of G(s) has the following asymptotic relation\nwhere the constant C = |(lim s→1 G(s) 2d (s − 1) ℓ−1 ) 1 2d | Γ((ℓ − 1)/(2d)) .\nAnd it gives the desired result."},{"title":"Corollary 1. 4 (","content":""},{"title":"Lemma 3. 1 .","content":""},{"title":"Corollary 3. 3 .","content":""},{"title":"( 1 +","content":""}],"string":"[\n {\n \"title\": \"INTRODUCTION\",\n \"content\": \"Let p be an odd prime, set q = p k for some positive integer k, and denote by F q the finite field with q elements. To (the smooth completion of) any hyperelliptic curve C over F q one associates a zeta function Z C (s). Weil has shown that Z C (s) = 0 implies that s = 1 2 + it for some t ∈ R. It is widely believed that for any fixed s = 1 2 + it, the 'vast majority' of (hyperelliptic) curves do not have s as a zero of their zeta function. For example, it follows from the work [6] of Chavdarov (and its improvement by Kowalski [13]) that for any fixed (large enough) g, the proportion of genus g hyperelliptic zeta functions vanishing at s tends to 0 as q → ∞.\\nHere we are concerned with the growing g regime. Namely, for fixed q (and s), we give an upper bound on (1.1) h q,s := sup\\nwhere H g (F q ) is the family of genus g hyperelliptic curves over F q . Our bound is better once q is large, as given by our main result. Theorem 1.2 (Theorem 3.2). Fix a prime p, a real number t, and set s = 1 2 + it. Then as k → ∞ we have (1.3) h p k ,s ≪ p −k/276 .\\nIn particular, h p k ,s tends to 0 as k tends to ∞.\\nThis complements (but does not quite match) lower bounds on h q,s obtained by Li in [15].\\nRestricting q to powers of a fixed prime p is not always necessary. In case s = 1 2 , one can show (see [19]) using transcendental number theory (six exponentials theorem [14, Chapter 2, Section 1]) that there are only finitely many p for which p −s is algebraic, so h q,s = 0 for any q not divisible by these p (as Z C (s) is a rational function in q −s ). Hence, it suffices to work with one characteristic at a time, as we do in the theorem above. For s = 1 2 , since the upper bound in Corollary 2.6 holds for any ℓ when q is a perfect square, we can conclude lim q→∞ h q,s = 0 ranging over q which is an even power of a prime. 1 Additional motivation for Theorem 1.2 comes from the ability to write Z C (s) as a rational function in q −s , with the numerator being a quadratic Dirichlet L-function. Interpreted in this language of Dirichlet characters, Theorem 1.2 improves (for all sufficiently large q) upon [5, Corollary 2.1] of Bui and Florea (they give a lower bound of more than 94.27% nonvanishing at s = 1 2 ). Regarding the analogous vanishing problem for quadratic Dirichlet L-functions over Z, we refer to the work [22] of Soundararajan and references therein 1 .\\nAs we explain in the last section, our theorem can be rephrased as an upper bound for the number of quadratic twists of a constant abelian variety which have positive rank. Motivated by [5,Corollary 2.2] and the analogous results over Z of Conrey, Ghosh and Gonek from [7], we bound the multiplicity of the zeros of Z C , and obtain further information on nonvanishing at s = 1 2 . Theorem 1.5. Let C be a hyperelliptic curve of genus at least 2 over F q and S be the set of Weierstrass points of C. The Frobenius acts on S by permuting the 2g + 2 Weierstrass points via some permutation π. Suppose that either • g is even and π is a (2g + 2)-cycle; or • π is the product of two disjoint cycles of odd length. Then:\\n(1) The point s = 1 2 is not a zero of Z C . (2) All zeros of Z C are of multiplicity at most 2. Moreover, if π is the product of two disjoint cycles of coprime lengths, all zeros of Z C are simple.\\nIn the language of Dirichlet characters, this implies in particular the nonvanishing (at the central point) in the case of prime conductor of degree not divisible by 4 and therefore gives an explicit set of size on order X/ log X of Dirichlet characters of conductor at most X which have L-functions nonvanishing at the critical point. See the statement below.\\nIn particular, the number of quadratic characters with irreducible conductor of size at most X whose L-function does not vanish at s = 1/2 is ≫ X/ log X as X → ∞. This result improves on [ In fact, there is nothing special about hyperelliptic curves in Theorem 1.5. A similar \\\"genus-theory\\\" argument allows us to handle the case of cyclic ℓ-covers of P 1 for an odd prime ℓ. Theorem 1.7. Let ℓ be an odd prime different from the characteristic of F q . Let C be a smooth projective curve over F q which admits a degree ℓ map f : C → P 1 F q such that f is Galois over F q with Galois group isomorphic to Z/ℓZ. Let S ⊂ P 1 (F q ) be the set of branch points of f . Let π be the permutation induced by the Frobenius action on S, and suppose that π is the composition of disjoint cycles of orders k 1 , k 2 , . . . , k r , all prime to ℓ.\\n(1) Suppose the k i are mutually coprime and r ≤ 3. Then every zero of Z C has degree ℓ − 1.\\n(2) Define κ i to be k i if k i is odd and k i /2 if k i is even. Suppose that either • q is congruent to 1 modulo ℓ and r = 2, with both cycles of odd length; or • There is no i such that q κ i is congruent to 1 modulo ℓ. Then the point s = 1 2 is not a zero of Z C . We remark that this theorem, like Theorem 1.5 above, can be used to produce a set of size on order X/ log(X) a of order-ℓ Dirichlet characters of conductor at most X whose zeta functions are non-vanishing at s = 1/2, for some power a ∈ (0, 1]. See Corollary 1.8 for example. This lower bound improves, for ℓ = 3, upon Corollary 1.3 of recent work by David-Florea-Lalin [8] which gives a lower bound of the form X 1−ǫ (for any ǫ > 0). Corollary 1.8. Let ℓ be an odd prime different from the characteristic of F q . Let d be the order of (q mod ℓ) in (Z/ℓZ) * . Let N(X) be the number of primitive degree ℓ Dirichlet characters\\nWe show the following:\\nfor X → ∞. In particular, since d | ℓ − 1, we have N(X) ≫ X (log X) 1/2 for X → ∞. We do not get a lower bound of this quality in the case where q has odd order modulo ℓ.\\nThe main idea that connects all the theorems in this paper is the study of L-functions modulo ℓ. The value of an L-function over F q (x) at a complex number s can be expressed as a polynomial P(T) ∈ Z[T] evaluated at T = q −s . So if we want to prove that P(T) is nonvanishing, it suffices to prove that P(T) is nonvanishing modulo ℓ for some prime ℓ. For Theorem 1.2, we will show that, for suitably chosen ℓ, the vanishing mod ℓ of the L-function is related to the dimension of a certain Frobenius eigenspace in the ℓ-torsion of a hyperelliptic Jacobian over F q ; the average size of this eigenspace can then be controlled by a point count on a moduli space over a finite field, which is a modest generalization and explication of the arguments in [10] and [16] respectively. For Theorem 1.5, on the other hand, we argue that under the given condition on Weierstrass points the L-function of χ f is nonvanishing mod 2 at s = 1/2. For the similar Theorem 1.7, the ℓ is again the order of the Dirichlet character in question. author was partially supported by NSF grant DMS-1700885 and by a fellowship from the Simons Foundation. We thank the referees for the valuable feedback and comments.\"\n },\n {\n \"title\": \"MAIN THEOREM AND PROOF\",\n \"content\": \"2.1. Setup and Notations. Throughout the paper, F q is a finite field of odd characteristic p. Let Q n,q be the set of squarefree polynomials over F q of degree n. For each f ∈ Q n,q , write J f for the Jacobian of the hyperelliptic curve\\nfor the characteristic polynomial of geometric Frobenius acting on the ℓ-adic Tate module of J f . Let ℓ be a prime not equal to the characteristic of F q and let a be an element of (Z/ℓZ) * . The elements R of J f [ℓ](F q ) which satisfy\\nform a finite-dimensional vector space over Z/ℓZ and we denote by m a ( f ) the number of nonzero elements of this vector space. Note that m 1 ( f ) is just the number of F q -rational nontrivial ℓ-torsion points of J f . Let Q a,ℓ n,q be the set of squarefree polynomials f over F q of degree n such that m a ( f ) is greater than 0.\\nLet α be a q-Weil number of weight 1 with minimal polynomial g α (x) ∈ Z[x]. Namely, it is an algebraic integer whose absolute values under all complex embeddings equal √ q. Let Q α n,q be the subset of Q n,q defined by { f ∈ Q n,q | P f (α −1 ) = 0}. With notation introduced as above, if g α (a) = 0 mod ℓ, then |Q α n,q | ≤ |Q a,ℓ n,q |. 2.2. Rational points on twisted Hurwitz spaces over finite fields. Our main tool will be the following result about the average size of the subspace of Jac(C)[ℓ](F q ) on which Frobenius acts by some specified scalar a, as C ranges over hyperelliptic curves over F q . More precisely, we study the variation as we range over y 2 = f (x) with f ranging over squarefree polynomials in F q [x]; this amounts to the same, since each isomorphism class of hyperelliptic curves is represented in this form the same number of times (assuming, of course, that the isomorphism classes are weighted inversely to the number of automorphisms they possess.) Proposition 2.1. Let a ∈ (Z/ℓZ) × . With notation as in Section 2.1, there exist constants C ℓ , N ℓ , Q ℓ only depending on ℓ such that\\nRemark 2.2. While this paper was in proof, we learned that Proposition 2.1 follows from the proof of Theorem 1.1 of [16], which is in fact more general, and the arguments used are essentially the same as those here. We have left the proof of Proposition 2.1 in the present paper because the form in which we present the proof here is conducive to proving the explicit bounds for the stable range obtained in Proposition 2.7 below. It would be interesting to address the questions of explicit bounds for the stable range in the more general situations considered by [16], where the group-theoretic part of the proof of Proposition 2.7 would presumably be more complicated.\\nWhen a = 1 and n is odd, Proposition 2.1 is essentially Theorem 8.8 of [10], and indeed the proof here is a modification of the proof of that theorem.\\nThe reader may note that [10, Thm 8.8] requires not only that q is not a multiple of ℓ but that q is not congruent to 1 modulo ℓ. We face no such restriction here. That's because [10,Thm 8.8] computes arbitrary moments of the Cohen-Lenstra distribution, whereas we are only studying the analogue of the average size of the ℓ-part of the class group. In the language of [10, Thm 8.8], we are only considering the case A = Z/ℓZ. The difference is as follows. In the proof, we will end up estimating the number of F q -points on a moduli space over F p , and the result will depend on that space having just one geometrically irreducible component defined over F q . In the more general setting treated in [10,Thm 8.8], that space has many geometric components, all but one of which have fields of definition containing µ ℓ ; so when q is congruent to 1 mod ℓ there are multiple F q -rational components. In the case treated here, the moduli space in question is geometrically irreducible, so this issue does not arise.\\nProof. We begin by observing that ∑ f ∈Q n,q m a ( f ) can be interpreted as the number of F qrational points of a certain moduli space.\\nTo this end we briefly recall the setup of [10, Section 7]. Let k be a field, let G be a finite group with trivial center, denote by e the identity element of G, and let c be a conjugacy-closed subset of G\\\\e. By a tame G-cover of P 1 with monodromy type c we mean a triple (X, f , φ) where • X is a smooth proper geometrically connected curve X/k; • f : X → P 1 is a tamely ramified finite cover;\\n• The image of tame inertia at each branch point of f excepting ∞ lies in c;\\n• f is Galois with group G; that is, Aut( f ) acts transitively on the geometric fibers of f and φ is an automorphism from G to Aut( f ). Here by an isomorphism between two covers f : X → P 1 and f ′ : X ′ → P 1 we mean a morphism ψ :\\nof P 1 and f ′ • ψ = ι • f . In other words, our P 1 is \\\"labeled\\\". Then, as in [10, Section 7] (more or less immediate from a theorem of Romagny and Wewers [20]), there is a scheme Hn c G,n over Z[1/|G|] whose k-points (as long as k has characteristic prime to |G|) are in bijection with the isomorphism classes of tame G-covers of P 1 which have n branch points on A 1 with monodromy type c. (We do not specify whether or how the cover is branched at ∞.) 2 In fact ([20, Theorem 2.1]), for a scheme S, the set Hn c G,n (S) corresponds to isomorphism classes of tame G-covers over S, suitably defined; we will not need to spell out that definition here. Once the n branch points are chosen on A 1 there are finitely many choices for f and φ. Thus the dimension of Hn c G,n equals to n.\\nFrom now on, we suppose that k is F q , that G is the dihedral group Z/ℓZ ⋊ Z/2Z, and that c is the conjugacy class of an involution in G. We will now explain the relationship between the space of G-covers and the ℓ-torsion in the Jacobian of hyperelliptic curves.\\nThe key point is that, for any algebraic curve C, the set of surjections Jac(C)[ℓ] → (Z/ℓZ) k is naturally identified with the set ofétale (Z/ℓZ) k covers of C. For details, see Section 3.9 of [17].\\nIf f : X → P 1 is a G-cover, the product structure of G allows us to factor f as\\nwhere h is a hyperelliptic cover and g is a Galois cover with group Z/ℓZ; that is, g is endowed with an isomorphism φ : Z/ℓZ → Aut(g). What's more, the fact that the monodromy in f is of type c implies that g is anétale cover, at least away from the points of C over ∞ ∈ P 1 . What happens over ∞ is slightly more delicate. The double cover h is branched at n points on A 1 , but the total number of branch points of h must be even as C is a smooth proper hyperelliptic curve. Thus, if n is odd, h is branched at ∞. The monodromy around ∞ in the cover X → P 1 is thus an element of G projecting to the nontrivial element of Z/2Z. Such an element must be an involution, and it follows that g is unramified at ∞. If n is even, on the other hand, it is possible for g to be ramified. We thus wish to restrict our attention to those G-covers X → C which are unramified over ∞. These are parametrized by a closed and open subscheme of Hn c G,n (indeed, it is the second term in the disjoint union in the paragraph following (7.3.1) of [10]). Let X n be this subscheme of Hn c G,n when n is even, and Hn c G,n when n is odd. In both cases, dim X n = n. We have explained how every point of X n (k) gives rise to a triple (g, φ, h)/k up to isomorphism, and in fact it is not hard to check that the converse holds as well. (This is essentially the last paragraph of the proof of [10,Proposition 8.7].)\\nIf a is an element of (Z/ℓZ) * , we denote by a the automorphism of X n which sends (g, φ, h) to (g, aφ, h). We then write X a n for the twist of X n by the homomorphism\\nThe fact that the isomorphism class of h is fixed by Frobenius implies that the branch locus of h is an\\nbe the unique monic squarefree polynomial which vanishes precisely at the branch locus of h. Then C is isomorphic (over F q ) to the smooth completion of the hyperelliptic curve defined by y 2 = f (x).\\nFixing such an h, and thus such a C, we now consider the set of points of X a n (F q ) lying over this h. First of all, the choices of (g, φ) such that (g, φ, h) ∈ X a n (F q ) = X n (F q ) for a specified h are in bijection with the ℓ 2g(C) − 1 surjections from J(C)[ℓ](F q ) to Z/ℓZ. Two such surjections s, s ′ are isomorphic (that is, are parametrized by the same point of X a n (F q )) if and only if s = ±s ′ . The action of Frobenius on the set of surjections sends s to a −1 Frob q s; so s descends to a point of X a n (F q ) if and only if Frob q ·s = ±as. We conclude that the number of points of X a n (F q ) lying over h is (1/2)(m a ( f ) + m −a ( f )).\\nNow if f in Q n,q is not monic then f = ǫF for some ǫ ∈ F * q and some monic F. The curve C f is isomorphic to C F if ǫ is a quadratic residue and to the nontrivial quadratic twist of C F otherwise. In the former case, m a ( f ) = m a (F), and in the latter, m a ( f ) = m −a (F). In particular, the quantity (1/2)(m a ( f ) + m −a ( f )) is the same for all q − 1 nonzero multiples of F. We conclude that\\nWe now argue exactly as in the proof of [10, Theorem 8.8].\\nSince |Q n,q | = (q − 1)(q n − q n−1 ), it suffices to prove that (2.4) |q −n |X a n (F q )| − 1| ≤ C ℓ q −1/2 for some C ℓ depending only on ℓ and for all n > N ℓ , q > Q ℓ . Via the Grothendieck-Lefschetz trace formula, we have\\nwhere λ is a prime greater than max{2ℓ, q, n}. Note that theétale cohomology is that of the base change of X a n to F q , where it becomes isomorphic to the untwisted space X n ; in particular, the choice of a affects the action of Frobenius on theétale cohomology, but not theétale Betti numbers, bounds on which are the main engine of the argument.\\nWe begin by computing the main term:\\nThis follows immediately from the fact that (X a n ) F q ∼ = (X n ) F q is irreducible. When n is odd, this is shown in the proof of [10, Theorem 8.8] as a consequence of a big monodromy theorem of J.K. Yu. (This is actually the only place where we need n to be large, and indeed n = 3 would be enough.) When n is even, we argue as follows. The map from X n to the configuration space Conf n A 1 sending a G-cover to its branch locus is a finite cover ([10, Section 2.2]), and irreducibility of X n is equivalent to the monodromy group of this cover acting transitively on the fiber. It suffices to check that this holds on a closed subvariety of the base. So write Z for the subvariety of Conf n A 1 consisting of those configurations containing some specified point p 0 ∈ P 1 (F q ), and let Y be the preimage of Z in X n . An automorphism of P 1 taking p 0 to ∞ now identifies Y with Hn c G,n−1 , which we know to be irreducible since n − 1 is odd. This implies that X n is irreducible.\\nWe now turn to the error term. The moduli space X n is a closed and open subscheme of Hn c G,n , so its Betti numbers are bounded by those of Hn c G,n ; by [10, (7.8.1)] we have dim H 2n−i c,ét ((X a n ) F q ; Q λ ) ≤ K ℓ (B ℓ ) i where K ℓ , B ℓ are constants depending only on ℓ.\\nUsing the Deligne bound [9], the eigenvalue of Frobenius on H i c,ét ((X a n ) F q ; Q λ ) is bounded in absolute value by q i/2 ; so the absolute value of the contribution of all i < 2n to (2.5) is bounded above by the sum of a geometric series which converges for all q > B 2 ℓ . In particular, as in [10, Section 1.8], this contribution is at most\\nSo if we take Q ℓ = 4B 2 ℓ and q > Q ℓ , we may take C ℓ = 2K ℓ B ℓ and conclude |X a\\nwhich proves (2.4) and thus the desired result.\\nProposition 2.1 allows us to bound the proportion of hyperelliptic curves whoseétale cohomology has a Frobenius eigenvalue congruent to a mod ℓ. Recall from Section 2.1 that Q a,ℓ n,q is the set of squarefree polynomials over F q of degree n such that m a ( f ) is greater than 0. Corollary 2.6. There are constants C ′ ℓ , Q ℓ , N ℓ such that for any a ∈ (Z/ℓZ) × we have |Q a,ℓ n,q | |Q n,q |\\nProof. Write δ for the quantity |Q n,q | −1 |Q a,ℓ n,q | to be bounded. Since m a ( f ) is the number of nonzero elements of a vector space over Z/ℓZ, it is at least ℓ − 1 if it is greater than 0. In particular,\\nfor all sufficiently large n, q, which yields the desired result by taking C ′ ℓ = C ℓ /(ℓ − 1). So far, we have used the results of [10] as they appear in that paper. However, for the present application, it is useful to compute explicit values C ℓ , Q ℓ for which Corollary 2.6 holds. We do so by going back to the main body of [10] and working out explicit bounds for quantities that are given in [10, (7.8.1)] only as unspecified constants. Proposition 2.7. Corollary 2.6 holds with C ′ ℓ = 2(ℓ − 1) −1 (4ℓ) 138 and Q ℓ = 4 · (4ℓ) 156 .\\nProof. By the proof of Corollary 2.6, we may take C ′ ℓ to be C ℓ /(ℓ − 1), where C ℓ is the constant appearing in the statement of Proposition 2.1. Moreover, we may take C ℓ to be 2K ℓ B ℓ and Q ℓ to be 4B 2 ℓ , where B ℓ , K ℓ are the constants appearing in the proof of Proposition 2.1 controlling the exponential growth of the Betti numbers of the relevant Hurwitz space. We now explain how to bound B ℓ explicitly.\\nIn [10, 7.8.1], the bound dim H ié t ((X a n ) F q ; Q λ ) ≤ K ℓ (B ℓ ) i arises from two facts. First, there is a stability theorem [10, 6.2], which tells us in this context that A, B, and D are constants we shall specify. Second, there is an absolute bound [10, 2.5] which tells us that dim H ié t ((X a n ) F q ; Q λ ) ≤ (4ℓ) n . These two facts together imply that dim H ié t ((X a n ) F q ; Q λ ) ≤ (4ℓ) Ai+B+D so we may take B ℓ = (4ℓ) A and K ℓ = (4ℓ) B+D . It remains to compute A, B, and D.\\nThe key object of computation is the ring R defined in [10, §3]. This ring is defined for any finite group G and any conjugacy-closed subset of G; we will consider here just the case relevant to us, which is that where G is the dihedral group of order 2ℓ and c is the class of involutions in G. The set of n-tuples of involutions (τ 1 , . . . , τ n ) ∈ G n carries a natural action of the n-strand braid group; the ring R is a graded Q-algebra whose degree-n part is spanned by the set of orbits of that action, which set we denote Σ n . The multiplication in R is given by concatenation of n-tuples.\\nThe key fact about R is that it contains a central element U with the property that R[U] and R/UR both have finite degree (that is, they are supported in only finitely many grades.) In the dihedral case, R and U are particularly easy to describe. For any n, there is map from Σ n to G sending (τ 1 , . . . , τ n ) to the product τ 1 . . . , τ n , which is called the boundary monodromy. Each n-tuple in Σ n also has a monodromy group; namely, the group generated by τ 1 , . . . , τ n . The possible monodromy groups are just the order-2 subgroups of G and G itself. It is not hard to check that, for all n ≥ 4, the elements of Σ n are determined by their boundary monodromy and their monodromy group; to be precise, Σ n consists of ℓ orbits consisting of the single element (τ, τ, . . . , τ) as τ ranges over the ℓ involutions, and ℓ more orbits, each of which consists of all n-tuples with monodromy group G and boundary monodromy g, as g ranges over the index-2 cyclic subgroup of G (when n is even) or its nontrivial coset (when n is odd.) In particular, dim R n = 2ℓ for all n ≥ 4. We may take U to be the degree-2 central operator\\nand check that U induces an isomorphism from R n to R n+2 for all n ≥ 4. In particular, deg R[U] and deg R/UR are both at most 4, where by the degree of a graded ring we mean the highest grade represented in its support.\\nThis combinatorial information about the dihedral group is what goes into the computation of constants in [10]. The constant D in [10, 6.1] is just the degree of U, which is 2. The stability result in [10, 6.1] is derived from a general theorem [10, 4.2] about R-modules. The R-module M governing the H i of Hurwitz space, to which we apply [10, 4.2] is the one called M i in [10, 6.1], So (using the constants appearing in those theorems) stability begins when n = max(h 0 , h 1 ) + A 0 , where h j is the quantity denoted deg H j (K(M i ) in [10, 6.1]. In turn, as asserted in the first paragraph of the proof of [10,\\nSo we find that (2.8) holds for all n ≤ A 2 + A 0 (3i + 1)\\nIn other words, we may take A = 3A 0 and B = 2A 0 + A 2 . Finally, the values of A 0 and A 2 are given in [10, 4.5.3]. They are defined in terms of\\nThus, A = 78 and B = 58. Since D = 2, we conclude that we may take B ℓ = (4ℓ) 78 and K ℓ = (4ℓ) 60 . So we have Q ℓ = 4 · (4ℓ) 156 and C ′ ℓ = 2(ℓ − 1) −1 (4ℓ) 138 , as claimed.\"\n },\n {\n \"title\": \"APPLICATION TO NONVANISHING OF L-FUNCTIONS\",\n \"content\": \"We can use the above reasoning to bound the number of quadratic L-functions over function fields which vanish at a specified point on the critical line. For the rest of this section we fix an odd prime p and consider only fields of characteristic p. We note that, if χ f is a quadratic character of F q (x), then L(s, χ f ) can vanish only at a point s such that q s is a q-Weil number of weight 1. We first recall the following lemma relating the vanishing of the L-function of a quadratic character in terms of the Frobenius eigenvalues of a hyperelliptic curve; This is immediate from the description of P as the numerator of the zeta function of C, and the connection of the latter to L(s, χ f ) (see, for instance, [21, Section 2]).\"\n },\n {\n \"title\": \"Theorem 3.2.\",\n \"content\": \"For any squarefree polynomial f ∈ Q n,q , let L(s, χ f ) be the Dirichlet L-function associated to the quadratic character χ f as was defined in Section 2.1. Then for any s = 0,\\nwhere the limit is taken over all powers q of a fixed odd prime number p.\\nProof. Fix an odd prime number p, and let q be a power of p. By Lemma 3.1, L(s, χ f ) = 0 is equivalent to P(q −s ) = 0 where P(x) ∈ Z[x] is the characteristic polynomial of Frobenius acting on the Jacobian of the hyperelliptic curve defined by y 2 = f (x). Thus, the set { f ∈ Q n,q | L(s, χ f ) = 0} is the same as Q q s n,q .\\nBy Chebotarev's density theorem, we can (for large enough q) find a prime ℓ = 1 4 q 4 1/276 (1 + o(1)) mod which g p s , the minimal polynomial of p s , splits completely. Let a ∈ Z/lZ such that g p s (a) = 0 mod ℓ. If q = p t , then any f with L(χ f , s) = 0 has m a t ( f ) > 0. So |Q q s n,q | |Q n,q | ≤ |Q a t ,ℓ n,q | |Q n,q | and now we can apply Corollary 2.6 to conclude using the second equation of 2.1 that, for all sufficiently large t, we have\\nThe required bound follows from Proposition 2.7.\\nResults on the vanishing of quadratic L-functions over function fields can be used to study the rank distribution of quadratic twist families of constant abelian varieties. In the following corollary, we show that as the constant field grows (so the characteristic is not changing), the probability for a quadratic twist of a constant abelian variety to have positive rank goes to 0. In the elliptic curve case, this agrees with the general \\\"Minimalist Conjecture\\\" philosophy, which holds that positive ranks should be a density-0 phenomenon except when forced by parity considerations from the functional equation (in this setting the functional equation never forces positive rank, and the rank is always even.) We now prove Theorem 1.5, which makes use of the mod 2 Galois representations on J(C) rather than the representations modulo odd primes.\\nProof of Theorem 1.5. Let x 1 , ..., x 2g+2 be the set of Weierstrass points of C. The 2-torsion subgroup J(C) [2] is spanned by the degree-0 2-torsion divisors x i − x j . That is, the group of divisors of the form ∑ a i x i with ∑ a i = 0 surjects onto J(C) [2]. Note also that x 1 + . . . + x 2g+2 − (2g + 2)x 1 is a principal divisor and thus x 1 + . . . + x 2g+2 is 0 in J(C) [2]. See [11,Secion 4] for detailed discussion. This identifies J(C) [2] with an explicit subquotient of F 2g+2 2 ; namely, J(C) [2] is the quotient of the subspace (a 1 , . . . , a 2g+2 ) : ∑ a i = 0 by the 1-dimensional subspace spanned by (1, . . . , 1). This identification is equivariant for the Frobenius action on both sides, so it allows us to describe the mod 2 Galois representation afforded by J(C) in terms of the permutation π which Frobenius induces on x 1 , . . . , x 2g+2 . To be precise, the action of S 2g+2 on J(C) [2] is a representation ρ : S 2g+2 → Sp 2g (Z/2Z), and the action of Frobenius on J(C) [2] is given by ρ(π).\\nThe conditions on π given in Theorem 1.5 are equivalent to the condition that π 2 is a product of two disjoint odd cycles. Thus, the action of π 2 in its permutation representation F 2g+2 2 has eigenvalues given by µ k and µ 2g+2−k for some odd 1 ≤ k ≤ 2g + 1; passing to the subquotient J(C) [2] removes two eigenspaces of ρ(π 2 ) with the eigenvalue 1. So the eigenvalues of Frob 2 on J(C) [2] are the multiset µ ′ k ∪ µ ′ 2g+2−k where µ ′ n denotes the nontrivial nth roots of unity. We see in particular that ρ(π 2 ) does not have 1 as an eigenvalue. But if the zeta function Z C had a zero at 1/2, then √ q would be a Frobenius eigenvalue on C, which would mean that q was an eigenvalue of Frob 2 ; we have shown that Frob 2 has no eigenvalue congruent to 1 mod 2, which rules this out. This proves (1). What's more, the multiset µ ′ k ∪ µ ′ 2g+2−k contains any eigenvalue at most twice, and if (k, 2g + 2 − k) = 1, no eigenvalue appears more than once. This proves (2) (or rather, it proves (2) for the zeta function of C/F q 2 , from which (2) is immediate.)\\nProof of Corollary 1.6. By assumption, f is an irreducible polynomial over F q . So when deg f = n is even, Frobenius acts on the set of Weierstrass points of C f : y 2 = f (x) as a n-cycle. If deg f = n is odd, then Frobenius acts on the set of Weierstrass points of C f as a disjoint union of a n-cycle and a 1-cycle. In either case, when 4 ∤ deg f we can apply Theorem 1.5 to conclude that Z C f does not vanish at 1/2 and so behaves L(s, χ f ).\\nFor any X = q 2g+2 , the set of irreducible polynomials of odd degree at most 2g + 1 gives quadratic characters with bounded conductor whose L-function does not vanish at the central point s = 1/2. By the prime number theorem for function fields, the number of irreducible polynomials in F q [x] of degree at most n is ≫ q n /n. This gives the lower bound in statement.\\nThe proof of Theorem 1.7 is very similar to that of Theorem 1.5, but we treat it separately in order to make the hyperelliptic case above more readable.\\nProof of Theorem 1.7. Let x 1 , . . . , x m be the ramification points of the (Z/ℓZ)-cover of P 1 in S, where m = k 1 + · · · + k r . The Jacobian J(C) of C carries an action of Z[(Z/ℓZ)]; write λ ∈ Z[(Z/ℓZ)] for ζ ℓ − 1, where ζ ℓ is a generator of (Z/ℓZ). A Riemann-Hurwitz computation shows that the genus of C is (m − 2)(ℓ − 1)/2, so the Tate module T ℓ J(C) is a free Z ℓ [ζ ℓ ]-module of rank m − 2, and J(C) [λ] has dimension m − 2.\\nThe λ-torsion subgroup of J(C) is spanned by the degree-0 λ-torsion divisors x i − x j . That is, the group of divisors of the form ∑ a i x i with ∑ a i = 0 surjects onto J(C) [λ]. This surjection is not an isomorphism; there is a 1-dimensional kernel, which we can describe as follows. Over F q , the curve C has an affine model of the form y ℓ = f (x) with f a rational function with no zeroes or poles at ∞. Then the principal divisor associated to y is ∑ a i x i where a i = ord x i f . We have now expressed J(C) [λ] as an explicit subquotient of F m ℓ . This identification is equivariant for the Frobenius action on both sides, so it allows us to describe the mod ℓ Galois representation afforded by J(C) in terms of the permutation π which Frobenius induces on x 1 , . . . , x m .\\nThe action of π splits x 1 , . . . , x m into cycles of length k 1 , . . . , k r , which by hypothesis are prime to ℓ, and which must be multiples of d, where d is the order of q in F * ℓ . So the eigenvalues of π in its action on F m ℓ are the union (as multisets) r j=1 µ k j . Now the composition factors of F m ℓ as a representation of the cyclic group π are J(C) [λ], F ℓ div(y), and the π-trivial one-dimensional representation onto which F m ℓ maps by summing coordinates. The action of π on the latter factor is trivial, while π acts on F ℓ div(y) as multiplication by q. If the zeta function Z C had a zero at 1/2, then √ q would be a Frobenius eigenvalue of C, which would mean that some eigenvalue µ of the action of π on J(C)[λ] satisfied µ 2 = q.\\nIn case q is congruent to 1 modulo ℓ (i.e., d = 1) the eigenvalues of π in its action on J(C) [λ] are the multiset r j=1 µ ′ k j together with r − 2 copies of 1, where µ ′ n denotes the nontrivial nth roots of unity. The hypotheses r = 2 and k j odd now guarantee that the eigenvalues of π on J(C) [λ] contain no copies of either 1 or −1, completing the proof in this case.\\nIf d > 1, we note that our condition on q κ j can be satisfied only when d is even. (If d is odd, then κ j is always a multiple of d, so q κ j = 1.) When d is even, our condition in fact says precisely that each k i is a multiple of d but not of 2d. The two square roots of q inF * ℓ both have order 2d, and thus neither can appear among the eigenvalues of Frobenius on J(C) [λ].\\nWe now turn to the first assertion of the theorem. We note that the coprimality of the k i implies that d = 1, or in other words that q is conguent to 1 modulo ℓ. The fact that J(C) carries an action of Z/ℓZ defined over F q with trivial invariant subspace implies that the Frobenius eigenvalues on J(C) all appear with multiplicity a multiple of ℓ − 1, and that a root of multiplicity k(ℓ − 1) reduces to a root of multiplicity k in the action of Frobenius on J(C) [λ]. So we just need to show that the action of Frobenius on J(C) [λ] has no repeated eigenvalues. The coprimality of the k i guarantees that the union r j=1 µ ′ k j is disjoint; the remaining eigenvalues are r − 2 copies of 1, so since r ≤ 3 we are done.\\nProof of Corollary 1.8. We first consider the case d = 1 (i.e., q is congruent to 1 modulo ℓ).\\nLet f 1 , f 2 be two distinct monic irreducible polynomials in F q [t] of degrees being odd and prime to ℓ. Let d 1 = deg f 1 and d 2 = deg f 2 . Let e ∈ {1, . . . , ℓ − 1} be such that ℓ | d 1 + ed 2 . The smooth projective curve C with affine model y ℓ = f 1 f e 2 admits a cyclic degree ℓ map to P 1 F q where the set of branch points are roots of f = f 1 f 2 in the affine line (the condition on d 1 + ed 2 guarantees there is no branching at ∞.) By Theorem 1.7(2), the zeta function Z C does not vanish at s = 1/2. Thus, all degree ℓ Dirichlet characters with conductor f have L-functions nonvanishing at the central point. Evidently, the number of such pairs f 1 , f 2 with d 1 + d 2 ≤ n is at least the number of irreducible polynomials of degree n − 3; the number of characters with non-vanishing L-functions is thus bounded below by a constant multiple of q n /n = X/(log X). This concludes the proof in the d = 1 case.\\nWe now turn to the case where q ≡ 1 mod ℓ. We recall that d is the order of q mod ℓ in (Z/ℓZ) * . Let Σ d,2d be the set of monic squarefree polynomials such that all of their irreducible factors have degree divisible by d but not divisible by 2d. For any f ∈ Σ d,2d there exists a (Z/ℓZ) field extension K/F q (x) with conductor f . The field K is the function field of a curve C/F q which admits a cyclic degree ℓ map to P 1 F q whose set of branch points are roots of f . By Theorem 1.7(2), the zeta function Z C does not vanish at s = 1/2. Thus, all degree ℓ Dirichlet characters with conductor f have their L-functions do not vanish at the central point. The number of such characters is (ℓ − 1) ω( f ) where ω( f ) denotes the number of irreducible factors of f .\\nSo it remains to count the number of degree ℓ Dirichlet characters whose conductor is in Σ d,2d , which by the above discussion is given by\\nTo estimate this sum, we start by considering the Dirichlet series (1 + (ℓ − 1)|P| −s ). (1 + |P| −s ) d so that H(s) = (Z d (s)) 2 /Z 2d (s). Since Z d (s) and Z 2d (s) each have a simple pole at s = 1, so does H(s).\\nWe conclude that G(s) 2d has a pole at s = 1 with order ℓ − 1 and is absolutely convergent for ℜ(s) > 1. By [12,Theorem 3.1], the sum of coefficients of G(s) has the following asymptotic relation\\nwhere the constant C = |(lim s→1 G(s) 2d (s − 1) ℓ−1 ) 1 2d | Γ((ℓ − 1)/(2d)) .\\nAnd it gives the desired result.\"\n },\n {\n \"title\": \"Corollary 1. 4 (\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 3. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Corollary 3. 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"( 1 +\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Corollary 3.3). Let A be a constant abelian variety defined overF q (x). For each f ∈ F q m [x], denote by A f the quadratic twist of A ⊗ F q m (x) by f . Let R n,m be the set { f ∈ F q m [x],squarefree, of deg n : A f has positive rank}. Then, lim m→∞ lim sup n→∞ |R n,m | q m(n+1) = 0.","content":"Corollary 1. 4 (\n4Corollary 3.3). Let A be a constant abelian variety defined overF q (x). For each f ∈ F q m [x], denote by A f the quadratic twist of A ⊗ F q m (x) by f . Let R n,m be the set { f ∈ F q m [x],squarefree, of deg n : A f has positive rank}. Then, lim m→∞ lim sup n→∞ |R n,m | q m(n+1) = 0."},{"id":"","caption":"Let f be a monic squarefree polynomial in F q [x] and χ f be the quadratic character with conductor f . Let C be the hyperelliptic curve defined by y 2 = f (x) and let P ∈ Z[x] be the characteristic polynomial of geometric Frobenius acting on the Jacobian of C. Then for any s = 0, L(s, χ f ) = 0 if and only if P(q s ) = 0.","content":"Lemma 3. 1 .\n1Let f be a monic squarefree polynomial in F q [x] and χ f be the quadratic character with conductor f . Let C be the hyperelliptic curve defined by y 2 = f (x) and let P ∈ Z[x] be the characteristic polynomial of geometric Frobenius acting on the Jacobian of C. Then for any s = 0, L(s, χ f ) = 0 if and only if P(q s ) = 0."},{"id":"","caption":"Let A be an abelian variety defined over a finite field F q of odd characteristic. For each f ∈ Q n,q m , denote by A f the quadratic twist of A × F q F q m (x) by f . Let R n,m be the set { f ∈ Q n,q m : A f has positive rank}. Then lim m→∞ lim sup n→∞ |R n,m | |Q n,q m | = 0. Proof. Let P(x) be the characteristic polynomial of Frobenius acting on the Tate module of A and let q −s be one of its roots. Then rank A f > 0 is equivalent to L(s, χ f ) = 0. (See [15, Proposition 4.6] for a similar statement with the same proof.) Thus, the statement is a direct application of Theorem 3.2.","content":"Corollary 3. 3 .\n3Let A be an abelian variety defined over a finite field F q of odd characteristic. For each f ∈ Q n,q m , denote by A f the quadratic twist of A × F q F q m (x) by f . Let R n,m be the set { f ∈ Q n,q m : A f has positive rank}. Then lim m→∞ lim sup n→∞ |R n,m | |Q n,q m | = 0. Proof. Let P(x) be the characteristic polynomial of Frobenius acting on the Tate module of A and let q −s be one of its roots. Then rank A f > 0 is equivalent to L(s, χ f ) = 0. (See [15, Proposition 4.6] for a similar statement with the same proof.) Thus, the statement is a direct application of Theorem 3.2."},{"id":"","caption":"f . Then G(s) has a Euler product expansion G(s) = ∏ P∈Σ d,2d , irrd.","content":"f . Then G(s) has a Euler product expansion G(s) = ∏ P∈Σ d,2d , irrd."},{"id":"","caption":"|P| −s ) 2d , we see that G(s) 2d /H(s) ℓ−1 is holomorphic at s = 1. Now define Z d (s) = ∏ d|deg P, P irrd.","content":"( 1 +\n1|P| −s ) 2d , we see that G(s) 2d /H(s) ℓ−1 is holomorphic at s = 1. Now define Z d (s) = ∏ d|deg P, P irrd."},{"id":"","caption":"3, Corollary 2.6] of Andrade and Keating and on [2, Corollary 2.8] of Andrade, Bae, and Jung, which give a proportion on order (log X) −2 , and goes beyond the methods of [1] by Andrade and Baluyot. For the analogous problem over Z, we refer to the recent work [4] of Baluyot and Pratt.","content":"3, Corollary 2.6] of Andrade and Keating and on [2, Corollary 2.8] of Andrade, Bae, and Jung, which give a proportion on order (log X) −2 , and goes beyond the methods of [1] by Andrade and Baluyot. For the analogous problem over Z, we refer to the recent work [4] of Baluyot and Pratt."},{"id":"","caption":"sends Frob q to a. (Reference: [18, §4.5].) Lemma 2.3. With notation as in Section 2.1,","content":"sends Frob q to a. (Reference: [18, §4.5].) Lemma 2.3. With notation as in Section 2.1,"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Corollary 3.3). Let A be a constant abelian variety defined overF q (x). For each f ∈ F q m [x], denote by A f the quadratic twist of A ⊗ F q m (x) by f . Let R n,m be the set { f ∈ F q m [x],squarefree, of deg n : A f has positive rank}. Then, lim m→∞ lim sup n→∞ |R n,m | q m(n+1) = 0.\",\n \"content\": \"Corollary 1. 4 (\\n4Corollary 3.3). Let A be a constant abelian variety defined overF q (x). For each f ∈ F q m [x], denote by A f the quadratic twist of A ⊗ F q m (x) by f . Let R n,m be the set { f ∈ F q m [x],squarefree, of deg n : A f has positive rank}. Then, lim m→∞ lim sup n→∞ |R n,m | q m(n+1) = 0.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let f be a monic squarefree polynomial in F q [x] and χ f be the quadratic character with conductor f . Let C be the hyperelliptic curve defined by y 2 = f (x) and let P ∈ Z[x] be the characteristic polynomial of geometric Frobenius acting on the Jacobian of C. Then for any s = 0, L(s, χ f ) = 0 if and only if P(q s ) = 0.\",\n \"content\": \"Lemma 3. 1 .\\n1Let f be a monic squarefree polynomial in F q [x] and χ f be the quadratic character with conductor f . Let C be the hyperelliptic curve defined by y 2 = f (x) and let P ∈ Z[x] be the characteristic polynomial of geometric Frobenius acting on the Jacobian of C. Then for any s = 0, L(s, χ f ) = 0 if and only if P(q s ) = 0.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let A be an abelian variety defined over a finite field F q of odd characteristic. For each f ∈ Q n,q m , denote by A f the quadratic twist of A × F q F q m (x) by f . Let R n,m be the set { f ∈ Q n,q m : A f has positive rank}. Then lim m→∞ lim sup n→∞ |R n,m | |Q n,q m | = 0. Proof. Let P(x) be the characteristic polynomial of Frobenius acting on the Tate module of A and let q −s be one of its roots. Then rank A f > 0 is equivalent to L(s, χ f ) = 0. (See [15, Proposition 4.6] for a similar statement with the same proof.) Thus, the statement is a direct application of Theorem 3.2.\",\n \"content\": \"Corollary 3. 3 .\\n3Let A be an abelian variety defined over a finite field F q of odd characteristic. For each f ∈ Q n,q m , denote by A f the quadratic twist of A × F q F q m (x) by f . Let R n,m be the set { f ∈ Q n,q m : A f has positive rank}. Then lim m→∞ lim sup n→∞ |R n,m | |Q n,q m | = 0. Proof. Let P(x) be the characteristic polynomial of Frobenius acting on the Tate module of A and let q −s be one of its roots. Then rank A f > 0 is equivalent to L(s, χ f ) = 0. (See [15, Proposition 4.6] for a similar statement with the same proof.) Thus, the statement is a direct application of Theorem 3.2.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"f . Then G(s) has a Euler product expansion G(s) = ∏ P∈Σ d,2d , irrd.\",\n \"content\": \"f . Then G(s) has a Euler product expansion G(s) = ∏ P∈Σ d,2d , irrd.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"|P| −s ) 2d , we see that G(s) 2d /H(s) ℓ−1 is holomorphic at s = 1. Now define Z d (s) = ∏ d|deg P, P irrd.\",\n \"content\": \"( 1 +\\n1|P| −s ) 2d , we see that G(s) 2d /H(s) ℓ−1 is holomorphic at s = 1. Now define Z d (s) = ∏ d|deg P, P irrd.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"3, Corollary 2.6] of Andrade and Keating and on [2, Corollary 2.8] of Andrade, Bae, and Jung, which give a proportion on order (log X) −2 , and goes beyond the methods of [1] by Andrade and Baluyot. For the analogous problem over Z, we refer to the recent work [4] of Baluyot and Pratt.\",\n \"content\": \"3, Corollary 2.6] of Andrade and Keating and on [2, Corollary 2.8] of Andrade, Bae, and Jung, which give a proportion on order (log X) −2 , and goes beyond the methods of [1] by Andrade and Baluyot. For the analogous problem over Z, we refer to the recent work [4] of Baluyot and Pratt.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"sends Frob q to a. (Reference: [18, §4.5].) Lemma 2.3. With notation as in Section 2.1,\",\n \"content\": \"sends Frob q to a. (Reference: [18, §4.5].) Lemma 2.3. With notation as in Section 2.1,\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Small zeros of Dirichlet L-functions of quadratic characters of prime modulus. J Andrade, S Baluyot, arXiv preprintJ. Andrade and S. Baluyot, Small zeros of Dirichlet L-functions of quadratic characters of prime modulus, (2018), arXiv preprint.","doi":""},{"text":"Average values of L-series for real characters in function fields. J C Andrade, S Bae, H Jung, 47. MR 3567720Res. Math. Sci. 3Paper No. 38J. C. Andrade, S. Bae, and H. Jung, Average values of L-series for real characters in function fields, Res. Math. Sci. 3 (2016), Paper No. 38, 47. MR 3567720","doi":"47. MR 3567720"},{"text":"Mean value theorems for L-functions over prime polynomials for the rational function field. J C Andrade, J P Keating, MR 3150889Acta Arith. 1614J. C. Andrade and J. P. Keating, Mean value theorems for L-functions over prime polynomials for the rational function field, Acta Arith. 161 (2013), no. 4, 371-385. MR 3150889","doi":"MR 3150889"},{"text":"Dirichlet L-functions of quadratic characters of prime conductor at the central point. S Baluyot, K Pratt, arXiv preprintS. Baluyot and K. Pratt, Dirichlet L-functions of quadratic characters of prime conductor at the central point, (2018), arXiv preprint.","doi":""},{"text":"Zeros of quadratic Dirichlet L-functions in the hyperelliptic ensemble. H M Bui, A Florea, 8013-8045. MR 3852456Trans. Amer. Math. Soc. 37011H. M. Bui and A. Florea, Zeros of quadratic Dirichlet L-functions in the hyperelliptic ensemble, Trans. Amer. Math. Soc. 370 (2018), no. 11, 8013-8045. MR 3852456","doi":"8013-8045. MR 3852456"},{"text":"The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy. N Chavdarov, MR 1440067Duke Math. J. 871N. Chavdarov, The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy, Duke Math. J. 87 (1997), no. 1, 151-180. MR 1440067","doi":"MR 1440067"},{"text":"Simple zeros of the Riemann zeta-function. J B Conrey, A Ghosh, S M Gonek, MR 1616809Proc. London Math. Soc. 3J. B. Conrey, A. Ghosh, and S. M. Gonek, Simple zeros of the Riemann zeta-function, Proc. London Math. Soc. (3) 76 (1998), no. 3, 497-522. MR 1616809","doi":"MR 1616809"},{"text":"The mean values of cubic L-functions over function fields. C David, A Florea, M Lalin, arXiv preprintC. David, A. Florea, and M. Lalin, The mean values of cubic L-functions over function fields, (2019), arXiv preprint, https://arxiv.org/abs/1901.00817.","doi":""},{"text":". Pierre Deligne, La Weil, MR 601520Inst. HautesÉtudes Sci. Publ. Math. II52Pierre Deligne, La conjecture de Weil. II, Inst. HautesÉtudes Sci. Publ. Math. (1980), no. 52, 137-252. MR 601520","doi":"MR 601520"},{"text":"Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields. J S Ellenberg, A Venkatesh, C Westerland, MR 3488737Ann. of Math. 2J. S. Ellenberg, A. Venkatesh, and C. Westerland, Homological stability for Hurwitz spaces and the Cohen- Lenstra conjecture over function fields, Ann. of Math. (2) 183 (2016), no. 3, 729-786. MR 3488737","doi":"MR 3488737"},{"text":"Hanoi lectures on the arithmetic of hyperelliptic curves. B H Gross, MR 3058664Acta Math. Vietnam. 374B. H. Gross, Hanoi lectures on the arithmetic of hyperelliptic curves, Acta Math. Vietnam. 37 (2012), no. 4, 579-588. MR 3058664","doi":"MR 3058664"},{"text":"A remark on the Wiener-Ikehara Tauberian theorem. Ryo Kato, MR 3410120Comment. Math. Univ. St. Pauli. 641Ryo Kato, A remark on the Wiener-Ikehara Tauberian theorem, Comment. Math. Univ. St. Pauli 64 (2015), no. 1, 47-58. MR 3410120","doi":"MR 3410120"},{"text":"The large sieve, monodromy and zeta functions of curves. E Kowalski, MR 2289204J. Reine Angew. Math. 601E. Kowalski, The large sieve, monodromy and zeta functions of curves, J. Reine Angew. Math. 601 (2006), 29-69. MR 2289204","doi":"MR 2289204"},{"text":"Introduction to transcendental numbers, addison-wesley pub. S Lang, S. Lang, Introduction to transcendental numbers, addison-wesley pub. co., (1966).","doi":""},{"text":"Vanishing of hyperelliptic L-functions at the central point. W Li, MR 3825462J. Number Theory. 191W. Li, Vanishing of hyperelliptic L-functions at the central point, J. Number Theory 191 (2018), 85-103. MR 3825462","doi":"MR 3825462"},{"text":"Cohen-lenstra heuristics forétale group schemes and symplectic pairings. Michael Lipnowski, Jacob Tsimerman, Compositio Mathematica. 1554Michael Lipnowski and Jacob Tsimerman, Cohen-lenstra heuristics forétale group schemes and symplectic pairings, Compositio Mathematica 155 (2019), no. 4, 758-775.","doi":""},{"text":"Abelian varieties. J S Milne, v2.00J. S. Milne, Abelian varieties (v2.00), 2008, Available at www.jmilne.org/math/.","doi":"v2.00"},{"text":"Rational points on varieties. B Poonen, MR 3729254Graduate Studies in Mathematics. 186American Mathematical SocietyB. Poonen, Rational points on varieties, Graduate Studies in Mathematics, vol. 186, American Mathemat- ical Society, Providence, RI, 2017. MR 3729254","doi":"MR 3729254"},{"text":"Algebraic exponential values. A Ray, A. Ray, Algebraic exponential values, https://mathoverflow.net/questions/314098/algebraic- exponential-values.","doi":""},{"text":"Hurwitz spaces. M Romagny, S Wewers, MR 2316356Groupes de Galois arithmétiques et différentiels. ParisSoc. Math. France13M. Romagny and S. Wewers, Hurwitz spaces, Groupes de Galois arithmétiques et différentiels, Sémin. Congr., vol. 13, Soc. Math. France, Paris, 2006, pp. 313-341. MR 2316356","doi":"MR 2316356"},{"text":"Traces of high powers of the frobenius class in the hyperelliptic ensemble. Z Rudnick, Acta Arith. 143Z. Rudnick, Traces of high powers of the frobenius class in the hyperelliptic ensemble, Acta Arith. 143 (2010), 81-99.","doi":""},{"text":"Nonvanishing of quadratic Dirichlet L-functions at s = 1 2. K Soundararajan, MR 1804529Ann. of Math. 2K. Soundararajan, Nonvanishing of quadratic Dirichlet L-functions at s = 1 2 , Ann. of Math. (2) 152 (2000), no. 2, 447-488. MR 1804529","doi":"MR 1804529"}],"string":"[\n {\n \"text\": \"Small zeros of Dirichlet L-functions of quadratic characters of prime modulus. J Andrade, S Baluyot, arXiv preprintJ. Andrade and S. Baluyot, Small zeros of Dirichlet L-functions of quadratic characters of prime modulus, (2018), arXiv preprint.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Average values of L-series for real characters in function fields. J C Andrade, S Bae, H Jung, 47. MR 3567720Res. Math. Sci. 3Paper No. 38J. C. Andrade, S. Bae, and H. Jung, Average values of L-series for real characters in function fields, Res. Math. Sci. 3 (2016), Paper No. 38, 47. MR 3567720\",\n \"doi\": \"47. MR 3567720\"\n },\n {\n \"text\": \"Mean value theorems for L-functions over prime polynomials for the rational function field. J C Andrade, J P Keating, MR 3150889Acta Arith. 1614J. C. Andrade and J. P. Keating, Mean value theorems for L-functions over prime polynomials for the rational function field, Acta Arith. 161 (2013), no. 4, 371-385. MR 3150889\",\n \"doi\": \"MR 3150889\"\n },\n {\n \"text\": \"Dirichlet L-functions of quadratic characters of prime conductor at the central point. S Baluyot, K Pratt, arXiv preprintS. Baluyot and K. Pratt, Dirichlet L-functions of quadratic characters of prime conductor at the central point, (2018), arXiv preprint.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Zeros of quadratic Dirichlet L-functions in the hyperelliptic ensemble. H M Bui, A Florea, 8013-8045. MR 3852456Trans. Amer. Math. Soc. 37011H. M. Bui and A. Florea, Zeros of quadratic Dirichlet L-functions in the hyperelliptic ensemble, Trans. Amer. Math. Soc. 370 (2018), no. 11, 8013-8045. MR 3852456\",\n \"doi\": \"8013-8045. MR 3852456\"\n },\n {\n \"text\": \"The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy. N Chavdarov, MR 1440067Duke Math. J. 871N. Chavdarov, The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy, Duke Math. J. 87 (1997), no. 1, 151-180. MR 1440067\",\n \"doi\": \"MR 1440067\"\n },\n {\n \"text\": \"Simple zeros of the Riemann zeta-function. J B Conrey, A Ghosh, S M Gonek, MR 1616809Proc. London Math. Soc. 3J. B. Conrey, A. Ghosh, and S. M. Gonek, Simple zeros of the Riemann zeta-function, Proc. London Math. Soc. (3) 76 (1998), no. 3, 497-522. MR 1616809\",\n \"doi\": \"MR 1616809\"\n },\n {\n \"text\": \"The mean values of cubic L-functions over function fields. C David, A Florea, M Lalin, arXiv preprintC. David, A. Florea, and M. Lalin, The mean values of cubic L-functions over function fields, (2019), arXiv preprint, https://arxiv.org/abs/1901.00817.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Pierre Deligne, La Weil, MR 601520Inst. HautesÉtudes Sci. Publ. Math. II52Pierre Deligne, La conjecture de Weil. II, Inst. HautesÉtudes Sci. Publ. Math. (1980), no. 52, 137-252. MR 601520\",\n \"doi\": \"MR 601520\"\n },\n {\n \"text\": \"Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields. J S Ellenberg, A Venkatesh, C Westerland, MR 3488737Ann. of Math. 2J. S. Ellenberg, A. Venkatesh, and C. Westerland, Homological stability for Hurwitz spaces and the Cohen- Lenstra conjecture over function fields, Ann. of Math. (2) 183 (2016), no. 3, 729-786. MR 3488737\",\n \"doi\": \"MR 3488737\"\n },\n {\n \"text\": \"Hanoi lectures on the arithmetic of hyperelliptic curves. B H Gross, MR 3058664Acta Math. Vietnam. 374B. H. Gross, Hanoi lectures on the arithmetic of hyperelliptic curves, Acta Math. Vietnam. 37 (2012), no. 4, 579-588. MR 3058664\",\n \"doi\": \"MR 3058664\"\n },\n {\n \"text\": \"A remark on the Wiener-Ikehara Tauberian theorem. Ryo Kato, MR 3410120Comment. Math. Univ. St. Pauli. 641Ryo Kato, A remark on the Wiener-Ikehara Tauberian theorem, Comment. Math. Univ. St. Pauli 64 (2015), no. 1, 47-58. MR 3410120\",\n \"doi\": \"MR 3410120\"\n },\n {\n \"text\": \"The large sieve, monodromy and zeta functions of curves. E Kowalski, MR 2289204J. Reine Angew. Math. 601E. Kowalski, The large sieve, monodromy and zeta functions of curves, J. Reine Angew. Math. 601 (2006), 29-69. MR 2289204\",\n \"doi\": \"MR 2289204\"\n },\n {\n \"text\": \"Introduction to transcendental numbers, addison-wesley pub. S Lang, S. Lang, Introduction to transcendental numbers, addison-wesley pub. co., (1966).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Vanishing of hyperelliptic L-functions at the central point. W Li, MR 3825462J. Number Theory. 191W. Li, Vanishing of hyperelliptic L-functions at the central point, J. Number Theory 191 (2018), 85-103. MR 3825462\",\n \"doi\": \"MR 3825462\"\n },\n {\n \"text\": \"Cohen-lenstra heuristics forétale group schemes and symplectic pairings. Michael Lipnowski, Jacob Tsimerman, Compositio Mathematica. 1554Michael Lipnowski and Jacob Tsimerman, Cohen-lenstra heuristics forétale group schemes and symplectic pairings, Compositio Mathematica 155 (2019), no. 4, 758-775.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Abelian varieties. J S Milne, v2.00J. S. Milne, Abelian varieties (v2.00), 2008, Available at www.jmilne.org/math/.\",\n \"doi\": \"v2.00\"\n },\n {\n \"text\": \"Rational points on varieties. B Poonen, MR 3729254Graduate Studies in Mathematics. 186American Mathematical SocietyB. Poonen, Rational points on varieties, Graduate Studies in Mathematics, vol. 186, American Mathemat- ical Society, Providence, RI, 2017. MR 3729254\",\n \"doi\": \"MR 3729254\"\n },\n {\n \"text\": \"Algebraic exponential values. A Ray, A. Ray, Algebraic exponential values, https://mathoverflow.net/questions/314098/algebraic- exponential-values.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hurwitz spaces. M Romagny, S Wewers, MR 2316356Groupes de Galois arithmétiques et différentiels. ParisSoc. Math. France13M. Romagny and S. Wewers, Hurwitz spaces, Groupes de Galois arithmétiques et différentiels, Sémin. Congr., vol. 13, Soc. Math. France, Paris, 2006, pp. 313-341. MR 2316356\",\n \"doi\": \"MR 2316356\"\n },\n {\n \"text\": \"Traces of high powers of the frobenius class in the hyperelliptic ensemble. Z Rudnick, Acta Arith. 143Z. Rudnick, Traces of high powers of the frobenius class in the hyperelliptic ensemble, Acta Arith. 143 (2010), 81-99.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nonvanishing of quadratic Dirichlet L-functions at s = 1 2. K Soundararajan, MR 1804529Ann. of Math. 2K. Soundararajan, Nonvanishing of quadratic Dirichlet L-functions at s = 1 2 , Ann. of Math. (2) 152 (2000), no. 2, 447-488. MR 1804529\",\n \"doi\": \"MR 1804529\"\n }\n]","isConversational":false}}},{"rowIdx":489,"cells":{"text":{"kind":"string","value":"\n\n26 Oct 2007\n26 Oct 2007arXiv:math/0605701v2 [math.AG] VANISHING RESULTS FOR TORIC VARIETIES ASSOCIATED TO GL n AND G 2 QËNDRIM R. GASHI Dedicated to Scarlett MccGwire and Dr. Christian Duhamel\nToric varieties associated with root systems appeared very naturally in the theory of group compactifications. Here they are considered in a very different context.We prove the vanishing of higher cohomology groups for certain line bundles on toric varieties associated to GLn and G 2 . This can be considered of general interest and it improves the previously known results for these varieties. We also show how these results give a simple proof of a converse to Mazur's inequality for GLn and G 2 respectively. It is known that the latter imply the non-emptiness of some affine Deligne-Lusztig varieties.\n\nIntroduction\n\nFix an isocrystal and the corresponding Newton vector. Mazur's inequality (see [10], pg.658, and the group-theoretic generalization in [13], Theorem 4.2) states that, given a lattice in our isocrystal, the Hodge vector associated to it is greater than or equal to the Newton vector. Conversely, any vector satisfying Mazur's inequality is the Hodge vector for some lattice (see e.g. [12], Proposition 4.2, and [3], Theorem 2). The last result can be regarded as a statement for GL n , but, in [8], §4, R. Kottwitz and M. Rapoport formulated an analogous version for other groups and they proved it for GSp 2n and GL n . They also formulated a combinatorial statement involving respective root systems that would imply the converse to Mazur's inequality in each case (see also sections 4.3 and 4.4 in [6], for a detailed version). C. Lucarelli in [9], Theorem 0.2, showed that this combinatorial statement is true (see Proposition 1.1 below), and therefore a converse to Mazur's inequality was deduced, for all split classical groups. We prove this combinatorial statement for G 2 (Theorem A) and hence deduce a converse to Mazur's inequality for G 2 . We also generalize the combinatorial result for GL n (Theorem B), which in particular gives a new and simple proof of the converse to Mazur's inequality for GL n .\n\nThe results obtained here are proved using the theory of toric varieties. In fact, part of our aim is to show how the combinatorial result mentioned above can be naturally treated as a vanishing result for higher cohomology groups of certain line bundles on toric varieties associated to root systems (theorems C and D on Section 1), which improves the previously known results for these varieties. This raises the question of new vanishing results for toric varieties in general (see [5] for work in this direction concerning toric varieties associated with root systems). It is worth mentioning that these toric varieties, where the fan of the variety is the Weyl fan, appear naturally in the work of group compactifications (see e.g. [2], pp.187-206, and references therein).\n\nTo give a flavor of the type of problem that we deal with here, it is very instructive to use a picture (see below) to illustrate the statement of Theorem E for n=3. Consider a toric variety V = V SL3 with initial lattice (i.e., lattice of co-characters) L = Z 3 /Z(1, 1, 1) and the fan consisting of six maximal cones as pictured below in dotted lines. The dual lattice (of characters) is\nL ∨ = {(x 1 , x 2 , x 3 ) ∈ Z 3 |x 1 + x 2 + x 3 = 0}. α y Dα P z ′ z\nLet L be a globally-generated torus-equivariant line bundle on V . Then L is completely determined by a set of characters (one for each maximal cone in the fan), such that if σ and σ ′ are two \"neighboring\" maximal cones, and u(σ) and u(σ ′ ) are their corresponding characters, then u(σ) − u(σ ′ ), with respect to the canonical pairing of L ∨ and L, is perpendicular to the common face of σ and σ ′ , non-negative on σ, and non-positive on σ ′ . This way L determines a hexagon P where the vertices of P are the characters corresponding to L, one for each maximal cone of the fan. (In our picture all these characters are distinct and we have a true hexagon, but in general we may get some degenerate version of P , where some of the vertices coincide.) Then dim H 0 (V, L) is given by the number of L ∨ -points in the convex hull Conv(P ) of P . Note that Conv(P ) lies in L ∨ ⊗ Z R. This should explain why we are using dotted lines for the cones of our initial fan -they lie in the initial world, not the dual one.\n\nFix a root α as in the picture. Then the \"line-segment\" D α corresponds to a (non-torus-invariant) divisor on V . If we denote by i the embedding D α ֒→ V , then dim H 0 (V, i * (L| Dα )) equals the number of points that are both on D α and that are projections along the root α of points in L ∨ .\n\nOur question is as follows: For any point y on the line segment D α that is obtained by the projection along α of some point z ∈ L ∨ (z is not necessarily in Conv(P )), can we always find a point z ′ that is in both L ∨ and Conv(P ), and which maps to the initial point y? The answer is \"yes\" and, because of the long-exact sequence\n· · · −→ H 0 (V, L) ϕ −→ H 0 (V, i * (L| Dα )) −→ H 1 (V, J Dα ⊗ L) −→ 0,\nwhere J Dα stands for the ideal sheaf of D α , it is equivalent to the statement that H 1 (V, J Dα ⊗ L) = 0. We should mention here that the above long-exact sequence has trivial entries to the right of H 1 (V, J Dα ⊗L), because higher cohomology groups vanish for globally-generated line bundles on complete toric varieties, and L and L| Dα are globally-generated on (the complete toric varieties) V and D α , respectively. Also, note that the map ϕ is induced by projecting along α.\n\nIn this paper we precisely formulate the above question, using co-characters and co-roots, for a split connected reductive group G over a finite extension of Q p , or equivalently, by passing to the (Langlands) complex dual world, using characters and roots, for the corresponding groupĜ over C. Then we reformulate the question using certain toric varieties associated to the root system ofĜ (where the fan of our variety is the Weyl fan and the dual lattice (i.e., lattice of characters) is that of characters forĜ). We answer the question in the affirmative for G = GL n and for a weaker version (where only line bundles L arising from Weyl orbits are considered) for G = G 2 .\n\nAs mentioned at the beginning, this type of combinatorial question, without reference to toric varieties, was investigated by Kottwitz and Rapoport (and others). If we use the language of toric varieties to describe what they did, then they considered only globally generated line bundles L arising from Weyl-orbits, i.e., where the corresponding polygon P is a Weyl-orbit for some element in the character lattice.\n\nLet us now describe how our paper is organized. In the first section we introduce notation, set up the problem and state our main results (theorems A through E). We also show that Theorem C is equivalent to Theorem A. Sections 2 and 3 are devoted to some useful results in computing cohomology groups for line bundles on our toric varieties using the so-called piece-wise linear continuous functions on the support of the respective fans. We use these results in the next section, where we prove theorems B, D, and E, and where we show how E implies D and D implies B. Finally, the last section is devoted to the proof of Theorem C.\n\nAcknowledgments: The author is greatly indebted to his thesis adviser, Professor Robert Kottwitz, for suggesting that he work on the problems dealt with in this paper, for numerous helpful comments, and for continuous encouragement and support.\n\n\nSet Up and Main Results\n\nPrior to [9], a converse to Mazur's inequality for GL n and GSp 2n was proved in [8], Theorem 4.11 (see also [3], Theorem 2, for the GL n case), whose notation we follow. Once a notation is introduced, it will be fixed for the rest of the paper.\n\nLet F be a finite extension of Q p . Denote by O F the ring of integers of F . Suppose G is a split connected reductive group, B a Borel subgroup and T a maximal torus in B, all defined over O F . Let P = M N be a parabolic subgroup of G which contains B, where M is the unique Levi subgroup of P containing T .\n\nWe write X for the set of co-characters X * (T ). Then X G and X M will stand for the quotient of X by the co-root lattice for G and M , respectively. Also, we let ϕ G : X → X G and ϕ M : X → X M denote the respective natural projection maps.\n\nLet µ ∈ X be G-dominant and let W be the Weyl group of T in G. The group W acts on X and so we consider W µ := {w(µ) : w ∈ W } and the convex hull of W µ in a := X ⊗ Z R, which we denote by Conv (W µ). Define\nP µ = {ν ∈ X : (i) ϕ G (ν) = ϕ G (µ); and (ii) ν ∈ Conv (W µ)} .\nLet a M := X M ⊗ Z R and write pr M : a → a M for the natural projection induced by ϕ M . Note that although X M is a quotient of X, after tensoring with R any possible torsion is lost and we can therefore consider a M as a subspace of a; we shall do so throughout the paper.\n\nThe aim is to generalize the proposition below for G = GL n and prove an analogous result for G = G 2 . Kottwitz in [6] (sections 4.3 and 4.4) explains how, in each case, a converse to Mazur's inequality follows from the result below and therefore from our results. [9]) Let G be a split connected reductive group over F with every irreducible component of its Dynkin diagram of type A n , B n , C n , or D n . With the notation as above, we have\nProposition 1.1. (cf. Theorem 0.2 inϕ M (P µ ) = {ν 1 ∈ X M : (i) ν 1 , µ have the same image in X G ; (ii) the image of ν 1 in a M lies in pr M (Conv W µ)} .\nWe are first going to phrase our results using Arthur's notion of (G, T )-orthogonal sets (see e.g. [1], pg. 217). For more on these sets see for example [7], pp.441-447, whose notation we follow. A family of points x B in X, one for every Borel subgroup B of G that contains T , is called a (G, T )-orthogonal set in X if for every two adjacent Borel subgroups B, B ′ , there exists an integer n such that\nx B − x B ′ = nα ∨ ,\nwhere α ∨ is the unique co-root for T that is positive for B and negative for B ′ . Similarly, if we have a point x P ∈ X M for each parabolic subgroup P of G that admits M as a Levi component, then the family (x P ) is called a (G, M )-orthogonal set in X M (see e.g. [7], pg.442) if for every two adjacent parabolic subgroups P, P ′ that admit M as a Levi component, there exists an integer m so that\nx P − x P ′ = mβ ∨ P,P ′ .\nHere β ∨ P,P ′ is the unique element in Γ with the property that all the other elements of Γ are positive multiples of it; Γ consists of the images in X M of the co-roots α ∨ where α is a root occurring in Lie(N ) ∩ Lie(N ′ ), where P = M N andP ′ = MN ′ , withP ′ being the parabolic subgroup containing M opposite P .\n\nIf all the numbers n (resp. m) above are non-negative then we say that (x B ) is a positive (G, T )-orthogonal set (resp. (x P ) is a positive (G, M )-orthogonal set).\n\nAn important example of a positive (G, T )-orthogonal set arises from Weyl orbits W µ (see e.g. [7], pg.443). Let ν ∈ X, then we say that ν is dominant with respect to a Borel group B = T N if for every root α in Lie(N ) we have that α, ν ≥ 0. We get a positive (G, T )-orthogonal set by associating to every Borel group B the unique element x B ∈ X that is both dominant with respect to B and lies in W µ.\n\nIf (x B ) is a (G, T )-orthogonal set in X, then we get a (G, M )-orthogonal set as follows. The points (x B ), where B is a Borel subgroup containing T and B ⊂ P , form an (M, T )-orthogonal set in X, and we get a point x P ∈ X M as the common image in X M of all the points in {x B : B ⊂ P }. The set of all such points x P , where P is a parabolic that contains B and admits M as a Levi component, is a (G, M )-orthogonal set in X M . Moreover, if (x B ) is positive, then (x P ) obtained in this way is positive as well.\n\nWe can now state our main results.\n\nTheorem A. Let G = G 2 . With notation as above we have that\npr M (Conv(x B ) ∩ X) = Conv(x P ) ∩ pr M (X),\nfor every (G, T )-orthogonal set (x B ) that arises from a Weyl orbit and its corresponding (G, M )-orthogonal set (x P ).\n\nTheorem B. Let G = GL n . With notation as above we have that\npr M (Conv(x B ) ∩ X) = Conv(x P ) ∩ pr M (X),\nfor every positive (G, T )-orthogonal set (x B ) and its corresponding (G, M )-orthogonal set (x P ).\n\nIt is clear that the GL n case of Proposition 1.1 becomes a special case of Theorem B when (x B ) arises from a Weyl orbit. Also note that in both theorems A and B, the left-hand side is obviously contained in the right-hand side. The non-trivial part is to show the other containment.\n\nIt is important to mention that, while we believe that Theorem A is also true for the rest of the exceptional groups, Theorem B is probably only true for a split connected reductive group with irreducible components of Dynkin diagram of type A n . More precisely, it is easy to construct counterexamples to Theorem B for other classical groups and for the G 2 case. At the end of section 5 we do so explicitly, but only for G 2 , since the construction for classical groups is completely analogous. We believe that the same construction should also provide counterexamples to Theorem B for the other exceptional groups.\n\nThe proof of these two theorems will involve the theory of toric varieties and we will freely use standard terminology and basic facts from this theory which appear in [4]. Let us first make clear the correspondence between (G, T )-orthogonal sets and line bundles on a certain projective nonsingular toric variety V G , which we now define; we follow the exposition in [7], §23, and keep the same notation as above. LetĜ andT be the (Langlands) complex dual group for G and T , respectively. Let Z(Ĝ) be the center ofĜ. Then the fan of our toric variety V G is the Weyl fan in X * (T /Z(Ĝ)) ⊗ Z R and the torus isT /Z(Ĝ). (We would like to remark that the toric variety V G appears naturally in the theory of group compactifications-see e.g. [2], pp.187-206.)\n\nClearlyT /Z(Ĝ) acts on V G , but we are interested in the action ofT on V G . The latter action is obtained using the canonical surjectionT ։T /Z(Ĝ) and the action ofT /Z(Ĝ) on V G .\n\nAs with any toric variety, there is a one-to-one correspondence betweenT -orbits in V G and cones in the fan of V G . In our case, because the fan is the Weyl fan, this means that we have a one-to-one correspondence betweenT -orbits in V G and parabolic subgroups of G that contain T . According to this identification, if P is a parabolic subgroup of G that contains T , then the correspondingT -orbit is given byT /Z(M ), where M is the unique Levi component of P containing T andM is the corresponding Levi subgroup ofĜ containingT .\n\nAgain, as with any toric variety, there is a one-to-one correspondence between maximal cones in the fan of V G andT -fixed points in V G . In our case, since maximal cones correspond to Borel subgroups of G that contain T , we get a one-to-one correspondence between Borel subgroups of G containing T andT -fixed points in V G .\n\nIt is very important for us to note that there is a one-to-one correspondence between isomorphism classes ofT -equivariant line bundles on V G and (G, T )-orthogonal sets in X (see e.g. [7], §23.1). We describe the map that gives this correspondence.\n\nLet L be aT -equivariant line bundle on V G . Then at eachT -fixed point v in V G the torusT acts by a character, say, x B on the line in L at v, where B is the Borel subgroup corresponding to the fixed point v. In this way, for every Borel subgroup B (containing T ) we get a character x B ∈ X * (T ), i.e., a co-character x B ∈ X. The fact that (x B ) is a (G, T )-orthogonal set comes from the fact that the characters defining L must agree on the overlaps.\n\nA very useful remark is that positive (G, T )-orthogonal sets correspond to line bundles on V G which are generated by their sections, and hence their higher cohomology groups vanish. This last result follows from the more general fact that if the support of the fan (i.e., the union of all the cones in the fan) of a toric variety is a convex set, then the higher cohomology groups vanish for line bundles (on this variety) which are generated by their sections (see e.g. [4], pg.74).\n\nRemember that we are trying to reformulate theorems A and B in terms of toric varieties; we have already described the toric analog of (G, T )-orthogonal sets; the map pr M corresponds to the map\nX * (T ) ։ X * (T )/RM ,\nwhere RM stands for the root lattice forM (note that the codomain of the last map is just X * (Z(M ))); now we need to describe the toric analog of (G, M )-orthogonal sets.\n\nWe continue to have the same notation. Then for M as above (a Levi subgroup containing T ), we will need a toric variety Y G M for the torus Z(M )/Z(Ĝ) (see e.g. [7], §23.2). First assume that the groupĜ is adjoint; we can do this since the center of M /Z(Ĝ) inĜ/Z(Ĝ) is equal to Z(M )/Z(Ĝ)). Then Z(M ) is a subtorus ofT and so X * (Z(M )) is a subgroup of X * (T ). The collection of cones from the Weyl fan inside X * (T ) ⊗ Z R that lie in the subspace X * (Z(M )) ⊗ Z R gives a fan. This is the fan for the complete, nonsingular, projective toric variety Y G M . Now we go back to the general case where we no longer assume thatĜ is adjoint. Denote by G 0 the simply connected cover of the derived group of G and by M 0 the Levi subgroup in G 0 that is obtained as the inverse image of M under the map G 0 → G. Then we have thatĜ\n0 =Ĝ/Z(Ĝ),M 0 =M /Z(Ĝ), and Z(M 0 ) = Z(M )/Z(Ĝ). From the adjoint case, we can define the toric variety Y G0 M0 for Z(M 0 ) = Z(M )/Z(Ĝ). We can therefore view Y G0\nM0 as a space on which Z(M ) acts using the canonical map Z(M ) ։ Z(M 0 ). It is this toric variety, which sits inside V G as a closed Z(M )-stable subspace, that we denote by Y G M . As with the toric variety V G , it is easily seen that Z(M )-fixed points in Y G M are in one-to-one correspondence with parabolic subgroups of G that admit M as a Levi component. Also, there is a one-to-one correspondence between isomorphism classes of Z(M )-equivariant line bundles on Y G M and (G, M )-orthogonal sets in X M . This correspondence is defined is the same way as the one for V G (see e.g. [7], §23.4).\n\nWe note that (see e.g. [7], §23.4 ) restricting aT -equivariant line bundle L on V G to Y G M corresponds the procedure described earlier in this section of obtaining a (G, M )-orthogonal set (the one corresponding to the restriction L| Y G M ) from a (G, T )-orthogonal set (the one corresponding to L).\n\nLet us now assume that our parabolic subgroup P is of semisimple rank 1. This implies that the root lattice RM is just Zα for a unique, up to a sign, root α ofĜ, and that the toric variety Y G M , which we now denote by D α , is a non-torus-invariant divisor in V G . The map pr M will now be denoted by p α . By tensoring with R we get a map from p α , which we still denote by\np α : X * (T ) ⊗ Z R ։ (X * (T )/Zα) ⊗ Z R.\nSince tensoring with R will loose any possible torsion, we can identify the codomain of the last map with the co-root hyperplane\n[α ∨ = 0] := {x ∈ X * (T ) ⊗ Z R : α ∨ , x = 0},\nwhere , is the canonical pairing between co-characters and characters, and α ∨ stands for the co-root ofĜ corresponding to α. We will therefore say that the map p α is the projection along α onto the co-root hyperplane [α ∨ = 0]. Also, the fan of D α is contained in the root hyperplane\n[α = 0] := {x ∈ X * (T ) ⊗ Z R : x, α = 0}.\nNow let L be aT -line bundle on V G that is generated by its sections. Then we have a short-exact sequence of sheaves on V G :\n0 −→ J Dα ⊗ L −→ L −→ i * (L| Dα ) −→ 0, where J Dα is the ideal sheaf of D α and i is the inclusion map D α ֒→ V G . Note that H i (V G , L)=0\n, for all i > 0, since L is generated by its sections and V G is projective, and also\nH i (V G , i * (L| Dα )) = H i (D α , L| Dα ) = 0,\nfor all i > 0, since L| Dα is generated by its sections and D α is a projective toric variety. Therefore the short-exact sequence gives rise to the long-exact sequence\n( * ) ... −→ H 0 (V G , L) ϕ −→ H 0 (V G , i * (L| Dα )) −→ H 1 (V G , J Dα ⊗ L) −→ 0.\nSo, we see that surjectivity of the map ϕ is equivalent to\nH 1 (V G , J Dα ⊗ L) = 0. Let G = G 2 .\nWe claim that the following result is equivalent to Theorem A.\n\nTheorem C. With notation as above, we have that\nH 1 (V G2 , J Dα ⊗ L) = 0,\nwhenever L arises from a Weyl orbit.\n\nRemark 1.2. Theorem C is true for split classical groups. In fact, in the same way that we will show the equivalence of theorems A and C one can show that theorem C for a split classical group is equivalent to Proposition 1.1 for the case of parabolic subgroups P of semisimple rank 1.\n\nLet us demonstrate the equivalence between theorems A and C. Put G = G 2 . Suppose that L corresponds to the Weyl-orbit orthogonal set (x B ) in X, i.e. in X * (T ), and, as before, denote by (x P ) the corresponding (G, M )-orthogonal set in X M , i.e. in X * (Z(M )). Write P L for the intersection of X * (T ) with the convex hull Conv{x B } of (x B ). Then we have (see e.g. [4], pg.66)\nH 0 (V G , L) = u∈PL C χ u ,\nwhere the section χ u is an eigenvector corresponding to the character u. (Recall thatT acts on H 0 (V G , L) and so H 0 (V G , L) decomposes according to the characters ofT .)\n\nAssume that α is the root ofĜ, up to a sign, that corresponds toM , in the sense that the projection map p α : X * (T ) → [α ∨ = 0] along the root α corresponds to the projection map pr M . Then we write P L,α for the intersection of Conv{x P } with p α (X * (T )) and we get\nH 0 (V G , i * (L| Dα )) = u∈PL,α C χ u . The map ϕ : H 0 (V G , L) → H 0 (V G , i * (L| Dα )\n) from the long-exact sequence above is given by ϕ(χ u ) = χ pα(u) . This explains why Theorem C is equivalent to Theorem A, since p α corresponds to pr M . Now let G = GL n . Theorem B will follow from the following result.\n\nTheorem D. With notation as above, we have that\nH 1 (V GLn , J Dα ⊗ L) = 0,\nwhenever L is generated by its sections.\n\nTheorem D itself will be deduced (in Section 4.5) from the result for G = SL n : Theorem E. With notation as above, we have that\nH 1 (V SLn , J Dα ⊗ L) = 0,\nwhenever L is generated by its sections.\n\nWe note that while theorems A and C were seen to be equivalent, to show that theorems B and D imply each other will take a little more work (because we are no longer working with a two-dimensional space!); however, the same argument above for G = G 2 applies in our case to show that Theorem B for the case of parabolic subgroups P of semisimple rank 1 is equivalent to Theorem D. But, we will deduce our Theorem B from Theorem D, where the latter will be proved using toric geometry.\n\nWe explain in Section 4.6 how the general case is handled, but here let us just show, using a similar method used there, how Theorem D, i.e. surjectivity of ϕ, implies Theorem B for all M = GL n1 × GL n2 × · · · × GL nr , where r i=1 n i = n and only one of n i 's is equal to 2 and the rest are equal to 1.\n\nAssume that n k = 2 for some 1 ≤ k ≤ n r and that n i = 1 for all i such that i = k and 1 ≤ i ≤ n r . Keeping the same notation as above, we have in this case a = R n and\na M = {(x 1 , . . . , x n ) ∈ R n : x k = x k+1 },\nwhere, as mentioned earlier, we have identified a M as a subspace of a. Also, the map pr M , which in general is given by averaging over each of the r \"batches\", now agrees with p α where α = (0, . . . , 0, 1, −1, 0, . . . , 0) (with 1 in the k-th place and -1 in the (k + 1)-st place):\np α (x 1 , . . . , x n ) = (x 1 , . . . , x k−1 , x k + x k+1 2 , x k + x k+1 2 , x k+2 , . . . , x n ).\n(In general pr M does not correspond to one p α , rather, it is a composition of a number of p α 's, for distinct roots α.) As in the G 2 case, we find that\nH 0 (V G , L) = u∈PL C χ u , H 0 (V G , i * (L| Dα )) = u∈PL,α C χ u ,\nand ϕ(χ u ) = χ pα(u) . Then surjectivity of ϕ clearly implies Theorem B for this particular case. As we mentioned in the Introduction, the next sections are devoted to proving our results. It is now safe for the reader to assume that, for the rest of the paper, we are working over the complex numbers-we will be in the toric geometry setting. Also, since we will be working in the complex dual world, when we use terms root, co-root, character, and co-character, they will always refer to those terms for the complex dual groupsĜ andT .\n\nBefore we end this section it is worth mentioning that vanishing results like the ones in theorems C and D (and E) are also of independent interest just from a toric-variety point of view. A very important vanishing result for toric varieties has been proved by Mustaţȃ (see e.g. [11], Theorem 0.1) and for the particular toric varieties arising from GL n and G 2 theorems C and D (and E) give vanishings of higher cohomology groups for more line bundles on these varieties. (See also [5] for more vanishing results for toric varieties associated with root systems.)\n\n\nSome useful results\n\nWe keep the same notation as in the previous section. Let V = V G be the non-singular, projective toric variety corresponding to G as described in Section 1 and let ∆ be the fan of V . Each element of ∆ corresponds to a unique torus orbit in V , and under this identification, if we denote by ∆(1) the set of rays in ∆ and if ρ ∈ ∆(1), we write D ρ for the closure of the orbit of ρ, aT -Cartier divisor (cf. [4], §3.1 and §3.3). In fact,T -Cartier divisors on V can be written as a linear combination of D ρ 's. From now on let us agree to refer toT -Cartier divisors simply as divisors. (No confusion should arise from this agreement even though our divisor D α is not aT -divisor; in fact, thanks to Lemma 2.3 below, when computing cohomology groups, we will deal with aT -Cartier divisor instead of D α .)\n\nLet us mention another way of characterizing divisors on toric varieties (see e.g. [4], §3.4, but be aware of a sign difference). A divisor D on a toric variety V , with fan ∆, is completely determined by a set of characters u(σ), one for every element σ ∈ Max(∆), such that they \"agree\" on the overlaps, i.e., such that they form a (G, T )-orthogonal set (we will usually call these just orthogonal sets; here and throughout this paper, Max(∆) stands for the set of maximal cones in ∆). This orthogonal set in turn gives a continuous (integral) piece-wise linear function ψ D on the support |∆| := ∪ σ∈|∆| σ of the fan, where ψ D (v) = u(σ), v , for all v ∈ σ. And conversely, given a continuous (integral) piece-wise linear function ψ on |∆|, i.e., a continuous function that is linear and given by an element of the character lattice on each cone of ∆, we get a divisor by considering the characters u(σ), σ ∈ Max(∆), which define ψ on the maximal affine pieces of V . (The continuity of ψ guarantees that the characters defining it agree on the overlaps and hence give a divisor.)\n\nFrom what we said, it is easy to see that if D = ρ∈∆ a ρ D ρ , then the corresponding piece-wise linear function is given by ψ D (v) = u(σ), v (v ∈ σ) where, for each σ, the character u(σ) is found by solving the system of equations\nu(σ), v ρ = a ρ ,\nwhere ρ varies through the rays in the cone σ and v ρ stands for the primitive lattice element in the ray ρ.\n\nLet us mention that ψ D is called convex or lower convex if for any v, v ′ ∈ |∆| and 0 ≤ p ≤ 1, we have that ψ D (pv\n+ (1 − p)v ′ ) ≤ pψ D (v) + (1 − p)ψ D (v ′ ).\nThe following result is going to be very useful. where H i (V, O(D)) u denotes the u-eigenspace that is obtained by the action ofT on H i (V, O(D)) and Z(u) = {v ∈ |∆| : ψ(v) ≥ u, v }.\n\nKeep the same notation as above and let α be a root ofĜ. The following result was suggested to us by R. Kottwitz. Proof. This follows from considering the rational function χ α −1 on V G and computing the corresponding principal divisor. Note that the divisor of zeros corresponds to D α .\n\nSince we will be interested in computing the cohomology group H 1 (V G , O(D − D α )), the last lemma allows us to focus instead on\nH 1 (V G , O(D − ρ∈∆(1), α,vρ >0 α, v ρ D ρ )).\nThe lemma below describes the orthogonal set corresponding to theT -equivariant divisor ρ∈∆(1), α,vρ >0 α, v ρ D ρ , which, due to the previous lemma, will be referred to as the orthogonal set corresponding to D α . Lemma 2.3. Let α be a root ofĜ and consider V G . The corresponding orthogonal set for D α is given by (u(σ)) σ∈Max(∆) , where u(σ) = α if α ≥ 0 on σ, and u(σ) = 0 if α ≤ 0 on σ.\n\nProof. The result follows at once from Lemma 2.2 and the fact that, as we discussed earlier, if D = ρ∈∆ a ρ D ρ , then u(σ)'s can be found by solving the system of equations u(σ), v ρ = a ρ , where ρ ∈ ∆(1) varies through the rays of σ. Proof. First note that using Proposition 2.1, part b), it is sufficient (and necessary) to show that\n\n\nAn Important Lemma\n(1) H 1 (|∆| , U D,α (0); C) = 0, where U D,α (0) = {v ∈ |∆| : ψ D−Dα (v) < 0} .\nLooking at the long-exact sequence we get from the pair (|∆| , U D,α (0)), since |∆| is a vector space (V G is complete), we only need to show that U D,α (0) is path-wise connected. Let us do that.\n\nAccording to Lemma 2.3, U D,α (0) is equal to the union of sets\n(2) U D,α (0) ∩ [α ≤ 0] = {v ∈ |∆| : ψ D (v) < 0 and α, v ≤ 0} and (3) U D,α (0) ∩ [α ≥ 0] = {v ∈ |∆| : ψ D−Dα (v) < 0 and α, v ≥ 0} .\nHere we have written [α ≥ 0] and [α ≤ 0] for the sets {v ∈ |∆| : α, v ≥ 0} and {v ∈ |∆| : α, v ≤ 0}, respectively. Proposition 2.1, part a), implies that sets (2) and (3) are convex. Let us see how this works in the case of (3). Let v and v ′ be in (3) and let 0 ≤ p ≤ 1. Then using Lemma 2.3. we have\nψ D−Dα (pv + (1 − p)v ′ ) = ψ D (pv + (1 − p)v ′ ) − α, pv + (1 − p)v ′ ,\nwhich, since ψ D is convex, is not greater than\npψ D (v) + (1 − p)ψ D (v ′ ) − p α, v − (1 − p) α, v ′ = pψ D−Dα (v) + (1 − p)ψ D−Dα (v ′ ),\nand the last expression is less than zero since ψ D−Dα (v) < 0, ψ D−Dα (v ′ ) < 0 and 0 ≤ p ≤ 1. Clearly, from our assumptions, we also have that pv +(1−p)v ′ ∈ [α ≥ 0], so (3) is a convex set, as desired.\n\nIf condition (i) of the lemma is satisfied, then U D,α (0) equals the set (3) and hence (1) holds, because the set (3) is convex.\n\nSuppose that (ii) is satisfied. Since {v ∈ |∆| : ψ D (v) < 0} is convex, this implies that there exists a (non-empty, convex) subset U in [α ≤ 0] ∩ [α ≥ 0] that is contained in both sets appearing in (ii). We claim that U is then contained in both sets (2) and (3). This claim is sufficient to prove that U D,α (0) is path-wise connected (and hence (1) holds) since we already saw that (2) and (3) are convex.\n\nFirst, U is contained in (2) by our assumption that it is contained in both sets appearing in (ii), and in particular the second one. Second, since U is contained in the first set appearing in (ii), we see that ψ D (v) < 0 and α, v ≥ 0, for all v ∈ U . Therefore, using Lemma 2.3, we see that\nψ D−Dα (v) = ψ D (v) − α, v < 0.\nThus U is contained in (3) as well, which ends the proof of the claim and hence of the lemma.\n\nNote that if we knew that (1) is true for all globally generated divisors D, then we could conclude that for those divisors we have H 1 (V G , O(D − D α )) u = 0, for all u. Indeed, to prove this, consider the (globally generated) divisor D ′ associated to the orthogonal set {u(σ) − u}, where {u(σ)} corresponds to D. The result follows by using (1) (which is assumed to be true for all globally generated divisors), where instead of U D,α (0) we have U D ′ ,α (0).\n\nUsing Proposition 2.1, part b), we deduce that if (1) holds for all globally generated divisors D, then\nH 1 (V G , J Dα ⊗ O(D)) = 0.\nAn important consequence of this is that Theorem D needs to be proved only in the case when the set (2) is non-empty and\n(4) {v ∈ |∆| : ψ D (v) < 0} ∩ [α ≥ 0] = ∅.\nMore concretely, it suffices to show that, under these conditions,\n(5) U D,α (0) ∩ [α ≥ 0] = ∅,\nbecause then we get that U D,α (0) is equal to the set (2), which we know is a convex set, and we can apply the same reasoning as in Lemma 3.1 to conclude that (1) is true, and by what we just wrote, H 1 (V G , J Dα ⊗ O(D) = 0 is true as well.\n\n\nProofs of Theorems B, D, and E\n\nConsider V G , where G = SL n (and soĜ = P GL(n, C)). We want to concretely describe the fan ∆ of V G . But first, the initial lattice (i.e. lattice of characters) for V G is L = Z n /Z (1, 1, . . . , 1) and so the dual one is L ∨ = {(x 1 , . . . , x n ) ∈ Z n |x 1 + . . . + x n = 0}. For every i = 1, . . . , n we put L i for (the element represented by) (0, . . . , 0, 1, 0, . . . , 0), where 1 appears only on the i-th place. Identify the rays in ∆ with their corresponding minimal lattice points. Then ∆(1) consists of the sums of the form k j=1 L ij , where i j 's are distinct elements from {1, 2, . . . , n} and k = 1, ..., n − 1. The maximal cones in V G are the n-dimensional cones whose rays are of the type\nL i1 , L i1 + L i2 , . . . , L i1 + L i2 + . . . + L in−1 ,\nwhere i 1 , i 2 , . . . , i n−1 are distinct elements from {1, 2, . . . , n}. Let α be a root of G. We can assume that α is simple (for a choice of a Borel subgroup) and, with a change of coordinates, we can always ensure that α is written as L 1 −L 2 . Therefore, without loss of generality, throughout this section we let α = L 1 − L 2 .\n\nThe proof of Theorem E will be proved by induction, although we begin by giving the proof for the first two nontrivial cases, n = 3 and n = 4. This may seem a little unusual, but we do so because in order to apply the induction process we will need \"many\" root hyperplanes and for n = 4 we don't have \"enough\" of them (see Case 2) of the proof in Section 4.3 where we use n > 4).\n\n\n4.1.\n\nProof of Theorem E for n = 3. In this case the set of maximal cones Max(∆) in the fan ∆ of V G consists of six cones σ 1 , . . . , σ 6 , generated respectively by L 1 and −L 3 ; −L 3 and L 2 ; L 2 and −L 1 ; −L 1 and L 3 ; L 3 and −L 2 ; and −L 2 and L 1 .\n\nLet D be a divisor on V G such that O(D) is generated by its sections. If u(σ i ), i = 1, . . . , 6 form the orthogonal set corresponding to D, where u(σ i ) is assigned to the maximal cone σ i , then, according to Lemma 2.3, the orthogonal set corresponding to the divisor D − D α consists of u(σ i ) − α for i = 1, 5, 6 and u(σ i ) for i = 2, 3, 4.\n\nFrom our discussion in Section 3, we only need to prove that (5) is true, under the assumption that the set (2) is non-empty and (4) is true. Suppose, for a contradiction, that under these assumptions there exists v 0 ∈ |∆| such that α, v 0 ≥ 0 and ψ D (v 0 ) − α, v 0 < 0. Since ψ D is continuous and piecewise linear, there are only two cases we need to consider: (i) v 0 = L 1 , and (ii) v 0 = L 1 + L 3 = −L 2 . We can use the symmetry (x, y, z) → (−y, −x, −z) of the root system, which preserves the half-space [α ≥ 0], to see that we only need to consider the case (i).\n\nLet\nv 0 = L 1 . Suppose that u(σ 1 ) = (a, b, −a − b), where a, b ∈ Z. Since ψ D ≥ 0 on [α ≥ 0], we find that u(σ 1 ), L 1 ≥ 0 and u(σ 1 ), −L 3 ≥ 0. Therefore a ≥ 0 and a + b ≥ 0. But, by assumption u(σ 1 ) − α, L 1 < 0, hence a − 1 < 0. So u(σ 1 ) = (0, b, −b) and b ≥ 0. Now, since O(D)\nis generated by its sections, u(σ i )'s form a positive orthogonal set and therefore there exists a non-negative number n ∈ Z such that u(σ 2 ) = u(σ 1 ) + n(L 2 − L 1 ). We get ψ D (L 2 ) = u(σ 2 ), L 2 = b + n ≥ 0. For the same reasons, there exists a non-negative integer m such that u(σ 3 ) = u(σ 2 ) + m(L 3 − L 1 ), and therefore ψ D (−L 1 ) = u(σ 3 ), −L 1 = n + m ≥ 0. Thus, since ψ D is continuous and piece-wise linear, ψ D takes non-negative values on all of [α ≤ 0], and this contradicts the non-emptiness of (2). Theorem E follows for n = 3.\n\nRemark 4.1. If we are working with V SL4 , then we can consider sub-fans of the initial fan ∆ that are contained in the root hyperplanes. For example, if our given root is β = L 3 − L 4 , then the sub-fan corresponding to the root hyperplane [β = 0] has rays L 1 , −L 3 − L 4 , L 2 , −L 1 , L 3 + L 4 , −L 2 and the maximal-dimensional cones are the 2-dimensional cones in ∆ obtained from these rays. This is just a \"copy\" of the fan of V SL3 and indeed we can carry out the same calculations as above to see that Theorem E holds for this \"copy\" of V SL3 inside V SL4 . This remark is important for our induction process and is part of a more general story which we tell in Section 4.4.\n\n\n4.2.\n\nProof of Theorem E for n = 4. Note that we are still assuming that α = L 1 − L 2 . To simplify notation, write σ i,j,k for the maximal cone whose rays are L i , L i + L j , L i + L j + L k , where i, j, k are distinct elements of {1, 2, 3, 4}.\n\nLet D be a divisor on V G with O(D) generated by its sections. Denote by u(σ i,j,k ) the character corresponding to D for the cone σ i,j,k . Then these characters form an orthogonal set and, according to Lemma 2.3, the orthogonal set corresponding to D − D α consists of u(σ i,j,k ) − α for the maximal cones σ i,j,k contained in the half-space [α ≥ 0] and u(σ i,j,k ) for the ones contained in [α ≤ 0]. Following the discussion in Section 3, we assume that the set (2) is non-empty and that (4) is true. We want to prove (5) and, for a contradiction, suppose that there exists a point v 0 in the half-space [α ≥ 0] such that ψ D (v 0 ) − α, v 0 < 0. Since ψ D is continuous and piece-wise linear on |∆|, there are only four cases we need to consider, corresponding to the rays whose primitive lattice points are contained in\n[α > 0]: (i) v 0 = L 1 , (ii) v 0 = L 1 +L 3 , (iii) v 0 = L 1 +L 4 , and (iv) v 0 = L 1 +L 3 +L 4 .\nBut, we can use the symmetry (x, y, z, w) → (−y, −x, −z, −w) of the root system, which preserves the half-space [α ≥ 0], to see that we only need to consider cases (i) and (ii).\n\nCase (i): Our aim is to show that ψ D takes non-negative values at L 2 , L 2 + L 3 , L 2 + L 4 and L 2 + L 3 + L 4 , which would contradict our assumption that the set (2) is non-empty. First, if we look at the hyperplane corresponding to the root L 2 − L 3 , as mentioned in Remark 4.1, we get a copy of V SL3 , where the rays of the sub-fan are L 1 , L 1 + L 4 , L 4 , L 2 + L 3 + L 4 , L 2 + L 3 , L 1 + L 2 + L 3 . These are precisely the rays of ∆ contained in our hyperplane. Apply Theorem E, case n=3, to see that ψ D (L 2 + L 3 ) ≥ 0 and ψ D (\nL 2 + L 3 + L 4 ) ≥ 0.\nSimilarly, by looking at the hyperplanes corresponding to L 3 − L 4 and L 2 − L 4 and applying Theorem E, for n=3, we get that ψ D (L 2 ) ≥ 0 and ψ D (L 2 + L 4 ) ≥ 0 respectively, which is what we intended to prove.\n\nCase (ii): Completely similarly as in Case (i) we see that ψ D (L 2 ) ≥ 0 and ψ D (L 2 + L 4 ) ≥ 0 (use the root L 1 − L 3 ), and ψ D (L 2 + L 3 + L 4 ) ≥ 0 (use the root L 2 − L 4 ). The only non-trivial case is when we want to show that ψ D (L 2 + L 3 ) ≥ 0, because there is no root-hyperplane which contains both L 1 + L 3 and L 2 + L 3 . We proceed as follows. From our assumptions we already know that ψ D (L 1 + L 3 ) ≥ 0, ψ D (L 1 +L 3 )−1 < 0, ψ D (L 3 ) ≥ 0 and ψ D (L 1 +L 2 +L 3 ) ≥ 0. Therefore if u(σ 3,1,2 ) = (a, b, c, −a−b−c) where a, b, c ∈ Z, then a+c ≥ 0, a+c−1 < 0, c ≥ 0 and a+b+c ≥ 0, i.e., a = −c, c ≥ 0 and b ≥ 0. But since u(σ i,j,k )'s form a positive orthogonal set, we can find a non-negative integer m so that u(σ 3,2,1 ) = m(L 2 − L 1 ) + u(σ 3,1,2 ). Thus u(σ 3,2,1 ), L 2 + L 3 = b + c + m ≥ 0, i.e., ψ D (L 2 + L 3 ) ≥ 0, and this concludes the proof.\n\n\n4.3.\n\nProof of Theorem E for n > 4. We use induction to prove the theorem in the general case. Assume that the theorem is true for all V SLn (as well as for the root-hyperplane copies of V SLn inside V SLn+1 ; again see Section 4.4), and we want to prove it for V SLn+1 .\n\nLet D be a divisor on V SLn+1 such that O(D) is generated by its sections. Once more we consider D − D α , where we have taken without loss of generality α = L 1 − L 2 . Following the discussion in Section 3, we assume that the set (2) is nonempty and that (4) is true. We want to prove that (5) holds. For a contradiction, suppose that there exists a point v 0 in the half-space [α ≥ 0] such that ψ D (v 0 ) − α, v 0 < 0. Since the function ψ D is continuous and piece-wise linear, we would get a contradiction if we showed that ψ D takes non-negative values on all the primitive lattice points along the rays of the fan of V SLn+1 which are contained in the halfspace [α ≤ 0]. Moreover, due to the same properties of ψ D , we may assume that (6) v\n0 = L 1 + L ij ,\nwhere i j 's are distinct elements of {3, 4, . . . , n + 1}. We want to prove that for all the primitive lattice points of the form\n(7) L 2 + L pq ,\nwhere p q 's are distinct elements of {3, 4, . . . , n + 1}, we have\n(8) ψ D (L 2 + L pq ) ≥ 0.\nFor this, it suffices to show that there is a root β ofĜ such that both v 0 = L 1 + L ij and L 2 + L pq are contained in the root hyperplane corresponding to β, because we can then apply the induction hypothesis to conclude that (8) is true.\n\nIf there exists an i j from (6) which is not equal to any of p q 's appearing in (7), then put β = L 1 − L ij . If there exists a p q from (7) which is not equal to any of i j 's from (6), then we put β = L 2 − L pq . In both cases, (8) follows at once.\n\nTherefore we are only left with the possibility that, up to permutation, i j 's are equal to p q 's. If this is the case, then we distinguish two cases:\n\nCase 1). The number of L ij 's is greater than one, and so if, say, L ij 1 and L ij 2 appear in (6) we put β = L ij 1 − L ij 2 to get (8).\n\nCase 2). The number of L ij 's is less than two, and so, because n > 4, we can find L km and L kn which don't appear in (6) and therefore in (7) either. Put β = L k1 − L k2 to see that v 0 and L 2 + L pq belong to the root hyperplane corresponding to β. So we get (8).\n\nThis proves Theorem E for all n. The dual lattice for D α is p α (X * (T )). More concretely, the dual lattice for D α is\n{( x 2 2 , x 2 2 , x 3 , . . . , x n+1 ) : n+1 i=2 x i = 0; x i ∈ Z, ∀i},\nbecause the map p α is given by p α (x 1 , x 2 , . . . , x n+1 ) = ( x1+x2 2 , x1+x2 2 , x 3 , . . . , x n+1 ). The claim is that we can treat D α as a copy of V SLn inside V SLn+1 .\n\nFor ease of notation, let L ∨ α := p α (X * (T )) and L ∨ n be the dual lattice of V SLn . Note that we have a bijection Ψ : L ∨ α → L ∨ n , given by Ψ(x 2 /2, x 2 /2, x 3 , . . . , x n+1 ) = (x 2 , x 3 , . . . , x n+1 ). Now let β = α be a root ofĜ. We only need to consider two cases: I) β = L 2 − L 3 ; II) β = L 3 − L 4 , to prove that we can treat D α as a copy of V SLn . I) Let β = L 2 − L 3 . Define p β , the projection using β, in the same way as p α . We project L ∨ α via p β to get the lattice\nL ∨ α,β := {( x 3 3 , x 3 3 , x 3 3 , x 4 , . . . , x n+1 ) : n+1 i=3 x i = 0; x i ∈ Z, ∀i}.\nIf we project L ∨ n via the corresponding p β , we get the lattice\nL ∨ n,β := {( x 3 2 , x 3 2 , x 4 , . . . , x n+1 ) : n+1 i=3 x i = 0; x i ∈ Z, ∀i},\nand we also have a natural bijection Ψ β : L ∨ α,β → L ∨ n,β , given by\nΨ β ( x 3 3 , x 3 3 , x 3 3 , x 4 , . . . , x n+1 ) = ( x 3 2 , x 3 2 , x 4 , . . . , x n+1 ).\nFurther, if we have an orthogonal set (ξ i ) in V SLn , then we get an orthogonal set (Ψ −1 (ξ i )) in D α by applying Ψ −1 to the former. And conversely, we apply Ψ to an orthogonal set in D α in order to get an orthogonal set in V SLn . Also, points in p β (Conv(ξ i ) ∩ L ∨ α ) are identified with points in p β (Conv(Ψ −1 (ξ i ) ∩ L ∨ α ) via the map Ψ β and, under the same map, points in p β (Conv(ξ i )) ∩ p β (L ∨ α ) are identified with those in p β (Conv(Ψ −1 (ξ i ))) ∩ p β (L ∨ n ). The proof of the following Lemma then solely involves unravelling the definitions of the maps involved and the definition of a convex hull. Incidentally, it proves, as we wanted, that we can treat D α as a copy of V SLn inside V SLn+1 , for which Theorem E holds.\n\n\nLemma 4.2.\n\nFor the corresponding orthogonal sets (ξ i ) and (Ψ −1 (ξ i )) we have that\np β (Conv(ξ i ) ∩ L ∨ α ) = p β (Conv(ξ i )) ∩ p β (L ∨ α ) if and only if p β (Conv(Ψ −1 (ξ i ) ∩ L ∨ α ) = p β (Conv(Ψ −1 (ξ i ))) ∩ p β (L ∨ n ).\nII) Let β = L 3 − L 4 . This is handled analogously to I) and the lemma above holds in this case, too. Let us mention that in this case we have x i = 0; x i ∈ Z, ∀i}.\nL ∨ α,β = {( x 3 2 , x 3 2 , x 4 2 , x 4 2 , x 5 , . . . , x n+1 ) :\n\n4.5.\n\nProof of Theorem D. Let G = GL n and, as earlier, take without loss of generality α = L 1 − L 2 . We have that X * (T ) = Z n . Suppose D is a divisor on V G such that O(D) is globally generated and let {u(σ i )} i∈I be the corresponding orthogonal set, where for each i ∈ I, u(σ i ) = (a\n(i) 1 , a (i) 2 , . . . , a (i)\nn ) ∈ Z n corresponds to the maximal cone σ i of the fan ∆ of V G . Then, the fact that {u(σ i )} i∈I is an orthogonal set means in this case that there exists s ∈ Z such that a\n(i) 1 + a (i) 2 + . . . + a (i) n = s, ∀i ∈ I.\nIf we denote by P D the intersection of X * (T ) with the convex hull of the orthogonal set {u(σ i )} i∈I , then we have that (see e.g. [4], pg.66)\nP D = {u ∈ Z n : u, v ≤ u(σ i ), v , ∀v ∈ σ i , ∀i ∈ I} and H 0 (V G , O(D)) = u∈PD C χ u .\nSimilarly, we get that\nH 0 (V G , i * (O(D)| Dα )) = u∈PD α C χ u ,\nwhere P Dα = p α (Z n ) ∩ P α , with P α standing for the image of the convex hull of the orthogonal set {u(σ i )} i∈I under the map p α . (Recall that the map p α was defined in Section 1). Using the long-exact sequence (*) from Section 1, Theorem D can be re-written in the following way:\nu∈PD C χ u ϕ −→ u∈PD α C χ u\nis surjective, for all globally generated line bundles O(D) on V G , where ϕ(χ u ) = χ pα(u) . Let us prove Theorem D using Theorem E. Suppose x ∈ P Dα . It suffices to find a point z in P D such that x = p α (z).\n\nFirst \"shift\" the orthogonal set {u(σ i )} i∈I by u(σ 1 ), i.e., consider the orthogonal set {u(σ i ) − u(σ 1 )} i∈I , which gives a new divisor D ′ on V G . Then, if we write\nu(σ i ) − u(σ 1 ) = (b (i) 1 , b (i) 2 , . . . , b (i) n ), we see that b (i) 1 + b (i) 2 + . . . + b (i) n = 0, ∀i. So {u(σ i ) − u(σ 1 )} i∈I gives an orthogonal set in V SLn .\nNote that p α (x − u(σ 1 )) lies in the intersection of the image of the set\n{u ∈ R n : u, v ≤ u(σ) − u(σ 1 ), v , ∀v ∈ σ, ∀σ ∈ Max(∆)}\nunder the map p α and the set p α (Z n ).\n\nUsing Theorem E, we can then conclude that there exists z ′ ∈ P D ′ such that p α (z ′ ) = p α (x − u(σ 1 )), where P D ′ := {u ∈ Z n : u, v ≤ u(σ) − u(σ 1 ), v , ∀v ∈ σ, ∀σ ∈ Max(∆)}.\n\nOne can easily check that p α (z ′ + u(σ 1 )) = x and z ′ + u(σ 1 ) ∈ P D . This means that we found z = z ′ + u(σ 1 ) in P D such that x = p α (z), which proves Theorem D. 4.6. Proof of Theorem B. Now it is time to explain how Theorem B follows from Theorem D. We keep the same notation as in Section 1, with the difference that we consider G = GL n+1 instead of G = GL n . Then M must be of the form M = GL n1 × GL n2 × · · · × GL nr where n+1 i=1 n i = n + 1. We can also see that a = R n+1 ,\na M = {(x 1 , . . . , x n+1 ) ∈ R n+1 : x 1 = . . . = x n1 , x n1+1 = . . . = x n1+n2 , · · · , x n1+...+nr−1 = . . . = x n+1 },\nand pr M is given by averaging over each of the r \"batches\". (We remind the reader that here, as before, we are considering a M as a subspace of a.)\n\nThe simplest case is when only one of n i 's is equal to 2 and the rest are equal to 1, but we saw in Section 1 how Theorem B is deduced from Theorem D in that case. Now assume that only two of the n i 's are equal to 2 and the rest equal 1. We can assume without loss of generality that n 1 = n 2 = 2 and n 3 = . . . = n r = 1, with r = n − 3. Then\npr M (x 1 , . . . , x n+1 ) = ( x 1 + x 2 2 , x 1 + x 2 2 , x 3 + x 4 2 , x 3 + x 4 2 , x 5 , . . . , x n+1 )\nand so pr M is just the composition p L3−L4 • p L1−L2 . Therefore when we apply Theorem D to D L1−L2 , where we consider D L1−L2 as a copy of V GLn and α = L 3 − L 4 , we find that Theorem B follows in this case, too. The next case would be to consider, without loss of generality, n 1 = 3 and n 2 = . . . = n r = 1, where r = n − 2. Now we see that pr M is the composition p L2−L3 • p L1−L2 and we get our desired conclusion again using Theorem D.\n\nIn general we write pr M as a composition of a finite number of p α 's for distinct roots α and then apply Theorem D as many times as there are roots α appearing in that composition. If we want to be very specific, then, for M = GL n1 × GL n2 × · · · × GL nr we can see that pr M agrees with the composition of maps q 1 , q 2 , . . . , q r where q k = p L (n k −1) −Ln k • p L (n k −2) −L (n k −1) • . . . • p L (n k−1 +1) −L (n k−1 +2) , with the convention n 0 = 0. This concludes the proof of Theorem B.\n\n\nProof of Theorem C\n\nConsider V G for G = G 2 . Just as in the case of SL n , we would like to describe the fan ∆ of V . But first, the initial lattice is L = Z 3 /Z(1, 1, 1) and so the dual one is L ∨ = {(x 1 , x 2 , x 3 ) ∈ Z 3 : x 1 + x 2 + x 3 = 0}. Identify the rays in ∆ with their corresponding minimal lattice points. Then ∆(1) consists of twelve points: v 1 = (1, 0, 0), v 2 = (1, 0, −1), v 3 = (0, 0, −1), v 4 = (0, 1, −1), v 5 = (0, 1, 0), v 6 = (−1, 1, 0) and v i+6 = −v i , for i = 1, . . . , 6. There are twelve maximal cones: σ i , i = 1, 2, . . . , 12, where the rays of σ i are v i and v i+1 (with v 13 := v 1 ). Let D be aT -Cartier divisor on V corresponding to a Weyl orbit (see Section 1 for the precise definition) and α a root ofĜ. For every maximal cone σ i denote by u(σ i ) the character corresponding to the divisor D. Obviously, u(σ i )'s form a positive orthogonal set. We want to prove that\nH 1 (V G , J Dα ⊗ O(D)) = 0.\nThis is true if the orthogonal set corresponding to D is strictly positive, i.e., if u(σ i )'s are distinct and they form a positive orthogonal set. Indeed, use Lemma 2.3 and the definition of orthogonal sets to conclude that the orthogonal set corresponding to D − D α is still positive (but not necessarily strictly positive), i.e., J Dα ⊗ O(D) is globally generated. It is also obvious that if all u(σ i )'s are equal to each other, then we get a degenerate case for which the result is true as well.\n\nTherefore the only problem might arise if only some but not all of the u(σ i )'s are the same. Using symmetries of the current root system, there are only four cases we need to consider, two for a short root and two for a long root: Let us mention again that we want to prove that H 1 (V G , J Dα ⊗ O(D)) = 0. As we explained in Section 1, we have a long-exact sequence ... −→ H 0 (V G , O(D)) ϕ the Weyl-orbit ones. More concretely, in Section 3, after Lemma 3.1, we saw that the vanishing of the first cohomology group there was implied by the vanishing of its corresponding 0-eigenspace (under the action of our torus). This is not true for G = G 2 . The crucial step was to note that by \"shifting\" a positive orthogonal set, say {u(σ)}, by a character, say, u, we get a positive orthogonal set {u(σ) − u}; however, it is clear that a shift of a Weyl-orbit is in general not a Weyl-orbit. 5.2. At the very end, we give an example to show that Theorem B fails in the case of G 2 . Consider V G2 and let α = (1, −1, 0). Let D be the positive orthogonal set (i.e., the globally generated torus-equivariant divisor on V G2 ) given by u(σ i ) = (1, 0, −1) for i ∈ {1, 2, 3, 4, 5, 12} and u(σ i ) = (0, −1, 1) for i ∈ {6, 7, 8, 9, 10, 11}. Following the same notation as before, we see that in our case \n\n\ncf. [4], pg. 68) Let D be a Cartier divisor on V . Then O(D) is generated by its sections if and only if ψ D is (lower) convex. b) (cf. [4], pg.74) For every i ≥ 0 there exist canonical isomorphisms H i (V, O(D)) ∼ = u∈X * (T ) H i (V, O(D)) u , and H i (V, O(D)) u ∼ = H i (|∆| , |∆| \\ Z(u); C),\n\nLemma 2 . 2 .\n22For the toric variety V = V G , we have that D α and ρ∈∆(1), α,vρ >0 α, v ρ D ρ are linearly equivalent divisors.\n\nLemma 3 . 1 .\n31Consider V G and let α be a root ofĜ. Suppose O(D) is generated by its sections. If one of the conditions (i) or (ii) below is satisfied, then the 0-th eigenspace H 1 (V G , J Dα ⊗ O(D)) 0 = 0. (i) {v ∈ |∆| : ψ D (v) < 0 and α, v ≤ 0} = ∅; (ii) {v ∈ |∆| : ψ D (v) < 0 and α, v ≥ 0} = ∅ and {v ∈ |∆| : ψ D (v) < 0 and α, v ≤ 0} = ∅.\n\n4. 4 .\n4Induction process in Proof of Theorem E. Let us justify the induction process we used in the proof of Theorem E. Suppose G = SL n+1 and consider V G , with the initial lattice Z n+1 /Z(1, 1, . . . , 1) and with dual lattice X * (T ) = {(x 1 , . . . , x n+1 ) ∈ Z n+1 |x 1 + . . . + x n+1 = 0}. Take without loss of generality α = L 1 − L 2 . Then the fan of D α consists of the (n − 1)-dimensional cones of the fan ∆ of V SLn+1 which lie inside the hyperplane [α = 0] = {v ∈ R n+1 : α, v = 0}.\n\n\n= 0; x i ∈ Z, ∀i} and L ∨ n,β = {(x 3 , x 4 2 , x 4 2 , x 5 , . . . , x n+1 ) : n+1 i=3\n\n\n(a) α = (1, −1, 0) and there exists a natural number n so thatu(σ 2 ) = u(σ 3 ) = −u(σ 8 ) = −u(σ 9 ) = n(1, 1, −2),u(σ 4 ) = u(σ 5 ) = −u(σ 10 ) = −u(σ 11 ) = n(−1, 2, −1), u(σ 6 ) = u(σ 7 ) = −u(σ 12 ) = −u(σ 1 ) = n(−2, 1, 1); (b) α = (2, −1, −1) and there exists a natural number n so that u(σ 2 ) = u(σ 3 ) = −u(σ 8 ) = −u(σ 9 ) = n(1, 1, −2), u(σ 4 ) = u(σ 5 ) = −u(σ 10 ) = −u(σ 11 ) = n(−1, 2, −1), u(σ 6 ) = u(σ 7 ) = −u(σ 12 ) = −u(σ 1 ) = n(−2, 1, 1); (c) α = (1, −1, 0) and there exists a natural number n so that u(σ 1 ) = u(σ 2 ) = −u(σ 7 ) = −u(σ 8 ) = n(1, 0, −1), u(σ 3 ) = u(σ 4 ) = −u(σ 9 ) = −u(σ 10 ) = n(0, 1, −1), u(σ 5 ) = u(σ 6 ) = −u(σ 11 ) = −u(σ 12 ) = n(−1, 1, 0); (d) α = (2, −1, −1) and there exists a natural number n so that u(σ 1 ) = u(σ 2 ) = −u(σ 7 ) = −u(σ 8 ) = n(1, 0, −1), u(σ 3 ) = u(σ 4 ) = −u(σ 9 ) = −u(σ 10 ) = n(0, 1, −1), u(σ 5 ) = u(σ 6 ) = −u(σ 11 ) = −u(σ 12 ) = n(−1, 1, 0).\n\n\nH 0 (V G2 , O(D)) = 2, but dim H 0 (V G2 , i * (O(D)| Dα )) = 3. Hence the map ϕ : H 0 (V G2 , O(D)) → H 0 (V G2 , i * (O(D)| Dα )), induced by the projection p α along α, is not surjective, and Theorem B is no longer true if we put G 2 instead of GL n .\nwhere terms on the right vanish because O(D) is generated by its sections. Therefore it suffices to prove that ϕ is surjective in each of the cases above.Case (a): We have that (cf.[4], pg.66)for all odd integers k between −2n and 2n. But this is easily seen to be true by checking that ( k+1where v j 's are the respective primitive lattice points along the rays of our fan ∆, written explicitly at the beginning of this section.Case (b): Keeping the same notation as in Case (a) we now have that P Dα =. It is an easy calculation to see that for all odd k between −3n and 3n we get (1, k−1 2 , − k+1 2 ) ∈ P D and therefore ϕ is surjective, which we wanted to prove.Case (c): This time we have P Dα = {( k 2 , k 2 , −k) : k ∈ [−n, n] ∩ Z} and ϕ is the same as in case (a). Clearly ( k 2 , k 2 , −k) is in P D and χ ( k 2 , k 2 ,−k) is fixed by ϕ, whenever k is even. If k is odd then consider ( k+1 2 , k−1 2 , −k), which is easily seen to lie in P D . ϕ is surjective because it maps χ ( k+1ϕ is the same as in Case (b) and it fixes χ (0, k 2 ,− k 2 ) for k even. Clearly (0, k 2 , − k 2 ) ∈ P D and it is not difficult to see that if k is odd then (1, k−1 2 , − k+1 2 ) ∈ P D as well. Moreover, still for k odd, ϕ(χ (1, k−1. Hence ϕ is surjective. This concludes the proof of Theorem C and therefore, as explained in Section 1, of Theorem A, too.5.1. We would like to note that the proof of Theorem D was different from that of Theorem C because in the former we used all positive orthogonal sets whereas in the latter we only used some and not all positive orthogonal sets, namely only\nThe characters of discrete series as orbital integrals. J Arthur, Invent. Math. 323J. ARTHUR, The characters of discrete series as orbital integrals, Invent. Math. 32 1976, no.3, 205-261.\n\nFrobenius splitting methods in geometry and representation theory. M Brion, S Kumar, BirkhäuserBostonM. BRION and S. KUMAR, Frobenius splitting methods in geometry and representation theory, Birkhäuser Boston (2004).\n\nExistence de filtrations admissible sur des isocristaux. J.-M Fontaine, M Rapoport, Bull. Soc. Math. France. 1331J.-M. FONTAINE and M. RAPOPORT, Existence de filtrations admissible sur des isocristaux, Bull. Soc. Math. France 133 (2005), no.1, 73-86.\n\nIntroduction to Toric Varieties. W Fulton, Ann. of Math. Stud. 131Princeton Univ. PressW. FULTON, Introduction to Toric Varieties, Ann. of Math. Stud. 131, Princeton Univ. Press, Princeton, NJ, 1993.\n\nVanishing results for toric varieties associated with root systems. Q R Gashi, in preparationQ. R. GASHI, Vanishing results for toric varieties associated with root systems, in prepara- tion.\n\nOn the Hodge-Newton decomposition for split groups. R Kottwitz, Int. Math. Res. Not. 26R. KOTTWITZ, On the Hodge-Newton decomposition for split groups, Int. Math. Res. Not. 2003, no.26, 1433-1447.\n\nR Kottwitz, Harmonic analysis on reductive p-adic groups and Lie algebras, in Harmonic Analysis, the Trace Formula, and Shimura Varieties. Providence, RIAmer. Math. Soc4R. KOTTWITZ, Harmonic analysis on reductive p-adic groups and Lie algebras, in Harmonic Analysis, the Trace Formula, and Shimura Varieties, 393-522, Clay Math. Proc., 4, Amer. Math. Soc., Providence, RI, 2005.\n\nOn the existence of F-isocrystals. R Kottwitz, M Rapoport, Comment. Math. Helv. 78R. KOTTWITZ and M. RAPOPORT, On the existence of F-isocrystals, Comment. Math. Helv. 78 (2003), 153-184.\n\nA Converse to Mazur's inequality for split classical groups. C Lucarelli, Journal of the Inst. Math. Jussieu. 32C. LUCARELLI, A Converse to Mazur's inequality for split classical groups, Journal of the Inst. Math. Jussieu (2004) 3 (2), 165-183.\n\nFrobenius and the Hodge filtration. B Mazur, Bull. Amer. Math. Soc. 78B. MAZUR, Frobenius and the Hodge filtration, Bull. Amer. Math. Soc. 78 (1972) 653-667.\n\nVanishing theorems on toric varieties. M Mustaţȃ, Tohoku Math. J. 2M. MUSTAŢȂ, Vanishing theorems on toric varieties, Tohoku Math. J. (2) 54 (2002), no.3, 451-470.\n\nA positivity property of the Satake isomorphism. M Rapoport, Manuscripta Math. 1012M. RAPOPORT, A positivity property of the Satake isomorphism, Manuscripta Math. 101 (2000), no.2, 153-166.\n\nOn the classification and specialization of F-isocrystals with additional structure. M Rapoport, M Richartz, Composito Math. 1032M. RAPOPORT and M. RICHARTZ, On the classification and specialization of F-isocrystals with additional structure, Composito Math. 103 (1996), no.2, 153-181\n"},"corpus_id":{"kind":"number","value":16153264,"string":"16,153,264"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"math/0605701\",\n \"mag\": 2950849610,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1007/s00031-008-9002-4\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/math/0605701v2.pdf\"\n ],\n \"pdf_sha\": \"35921a21b512a70e88b8c8c408e3e50f901da599\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/math/0605701\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":""},"authors":{"kind":"list like","value":[],"string":"[]","isConversational":false},"abstract":{"kind":"string","value":"Toric varieties associated with root systems appeared very naturally in the theory of group compactifications. Here they are considered in a very different context.We prove the vanishing of higher cohomology groups for certain line bundles on toric varieties associated to GLn and G 2 . This can be considered of general interest and it improves the previously known results for these varieties. We also show how these results give a simple proof of a converse to Mazur's inequality for GLn and G 2 respectively. It is known that the latter imply the non-emptiness of some affine Deligne-Lusztig varieties."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Fix an isocrystal and the corresponding Newton vector. Mazur's inequality (see [10], pg.658, and the group-theoretic generalization in [13], Theorem 4.2) states that, given a lattice in our isocrystal, the Hodge vector associated to it is greater than or equal to the Newton vector. Conversely, any vector satisfying Mazur's inequality is the Hodge vector for some lattice (see e.g. [12], Proposition 4.2, and [3], Theorem 2). The last result can be regarded as a statement for GL n , but, in [8], §4, R. Kottwitz and M. Rapoport formulated an analogous version for other groups and they proved it for GSp 2n and GL n . They also formulated a combinatorial statement involving respective root systems that would imply the converse to Mazur's inequality in each case (see also sections 4.3 and 4.4 in [6], for a detailed version). C. Lucarelli in [9], Theorem 0.2, showed that this combinatorial statement is true (see Proposition 1.1 below), and therefore a converse to Mazur's inequality was deduced, for all split classical groups. We prove this combinatorial statement for G 2 (Theorem A) and hence deduce a converse to Mazur's inequality for G 2 . We also generalize the combinatorial result for GL n (Theorem B), which in particular gives a new and simple proof of the converse to Mazur's inequality for GL n .\nThe results obtained here are proved using the theory of toric varieties. In fact, part of our aim is to show how the combinatorial result mentioned above can be naturally treated as a vanishing result for higher cohomology groups of certain line bundles on toric varieties associated to root systems (theorems C and D on Section 1), which improves the previously known results for these varieties. This raises the question of new vanishing results for toric varieties in general (see [5] for work in this direction concerning toric varieties associated with root systems). It is worth mentioning that these toric varieties, where the fan of the variety is the Weyl fan, appear naturally in the work of group compactifications (see e.g. [2], pp.187-206, and references therein).\nTo give a flavor of the type of problem that we deal with here, it is very instructive to use a picture (see below) to illustrate the statement of Theorem E for n=3. Consider a toric variety V = V SL3 with initial lattice (i.e., lattice of co-characters) L = Z 3 /Z(1, 1, 1) and the fan consisting of six maximal cones as pictured below in dotted lines. The dual lattice (of characters) is\nLet L be a globally-generated torus-equivariant line bundle on V . Then L is completely determined by a set of characters (one for each maximal cone in the fan), such that if σ and σ ′ are two \"neighboring\" maximal cones, and u(σ) and u(σ ′ ) are their corresponding characters, then u(σ) − u(σ ′ ), with respect to the canonical pairing of L ∨ and L, is perpendicular to the common face of σ and σ ′ , non-negative on σ, and non-positive on σ ′ . This way L determines a hexagon P where the vertices of P are the characters corresponding to L, one for each maximal cone of the fan. (In our picture all these characters are distinct and we have a true hexagon, but in general we may get some degenerate version of P , where some of the vertices coincide.) Then dim H 0 (V, L) is given by the number of L ∨ -points in the convex hull Conv(P ) of P . Note that Conv(P ) lies in L ∨ ⊗ Z R. This should explain why we are using dotted lines for the cones of our initial fan -they lie in the initial world, not the dual one.\nFix a root α as in the picture. Then the \"line-segment\" D α corresponds to a (non-torus-invariant) divisor on V . If we denote by i the embedding D α ֒→ V , then dim H 0 (V, i * (L| Dα )) equals the number of points that are both on D α and that are projections along the root α of points in L ∨ .\nOur question is as follows: For any point y on the line segment D α that is obtained by the projection along α of some point z ∈ L ∨ (z is not necessarily in Conv(P )), can we always find a point z ′ that is in both L ∨ and Conv(P ), and which maps to the initial point y? The answer is \"yes\" and, because of the long-exact sequence\nwhere J Dα stands for the ideal sheaf of D α , it is equivalent to the statement that H 1 (V, J Dα ⊗ L) = 0. We should mention here that the above long-exact sequence has trivial entries to the right of H 1 (V, J Dα ⊗L), because higher cohomology groups vanish for globally-generated line bundles on complete toric varieties, and L and L| Dα are globally-generated on (the complete toric varieties) V and D α , respectively. Also, note that the map ϕ is induced by projecting along α.\nIn this paper we precisely formulate the above question, using co-characters and co-roots, for a split connected reductive group G over a finite extension of Q p , or equivalently, by passing to the (Langlands) complex dual world, using characters and roots, for the corresponding groupĜ over C. Then we reformulate the question using certain toric varieties associated to the root system ofĜ (where the fan of our variety is the Weyl fan and the dual lattice (i.e., lattice of characters) is that of characters forĜ). We answer the question in the affirmative for G = GL n and for a weaker version (where only line bundles L arising from Weyl orbits are considered) for G = G 2 .\nAs mentioned at the beginning, this type of combinatorial question, without reference to toric varieties, was investigated by Kottwitz and Rapoport (and others). If we use the language of toric varieties to describe what they did, then they considered only globally generated line bundles L arising from Weyl-orbits, i.e., where the corresponding polygon P is a Weyl-orbit for some element in the character lattice.\nLet us now describe how our paper is organized. In the first section we introduce notation, set up the problem and state our main results (theorems A through E). We also show that Theorem C is equivalent to Theorem A. Sections 2 and 3 are devoted to some useful results in computing cohomology groups for line bundles on our toric varieties using the so-called piece-wise linear continuous functions on the support of the respective fans. We use these results in the next section, where we prove theorems B, D, and E, and where we show how E implies D and D implies B. Finally, the last section is devoted to the proof of Theorem C.\nAcknowledgments: The author is greatly indebted to his thesis adviser, Professor Robert Kottwitz, for suggesting that he work on the problems dealt with in this paper, for numerous helpful comments, and for continuous encouragement and support."},{"title":"Set Up and Main Results","content":"Prior to [9], a converse to Mazur's inequality for GL n and GSp 2n was proved in [8], Theorem 4.11 (see also [3], Theorem 2, for the GL n case), whose notation we follow. Once a notation is introduced, it will be fixed for the rest of the paper.\nLet F be a finite extension of Q p . Denote by O F the ring of integers of F . Suppose G is a split connected reductive group, B a Borel subgroup and T a maximal torus in B, all defined over O F . Let P = M N be a parabolic subgroup of G which contains B, where M is the unique Levi subgroup of P containing T .\nWe write X for the set of co-characters X * (T ). Then X G and X M will stand for the quotient of X by the co-root lattice for G and M , respectively. Also, we let ϕ G : X → X G and ϕ M : X → X M denote the respective natural projection maps.\nLet µ ∈ X be G-dominant and let W be the Weyl group of T in G. The group W acts on X and so we consider W µ := {w(µ) : w ∈ W } and the convex hull of W µ in a := X ⊗ Z R, which we denote by Conv (W µ). Define\nLet a M := X M ⊗ Z R and write pr M : a → a M for the natural projection induced by ϕ M . Note that although X M is a quotient of X, after tensoring with R any possible torsion is lost and we can therefore consider a M as a subspace of a; we shall do so throughout the paper.\nThe aim is to generalize the proposition below for G = GL n and prove an analogous result for G = G 2 . Kottwitz in [6] (sections 4.3 and 4.4) explains how, in each case, a converse to Mazur's inequality follows from the result below and therefore from our results. [9]) Let G be a split connected reductive group over F with every irreducible component of its Dynkin diagram of type A n , B n , C n , or D n . With the notation as above, we have\nWe are first going to phrase our results using Arthur's notion of (G, T )-orthogonal sets (see e.g. [1], pg. 217). For more on these sets see for example [7], pp.441-447, whose notation we follow. A family of points x B in X, one for every Borel subgroup B of G that contains T , is called a (G, T )-orthogonal set in X if for every two adjacent Borel subgroups B, B ′ , there exists an integer n such that\nwhere α ∨ is the unique co-root for T that is positive for B and negative for B ′ . Similarly, if we have a point x P ∈ X M for each parabolic subgroup P of G that admits M as a Levi component, then the family (x P ) is called a (G, M )-orthogonal set in X M (see e.g. [7], pg.442) if for every two adjacent parabolic subgroups P, P ′ that admit M as a Levi component, there exists an integer m so that\nHere β ∨ P,P ′ is the unique element in Γ with the property that all the other elements of Γ are positive multiples of it; Γ consists of the images in X M of the co-roots α ∨ where α is a root occurring in Lie(N ) ∩ Lie(N ′ ), where P = M N andP ′ = MN ′ , withP ′ being the parabolic subgroup containing M opposite P .\nIf all the numbers n (resp. m) above are non-negative then we say that (x B ) is a positive (G, T )-orthogonal set (resp. (x P ) is a positive (G, M )-orthogonal set).\nAn important example of a positive (G, T )-orthogonal set arises from Weyl orbits W µ (see e.g. [7], pg.443). Let ν ∈ X, then we say that ν is dominant with respect to a Borel group B = T N if for every root α in Lie(N ) we have that α, ν ≥ 0. We get a positive (G, T )-orthogonal set by associating to every Borel group B the unique element x B ∈ X that is both dominant with respect to B and lies in W µ.\nIf (x B ) is a (G, T )-orthogonal set in X, then we get a (G, M )-orthogonal set as follows. The points (x B ), where B is a Borel subgroup containing T and B ⊂ P , form an (M, T )-orthogonal set in X, and we get a point x P ∈ X M as the common image in X M of all the points in {x B : B ⊂ P }. The set of all such points x P , where P is a parabolic that contains B and admits M as a Levi component, is a (G, M )-orthogonal set in X M . Moreover, if (x B ) is positive, then (x P ) obtained in this way is positive as well.\nWe can now state our main results.\nTheorem A. Let G = G 2 . With notation as above we have that\nfor every (G, T )-orthogonal set (x B ) that arises from a Weyl orbit and its corresponding (G, M )-orthogonal set (x P ).\nTheorem B. Let G = GL n . With notation as above we have that\nfor every positive (G, T )-orthogonal set (x B ) and its corresponding (G, M )-orthogonal set (x P ).\nIt is clear that the GL n case of Proposition 1.1 becomes a special case of Theorem B when (x B ) arises from a Weyl orbit. Also note that in both theorems A and B, the left-hand side is obviously contained in the right-hand side. The non-trivial part is to show the other containment.\nIt is important to mention that, while we believe that Theorem A is also true for the rest of the exceptional groups, Theorem B is probably only true for a split connected reductive group with irreducible components of Dynkin diagram of type A n . More precisely, it is easy to construct counterexamples to Theorem B for other classical groups and for the G 2 case. At the end of section 5 we do so explicitly, but only for G 2 , since the construction for classical groups is completely analogous. We believe that the same construction should also provide counterexamples to Theorem B for the other exceptional groups.\nThe proof of these two theorems will involve the theory of toric varieties and we will freely use standard terminology and basic facts from this theory which appear in [4]. Let us first make clear the correspondence between (G, T )-orthogonal sets and line bundles on a certain projective nonsingular toric variety V G , which we now define; we follow the exposition in [7], §23, and keep the same notation as above. LetĜ andT be the (Langlands) complex dual group for G and T , respectively. Let Z(Ĝ) be the center ofĜ. Then the fan of our toric variety V G is the Weyl fan in X * (T /Z(Ĝ)) ⊗ Z R and the torus isT /Z(Ĝ). (We would like to remark that the toric variety V G appears naturally in the theory of group compactifications-see e.g. [2], pp.187-206.)\nClearlyT /Z(Ĝ) acts on V G , but we are interested in the action ofT on V G . The latter action is obtained using the canonical surjectionT ։T /Z(Ĝ) and the action ofT /Z(Ĝ) on V G .\nAs with any toric variety, there is a one-to-one correspondence betweenT -orbits in V G and cones in the fan of V G . In our case, because the fan is the Weyl fan, this means that we have a one-to-one correspondence betweenT -orbits in V G and parabolic subgroups of G that contain T . According to this identification, if P is a parabolic subgroup of G that contains T , then the correspondingT -orbit is given byT /Z(M ), where M is the unique Levi component of P containing T andM is the corresponding Levi subgroup ofĜ containingT .\nAgain, as with any toric variety, there is a one-to-one correspondence between maximal cones in the fan of V G andT -fixed points in V G . In our case, since maximal cones correspond to Borel subgroups of G that contain T , we get a one-to-one correspondence between Borel subgroups of G containing T andT -fixed points in V G .\nIt is very important for us to note that there is a one-to-one correspondence between isomorphism classes ofT -equivariant line bundles on V G and (G, T )-orthogonal sets in X (see e.g. [7], §23.1). We describe the map that gives this correspondence.\nLet L be aT -equivariant line bundle on V G . Then at eachT -fixed point v in V G the torusT acts by a character, say, x B on the line in L at v, where B is the Borel subgroup corresponding to the fixed point v. In this way, for every Borel subgroup B (containing T ) we get a character x B ∈ X * (T ), i.e., a co-character x B ∈ X. The fact that (x B ) is a (G, T )-orthogonal set comes from the fact that the characters defining L must agree on the overlaps.\nA very useful remark is that positive (G, T )-orthogonal sets correspond to line bundles on V G which are generated by their sections, and hence their higher cohomology groups vanish. This last result follows from the more general fact that if the support of the fan (i.e., the union of all the cones in the fan) of a toric variety is a convex set, then the higher cohomology groups vanish for line bundles (on this variety) which are generated by their sections (see e.g. [4], pg.74).\nRemember that we are trying to reformulate theorems A and B in terms of toric varieties; we have already described the toric analog of (G, T )-orthogonal sets; the map pr M corresponds to the map\nwhere RM stands for the root lattice forM (note that the codomain of the last map is just X * (Z(M ))); now we need to describe the toric analog of (G, M )-orthogonal sets.\nWe continue to have the same notation. Then for M as above (a Levi subgroup containing T ), we will need a toric variety Y G M for the torus Z(M )/Z(Ĝ) (see e.g. [7], §23.2). First assume that the groupĜ is adjoint; we can do this since the center of M /Z(Ĝ) inĜ/Z(Ĝ) is equal to Z(M )/Z(Ĝ)). Then Z(M ) is a subtorus ofT and so X * (Z(M )) is a subgroup of X * (T ). The collection of cones from the Weyl fan inside X * (T ) ⊗ Z R that lie in the subspace X * (Z(M )) ⊗ Z R gives a fan. This is the fan for the complete, nonsingular, projective toric variety Y G M . Now we go back to the general case where we no longer assume thatĜ is adjoint. Denote by G 0 the simply connected cover of the derived group of G and by M 0 the Levi subgroup in G 0 that is obtained as the inverse image of M under the map G 0 → G. Then we have thatĜ\nM0 as a space on which Z(M ) acts using the canonical map Z(M ) ։ Z(M 0 ). It is this toric variety, which sits inside V G as a closed Z(M )-stable subspace, that we denote by Y G M . As with the toric variety V G , it is easily seen that Z(M )-fixed points in Y G M are in one-to-one correspondence with parabolic subgroups of G that admit M as a Levi component. Also, there is a one-to-one correspondence between isomorphism classes of Z(M )-equivariant line bundles on Y G M and (G, M )-orthogonal sets in X M . This correspondence is defined is the same way as the one for V G (see e.g. [7], §23.4).\nWe note that (see e.g. [7], §23.4 ) restricting aT -equivariant line bundle L on V G to Y G M corresponds the procedure described earlier in this section of obtaining a (G, M )-orthogonal set (the one corresponding to the restriction L| Y G M ) from a (G, T )-orthogonal set (the one corresponding to L).\nLet us now assume that our parabolic subgroup P is of semisimple rank 1. This implies that the root lattice RM is just Zα for a unique, up to a sign, root α ofĜ, and that the toric variety Y G M , which we now denote by D α , is a non-torus-invariant divisor in V G . The map pr M will now be denoted by p α . By tensoring with R we get a map from p α , which we still denote by\nSince tensoring with R will loose any possible torsion, we can identify the codomain of the last map with the co-root hyperplane\nwhere , is the canonical pairing between co-characters and characters, and α ∨ stands for the co-root ofĜ corresponding to α. We will therefore say that the map p α is the projection along α onto the co-root hyperplane [α ∨ = 0]. Also, the fan of D α is contained in the root hyperplane\nNow let L be aT -line bundle on V G that is generated by its sections. Then we have a short-exact sequence of sheaves on V G :\n, for all i > 0, since L is generated by its sections and V G is projective, and also\nfor all i > 0, since L| Dα is generated by its sections and D α is a projective toric variety. Therefore the short-exact sequence gives rise to the long-exact sequence\nSo, we see that surjectivity of the map ϕ is equivalent to\nWe claim that the following result is equivalent to Theorem A.\nTheorem C. With notation as above, we have that\nwhenever L arises from a Weyl orbit.\nRemark 1.2. Theorem C is true for split classical groups. In fact, in the same way that we will show the equivalence of theorems A and C one can show that theorem C for a split classical group is equivalent to Proposition 1.1 for the case of parabolic subgroups P of semisimple rank 1.\nLet us demonstrate the equivalence between theorems A and C. Put G = G 2 . Suppose that L corresponds to the Weyl-orbit orthogonal set (x B ) in X, i.e. in X * (T ), and, as before, denote by (x P ) the corresponding (G, M )-orthogonal set in X M , i.e. in X * (Z(M )). Write P L for the intersection of X * (T ) with the convex hull Conv{x B } of (x B ). Then we have (see e.g. [4], pg.66)\nwhere the section χ u is an eigenvector corresponding to the character u. (Recall thatT acts on H 0 (V G , L) and so H 0 (V G , L) decomposes according to the characters ofT .)\nAssume that α is the root ofĜ, up to a sign, that corresponds toM , in the sense that the projection map p α : X * (T ) → [α ∨ = 0] along the root α corresponds to the projection map pr M . Then we write P L,α for the intersection of Conv{x P } with p α (X * (T )) and we get\n) from the long-exact sequence above is given by ϕ(χ u ) = χ pα(u) . This explains why Theorem C is equivalent to Theorem A, since p α corresponds to pr M . Now let G = GL n . Theorem B will follow from the following result.\nTheorem D. With notation as above, we have that\nwhenever L is generated by its sections.\nTheorem D itself will be deduced (in Section 4.5) from the result for G = SL n : Theorem E. With notation as above, we have that\nwhenever L is generated by its sections.\nWe note that while theorems A and C were seen to be equivalent, to show that theorems B and D imply each other will take a little more work (because we are no longer working with a two-dimensional space!); however, the same argument above for G = G 2 applies in our case to show that Theorem B for the case of parabolic subgroups P of semisimple rank 1 is equivalent to Theorem D. But, we will deduce our Theorem B from Theorem D, where the latter will be proved using toric geometry.\nWe explain in Section 4.6 how the general case is handled, but here let us just show, using a similar method used there, how Theorem D, i.e. surjectivity of ϕ, implies Theorem B for all M = GL n1 × GL n2 × · · · × GL nr , where r i=1 n i = n and only one of n i 's is equal to 2 and the rest are equal to 1.\nAssume that n k = 2 for some 1 ≤ k ≤ n r and that n i = 1 for all i such that i = k and 1 ≤ i ≤ n r . Keeping the same notation as above, we have in this case a = R n and\nwhere, as mentioned earlier, we have identified a M as a subspace of a. Also, the map pr M , which in general is given by averaging over each of the r \"batches\", now agrees with p α where α = (0, . . . , 0, 1, −1, 0, . . . , 0) (with 1 in the k-th place and -1 in the (k + 1)-st place):\n(In general pr M does not correspond to one p α , rather, it is a composition of a number of p α 's, for distinct roots α.) As in the G 2 case, we find that\nand ϕ(χ u ) = χ pα(u) . Then surjectivity of ϕ clearly implies Theorem B for this particular case. As we mentioned in the Introduction, the next sections are devoted to proving our results. It is now safe for the reader to assume that, for the rest of the paper, we are working over the complex numbers-we will be in the toric geometry setting. Also, since we will be working in the complex dual world, when we use terms root, co-root, character, and co-character, they will always refer to those terms for the complex dual groupsĜ andT .\nBefore we end this section it is worth mentioning that vanishing results like the ones in theorems C and D (and E) are also of independent interest just from a toric-variety point of view. A very important vanishing result for toric varieties has been proved by Mustaţȃ (see e.g. [11], Theorem 0.1) and for the particular toric varieties arising from GL n and G 2 theorems C and D (and E) give vanishings of higher cohomology groups for more line bundles on these varieties. (See also [5] for more vanishing results for toric varieties associated with root systems.)"},{"title":"Some useful results","content":"We keep the same notation as in the previous section. Let V = V G be the non-singular, projective toric variety corresponding to G as described in Section 1 and let ∆ be the fan of V . Each element of ∆ corresponds to a unique torus orbit in V , and under this identification, if we denote by ∆(1) the set of rays in ∆ and if ρ ∈ ∆(1), we write D ρ for the closure of the orbit of ρ, aT -Cartier divisor (cf. [4], §3.1 and §3.3). In fact,T -Cartier divisors on V can be written as a linear combination of D ρ 's. From now on let us agree to refer toT -Cartier divisors simply as divisors. (No confusion should arise from this agreement even though our divisor D α is not aT -divisor; in fact, thanks to Lemma 2.3 below, when computing cohomology groups, we will deal with aT -Cartier divisor instead of D α .)\nLet us mention another way of characterizing divisors on toric varieties (see e.g. [4], §3.4, but be aware of a sign difference). A divisor D on a toric variety V , with fan ∆, is completely determined by a set of characters u(σ), one for every element σ ∈ Max(∆), such that they \"agree\" on the overlaps, i.e., such that they form a (G, T )-orthogonal set (we will usually call these just orthogonal sets; here and throughout this paper, Max(∆) stands for the set of maximal cones in ∆). This orthogonal set in turn gives a continuous (integral) piece-wise linear function ψ D on the support |∆| := ∪ σ∈|∆| σ of the fan, where ψ D (v) = u(σ), v , for all v ∈ σ. And conversely, given a continuous (integral) piece-wise linear function ψ on |∆|, i.e., a continuous function that is linear and given by an element of the character lattice on each cone of ∆, we get a divisor by considering the characters u(σ), σ ∈ Max(∆), which define ψ on the maximal affine pieces of V . (The continuity of ψ guarantees that the characters defining it agree on the overlaps and hence give a divisor.)\nFrom what we said, it is easy to see that if D = ρ∈∆ a ρ D ρ , then the corresponding piece-wise linear function is given by ψ D (v) = u(σ), v (v ∈ σ) where, for each σ, the character u(σ) is found by solving the system of equations\nwhere ρ varies through the rays in the cone σ and v ρ stands for the primitive lattice element in the ray ρ.\nLet us mention that ψ D is called convex or lower convex if for any v, v ′ ∈ |∆| and 0 ≤ p ≤ 1, we have that ψ D (pv\nThe following result is going to be very useful. where H i (V, O(D)) u denotes the u-eigenspace that is obtained by the action ofT on H i (V, O(D)) and Z(u) = {v ∈ |∆| : ψ(v) ≥ u, v }.\nKeep the same notation as above and let α be a root ofĜ. The following result was suggested to us by R. Kottwitz. Proof. This follows from considering the rational function χ α −1 on V G and computing the corresponding principal divisor. Note that the divisor of zeros corresponds to D α .\nSince we will be interested in computing the cohomology group H 1 (V G , O(D − D α )), the last lemma allows us to focus instead on\nThe lemma below describes the orthogonal set corresponding to theT -equivariant divisor ρ∈∆(1), α,vρ >0 α, v ρ D ρ , which, due to the previous lemma, will be referred to as the orthogonal set corresponding to D α . Lemma 2.3. Let α be a root ofĜ and consider V G . The corresponding orthogonal set for D α is given by (u(σ)) σ∈Max(∆) , where u(σ) = α if α ≥ 0 on σ, and u(σ) = 0 if α ≤ 0 on σ.\nProof. The result follows at once from Lemma 2.2 and the fact that, as we discussed earlier, if D = ρ∈∆ a ρ D ρ , then u(σ)'s can be found by solving the system of equations u(σ), v ρ = a ρ , where ρ ∈ ∆(1) varies through the rays of σ. Proof. First note that using Proposition 2.1, part b), it is sufficient (and necessary) to show that"},{"title":"An Important Lemma","content":"Looking at the long-exact sequence we get from the pair (|∆| , U D,α (0)), since |∆| is a vector space (V G is complete), we only need to show that U D,α (0) is path-wise connected. Let us do that.\nAccording to Lemma 2.3, U D,α (0) is equal to the union of sets\nHere we have written [α ≥ 0] and [α ≤ 0] for the sets {v ∈ |∆| : α, v ≥ 0} and {v ∈ |∆| : α, v ≤ 0}, respectively. Proposition 2.1, part a), implies that sets (2) and (3) are convex. Let us see how this works in the case of (3). Let v and v ′ be in (3) and let 0 ≤ p ≤ 1. Then using Lemma 2.3. we have\nwhich, since ψ D is convex, is not greater than\nand the last expression is less than zero since ψ D−Dα (v) < 0, ψ D−Dα (v ′ ) < 0 and 0 ≤ p ≤ 1. Clearly, from our assumptions, we also have that pv +(1−p)v ′ ∈ [α ≥ 0], so (3) is a convex set, as desired.\nIf condition (i) of the lemma is satisfied, then U D,α (0) equals the set (3) and hence (1) holds, because the set (3) is convex.\nSuppose that (ii) is satisfied. Since {v ∈ |∆| : ψ D (v) < 0} is convex, this implies that there exists a (non-empty, convex) subset U in [α ≤ 0] ∩ [α ≥ 0] that is contained in both sets appearing in (ii). We claim that U is then contained in both sets (2) and (3). This claim is sufficient to prove that U D,α (0) is path-wise connected (and hence (1) holds) since we already saw that (2) and (3) are convex.\nFirst, U is contained in (2) by our assumption that it is contained in both sets appearing in (ii), and in particular the second one. Second, since U is contained in the first set appearing in (ii), we see that ψ D (v) < 0 and α, v ≥ 0, for all v ∈ U . Therefore, using Lemma 2.3, we see that\nThus U is contained in (3) as well, which ends the proof of the claim and hence of the lemma.\nNote that if we knew that (1) is true for all globally generated divisors D, then we could conclude that for those divisors we have H 1 (V G , O(D − D α )) u = 0, for all u. Indeed, to prove this, consider the (globally generated) divisor D ′ associated to the orthogonal set {u(σ) − u}, where {u(σ)} corresponds to D. The result follows by using (1) (which is assumed to be true for all globally generated divisors), where instead of U D,α (0) we have U D ′ ,α (0).\nUsing Proposition 2.1, part b), we deduce that if (1) holds for all globally generated divisors D, then\nAn important consequence of this is that Theorem D needs to be proved only in the case when the set (2) is non-empty and\nMore concretely, it suffices to show that, under these conditions,\nbecause then we get that U D,α (0) is equal to the set (2), which we know is a convex set, and we can apply the same reasoning as in Lemma 3.1 to conclude that (1) is true, and by what we just wrote, H 1 (V G , J Dα ⊗ O(D) = 0 is true as well."},{"title":"Proofs of Theorems B, D, and E","content":"Consider V G , where G = SL n (and soĜ = P GL(n, C)). We want to concretely describe the fan ∆ of V G . But first, the initial lattice (i.e. lattice of characters) for V G is L = Z n /Z (1, 1, . . . , 1) and so the dual one is L ∨ = {(x 1 , . . . , x n ) ∈ Z n |x 1 + . . . + x n = 0}. For every i = 1, . . . , n we put L i for (the element represented by) (0, . . . , 0, 1, 0, . . . , 0), where 1 appears only on the i-th place. Identify the rays in ∆ with their corresponding minimal lattice points. Then ∆(1) consists of the sums of the form k j=1 L ij , where i j 's are distinct elements from {1, 2, . . . , n} and k = 1, ..., n − 1. The maximal cones in V G are the n-dimensional cones whose rays are of the type\nwhere i 1 , i 2 , . . . , i n−1 are distinct elements from {1, 2, . . . , n}. Let α be a root of G. We can assume that α is simple (for a choice of a Borel subgroup) and, with a change of coordinates, we can always ensure that α is written as L 1 −L 2 . Therefore, without loss of generality, throughout this section we let α = L 1 − L 2 .\nThe proof of Theorem E will be proved by induction, although we begin by giving the proof for the first two nontrivial cases, n = 3 and n = 4. This may seem a little unusual, but we do so because in order to apply the induction process we will need \"many\" root hyperplanes and for n = 4 we don't have \"enough\" of them (see Case 2) of the proof in Section 4.3 where we use n > 4)."},{"title":"4.1.","content":"Proof of Theorem E for n = 3. In this case the set of maximal cones Max(∆) in the fan ∆ of V G consists of six cones σ 1 , . . . , σ 6 , generated respectively by L 1 and −L 3 ; −L 3 and L 2 ; L 2 and −L 1 ; −L 1 and L 3 ; L 3 and −L 2 ; and −L 2 and L 1 .\nLet D be a divisor on V G such that O(D) is generated by its sections. If u(σ i ), i = 1, . . . , 6 form the orthogonal set corresponding to D, where u(σ i ) is assigned to the maximal cone σ i , then, according to Lemma 2.3, the orthogonal set corresponding to the divisor D − D α consists of u(σ i ) − α for i = 1, 5, 6 and u(σ i ) for i = 2, 3, 4.\nFrom our discussion in Section 3, we only need to prove that (5) is true, under the assumption that the set (2) is non-empty and (4) is true. Suppose, for a contradiction, that under these assumptions there exists v 0 ∈ |∆| such that α, v 0 ≥ 0 and ψ D (v 0 ) − α, v 0 < 0. Since ψ D is continuous and piecewise linear, there are only two cases we need to consider: (i) v 0 = L 1 , and (ii) v 0 = L 1 + L 3 = −L 2 . We can use the symmetry (x, y, z) → (−y, −x, −z) of the root system, which preserves the half-space [α ≥ 0], to see that we only need to consider the case (i).\nLet\nis generated by its sections, u(σ i )'s form a positive orthogonal set and therefore there exists a non-negative number n ∈ Z such that u(σ 2 ) = u(σ 1 ) + n(L 2 − L 1 ). We get ψ D (L 2 ) = u(σ 2 ), L 2 = b + n ≥ 0. For the same reasons, there exists a non-negative integer m such that u(σ 3 ) = u(σ 2 ) + m(L 3 − L 1 ), and therefore ψ D (−L 1 ) = u(σ 3 ), −L 1 = n + m ≥ 0. Thus, since ψ D is continuous and piece-wise linear, ψ D takes non-negative values on all of [α ≤ 0], and this contradicts the non-emptiness of (2). Theorem E follows for n = 3.\nRemark 4.1. If we are working with V SL4 , then we can consider sub-fans of the initial fan ∆ that are contained in the root hyperplanes. For example, if our given root is β = L 3 − L 4 , then the sub-fan corresponding to the root hyperplane [β = 0] has rays L 1 , −L 3 − L 4 , L 2 , −L 1 , L 3 + L 4 , −L 2 and the maximal-dimensional cones are the 2-dimensional cones in ∆ obtained from these rays. This is just a \"copy\" of the fan of V SL3 and indeed we can carry out the same calculations as above to see that Theorem E holds for this \"copy\" of V SL3 inside V SL4 . This remark is important for our induction process and is part of a more general story which we tell in Section 4.4."},{"title":"4.2.","content":"Proof of Theorem E for n = 4. Note that we are still assuming that α = L 1 − L 2 . To simplify notation, write σ i,j,k for the maximal cone whose rays are L i , L i + L j , L i + L j + L k , where i, j, k are distinct elements of {1, 2, 3, 4}.\nLet D be a divisor on V G with O(D) generated by its sections. Denote by u(σ i,j,k ) the character corresponding to D for the cone σ i,j,k . Then these characters form an orthogonal set and, according to Lemma 2.3, the orthogonal set corresponding to D − D α consists of u(σ i,j,k ) − α for the maximal cones σ i,j,k contained in the half-space [α ≥ 0] and u(σ i,j,k ) for the ones contained in [α ≤ 0]. Following the discussion in Section 3, we assume that the set (2) is non-empty and that (4) is true. We want to prove (5) and, for a contradiction, suppose that there exists a point v 0 in the half-space [α ≥ 0] such that ψ D (v 0 ) − α, v 0 < 0. Since ψ D is continuous and piece-wise linear on |∆|, there are only four cases we need to consider, corresponding to the rays whose primitive lattice points are contained in\nBut, we can use the symmetry (x, y, z, w) → (−y, −x, −z, −w) of the root system, which preserves the half-space [α ≥ 0], to see that we only need to consider cases (i) and (ii).\nCase (i): Our aim is to show that ψ D takes non-negative values at L 2 , L 2 + L 3 , L 2 + L 4 and L 2 + L 3 + L 4 , which would contradict our assumption that the set (2) is non-empty. First, if we look at the hyperplane corresponding to the root L 2 − L 3 , as mentioned in Remark 4.1, we get a copy of V SL3 , where the rays of the sub-fan are L 1 , L 1 + L 4 , L 4 , L 2 + L 3 + L 4 , L 2 + L 3 , L 1 + L 2 + L 3 . These are precisely the rays of ∆ contained in our hyperplane. Apply Theorem E, case n=3, to see that ψ D (L 2 + L 3 ) ≥ 0 and ψ D (\nSimilarly, by looking at the hyperplanes corresponding to L 3 − L 4 and L 2 − L 4 and applying Theorem E, for n=3, we get that ψ D (L 2 ) ≥ 0 and ψ D (L 2 + L 4 ) ≥ 0 respectively, which is what we intended to prove.\nCase (ii): Completely similarly as in Case (i) we see that ψ D (L 2 ) ≥ 0 and ψ D (L 2 + L 4 ) ≥ 0 (use the root L 1 − L 3 ), and ψ D (L 2 + L 3 + L 4 ) ≥ 0 (use the root L 2 − L 4 ). The only non-trivial case is when we want to show that ψ D (L 2 + L 3 ) ≥ 0, because there is no root-hyperplane which contains both L 1 + L 3 and L 2 + L 3 . We proceed as follows. From our assumptions we already know that ψ D (L 1 + L 3 ) ≥ 0, ψ D (L 1 +L 3 )−1 < 0, ψ D (L 3 ) ≥ 0 and ψ D (L 1 +L 2 +L 3 ) ≥ 0. Therefore if u(σ 3,1,2 ) = (a, b, c, −a−b−c) where a, b, c ∈ Z, then a+c ≥ 0, a+c−1 < 0, c ≥ 0 and a+b+c ≥ 0, i.e., a = −c, c ≥ 0 and b ≥ 0. But since u(σ i,j,k )'s form a positive orthogonal set, we can find a non-negative integer m so that u(σ 3,2,1 ) = m(L 2 − L 1 ) + u(σ 3,1,2 ). Thus u(σ 3,2,1 ), L 2 + L 3 = b + c + m ≥ 0, i.e., ψ D (L 2 + L 3 ) ≥ 0, and this concludes the proof."},{"title":"4.3.","content":"Proof of Theorem E for n > 4. We use induction to prove the theorem in the general case. Assume that the theorem is true for all V SLn (as well as for the root-hyperplane copies of V SLn inside V SLn+1 ; again see Section 4.4), and we want to prove it for V SLn+1 .\nLet D be a divisor on V SLn+1 such that O(D) is generated by its sections. Once more we consider D − D α , where we have taken without loss of generality α = L 1 − L 2 . Following the discussion in Section 3, we assume that the set (2) is nonempty and that (4) is true. We want to prove that (5) holds. For a contradiction, suppose that there exists a point v 0 in the half-space [α ≥ 0] such that ψ D (v 0 ) − α, v 0 < 0. Since the function ψ D is continuous and piece-wise linear, we would get a contradiction if we showed that ψ D takes non-negative values on all the primitive lattice points along the rays of the fan of V SLn+1 which are contained in the halfspace [α ≤ 0]. Moreover, due to the same properties of ψ D , we may assume that (6) v\nwhere i j 's are distinct elements of {3, 4, . . . , n + 1}. We want to prove that for all the primitive lattice points of the form\nwhere p q 's are distinct elements of {3, 4, . . . , n + 1}, we have\nFor this, it suffices to show that there is a root β ofĜ such that both v 0 = L 1 + L ij and L 2 + L pq are contained in the root hyperplane corresponding to β, because we can then apply the induction hypothesis to conclude that (8) is true.\nIf there exists an i j from (6) which is not equal to any of p q 's appearing in (7), then put β = L 1 − L ij . If there exists a p q from (7) which is not equal to any of i j 's from (6), then we put β = L 2 − L pq . In both cases, (8) follows at once.\nTherefore we are only left with the possibility that, up to permutation, i j 's are equal to p q 's. If this is the case, then we distinguish two cases:\nCase 1). The number of L ij 's is greater than one, and so if, say, L ij 1 and L ij 2 appear in (6) we put β = L ij 1 − L ij 2 to get (8).\nCase 2). The number of L ij 's is less than two, and so, because n > 4, we can find L km and L kn which don't appear in (6) and therefore in (7) either. Put β = L k1 − L k2 to see that v 0 and L 2 + L pq belong to the root hyperplane corresponding to β. So we get (8).\nThis proves Theorem E for all n. The dual lattice for D α is p α (X * (T )). More concretely, the dual lattice for D α is\nbecause the map p α is given by p α (x 1 , x 2 , . . . , x n+1 ) = ( x1+x2 2 , x1+x2 2 , x 3 , . . . , x n+1 ). The claim is that we can treat D α as a copy of V SLn inside V SLn+1 .\nFor ease of notation, let L ∨ α := p α (X * (T )) and L ∨ n be the dual lattice of V SLn . Note that we have a bijection Ψ : L ∨ α → L ∨ n , given by Ψ(x 2 /2, x 2 /2, x 3 , . . . , x n+1 ) = (x 2 , x 3 , . . . , x n+1 ). Now let β = α be a root ofĜ. We only need to consider two cases: I) β = L 2 − L 3 ; II) β = L 3 − L 4 , to prove that we can treat D α as a copy of V SLn . I) Let β = L 2 − L 3 . Define p β , the projection using β, in the same way as p α . We project L ∨ α via p β to get the lattice\nIf we project L ∨ n via the corresponding p β , we get the lattice\nand we also have a natural bijection Ψ β : L ∨ α,β → L ∨ n,β , given by\nFurther, if we have an orthogonal set (ξ i ) in V SLn , then we get an orthogonal set (Ψ −1 (ξ i )) in D α by applying Ψ −1 to the former. And conversely, we apply Ψ to an orthogonal set in D α in order to get an orthogonal set in V SLn . Also, points in p β (Conv(ξ i ) ∩ L ∨ α ) are identified with points in p β (Conv(Ψ −1 (ξ i ) ∩ L ∨ α ) via the map Ψ β and, under the same map, points in p β (Conv(ξ i )) ∩ p β (L ∨ α ) are identified with those in p β (Conv(Ψ −1 (ξ i ))) ∩ p β (L ∨ n ). The proof of the following Lemma then solely involves unravelling the definitions of the maps involved and the definition of a convex hull. Incidentally, it proves, as we wanted, that we can treat D α as a copy of V SLn inside V SLn+1 , for which Theorem E holds."},{"title":"Lemma 4.2.","content":"For the corresponding orthogonal sets (ξ i ) and (Ψ −1 (ξ i )) we have that\nII) Let β = L 3 − L 4 . This is handled analogously to I) and the lemma above holds in this case, too. Let us mention that in this case we have x i = 0; x i ∈ Z, ∀i}."},{"title":"4.5.","content":"Proof of Theorem D. Let G = GL n and, as earlier, take without loss of generality α = L 1 − L 2 . We have that X * (T ) = Z n . Suppose D is a divisor on V G such that O(D) is globally generated and let {u(σ i )} i∈I be the corresponding orthogonal set, where for each i ∈ I, u(σ i ) = (a\nn ) ∈ Z n corresponds to the maximal cone σ i of the fan ∆ of V G . Then, the fact that {u(σ i )} i∈I is an orthogonal set means in this case that there exists s ∈ Z such that a\nIf we denote by P D the intersection of X * (T ) with the convex hull of the orthogonal set {u(σ i )} i∈I , then we have that (see e.g. [4], pg.66)\nSimilarly, we get that\nwhere P Dα = p α (Z n ) ∩ P α , with P α standing for the image of the convex hull of the orthogonal set {u(σ i )} i∈I under the map p α . (Recall that the map p α was defined in Section 1). Using the long-exact sequence (*) from Section 1, Theorem D can be re-written in the following way:\nis surjective, for all globally generated line bundles O(D) on V G , where ϕ(χ u ) = χ pα(u) . Let us prove Theorem D using Theorem E. Suppose x ∈ P Dα . It suffices to find a point z in P D such that x = p α (z).\nFirst \"shift\" the orthogonal set {u(σ i )} i∈I by u(σ 1 ), i.e., consider the orthogonal set {u(σ i ) − u(σ 1 )} i∈I , which gives a new divisor D ′ on V G . Then, if we write\nNote that p α (x − u(σ 1 )) lies in the intersection of the image of the set\nunder the map p α and the set p α (Z n ).\nUsing Theorem E, we can then conclude that there exists z ′ ∈ P D ′ such that p α (z ′ ) = p α (x − u(σ 1 )), where P D ′ := {u ∈ Z n : u, v ≤ u(σ) − u(σ 1 ), v , ∀v ∈ σ, ∀σ ∈ Max(∆)}.\nOne can easily check that p α (z ′ + u(σ 1 )) = x and z ′ + u(σ 1 ) ∈ P D . This means that we found z = z ′ + u(σ 1 ) in P D such that x = p α (z), which proves Theorem D. 4.6. Proof of Theorem B. Now it is time to explain how Theorem B follows from Theorem D. We keep the same notation as in Section 1, with the difference that we consider G = GL n+1 instead of G = GL n . Then M must be of the form M = GL n1 × GL n2 × · · · × GL nr where n+1 i=1 n i = n + 1. We can also see that a = R n+1 ,\nand pr M is given by averaging over each of the r \"batches\". (We remind the reader that here, as before, we are considering a M as a subspace of a.)\nThe simplest case is when only one of n i 's is equal to 2 and the rest are equal to 1, but we saw in Section 1 how Theorem B is deduced from Theorem D in that case. Now assume that only two of the n i 's are equal to 2 and the rest equal 1. We can assume without loss of generality that n 1 = n 2 = 2 and n 3 = . . . = n r = 1, with r = n − 3. Then\nand so pr M is just the composition p L3−L4 • p L1−L2 . Therefore when we apply Theorem D to D L1−L2 , where we consider D L1−L2 as a copy of V GLn and α = L 3 − L 4 , we find that Theorem B follows in this case, too. The next case would be to consider, without loss of generality, n 1 = 3 and n 2 = . . . = n r = 1, where r = n − 2. Now we see that pr M is the composition p L2−L3 • p L1−L2 and we get our desired conclusion again using Theorem D.\nIn general we write pr M as a composition of a finite number of p α 's for distinct roots α and then apply Theorem D as many times as there are roots α appearing in that composition. If we want to be very specific, then, for M = GL n1 × GL n2 × · · · × GL nr we can see that pr M agrees with the composition of maps q 1 , q 2 , . . . , q r where q k = p L (n k −1) −Ln k • p L (n k −2) −L (n k −1) • . . . • p L (n k−1 +1) −L (n k−1 +2) , with the convention n 0 = 0. This concludes the proof of Theorem B."},{"title":"Proof of Theorem C","content":"Consider V G for G = G 2 . Just as in the case of SL n , we would like to describe the fan ∆ of V . But first, the initial lattice is L = Z 3 /Z(1, 1, 1) and so the dual one is L ∨ = {(x 1 , x 2 , x 3 ) ∈ Z 3 : x 1 + x 2 + x 3 = 0}. Identify the rays in ∆ with their corresponding minimal lattice points. Then ∆(1) consists of twelve points: v 1 = (1, 0, 0), v 2 = (1, 0, −1), v 3 = (0, 0, −1), v 4 = (0, 1, −1), v 5 = (0, 1, 0), v 6 = (−1, 1, 0) and v i+6 = −v i , for i = 1, . . . , 6. There are twelve maximal cones: σ i , i = 1, 2, . . . , 12, where the rays of σ i are v i and v i+1 (with v 13 := v 1 ). Let D be aT -Cartier divisor on V corresponding to a Weyl orbit (see Section 1 for the precise definition) and α a root ofĜ. For every maximal cone σ i denote by u(σ i ) the character corresponding to the divisor D. Obviously, u(σ i )'s form a positive orthogonal set. We want to prove that\nThis is true if the orthogonal set corresponding to D is strictly positive, i.e., if u(σ i )'s are distinct and they form a positive orthogonal set. Indeed, use Lemma 2.3 and the definition of orthogonal sets to conclude that the orthogonal set corresponding to D − D α is still positive (but not necessarily strictly positive), i.e., J Dα ⊗ O(D) is globally generated. It is also obvious that if all u(σ i )'s are equal to each other, then we get a degenerate case for which the result is true as well.\nTherefore the only problem might arise if only some but not all of the u(σ i )'s are the same. Using symmetries of the current root system, there are only four cases we need to consider, two for a short root and two for a long root: Let us mention again that we want to prove that H 1 (V G , J Dα ⊗ O(D)) = 0. As we explained in Section 1, we have a long-exact sequence ... −→ H 0 (V G , O(D)) ϕ the Weyl-orbit ones. More concretely, in Section 3, after Lemma 3.1, we saw that the vanishing of the first cohomology group there was implied by the vanishing of its corresponding 0-eigenspace (under the action of our torus). This is not true for G = G 2 . The crucial step was to note that by \"shifting\" a positive orthogonal set, say {u(σ)}, by a character, say, u, we get a positive orthogonal set {u(σ) − u}; however, it is clear that a shift of a Weyl-orbit is in general not a Weyl-orbit. 5.2. At the very end, we give an example to show that Theorem B fails in the case of G 2 . Consider V G2 and let α = (1, −1, 0). Let D be the positive orthogonal set (i.e., the globally generated torus-equivariant divisor on V G2 ) given by u(σ i ) = (1, 0, −1) for i ∈ {1, 2, 3, 4, 5, 12} and u(σ i ) = (0, −1, 1) for i ∈ {6, 7, 8, 9, 10, 11}. Following the same notation as before, we see that in our case"},{"title":"Lemma 2 . 2 .","content":""},{"title":"Lemma 3 . 1 .","content":""},{"title":"4. 4 .","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Fix an isocrystal and the corresponding Newton vector. Mazur's inequality (see [10], pg.658, and the group-theoretic generalization in [13], Theorem 4.2) states that, given a lattice in our isocrystal, the Hodge vector associated to it is greater than or equal to the Newton vector. Conversely, any vector satisfying Mazur's inequality is the Hodge vector for some lattice (see e.g. [12], Proposition 4.2, and [3], Theorem 2). The last result can be regarded as a statement for GL n , but, in [8], §4, R. Kottwitz and M. Rapoport formulated an analogous version for other groups and they proved it for GSp 2n and GL n . They also formulated a combinatorial statement involving respective root systems that would imply the converse to Mazur's inequality in each case (see also sections 4.3 and 4.4 in [6], for a detailed version). C. Lucarelli in [9], Theorem 0.2, showed that this combinatorial statement is true (see Proposition 1.1 below), and therefore a converse to Mazur's inequality was deduced, for all split classical groups. We prove this combinatorial statement for G 2 (Theorem A) and hence deduce a converse to Mazur's inequality for G 2 . We also generalize the combinatorial result for GL n (Theorem B), which in particular gives a new and simple proof of the converse to Mazur's inequality for GL n .\\nThe results obtained here are proved using the theory of toric varieties. In fact, part of our aim is to show how the combinatorial result mentioned above can be naturally treated as a vanishing result for higher cohomology groups of certain line bundles on toric varieties associated to root systems (theorems C and D on Section 1), which improves the previously known results for these varieties. This raises the question of new vanishing results for toric varieties in general (see [5] for work in this direction concerning toric varieties associated with root systems). It is worth mentioning that these toric varieties, where the fan of the variety is the Weyl fan, appear naturally in the work of group compactifications (see e.g. [2], pp.187-206, and references therein).\\nTo give a flavor of the type of problem that we deal with here, it is very instructive to use a picture (see below) to illustrate the statement of Theorem E for n=3. Consider a toric variety V = V SL3 with initial lattice (i.e., lattice of co-characters) L = Z 3 /Z(1, 1, 1) and the fan consisting of six maximal cones as pictured below in dotted lines. The dual lattice (of characters) is\\nLet L be a globally-generated torus-equivariant line bundle on V . Then L is completely determined by a set of characters (one for each maximal cone in the fan), such that if σ and σ ′ are two \\\"neighboring\\\" maximal cones, and u(σ) and u(σ ′ ) are their corresponding characters, then u(σ) − u(σ ′ ), with respect to the canonical pairing of L ∨ and L, is perpendicular to the common face of σ and σ ′ , non-negative on σ, and non-positive on σ ′ . This way L determines a hexagon P where the vertices of P are the characters corresponding to L, one for each maximal cone of the fan. (In our picture all these characters are distinct and we have a true hexagon, but in general we may get some degenerate version of P , where some of the vertices coincide.) Then dim H 0 (V, L) is given by the number of L ∨ -points in the convex hull Conv(P ) of P . Note that Conv(P ) lies in L ∨ ⊗ Z R. This should explain why we are using dotted lines for the cones of our initial fan -they lie in the initial world, not the dual one.\\nFix a root α as in the picture. Then the \\\"line-segment\\\" D α corresponds to a (non-torus-invariant) divisor on V . If we denote by i the embedding D α ֒→ V , then dim H 0 (V, i * (L| Dα )) equals the number of points that are both on D α and that are projections along the root α of points in L ∨ .\\nOur question is as follows: For any point y on the line segment D α that is obtained by the projection along α of some point z ∈ L ∨ (z is not necessarily in Conv(P )), can we always find a point z ′ that is in both L ∨ and Conv(P ), and which maps to the initial point y? The answer is \\\"yes\\\" and, because of the long-exact sequence\\nwhere J Dα stands for the ideal sheaf of D α , it is equivalent to the statement that H 1 (V, J Dα ⊗ L) = 0. We should mention here that the above long-exact sequence has trivial entries to the right of H 1 (V, J Dα ⊗L), because higher cohomology groups vanish for globally-generated line bundles on complete toric varieties, and L and L| Dα are globally-generated on (the complete toric varieties) V and D α , respectively. Also, note that the map ϕ is induced by projecting along α.\\nIn this paper we precisely formulate the above question, using co-characters and co-roots, for a split connected reductive group G over a finite extension of Q p , or equivalently, by passing to the (Langlands) complex dual world, using characters and roots, for the corresponding groupĜ over C. Then we reformulate the question using certain toric varieties associated to the root system ofĜ (where the fan of our variety is the Weyl fan and the dual lattice (i.e., lattice of characters) is that of characters forĜ). We answer the question in the affirmative for G = GL n and for a weaker version (where only line bundles L arising from Weyl orbits are considered) for G = G 2 .\\nAs mentioned at the beginning, this type of combinatorial question, without reference to toric varieties, was investigated by Kottwitz and Rapoport (and others). If we use the language of toric varieties to describe what they did, then they considered only globally generated line bundles L arising from Weyl-orbits, i.e., where the corresponding polygon P is a Weyl-orbit for some element in the character lattice.\\nLet us now describe how our paper is organized. In the first section we introduce notation, set up the problem and state our main results (theorems A through E). We also show that Theorem C is equivalent to Theorem A. Sections 2 and 3 are devoted to some useful results in computing cohomology groups for line bundles on our toric varieties using the so-called piece-wise linear continuous functions on the support of the respective fans. We use these results in the next section, where we prove theorems B, D, and E, and where we show how E implies D and D implies B. Finally, the last section is devoted to the proof of Theorem C.\\nAcknowledgments: The author is greatly indebted to his thesis adviser, Professor Robert Kottwitz, for suggesting that he work on the problems dealt with in this paper, for numerous helpful comments, and for continuous encouragement and support.\"\n },\n {\n \"title\": \"Set Up and Main Results\",\n \"content\": \"Prior to [9], a converse to Mazur's inequality for GL n and GSp 2n was proved in [8], Theorem 4.11 (see also [3], Theorem 2, for the GL n case), whose notation we follow. Once a notation is introduced, it will be fixed for the rest of the paper.\\nLet F be a finite extension of Q p . Denote by O F the ring of integers of F . Suppose G is a split connected reductive group, B a Borel subgroup and T a maximal torus in B, all defined over O F . Let P = M N be a parabolic subgroup of G which contains B, where M is the unique Levi subgroup of P containing T .\\nWe write X for the set of co-characters X * (T ). Then X G and X M will stand for the quotient of X by the co-root lattice for G and M , respectively. Also, we let ϕ G : X → X G and ϕ M : X → X M denote the respective natural projection maps.\\nLet µ ∈ X be G-dominant and let W be the Weyl group of T in G. The group W acts on X and so we consider W µ := {w(µ) : w ∈ W } and the convex hull of W µ in a := X ⊗ Z R, which we denote by Conv (W µ). Define\\nLet a M := X M ⊗ Z R and write pr M : a → a M for the natural projection induced by ϕ M . Note that although X M is a quotient of X, after tensoring with R any possible torsion is lost and we can therefore consider a M as a subspace of a; we shall do so throughout the paper.\\nThe aim is to generalize the proposition below for G = GL n and prove an analogous result for G = G 2 . Kottwitz in [6] (sections 4.3 and 4.4) explains how, in each case, a converse to Mazur's inequality follows from the result below and therefore from our results. [9]) Let G be a split connected reductive group over F with every irreducible component of its Dynkin diagram of type A n , B n , C n , or D n . With the notation as above, we have\\nWe are first going to phrase our results using Arthur's notion of (G, T )-orthogonal sets (see e.g. [1], pg. 217). For more on these sets see for example [7], pp.441-447, whose notation we follow. A family of points x B in X, one for every Borel subgroup B of G that contains T , is called a (G, T )-orthogonal set in X if for every two adjacent Borel subgroups B, B ′ , there exists an integer n such that\\nwhere α ∨ is the unique co-root for T that is positive for B and negative for B ′ . Similarly, if we have a point x P ∈ X M for each parabolic subgroup P of G that admits M as a Levi component, then the family (x P ) is called a (G, M )-orthogonal set in X M (see e.g. [7], pg.442) if for every two adjacent parabolic subgroups P, P ′ that admit M as a Levi component, there exists an integer m so that\\nHere β ∨ P,P ′ is the unique element in Γ with the property that all the other elements of Γ are positive multiples of it; Γ consists of the images in X M of the co-roots α ∨ where α is a root occurring in Lie(N ) ∩ Lie(N ′ ), where P = M N andP ′ = MN ′ , withP ′ being the parabolic subgroup containing M opposite P .\\nIf all the numbers n (resp. m) above are non-negative then we say that (x B ) is a positive (G, T )-orthogonal set (resp. (x P ) is a positive (G, M )-orthogonal set).\\nAn important example of a positive (G, T )-orthogonal set arises from Weyl orbits W µ (see e.g. [7], pg.443). Let ν ∈ X, then we say that ν is dominant with respect to a Borel group B = T N if for every root α in Lie(N ) we have that α, ν ≥ 0. We get a positive (G, T )-orthogonal set by associating to every Borel group B the unique element x B ∈ X that is both dominant with respect to B and lies in W µ.\\nIf (x B ) is a (G, T )-orthogonal set in X, then we get a (G, M )-orthogonal set as follows. The points (x B ), where B is a Borel subgroup containing T and B ⊂ P , form an (M, T )-orthogonal set in X, and we get a point x P ∈ X M as the common image in X M of all the points in {x B : B ⊂ P }. The set of all such points x P , where P is a parabolic that contains B and admits M as a Levi component, is a (G, M )-orthogonal set in X M . Moreover, if (x B ) is positive, then (x P ) obtained in this way is positive as well.\\nWe can now state our main results.\\nTheorem A. Let G = G 2 . With notation as above we have that\\nfor every (G, T )-orthogonal set (x B ) that arises from a Weyl orbit and its corresponding (G, M )-orthogonal set (x P ).\\nTheorem B. Let G = GL n . With notation as above we have that\\nfor every positive (G, T )-orthogonal set (x B ) and its corresponding (G, M )-orthogonal set (x P ).\\nIt is clear that the GL n case of Proposition 1.1 becomes a special case of Theorem B when (x B ) arises from a Weyl orbit. Also note that in both theorems A and B, the left-hand side is obviously contained in the right-hand side. The non-trivial part is to show the other containment.\\nIt is important to mention that, while we believe that Theorem A is also true for the rest of the exceptional groups, Theorem B is probably only true for a split connected reductive group with irreducible components of Dynkin diagram of type A n . More precisely, it is easy to construct counterexamples to Theorem B for other classical groups and for the G 2 case. At the end of section 5 we do so explicitly, but only for G 2 , since the construction for classical groups is completely analogous. We believe that the same construction should also provide counterexamples to Theorem B for the other exceptional groups.\\nThe proof of these two theorems will involve the theory of toric varieties and we will freely use standard terminology and basic facts from this theory which appear in [4]. Let us first make clear the correspondence between (G, T )-orthogonal sets and line bundles on a certain projective nonsingular toric variety V G , which we now define; we follow the exposition in [7], §23, and keep the same notation as above. LetĜ andT be the (Langlands) complex dual group for G and T , respectively. Let Z(Ĝ) be the center ofĜ. Then the fan of our toric variety V G is the Weyl fan in X * (T /Z(Ĝ)) ⊗ Z R and the torus isT /Z(Ĝ). (We would like to remark that the toric variety V G appears naturally in the theory of group compactifications-see e.g. [2], pp.187-206.)\\nClearlyT /Z(Ĝ) acts on V G , but we are interested in the action ofT on V G . The latter action is obtained using the canonical surjectionT ։T /Z(Ĝ) and the action ofT /Z(Ĝ) on V G .\\nAs with any toric variety, there is a one-to-one correspondence betweenT -orbits in V G and cones in the fan of V G . In our case, because the fan is the Weyl fan, this means that we have a one-to-one correspondence betweenT -orbits in V G and parabolic subgroups of G that contain T . According to this identification, if P is a parabolic subgroup of G that contains T , then the correspondingT -orbit is given byT /Z(M ), where M is the unique Levi component of P containing T andM is the corresponding Levi subgroup ofĜ containingT .\\nAgain, as with any toric variety, there is a one-to-one correspondence between maximal cones in the fan of V G andT -fixed points in V G . In our case, since maximal cones correspond to Borel subgroups of G that contain T , we get a one-to-one correspondence between Borel subgroups of G containing T andT -fixed points in V G .\\nIt is very important for us to note that there is a one-to-one correspondence between isomorphism classes ofT -equivariant line bundles on V G and (G, T )-orthogonal sets in X (see e.g. [7], §23.1). We describe the map that gives this correspondence.\\nLet L be aT -equivariant line bundle on V G . Then at eachT -fixed point v in V G the torusT acts by a character, say, x B on the line in L at v, where B is the Borel subgroup corresponding to the fixed point v. In this way, for every Borel subgroup B (containing T ) we get a character x B ∈ X * (T ), i.e., a co-character x B ∈ X. The fact that (x B ) is a (G, T )-orthogonal set comes from the fact that the characters defining L must agree on the overlaps.\\nA very useful remark is that positive (G, T )-orthogonal sets correspond to line bundles on V G which are generated by their sections, and hence their higher cohomology groups vanish. This last result follows from the more general fact that if the support of the fan (i.e., the union of all the cones in the fan) of a toric variety is a convex set, then the higher cohomology groups vanish for line bundles (on this variety) which are generated by their sections (see e.g. [4], pg.74).\\nRemember that we are trying to reformulate theorems A and B in terms of toric varieties; we have already described the toric analog of (G, T )-orthogonal sets; the map pr M corresponds to the map\\nwhere RM stands for the root lattice forM (note that the codomain of the last map is just X * (Z(M ))); now we need to describe the toric analog of (G, M )-orthogonal sets.\\nWe continue to have the same notation. Then for M as above (a Levi subgroup containing T ), we will need a toric variety Y G M for the torus Z(M )/Z(Ĝ) (see e.g. [7], §23.2). First assume that the groupĜ is adjoint; we can do this since the center of M /Z(Ĝ) inĜ/Z(Ĝ) is equal to Z(M )/Z(Ĝ)). Then Z(M ) is a subtorus ofT and so X * (Z(M )) is a subgroup of X * (T ). The collection of cones from the Weyl fan inside X * (T ) ⊗ Z R that lie in the subspace X * (Z(M )) ⊗ Z R gives a fan. This is the fan for the complete, nonsingular, projective toric variety Y G M . Now we go back to the general case where we no longer assume thatĜ is adjoint. Denote by G 0 the simply connected cover of the derived group of G and by M 0 the Levi subgroup in G 0 that is obtained as the inverse image of M under the map G 0 → G. Then we have thatĜ\\nM0 as a space on which Z(M ) acts using the canonical map Z(M ) ։ Z(M 0 ). It is this toric variety, which sits inside V G as a closed Z(M )-stable subspace, that we denote by Y G M . As with the toric variety V G , it is easily seen that Z(M )-fixed points in Y G M are in one-to-one correspondence with parabolic subgroups of G that admit M as a Levi component. Also, there is a one-to-one correspondence between isomorphism classes of Z(M )-equivariant line bundles on Y G M and (G, M )-orthogonal sets in X M . This correspondence is defined is the same way as the one for V G (see e.g. [7], §23.4).\\nWe note that (see e.g. [7], §23.4 ) restricting aT -equivariant line bundle L on V G to Y G M corresponds the procedure described earlier in this section of obtaining a (G, M )-orthogonal set (the one corresponding to the restriction L| Y G M ) from a (G, T )-orthogonal set (the one corresponding to L).\\nLet us now assume that our parabolic subgroup P is of semisimple rank 1. This implies that the root lattice RM is just Zα for a unique, up to a sign, root α ofĜ, and that the toric variety Y G M , which we now denote by D α , is a non-torus-invariant divisor in V G . The map pr M will now be denoted by p α . By tensoring with R we get a map from p α , which we still denote by\\nSince tensoring with R will loose any possible torsion, we can identify the codomain of the last map with the co-root hyperplane\\nwhere , is the canonical pairing between co-characters and characters, and α ∨ stands for the co-root ofĜ corresponding to α. We will therefore say that the map p α is the projection along α onto the co-root hyperplane [α ∨ = 0]. Also, the fan of D α is contained in the root hyperplane\\nNow let L be aT -line bundle on V G that is generated by its sections. Then we have a short-exact sequence of sheaves on V G :\\n, for all i > 0, since L is generated by its sections and V G is projective, and also\\nfor all i > 0, since L| Dα is generated by its sections and D α is a projective toric variety. Therefore the short-exact sequence gives rise to the long-exact sequence\\nSo, we see that surjectivity of the map ϕ is equivalent to\\nWe claim that the following result is equivalent to Theorem A.\\nTheorem C. With notation as above, we have that\\nwhenever L arises from a Weyl orbit.\\nRemark 1.2. Theorem C is true for split classical groups. In fact, in the same way that we will show the equivalence of theorems A and C one can show that theorem C for a split classical group is equivalent to Proposition 1.1 for the case of parabolic subgroups P of semisimple rank 1.\\nLet us demonstrate the equivalence between theorems A and C. Put G = G 2 . Suppose that L corresponds to the Weyl-orbit orthogonal set (x B ) in X, i.e. in X * (T ), and, as before, denote by (x P ) the corresponding (G, M )-orthogonal set in X M , i.e. in X * (Z(M )). Write P L for the intersection of X * (T ) with the convex hull Conv{x B } of (x B ). Then we have (see e.g. [4], pg.66)\\nwhere the section χ u is an eigenvector corresponding to the character u. (Recall thatT acts on H 0 (V G , L) and so H 0 (V G , L) decomposes according to the characters ofT .)\\nAssume that α is the root ofĜ, up to a sign, that corresponds toM , in the sense that the projection map p α : X * (T ) → [α ∨ = 0] along the root α corresponds to the projection map pr M . Then we write P L,α for the intersection of Conv{x P } with p α (X * (T )) and we get\\n) from the long-exact sequence above is given by ϕ(χ u ) = χ pα(u) . This explains why Theorem C is equivalent to Theorem A, since p α corresponds to pr M . Now let G = GL n . Theorem B will follow from the following result.\\nTheorem D. With notation as above, we have that\\nwhenever L is generated by its sections.\\nTheorem D itself will be deduced (in Section 4.5) from the result for G = SL n : Theorem E. With notation as above, we have that\\nwhenever L is generated by its sections.\\nWe note that while theorems A and C were seen to be equivalent, to show that theorems B and D imply each other will take a little more work (because we are no longer working with a two-dimensional space!); however, the same argument above for G = G 2 applies in our case to show that Theorem B for the case of parabolic subgroups P of semisimple rank 1 is equivalent to Theorem D. But, we will deduce our Theorem B from Theorem D, where the latter will be proved using toric geometry.\\nWe explain in Section 4.6 how the general case is handled, but here let us just show, using a similar method used there, how Theorem D, i.e. surjectivity of ϕ, implies Theorem B for all M = GL n1 × GL n2 × · · · × GL nr , where r i=1 n i = n and only one of n i 's is equal to 2 and the rest are equal to 1.\\nAssume that n k = 2 for some 1 ≤ k ≤ n r and that n i = 1 for all i such that i = k and 1 ≤ i ≤ n r . Keeping the same notation as above, we have in this case a = R n and\\nwhere, as mentioned earlier, we have identified a M as a subspace of a. Also, the map pr M , which in general is given by averaging over each of the r \\\"batches\\\", now agrees with p α where α = (0, . . . , 0, 1, −1, 0, . . . , 0) (with 1 in the k-th place and -1 in the (k + 1)-st place):\\n(In general pr M does not correspond to one p α , rather, it is a composition of a number of p α 's, for distinct roots α.) As in the G 2 case, we find that\\nand ϕ(χ u ) = χ pα(u) . Then surjectivity of ϕ clearly implies Theorem B for this particular case. As we mentioned in the Introduction, the next sections are devoted to proving our results. It is now safe for the reader to assume that, for the rest of the paper, we are working over the complex numbers-we will be in the toric geometry setting. Also, since we will be working in the complex dual world, when we use terms root, co-root, character, and co-character, they will always refer to those terms for the complex dual groupsĜ andT .\\nBefore we end this section it is worth mentioning that vanishing results like the ones in theorems C and D (and E) are also of independent interest just from a toric-variety point of view. A very important vanishing result for toric varieties has been proved by Mustaţȃ (see e.g. [11], Theorem 0.1) and for the particular toric varieties arising from GL n and G 2 theorems C and D (and E) give vanishings of higher cohomology groups for more line bundles on these varieties. (See also [5] for more vanishing results for toric varieties associated with root systems.)\"\n },\n {\n \"title\": \"Some useful results\",\n \"content\": \"We keep the same notation as in the previous section. Let V = V G be the non-singular, projective toric variety corresponding to G as described in Section 1 and let ∆ be the fan of V . Each element of ∆ corresponds to a unique torus orbit in V , and under this identification, if we denote by ∆(1) the set of rays in ∆ and if ρ ∈ ∆(1), we write D ρ for the closure of the orbit of ρ, aT -Cartier divisor (cf. [4], §3.1 and §3.3). In fact,T -Cartier divisors on V can be written as a linear combination of D ρ 's. From now on let us agree to refer toT -Cartier divisors simply as divisors. (No confusion should arise from this agreement even though our divisor D α is not aT -divisor; in fact, thanks to Lemma 2.3 below, when computing cohomology groups, we will deal with aT -Cartier divisor instead of D α .)\\nLet us mention another way of characterizing divisors on toric varieties (see e.g. [4], §3.4, but be aware of a sign difference). A divisor D on a toric variety V , with fan ∆, is completely determined by a set of characters u(σ), one for every element σ ∈ Max(∆), such that they \\\"agree\\\" on the overlaps, i.e., such that they form a (G, T )-orthogonal set (we will usually call these just orthogonal sets; here and throughout this paper, Max(∆) stands for the set of maximal cones in ∆). This orthogonal set in turn gives a continuous (integral) piece-wise linear function ψ D on the support |∆| := ∪ σ∈|∆| σ of the fan, where ψ D (v) = u(σ), v , for all v ∈ σ. And conversely, given a continuous (integral) piece-wise linear function ψ on |∆|, i.e., a continuous function that is linear and given by an element of the character lattice on each cone of ∆, we get a divisor by considering the characters u(σ), σ ∈ Max(∆), which define ψ on the maximal affine pieces of V . (The continuity of ψ guarantees that the characters defining it agree on the overlaps and hence give a divisor.)\\nFrom what we said, it is easy to see that if D = ρ∈∆ a ρ D ρ , then the corresponding piece-wise linear function is given by ψ D (v) = u(σ), v (v ∈ σ) where, for each σ, the character u(σ) is found by solving the system of equations\\nwhere ρ varies through the rays in the cone σ and v ρ stands for the primitive lattice element in the ray ρ.\\nLet us mention that ψ D is called convex or lower convex if for any v, v ′ ∈ |∆| and 0 ≤ p ≤ 1, we have that ψ D (pv\\nThe following result is going to be very useful. where H i (V, O(D)) u denotes the u-eigenspace that is obtained by the action ofT on H i (V, O(D)) and Z(u) = {v ∈ |∆| : ψ(v) ≥ u, v }.\\nKeep the same notation as above and let α be a root ofĜ. The following result was suggested to us by R. Kottwitz. Proof. This follows from considering the rational function χ α −1 on V G and computing the corresponding principal divisor. Note that the divisor of zeros corresponds to D α .\\nSince we will be interested in computing the cohomology group H 1 (V G , O(D − D α )), the last lemma allows us to focus instead on\\nThe lemma below describes the orthogonal set corresponding to theT -equivariant divisor ρ∈∆(1), α,vρ >0 α, v ρ D ρ , which, due to the previous lemma, will be referred to as the orthogonal set corresponding to D α . Lemma 2.3. Let α be a root ofĜ and consider V G . The corresponding orthogonal set for D α is given by (u(σ)) σ∈Max(∆) , where u(σ) = α if α ≥ 0 on σ, and u(σ) = 0 if α ≤ 0 on σ.\\nProof. The result follows at once from Lemma 2.2 and the fact that, as we discussed earlier, if D = ρ∈∆ a ρ D ρ , then u(σ)'s can be found by solving the system of equations u(σ), v ρ = a ρ , where ρ ∈ ∆(1) varies through the rays of σ. Proof. First note that using Proposition 2.1, part b), it is sufficient (and necessary) to show that\"\n },\n {\n \"title\": \"An Important Lemma\",\n \"content\": \"Looking at the long-exact sequence we get from the pair (|∆| , U D,α (0)), since |∆| is a vector space (V G is complete), we only need to show that U D,α (0) is path-wise connected. Let us do that.\\nAccording to Lemma 2.3, U D,α (0) is equal to the union of sets\\nHere we have written [α ≥ 0] and [α ≤ 0] for the sets {v ∈ |∆| : α, v ≥ 0} and {v ∈ |∆| : α, v ≤ 0}, respectively. Proposition 2.1, part a), implies that sets (2) and (3) are convex. Let us see how this works in the case of (3). Let v and v ′ be in (3) and let 0 ≤ p ≤ 1. Then using Lemma 2.3. we have\\nwhich, since ψ D is convex, is not greater than\\nand the last expression is less than zero since ψ D−Dα (v) < 0, ψ D−Dα (v ′ ) < 0 and 0 ≤ p ≤ 1. Clearly, from our assumptions, we also have that pv +(1−p)v ′ ∈ [α ≥ 0], so (3) is a convex set, as desired.\\nIf condition (i) of the lemma is satisfied, then U D,α (0) equals the set (3) and hence (1) holds, because the set (3) is convex.\\nSuppose that (ii) is satisfied. Since {v ∈ |∆| : ψ D (v) < 0} is convex, this implies that there exists a (non-empty, convex) subset U in [α ≤ 0] ∩ [α ≥ 0] that is contained in both sets appearing in (ii). We claim that U is then contained in both sets (2) and (3). This claim is sufficient to prove that U D,α (0) is path-wise connected (and hence (1) holds) since we already saw that (2) and (3) are convex.\\nFirst, U is contained in (2) by our assumption that it is contained in both sets appearing in (ii), and in particular the second one. Second, since U is contained in the first set appearing in (ii), we see that ψ D (v) < 0 and α, v ≥ 0, for all v ∈ U . Therefore, using Lemma 2.3, we see that\\nThus U is contained in (3) as well, which ends the proof of the claim and hence of the lemma.\\nNote that if we knew that (1) is true for all globally generated divisors D, then we could conclude that for those divisors we have H 1 (V G , O(D − D α )) u = 0, for all u. Indeed, to prove this, consider the (globally generated) divisor D ′ associated to the orthogonal set {u(σ) − u}, where {u(σ)} corresponds to D. The result follows by using (1) (which is assumed to be true for all globally generated divisors), where instead of U D,α (0) we have U D ′ ,α (0).\\nUsing Proposition 2.1, part b), we deduce that if (1) holds for all globally generated divisors D, then\\nAn important consequence of this is that Theorem D needs to be proved only in the case when the set (2) is non-empty and\\nMore concretely, it suffices to show that, under these conditions,\\nbecause then we get that U D,α (0) is equal to the set (2), which we know is a convex set, and we can apply the same reasoning as in Lemma 3.1 to conclude that (1) is true, and by what we just wrote, H 1 (V G , J Dα ⊗ O(D) = 0 is true as well.\"\n },\n {\n \"title\": \"Proofs of Theorems B, D, and E\",\n \"content\": \"Consider V G , where G = SL n (and soĜ = P GL(n, C)). We want to concretely describe the fan ∆ of V G . But first, the initial lattice (i.e. lattice of characters) for V G is L = Z n /Z (1, 1, . . . , 1) and so the dual one is L ∨ = {(x 1 , . . . , x n ) ∈ Z n |x 1 + . . . + x n = 0}. For every i = 1, . . . , n we put L i for (the element represented by) (0, . . . , 0, 1, 0, . . . , 0), where 1 appears only on the i-th place. Identify the rays in ∆ with their corresponding minimal lattice points. Then ∆(1) consists of the sums of the form k j=1 L ij , where i j 's are distinct elements from {1, 2, . . . , n} and k = 1, ..., n − 1. The maximal cones in V G are the n-dimensional cones whose rays are of the type\\nwhere i 1 , i 2 , . . . , i n−1 are distinct elements from {1, 2, . . . , n}. Let α be a root of G. We can assume that α is simple (for a choice of a Borel subgroup) and, with a change of coordinates, we can always ensure that α is written as L 1 −L 2 . Therefore, without loss of generality, throughout this section we let α = L 1 − L 2 .\\nThe proof of Theorem E will be proved by induction, although we begin by giving the proof for the first two nontrivial cases, n = 3 and n = 4. This may seem a little unusual, but we do so because in order to apply the induction process we will need \\\"many\\\" root hyperplanes and for n = 4 we don't have \\\"enough\\\" of them (see Case 2) of the proof in Section 4.3 where we use n > 4).\"\n },\n {\n \"title\": \"4.1.\",\n \"content\": \"Proof of Theorem E for n = 3. In this case the set of maximal cones Max(∆) in the fan ∆ of V G consists of six cones σ 1 , . . . , σ 6 , generated respectively by L 1 and −L 3 ; −L 3 and L 2 ; L 2 and −L 1 ; −L 1 and L 3 ; L 3 and −L 2 ; and −L 2 and L 1 .\\nLet D be a divisor on V G such that O(D) is generated by its sections. If u(σ i ), i = 1, . . . , 6 form the orthogonal set corresponding to D, where u(σ i ) is assigned to the maximal cone σ i , then, according to Lemma 2.3, the orthogonal set corresponding to the divisor D − D α consists of u(σ i ) − α for i = 1, 5, 6 and u(σ i ) for i = 2, 3, 4.\\nFrom our discussion in Section 3, we only need to prove that (5) is true, under the assumption that the set (2) is non-empty and (4) is true. Suppose, for a contradiction, that under these assumptions there exists v 0 ∈ |∆| such that α, v 0 ≥ 0 and ψ D (v 0 ) − α, v 0 < 0. Since ψ D is continuous and piecewise linear, there are only two cases we need to consider: (i) v 0 = L 1 , and (ii) v 0 = L 1 + L 3 = −L 2 . We can use the symmetry (x, y, z) → (−y, −x, −z) of the root system, which preserves the half-space [α ≥ 0], to see that we only need to consider the case (i).\\nLet\\nis generated by its sections, u(σ i )'s form a positive orthogonal set and therefore there exists a non-negative number n ∈ Z such that u(σ 2 ) = u(σ 1 ) + n(L 2 − L 1 ). We get ψ D (L 2 ) = u(σ 2 ), L 2 = b + n ≥ 0. For the same reasons, there exists a non-negative integer m such that u(σ 3 ) = u(σ 2 ) + m(L 3 − L 1 ), and therefore ψ D (−L 1 ) = u(σ 3 ), −L 1 = n + m ≥ 0. Thus, since ψ D is continuous and piece-wise linear, ψ D takes non-negative values on all of [α ≤ 0], and this contradicts the non-emptiness of (2). Theorem E follows for n = 3.\\nRemark 4.1. If we are working with V SL4 , then we can consider sub-fans of the initial fan ∆ that are contained in the root hyperplanes. For example, if our given root is β = L 3 − L 4 , then the sub-fan corresponding to the root hyperplane [β = 0] has rays L 1 , −L 3 − L 4 , L 2 , −L 1 , L 3 + L 4 , −L 2 and the maximal-dimensional cones are the 2-dimensional cones in ∆ obtained from these rays. This is just a \\\"copy\\\" of the fan of V SL3 and indeed we can carry out the same calculations as above to see that Theorem E holds for this \\\"copy\\\" of V SL3 inside V SL4 . This remark is important for our induction process and is part of a more general story which we tell in Section 4.4.\"\n },\n {\n \"title\": \"4.2.\",\n \"content\": \"Proof of Theorem E for n = 4. Note that we are still assuming that α = L 1 − L 2 . To simplify notation, write σ i,j,k for the maximal cone whose rays are L i , L i + L j , L i + L j + L k , where i, j, k are distinct elements of {1, 2, 3, 4}.\\nLet D be a divisor on V G with O(D) generated by its sections. Denote by u(σ i,j,k ) the character corresponding to D for the cone σ i,j,k . Then these characters form an orthogonal set and, according to Lemma 2.3, the orthogonal set corresponding to D − D α consists of u(σ i,j,k ) − α for the maximal cones σ i,j,k contained in the half-space [α ≥ 0] and u(σ i,j,k ) for the ones contained in [α ≤ 0]. Following the discussion in Section 3, we assume that the set (2) is non-empty and that (4) is true. We want to prove (5) and, for a contradiction, suppose that there exists a point v 0 in the half-space [α ≥ 0] such that ψ D (v 0 ) − α, v 0 < 0. Since ψ D is continuous and piece-wise linear on |∆|, there are only four cases we need to consider, corresponding to the rays whose primitive lattice points are contained in\\nBut, we can use the symmetry (x, y, z, w) → (−y, −x, −z, −w) of the root system, which preserves the half-space [α ≥ 0], to see that we only need to consider cases (i) and (ii).\\nCase (i): Our aim is to show that ψ D takes non-negative values at L 2 , L 2 + L 3 , L 2 + L 4 and L 2 + L 3 + L 4 , which would contradict our assumption that the set (2) is non-empty. First, if we look at the hyperplane corresponding to the root L 2 − L 3 , as mentioned in Remark 4.1, we get a copy of V SL3 , where the rays of the sub-fan are L 1 , L 1 + L 4 , L 4 , L 2 + L 3 + L 4 , L 2 + L 3 , L 1 + L 2 + L 3 . These are precisely the rays of ∆ contained in our hyperplane. Apply Theorem E, case n=3, to see that ψ D (L 2 + L 3 ) ≥ 0 and ψ D (\\nSimilarly, by looking at the hyperplanes corresponding to L 3 − L 4 and L 2 − L 4 and applying Theorem E, for n=3, we get that ψ D (L 2 ) ≥ 0 and ψ D (L 2 + L 4 ) ≥ 0 respectively, which is what we intended to prove.\\nCase (ii): Completely similarly as in Case (i) we see that ψ D (L 2 ) ≥ 0 and ψ D (L 2 + L 4 ) ≥ 0 (use the root L 1 − L 3 ), and ψ D (L 2 + L 3 + L 4 ) ≥ 0 (use the root L 2 − L 4 ). The only non-trivial case is when we want to show that ψ D (L 2 + L 3 ) ≥ 0, because there is no root-hyperplane which contains both L 1 + L 3 and L 2 + L 3 . We proceed as follows. From our assumptions we already know that ψ D (L 1 + L 3 ) ≥ 0, ψ D (L 1 +L 3 )−1 < 0, ψ D (L 3 ) ≥ 0 and ψ D (L 1 +L 2 +L 3 ) ≥ 0. Therefore if u(σ 3,1,2 ) = (a, b, c, −a−b−c) where a, b, c ∈ Z, then a+c ≥ 0, a+c−1 < 0, c ≥ 0 and a+b+c ≥ 0, i.e., a = −c, c ≥ 0 and b ≥ 0. But since u(σ i,j,k )'s form a positive orthogonal set, we can find a non-negative integer m so that u(σ 3,2,1 ) = m(L 2 − L 1 ) + u(σ 3,1,2 ). Thus u(σ 3,2,1 ), L 2 + L 3 = b + c + m ≥ 0, i.e., ψ D (L 2 + L 3 ) ≥ 0, and this concludes the proof.\"\n },\n {\n \"title\": \"4.3.\",\n \"content\": \"Proof of Theorem E for n > 4. We use induction to prove the theorem in the general case. Assume that the theorem is true for all V SLn (as well as for the root-hyperplane copies of V SLn inside V SLn+1 ; again see Section 4.4), and we want to prove it for V SLn+1 .\\nLet D be a divisor on V SLn+1 such that O(D) is generated by its sections. Once more we consider D − D α , where we have taken without loss of generality α = L 1 − L 2 . Following the discussion in Section 3, we assume that the set (2) is nonempty and that (4) is true. We want to prove that (5) holds. For a contradiction, suppose that there exists a point v 0 in the half-space [α ≥ 0] such that ψ D (v 0 ) − α, v 0 < 0. Since the function ψ D is continuous and piece-wise linear, we would get a contradiction if we showed that ψ D takes non-negative values on all the primitive lattice points along the rays of the fan of V SLn+1 which are contained in the halfspace [α ≤ 0]. Moreover, due to the same properties of ψ D , we may assume that (6) v\\nwhere i j 's are distinct elements of {3, 4, . . . , n + 1}. We want to prove that for all the primitive lattice points of the form\\nwhere p q 's are distinct elements of {3, 4, . . . , n + 1}, we have\\nFor this, it suffices to show that there is a root β ofĜ such that both v 0 = L 1 + L ij and L 2 + L pq are contained in the root hyperplane corresponding to β, because we can then apply the induction hypothesis to conclude that (8) is true.\\nIf there exists an i j from (6) which is not equal to any of p q 's appearing in (7), then put β = L 1 − L ij . If there exists a p q from (7) which is not equal to any of i j 's from (6), then we put β = L 2 − L pq . In both cases, (8) follows at once.\\nTherefore we are only left with the possibility that, up to permutation, i j 's are equal to p q 's. If this is the case, then we distinguish two cases:\\nCase 1). The number of L ij 's is greater than one, and so if, say, L ij 1 and L ij 2 appear in (6) we put β = L ij 1 − L ij 2 to get (8).\\nCase 2). The number of L ij 's is less than two, and so, because n > 4, we can find L km and L kn which don't appear in (6) and therefore in (7) either. Put β = L k1 − L k2 to see that v 0 and L 2 + L pq belong to the root hyperplane corresponding to β. So we get (8).\\nThis proves Theorem E for all n. The dual lattice for D α is p α (X * (T )). More concretely, the dual lattice for D α is\\nbecause the map p α is given by p α (x 1 , x 2 , . . . , x n+1 ) = ( x1+x2 2 , x1+x2 2 , x 3 , . . . , x n+1 ). The claim is that we can treat D α as a copy of V SLn inside V SLn+1 .\\nFor ease of notation, let L ∨ α := p α (X * (T )) and L ∨ n be the dual lattice of V SLn . Note that we have a bijection Ψ : L ∨ α → L ∨ n , given by Ψ(x 2 /2, x 2 /2, x 3 , . . . , x n+1 ) = (x 2 , x 3 , . . . , x n+1 ). Now let β = α be a root ofĜ. We only need to consider two cases: I) β = L 2 − L 3 ; II) β = L 3 − L 4 , to prove that we can treat D α as a copy of V SLn . I) Let β = L 2 − L 3 . Define p β , the projection using β, in the same way as p α . We project L ∨ α via p β to get the lattice\\nIf we project L ∨ n via the corresponding p β , we get the lattice\\nand we also have a natural bijection Ψ β : L ∨ α,β → L ∨ n,β , given by\\nFurther, if we have an orthogonal set (ξ i ) in V SLn , then we get an orthogonal set (Ψ −1 (ξ i )) in D α by applying Ψ −1 to the former. And conversely, we apply Ψ to an orthogonal set in D α in order to get an orthogonal set in V SLn . Also, points in p β (Conv(ξ i ) ∩ L ∨ α ) are identified with points in p β (Conv(Ψ −1 (ξ i ) ∩ L ∨ α ) via the map Ψ β and, under the same map, points in p β (Conv(ξ i )) ∩ p β (L ∨ α ) are identified with those in p β (Conv(Ψ −1 (ξ i ))) ∩ p β (L ∨ n ). The proof of the following Lemma then solely involves unravelling the definitions of the maps involved and the definition of a convex hull. Incidentally, it proves, as we wanted, that we can treat D α as a copy of V SLn inside V SLn+1 , for which Theorem E holds.\"\n },\n {\n \"title\": \"Lemma 4.2.\",\n \"content\": \"For the corresponding orthogonal sets (ξ i ) and (Ψ −1 (ξ i )) we have that\\nII) Let β = L 3 − L 4 . This is handled analogously to I) and the lemma above holds in this case, too. Let us mention that in this case we have x i = 0; x i ∈ Z, ∀i}.\"\n },\n {\n \"title\": \"4.5.\",\n \"content\": \"Proof of Theorem D. Let G = GL n and, as earlier, take without loss of generality α = L 1 − L 2 . We have that X * (T ) = Z n . Suppose D is a divisor on V G such that O(D) is globally generated and let {u(σ i )} i∈I be the corresponding orthogonal set, where for each i ∈ I, u(σ i ) = (a\\nn ) ∈ Z n corresponds to the maximal cone σ i of the fan ∆ of V G . Then, the fact that {u(σ i )} i∈I is an orthogonal set means in this case that there exists s ∈ Z such that a\\nIf we denote by P D the intersection of X * (T ) with the convex hull of the orthogonal set {u(σ i )} i∈I , then we have that (see e.g. [4], pg.66)\\nSimilarly, we get that\\nwhere P Dα = p α (Z n ) ∩ P α , with P α standing for the image of the convex hull of the orthogonal set {u(σ i )} i∈I under the map p α . (Recall that the map p α was defined in Section 1). Using the long-exact sequence (*) from Section 1, Theorem D can be re-written in the following way:\\nis surjective, for all globally generated line bundles O(D) on V G , where ϕ(χ u ) = χ pα(u) . Let us prove Theorem D using Theorem E. Suppose x ∈ P Dα . It suffices to find a point z in P D such that x = p α (z).\\nFirst \\\"shift\\\" the orthogonal set {u(σ i )} i∈I by u(σ 1 ), i.e., consider the orthogonal set {u(σ i ) − u(σ 1 )} i∈I , which gives a new divisor D ′ on V G . Then, if we write\\nNote that p α (x − u(σ 1 )) lies in the intersection of the image of the set\\nunder the map p α and the set p α (Z n ).\\nUsing Theorem E, we can then conclude that there exists z ′ ∈ P D ′ such that p α (z ′ ) = p α (x − u(σ 1 )), where P D ′ := {u ∈ Z n : u, v ≤ u(σ) − u(σ 1 ), v , ∀v ∈ σ, ∀σ ∈ Max(∆)}.\\nOne can easily check that p α (z ′ + u(σ 1 )) = x and z ′ + u(σ 1 ) ∈ P D . This means that we found z = z ′ + u(σ 1 ) in P D such that x = p α (z), which proves Theorem D. 4.6. Proof of Theorem B. Now it is time to explain how Theorem B follows from Theorem D. We keep the same notation as in Section 1, with the difference that we consider G = GL n+1 instead of G = GL n . Then M must be of the form M = GL n1 × GL n2 × · · · × GL nr where n+1 i=1 n i = n + 1. We can also see that a = R n+1 ,\\nand pr M is given by averaging over each of the r \\\"batches\\\". (We remind the reader that here, as before, we are considering a M as a subspace of a.)\\nThe simplest case is when only one of n i 's is equal to 2 and the rest are equal to 1, but we saw in Section 1 how Theorem B is deduced from Theorem D in that case. Now assume that only two of the n i 's are equal to 2 and the rest equal 1. We can assume without loss of generality that n 1 = n 2 = 2 and n 3 = . . . = n r = 1, with r = n − 3. Then\\nand so pr M is just the composition p L3−L4 • p L1−L2 . Therefore when we apply Theorem D to D L1−L2 , where we consider D L1−L2 as a copy of V GLn and α = L 3 − L 4 , we find that Theorem B follows in this case, too. The next case would be to consider, without loss of generality, n 1 = 3 and n 2 = . . . = n r = 1, where r = n − 2. Now we see that pr M is the composition p L2−L3 • p L1−L2 and we get our desired conclusion again using Theorem D.\\nIn general we write pr M as a composition of a finite number of p α 's for distinct roots α and then apply Theorem D as many times as there are roots α appearing in that composition. If we want to be very specific, then, for M = GL n1 × GL n2 × · · · × GL nr we can see that pr M agrees with the composition of maps q 1 , q 2 , . . . , q r where q k = p L (n k −1) −Ln k • p L (n k −2) −L (n k −1) • . . . • p L (n k−1 +1) −L (n k−1 +2) , with the convention n 0 = 0. This concludes the proof of Theorem B.\"\n },\n {\n \"title\": \"Proof of Theorem C\",\n \"content\": \"Consider V G for G = G 2 . Just as in the case of SL n , we would like to describe the fan ∆ of V . But first, the initial lattice is L = Z 3 /Z(1, 1, 1) and so the dual one is L ∨ = {(x 1 , x 2 , x 3 ) ∈ Z 3 : x 1 + x 2 + x 3 = 0}. Identify the rays in ∆ with their corresponding minimal lattice points. Then ∆(1) consists of twelve points: v 1 = (1, 0, 0), v 2 = (1, 0, −1), v 3 = (0, 0, −1), v 4 = (0, 1, −1), v 5 = (0, 1, 0), v 6 = (−1, 1, 0) and v i+6 = −v i , for i = 1, . . . , 6. There are twelve maximal cones: σ i , i = 1, 2, . . . , 12, where the rays of σ i are v i and v i+1 (with v 13 := v 1 ). Let D be aT -Cartier divisor on V corresponding to a Weyl orbit (see Section 1 for the precise definition) and α a root ofĜ. For every maximal cone σ i denote by u(σ i ) the character corresponding to the divisor D. Obviously, u(σ i )'s form a positive orthogonal set. We want to prove that\\nThis is true if the orthogonal set corresponding to D is strictly positive, i.e., if u(σ i )'s are distinct and they form a positive orthogonal set. Indeed, use Lemma 2.3 and the definition of orthogonal sets to conclude that the orthogonal set corresponding to D − D α is still positive (but not necessarily strictly positive), i.e., J Dα ⊗ O(D) is globally generated. It is also obvious that if all u(σ i )'s are equal to each other, then we get a degenerate case for which the result is true as well.\\nTherefore the only problem might arise if only some but not all of the u(σ i )'s are the same. Using symmetries of the current root system, there are only four cases we need to consider, two for a short root and two for a long root: Let us mention again that we want to prove that H 1 (V G , J Dα ⊗ O(D)) = 0. As we explained in Section 1, we have a long-exact sequence ... −→ H 0 (V G , O(D)) ϕ the Weyl-orbit ones. More concretely, in Section 3, after Lemma 3.1, we saw that the vanishing of the first cohomology group there was implied by the vanishing of its corresponding 0-eigenspace (under the action of our torus). This is not true for G = G 2 . The crucial step was to note that by \\\"shifting\\\" a positive orthogonal set, say {u(σ)}, by a character, say, u, we get a positive orthogonal set {u(σ) − u}; however, it is clear that a shift of a Weyl-orbit is in general not a Weyl-orbit. 5.2. At the very end, we give an example to show that Theorem B fails in the case of G 2 . Consider V G2 and let α = (1, −1, 0). Let D be the positive orthogonal set (i.e., the globally generated torus-equivariant divisor on V G2 ) given by u(σ i ) = (1, 0, −1) for i ∈ {1, 2, 3, 4, 5, 12} and u(σ i ) = (0, −1, 1) for i ∈ {6, 7, 8, 9, 10, 11}. Following the same notation as before, we see that in our case\"\n },\n {\n \"title\": \"Lemma 2 . 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 3 . 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"4. 4 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"cf. [4], pg. 68) Let D be a Cartier divisor on V . Then O(D) is generated by its sections if and only if ψ D is (lower) convex. b) (cf. [4], pg.74) For every i ≥ 0 there exist canonical isomorphisms H i (V, O(D)) ∼ = u∈X * (T ) H i (V, O(D)) u , and H i (V, O(D)) u ∼ = H i (|∆| , |∆| \\ Z(u); C),","content":"cf. [4], pg. 68) Let D be a Cartier divisor on V . Then O(D) is generated by its sections if and only if ψ D is (lower) convex. b) (cf. [4], pg.74) For every i ≥ 0 there exist canonical isomorphisms H i (V, O(D)) ∼ = u∈X * (T ) H i (V, O(D)) u , and H i (V, O(D)) u ∼ = H i (|∆| , |∆| \\ Z(u); C),"},{"id":"","caption":"For the toric variety V = V G , we have that D α and ρ∈∆(1), α,vρ >0 α, v ρ D ρ are linearly equivalent divisors.","content":"Lemma 2 . 2 .\n22For the toric variety V = V G , we have that D α and ρ∈∆(1), α,vρ >0 α, v ρ D ρ are linearly equivalent divisors."},{"id":"","caption":"Consider V G and let α be a root ofĜ. Suppose O(D) is generated by its sections. If one of the conditions (i) or (ii) below is satisfied, then the 0-th eigenspace H 1 (V G , J Dα ⊗ O(D)) 0 = 0. (i) {v ∈ |∆| : ψ D (v) < 0 and α, v ≤ 0} = ∅; (ii) {v ∈ |∆| : ψ D (v) < 0 and α, v ≥ 0} = ∅ and {v ∈ |∆| : ψ D (v) < 0 and α, v ≤ 0} = ∅.","content":"Lemma 3 . 1 .\n31Consider V G and let α be a root ofĜ. Suppose O(D) is generated by its sections. If one of the conditions (i) or (ii) below is satisfied, then the 0-th eigenspace H 1 (V G , J Dα ⊗ O(D)) 0 = 0. (i) {v ∈ |∆| : ψ D (v) < 0 and α, v ≤ 0} = ∅; (ii) {v ∈ |∆| : ψ D (v) < 0 and α, v ≥ 0} = ∅ and {v ∈ |∆| : ψ D (v) < 0 and α, v ≤ 0} = ∅."},{"id":"","caption":"Induction process in Proof of Theorem E. Let us justify the induction process we used in the proof of Theorem E. Suppose G = SL n+1 and consider V G , with the initial lattice Z n+1 /Z(1, 1, . . . , 1) and with dual lattice X * (T ) = {(x 1 , . . . , x n+1 ) ∈ Z n+1 |x 1 + . . . + x n+1 = 0}. Take without loss of generality α = L 1 − L 2 . Then the fan of D α consists of the (n − 1)-dimensional cones of the fan ∆ of V SLn+1 which lie inside the hyperplane [α = 0] = {v ∈ R n+1 : α, v = 0}.","content":"4. 4 .\n4Induction process in Proof of Theorem E. Let us justify the induction process we used in the proof of Theorem E. Suppose G = SL n+1 and consider V G , with the initial lattice Z n+1 /Z(1, 1, . . . , 1) and with dual lattice X * (T ) = {(x 1 , . . . , x n+1 ) ∈ Z n+1 |x 1 + . . . + x n+1 = 0}. Take without loss of generality α = L 1 − L 2 . Then the fan of D α consists of the (n − 1)-dimensional cones of the fan ∆ of V SLn+1 which lie inside the hyperplane [α = 0] = {v ∈ R n+1 : α, v = 0}."},{"id":"","caption":"= 0; x i ∈ Z, ∀i} and L ∨ n,β = {(x 3 , x 4 2 , x 4 2 , x 5 , . . . , x n+1 ) : n+1 i=3","content":"= 0; x i ∈ Z, ∀i} and L ∨ n,β = {(x 3 , x 4 2 , x 4 2 , x 5 , . . . , x n+1 ) : n+1 i=3"},{"id":"","caption":"(a) α = (1, −1, 0) and there exists a natural number n so thatu(σ 2 ) = u(σ 3 ) = −u(σ 8 ) = −u(σ 9 ) = n(1, 1, −2),u(σ 4 ) = u(σ 5 ) = −u(σ 10 ) = −u(σ 11 ) = n(−1, 2, −1), u(σ 6 ) = u(σ 7 ) = −u(σ 12 ) = −u(σ 1 ) = n(−2, 1, 1); (b) α = (2, −1, −1) and there exists a natural number n so that u(σ 2 ) = u(σ 3 ) = −u(σ 8 ) = −u(σ 9 ) = n(1, 1, −2), u(σ 4 ) = u(σ 5 ) = −u(σ 10 ) = −u(σ 11 ) = n(−1, 2, −1), u(σ 6 ) = u(σ 7 ) = −u(σ 12 ) = −u(σ 1 ) = n(−2, 1, 1); (c) α = (1, −1, 0) and there exists a natural number n so that u(σ 1 ) = u(σ 2 ) = −u(σ 7 ) = −u(σ 8 ) = n(1, 0, −1), u(σ 3 ) = u(σ 4 ) = −u(σ 9 ) = −u(σ 10 ) = n(0, 1, −1), u(σ 5 ) = u(σ 6 ) = −u(σ 11 ) = −u(σ 12 ) = n(−1, 1, 0); (d) α = (2, −1, −1) and there exists a natural number n so that u(σ 1 ) = u(σ 2 ) = −u(σ 7 ) = −u(σ 8 ) = n(1, 0, −1), u(σ 3 ) = u(σ 4 ) = −u(σ 9 ) = −u(σ 10 ) = n(0, 1, −1), u(σ 5 ) = u(σ 6 ) = −u(σ 11 ) = −u(σ 12 ) = n(−1, 1, 0).","content":"(a) α = (1, −1, 0) and there exists a natural number n so thatu(σ 2 ) = u(σ 3 ) = −u(σ 8 ) = −u(σ 9 ) = n(1, 1, −2),u(σ 4 ) = u(σ 5 ) = −u(σ 10 ) = −u(σ 11 ) = n(−1, 2, −1), u(σ 6 ) = u(σ 7 ) = −u(σ 12 ) = −u(σ 1 ) = n(−2, 1, 1); (b) α = (2, −1, −1) and there exists a natural number n so that u(σ 2 ) = u(σ 3 ) = −u(σ 8 ) = −u(σ 9 ) = n(1, 1, −2), u(σ 4 ) = u(σ 5 ) = −u(σ 10 ) = −u(σ 11 ) = n(−1, 2, −1), u(σ 6 ) = u(σ 7 ) = −u(σ 12 ) = −u(σ 1 ) = n(−2, 1, 1); (c) α = (1, −1, 0) and there exists a natural number n so that u(σ 1 ) = u(σ 2 ) = −u(σ 7 ) = −u(σ 8 ) = n(1, 0, −1), u(σ 3 ) = u(σ 4 ) = −u(σ 9 ) = −u(σ 10 ) = n(0, 1, −1), u(σ 5 ) = u(σ 6 ) = −u(σ 11 ) = −u(σ 12 ) = n(−1, 1, 0); (d) α = (2, −1, −1) and there exists a natural number n so that u(σ 1 ) = u(σ 2 ) = −u(σ 7 ) = −u(σ 8 ) = n(1, 0, −1), u(σ 3 ) = u(σ 4 ) = −u(σ 9 ) = −u(σ 10 ) = n(0, 1, −1), u(σ 5 ) = u(σ 6 ) = −u(σ 11 ) = −u(σ 12 ) = n(−1, 1, 0)."},{"id":"","caption":"H 0 (V G2 , O(D)) = 2, but dim H 0 (V G2 , i * (O(D)| Dα )) = 3. Hence the map ϕ : H 0 (V G2 , O(D)) → H 0 (V G2 , i * (O(D)| Dα )), induced by the projection p α along α, is not surjective, and Theorem B is no longer true if we put G 2 instead of GL n .","content":"H 0 (V G2 , O(D)) = 2, but dim H 0 (V G2 , i * (O(D)| Dα )) = 3. Hence the map ϕ : H 0 (V G2 , O(D)) → H 0 (V G2 , i * (O(D)| Dα )), induced by the projection p α along α, is not surjective, and Theorem B is no longer true if we put G 2 instead of GL n ."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"cf. [4], pg. 68) Let D be a Cartier divisor on V . Then O(D) is generated by its sections if and only if ψ D is (lower) convex. b) (cf. [4], pg.74) For every i ≥ 0 there exist canonical isomorphisms H i (V, O(D)) ∼ = u∈X * (T ) H i (V, O(D)) u , and H i (V, O(D)) u ∼ = H i (|∆| , |∆| \\\\ Z(u); C),\",\n \"content\": \"cf. [4], pg. 68) Let D be a Cartier divisor on V . Then O(D) is generated by its sections if and only if ψ D is (lower) convex. b) (cf. [4], pg.74) For every i ≥ 0 there exist canonical isomorphisms H i (V, O(D)) ∼ = u∈X * (T ) H i (V, O(D)) u , and H i (V, O(D)) u ∼ = H i (|∆| , |∆| \\\\ Z(u); C),\"\n },\n {\n \"id\": \"\",\n \"caption\": \"For the toric variety V = V G , we have that D α and ρ∈∆(1), α,vρ >0 α, v ρ D ρ are linearly equivalent divisors.\",\n \"content\": \"Lemma 2 . 2 .\\n22For the toric variety V = V G , we have that D α and ρ∈∆(1), α,vρ >0 α, v ρ D ρ are linearly equivalent divisors.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Consider V G and let α be a root ofĜ. Suppose O(D) is generated by its sections. If one of the conditions (i) or (ii) below is satisfied, then the 0-th eigenspace H 1 (V G , J Dα ⊗ O(D)) 0 = 0. (i) {v ∈ |∆| : ψ D (v) < 0 and α, v ≤ 0} = ∅; (ii) {v ∈ |∆| : ψ D (v) < 0 and α, v ≥ 0} = ∅ and {v ∈ |∆| : ψ D (v) < 0 and α, v ≤ 0} = ∅.\",\n \"content\": \"Lemma 3 . 1 .\\n31Consider V G and let α be a root ofĜ. Suppose O(D) is generated by its sections. If one of the conditions (i) or (ii) below is satisfied, then the 0-th eigenspace H 1 (V G , J Dα ⊗ O(D)) 0 = 0. (i) {v ∈ |∆| : ψ D (v) < 0 and α, v ≤ 0} = ∅; (ii) {v ∈ |∆| : ψ D (v) < 0 and α, v ≥ 0} = ∅ and {v ∈ |∆| : ψ D (v) < 0 and α, v ≤ 0} = ∅.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Induction process in Proof of Theorem E. Let us justify the induction process we used in the proof of Theorem E. Suppose G = SL n+1 and consider V G , with the initial lattice Z n+1 /Z(1, 1, . . . , 1) and with dual lattice X * (T ) = {(x 1 , . . . , x n+1 ) ∈ Z n+1 |x 1 + . . . + x n+1 = 0}. Take without loss of generality α = L 1 − L 2 . Then the fan of D α consists of the (n − 1)-dimensional cones of the fan ∆ of V SLn+1 which lie inside the hyperplane [α = 0] = {v ∈ R n+1 : α, v = 0}.\",\n \"content\": \"4. 4 .\\n4Induction process in Proof of Theorem E. Let us justify the induction process we used in the proof of Theorem E. Suppose G = SL n+1 and consider V G , with the initial lattice Z n+1 /Z(1, 1, . . . , 1) and with dual lattice X * (T ) = {(x 1 , . . . , x n+1 ) ∈ Z n+1 |x 1 + . . . + x n+1 = 0}. Take without loss of generality α = L 1 − L 2 . Then the fan of D α consists of the (n − 1)-dimensional cones of the fan ∆ of V SLn+1 which lie inside the hyperplane [α = 0] = {v ∈ R n+1 : α, v = 0}.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"= 0; x i ∈ Z, ∀i} and L ∨ n,β = {(x 3 , x 4 2 , x 4 2 , x 5 , . . . , x n+1 ) : n+1 i=3\",\n \"content\": \"= 0; x i ∈ Z, ∀i} and L ∨ n,β = {(x 3 , x 4 2 , x 4 2 , x 5 , . . . , x n+1 ) : n+1 i=3\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(a) α = (1, −1, 0) and there exists a natural number n so thatu(σ 2 ) = u(σ 3 ) = −u(σ 8 ) = −u(σ 9 ) = n(1, 1, −2),u(σ 4 ) = u(σ 5 ) = −u(σ 10 ) = −u(σ 11 ) = n(−1, 2, −1), u(σ 6 ) = u(σ 7 ) = −u(σ 12 ) = −u(σ 1 ) = n(−2, 1, 1); (b) α = (2, −1, −1) and there exists a natural number n so that u(σ 2 ) = u(σ 3 ) = −u(σ 8 ) = −u(σ 9 ) = n(1, 1, −2), u(σ 4 ) = u(σ 5 ) = −u(σ 10 ) = −u(σ 11 ) = n(−1, 2, −1), u(σ 6 ) = u(σ 7 ) = −u(σ 12 ) = −u(σ 1 ) = n(−2, 1, 1); (c) α = (1, −1, 0) and there exists a natural number n so that u(σ 1 ) = u(σ 2 ) = −u(σ 7 ) = −u(σ 8 ) = n(1, 0, −1), u(σ 3 ) = u(σ 4 ) = −u(σ 9 ) = −u(σ 10 ) = n(0, 1, −1), u(σ 5 ) = u(σ 6 ) = −u(σ 11 ) = −u(σ 12 ) = n(−1, 1, 0); (d) α = (2, −1, −1) and there exists a natural number n so that u(σ 1 ) = u(σ 2 ) = −u(σ 7 ) = −u(σ 8 ) = n(1, 0, −1), u(σ 3 ) = u(σ 4 ) = −u(σ 9 ) = −u(σ 10 ) = n(0, 1, −1), u(σ 5 ) = u(σ 6 ) = −u(σ 11 ) = −u(σ 12 ) = n(−1, 1, 0).\",\n \"content\": \"(a) α = (1, −1, 0) and there exists a natural number n so thatu(σ 2 ) = u(σ 3 ) = −u(σ 8 ) = −u(σ 9 ) = n(1, 1, −2),u(σ 4 ) = u(σ 5 ) = −u(σ 10 ) = −u(σ 11 ) = n(−1, 2, −1), u(σ 6 ) = u(σ 7 ) = −u(σ 12 ) = −u(σ 1 ) = n(−2, 1, 1); (b) α = (2, −1, −1) and there exists a natural number n so that u(σ 2 ) = u(σ 3 ) = −u(σ 8 ) = −u(σ 9 ) = n(1, 1, −2), u(σ 4 ) = u(σ 5 ) = −u(σ 10 ) = −u(σ 11 ) = n(−1, 2, −1), u(σ 6 ) = u(σ 7 ) = −u(σ 12 ) = −u(σ 1 ) = n(−2, 1, 1); (c) α = (1, −1, 0) and there exists a natural number n so that u(σ 1 ) = u(σ 2 ) = −u(σ 7 ) = −u(σ 8 ) = n(1, 0, −1), u(σ 3 ) = u(σ 4 ) = −u(σ 9 ) = −u(σ 10 ) = n(0, 1, −1), u(σ 5 ) = u(σ 6 ) = −u(σ 11 ) = −u(σ 12 ) = n(−1, 1, 0); (d) α = (2, −1, −1) and there exists a natural number n so that u(σ 1 ) = u(σ 2 ) = −u(σ 7 ) = −u(σ 8 ) = n(1, 0, −1), u(σ 3 ) = u(σ 4 ) = −u(σ 9 ) = −u(σ 10 ) = n(0, 1, −1), u(σ 5 ) = u(σ 6 ) = −u(σ 11 ) = −u(σ 12 ) = n(−1, 1, 0).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"H 0 (V G2 , O(D)) = 2, but dim H 0 (V G2 , i * (O(D)| Dα )) = 3. Hence the map ϕ : H 0 (V G2 , O(D)) → H 0 (V G2 , i * (O(D)| Dα )), induced by the projection p α along α, is not surjective, and Theorem B is no longer true if we put G 2 instead of GL n .\",\n \"content\": \"H 0 (V G2 , O(D)) = 2, but dim H 0 (V G2 , i * (O(D)| Dα )) = 3. Hence the map ϕ : H 0 (V G2 , O(D)) → H 0 (V G2 , i * (O(D)| Dα )), induced by the projection p α along α, is not surjective, and Theorem B is no longer true if we put G 2 instead of GL n .\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"The characters of discrete series as orbital integrals. J Arthur, Invent. Math. 323J. ARTHUR, The characters of discrete series as orbital integrals, Invent. Math. 32 1976, no.3, 205-261.","doi":""},{"text":"Frobenius splitting methods in geometry and representation theory. M Brion, S Kumar, BirkhäuserBostonM. BRION and S. KUMAR, Frobenius splitting methods in geometry and representation theory, Birkhäuser Boston (2004).","doi":""},{"text":"Existence de filtrations admissible sur des isocristaux. J.-M Fontaine, M Rapoport, Bull. Soc. Math. France. 1331J.-M. FONTAINE and M. RAPOPORT, Existence de filtrations admissible sur des isocristaux, Bull. Soc. Math. France 133 (2005), no.1, 73-86.","doi":""},{"text":"Introduction to Toric Varieties. W Fulton, Ann. of Math. Stud. 131Princeton Univ. PressW. FULTON, Introduction to Toric Varieties, Ann. of Math. Stud. 131, Princeton Univ. Press, Princeton, NJ, 1993.","doi":""},{"text":"Vanishing results for toric varieties associated with root systems. Q R Gashi, in preparationQ. R. GASHI, Vanishing results for toric varieties associated with root systems, in prepara- tion.","doi":""},{"text":"On the Hodge-Newton decomposition for split groups. R Kottwitz, Int. Math. Res. Not. 26R. KOTTWITZ, On the Hodge-Newton decomposition for split groups, Int. Math. Res. Not. 2003, no.26, 1433-1447.","doi":""},{"text":"R Kottwitz, Harmonic analysis on reductive p-adic groups and Lie algebras, in Harmonic Analysis, the Trace Formula, and Shimura Varieties. Providence, RIAmer. Math. Soc4R. KOTTWITZ, Harmonic analysis on reductive p-adic groups and Lie algebras, in Harmonic Analysis, the Trace Formula, and Shimura Varieties, 393-522, Clay Math. Proc., 4, Amer. Math. Soc., Providence, RI, 2005.","doi":""},{"text":"On the existence of F-isocrystals. R Kottwitz, M Rapoport, Comment. Math. Helv. 78R. KOTTWITZ and M. RAPOPORT, On the existence of F-isocrystals, Comment. Math. Helv. 78 (2003), 153-184.","doi":""},{"text":"A Converse to Mazur's inequality for split classical groups. C Lucarelli, Journal of the Inst. Math. Jussieu. 32C. LUCARELLI, A Converse to Mazur's inequality for split classical groups, Journal of the Inst. Math. Jussieu (2004) 3 (2), 165-183.","doi":""},{"text":"Frobenius and the Hodge filtration. B Mazur, Bull. Amer. Math. Soc. 78B. MAZUR, Frobenius and the Hodge filtration, Bull. Amer. Math. Soc. 78 (1972) 653-667.","doi":""},{"text":"Vanishing theorems on toric varieties. M Mustaţȃ, Tohoku Math. J. 2M. MUSTAŢȂ, Vanishing theorems on toric varieties, Tohoku Math. J. (2) 54 (2002), no.3, 451-470.","doi":""},{"text":"A positivity property of the Satake isomorphism. M Rapoport, Manuscripta Math. 1012M. RAPOPORT, A positivity property of the Satake isomorphism, Manuscripta Math. 101 (2000), no.2, 153-166.","doi":""},{"text":"On the classification and specialization of F-isocrystals with additional structure. M Rapoport, M Richartz, Composito Math. 1032M. RAPOPORT and M. RICHARTZ, On the classification and specialization of F-isocrystals with additional structure, Composito Math. 103 (1996), no.2, 153-181","doi":""}],"string":"[\n {\n \"text\": \"The characters of discrete series as orbital integrals. J Arthur, Invent. Math. 323J. ARTHUR, The characters of discrete series as orbital integrals, Invent. Math. 32 1976, no.3, 205-261.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Frobenius splitting methods in geometry and representation theory. M Brion, S Kumar, BirkhäuserBostonM. BRION and S. KUMAR, Frobenius splitting methods in geometry and representation theory, Birkhäuser Boston (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Existence de filtrations admissible sur des isocristaux. J.-M Fontaine, M Rapoport, Bull. Soc. Math. France. 1331J.-M. FONTAINE and M. RAPOPORT, Existence de filtrations admissible sur des isocristaux, Bull. Soc. Math. France 133 (2005), no.1, 73-86.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Introduction to Toric Varieties. W Fulton, Ann. of Math. Stud. 131Princeton Univ. PressW. FULTON, Introduction to Toric Varieties, Ann. of Math. Stud. 131, Princeton Univ. Press, Princeton, NJ, 1993.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Vanishing results for toric varieties associated with root systems. Q R Gashi, in preparationQ. R. GASHI, Vanishing results for toric varieties associated with root systems, in prepara- tion.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the Hodge-Newton decomposition for split groups. R Kottwitz, Int. Math. Res. Not. 26R. KOTTWITZ, On the Hodge-Newton decomposition for split groups, Int. Math. Res. Not. 2003, no.26, 1433-1447.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"R Kottwitz, Harmonic analysis on reductive p-adic groups and Lie algebras, in Harmonic Analysis, the Trace Formula, and Shimura Varieties. Providence, RIAmer. Math. Soc4R. KOTTWITZ, Harmonic analysis on reductive p-adic groups and Lie algebras, in Harmonic Analysis, the Trace Formula, and Shimura Varieties, 393-522, Clay Math. Proc., 4, Amer. Math. Soc., Providence, RI, 2005.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the existence of F-isocrystals. R Kottwitz, M Rapoport, Comment. Math. Helv. 78R. KOTTWITZ and M. RAPOPORT, On the existence of F-isocrystals, Comment. Math. Helv. 78 (2003), 153-184.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Converse to Mazur's inequality for split classical groups. C Lucarelli, Journal of the Inst. Math. Jussieu. 32C. LUCARELLI, A Converse to Mazur's inequality for split classical groups, Journal of the Inst. Math. Jussieu (2004) 3 (2), 165-183.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Frobenius and the Hodge filtration. B Mazur, Bull. Amer. Math. Soc. 78B. MAZUR, Frobenius and the Hodge filtration, Bull. Amer. Math. Soc. 78 (1972) 653-667.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Vanishing theorems on toric varieties. M Mustaţȃ, Tohoku Math. J. 2M. MUSTAŢȂ, Vanishing theorems on toric varieties, Tohoku Math. J. (2) 54 (2002), no.3, 451-470.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A positivity property of the Satake isomorphism. M Rapoport, Manuscripta Math. 1012M. RAPOPORT, A positivity property of the Satake isomorphism, Manuscripta Math. 101 (2000), no.2, 153-166.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the classification and specialization of F-isocrystals with additional structure. M Rapoport, M Richartz, Composito Math. 1032M. RAPOPORT and M. RICHARTZ, On the classification and specialization of F-isocrystals with additional structure, Composito Math. 103 (1996), no.2, 153-181\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":490,"cells":{"text":{"kind":"string","value":"\nSome Hierarchies of QCB 0 -Spaces\n5 Apr 2013\n\nMatthias Schröder \nVictor Selivanov \nA P Ershov \n\nUniversity of Bundeswehr-Munich\nNeubiebergGermany\n\n\nInstitute of Informatics Systems SB RAS Novisibirsk\nRussia\n\nSome Hierarchies of QCB 0 -Spaces\n5 Apr 2013A6A1372DF7738BD473BE804CEAD5F652arXiv:1304.1647v1[cs.LO]\nWe define and study hierarchies of topological spaces induced by the classical Borel and Luzin hierarchies of sets.Our hierarchies are divided into two classes: hierarchies of countably based spaces induced by their embeddings into Pω, and hierarchies of spaces (not necessarily countably based) induced by their admissible representations.We concentrate on the non-collapse property of the hierarchies and on the relationships between hierarchies in the two classes.\n\nIntroduction\n\nA basic notion of Computable Analysis (CA) [Wei00] is the notion of an admissible representation of a topological space X.This is a partial continuous surjection δ from the Baire space N onto X satisfying a certain universality property (see Subsection 2.6 for some more details).Such a representation of X induces a reasonable computability theory on X, and the class of admissibly represented spaces is wide enough to include most spaces of interest for Analysis or Numerical Mathematics.As shown by the first author [Sch03], this class coincides with the class of the so-called QCB 0 -spaces, i.e.T 0 -spaces which are quotients of countably based spaces, and it forms a cartesian closed category (with the continuous functions as morphisms).Thus, among QCB 0 -spaces one meets many important function spaces including the Kleene-Kreisel continuous functionals [Kl959,Kr59] interesting for several branches of logic and computability theory.\n\nAlong with the mentioned nice properties of QCB 0 -spaces, this class seems to be too broad to admit a deep understanding.Hence, it makes sense to search for natural subclasses of this class which still include \"practically\" important spaces but are (hopefully) easier to study.Interesting examples of such subclasses are obtained if we consider, for each level Γ of the classical Borel or Luzin (projective) hierarchies of Descriptive Set Theory (DST) [Ke95], the class of spaces which have an admissible representation of the complexity Γ (below we make this precise).The study of the resulting Borel and Luzin hierarchies of QCB 0 -spaces is one of the aims of this paper.\n\nAlong with the hierarchies of QCB 0 -spaces, we will consider Borel and Luzin hierarchies of countably based T 0 -spaces (CB 0 -spaces for short) which are induced by the well-known fact that any CB 0 -space may be embedded in the algebraic domain Pω of all subsets of ω.The precise definition depends crucially on the possibility to find the \"right\" extensions of the classical hierarchies from the class of Polish spaces to the arbitrary spaces (including Pω).Such extensions were introduced by the second author in [Se04] and were studied in [Se05,Se05a,Se06,Se08,BY09,Br13].In particular, it was shown that many properties of these hierarchies in ω-continuous domains resemble the properties of classical hierarchies in Polish spaces [Ke95].\n\nHierarchies of spaces obtained in this way turn out to be closely related to the corresponding hierarchies of QCB 0 -spaces.Moreover, among the first levels of the Borel hierarchy of CB 0spaces we meet some classes of spaces which attracted attention of several researches in the field of quasi-metric spaces, in particular the class of quasi-Polish spaces.The class of quasi-Polish spaces identified and studied in [Br13] is a good solution to the problem from [Se08] of finding a natural class of spaces that includes the Polish spaces and the ω-continuous domains and has a reasonable DST.\n\nIn this paper we establish some basic properties of the above-mentioned hierarchies of spaces.In particular, we show that the hierarchies of spaces do not collapse, and that any level of any hierarchy is closed under retracts.As main technical tools to prove these results we use suitable generalizations of some classical facts (e.g. of the injectivity property of Pω and of Lavrentyev's Theorem on extending partial homeomorphisms in Polish spaces).Some of those generalizations might be also interesting in their own right.We also show that the class of all spaces in our hierarchies forms in a sense the smallest cartesian closed category of QCB 0 -spaces containing the discrete space ω of natural numbers, and establish the close relationship of the Luzin hierarchy of QCB 0 -spaces to the continuous functionals of finite type.Hence the class of all spaces in the Luzin hierarchy of QCB 0 -spaces seems to be a reasonable subclass of QCB 0 -spaces that contains most of spaces of interest for CA, including the Kleene-Kreisel continuous functionals.\n\nAfter recalling some notions and known facts in the next section, we establish the main technical facts in Section 3. In Sections 4 and 5 we introduce and study the mentioned hierarchies of CB 0spaces and of QCB 0 -spaces.In Section 6 we establish close relationships between hierarchies of CB 0 -spaces to those of QCB 0 -spaces (namely, any level of the CB 0 -hierarchies coincides with the class of CB 0 -spaces in the corresponding level of the corresponding QCB 0 -hierarchy).In Section 7 we relate the Luzin hierarchy of QCB 0 -spaces to the Kleene-Kreisel continuous functionals.In Section 8 we discuss the cartesian closedness of the corresponding categories, and we conclude in Section 9.\n\n\nNotation and Preliminaries\n\n\nNotation\n\nWe freely use the standard set-theoretic notation like dom(f ), rng (f ) and graph(f ) for the domain, range and graph of a function f , respectively, X × Y for the Cartesian product, X ⊔ Y for the disjoint union of sets X and Y , Y X for the set of functions f : X → Y (but in the case when X, Y are QCB 0 -spaces we use the same notation to denote the space of continuous functions from X to Y ), and P (X) for the set of all subsets of X.For A ⊆ X, A denotes the complement X \\ A of A in X.We identify the set of natural numbers with the first infinite ordinal ω.The first uncountable ordinal is denoted by ω 1 .The notation f : X → Y (resp.f :⊆ X → Y ) means that f is a total (resp.a partial) function from a set X to a set Y .\n\n\nTopological Spaces\n\nWe assume the reader to be familiar with the basic notions of topology.The collection of all open subsets of a topological space X (i.e. the topology of X) is denoted by τ X ; for the underlying set of X we will write X in abuse of notation.We will usually abbreviate \"topological space\" to \"space\".Remember that a space is zero-dimensional if it has a basis of clopen sets.\n\nA space Y is called a (continuous) retract of a space X if there are continuous functions s : Y → X and r : X → Y such that composition rs coincides with the identity function id Y on Y .Such a pair of functions (s, r) is called a section-retraction pair.Note that the section s is a homeomorphism between Y and the subspace s(Y ) = {x ∈ X | sr(x) = x} of X, and\ns −1 = r| s(Y ) .\nLet ω be the space of non-negative integers with the discrete topology.Of course, the spaces ω × ω = ω 2 , and ω ⊔ ω are homeomorphic to ω, the first homeomorphism is realized by the Cantor pairing function •, • .\n\nLet N = ω ω be the set of all infinite sequences of natural numbers (i.e., of all functions ξ : ω → ω).Let ω * be the set of finite sequences of elements of ω, including the empty sequence.For σ ∈ ω * and ξ ∈ N , we write σ ⊑ ξ to denote that σ is an initial segment of the sequence ξ.By σξ = σ • ξ we denote the concatenation of σ and ξ, and by σ • N the set of all extensions of σ in N .For x ∈ N , we can write x = x(0)x(1) . . .where x(i) ∈ ω for each i < ω.For x ∈ N and n < ω, let x <n = x(0) . . .x(n − 1) denote the initial segment of x of length n.Notations in the style of regular expressions like 0 ω , 0 * 1 or 0 m 1 n have the obvious standard meaning.\n\nBy endowing N with the product of the discrete topologies on ω, we obtain the so-called Baire space.The product topology coincides with the topology generated by the collection of sets of the form σ • N for σ ∈ ω * .The Baire space is of primary importance for DST and CA.The importance stems from the fact that many countable objects are coded straightforwardly by elements of N , and it has very specific topological properties.In particular, it is a perfect zero-dimensional space and the spaces N 2 , N ω , ω ×N = N ⊔N ⊔• • • (endowed with the product topology) are all homeomorphic to N .Let (x, y) → x, y be a homeomorphism between N 2 and N .The Baire space N has the following universality property for zero-dimensional CB 0 -spaces:\nProposition 2.1 [Ke95, Theorems 1.1 and 7.8] A topological space X embeds into N iff X is a zero-dimensional CB 0 -space.\nThe subspace C = 2 ω of N formed by the infinite binary strings (endowed with the relative topology inherited from N ) is known as the Cantor space.Along with N and C, the space Pω is of principal importance for this paper.This is the space of subsets of the natural numbers with the Scott topology on (P (ω); ⊆).The basic open sets of this topology are of the form {A ⊆ ω | F ⊆ A}, where F ranges over the finite subsets of ω.\n\nThe importance of Pω for this paper is explained by its following well-known properties.First, Pω is universal for CB 0 -spaces.\nProposition 2.2 A topological space X embeds into Pω iff X is a CB 0 -space.\nProof.Since Pω is a CB 0 -space, any space X homeomorphic to a subspace Pω is a CB 0 -space as well.Conversely, if {β i | i ∈ ω} is a base for a countably-based T 0 -space X, then the function e : X → Pω mapping\nx to {i ∈ ω | x ∈ β i } is a homeomorphic embedding of X into Pω. ✷\nThe second property shows that Pω is an injective object in the category of all topological spaces.\n\nProposition 2.3 [G+80, Proposition 3.5] Let Y be topological space and X be a subspace of Y .Then any continuous function f : X → Pω can be extended to a continuous function g : Y → Pω.\n\nCompletely metrisable spaces satisfy the following extension theorem by Kuratowski.\n\nProposition 2.4 [Ke95, Theorem 3.8] Let X be a metrisable space and Y be a completely metrisable space.Then any continuous function f : A → Y defined on a subset A of X can be extended to a continuous function g :\nG → Y , where G is a G δ -subset of Y with A ⊆ G.\n\nFamilies of Pointclasses\n\nHere we recall a useful technical notion of a family of pointclasses introduced in [Se11].\n\nA pointclass on X is simply a collection Γ(X) of subsets of X.We need the following \"parameterized\" version of the notion of pointclass.A family of pointclasses is a family Γ = {Γ(X)} indexed by arbitrary topological spaces X such that each Γ(X) is a pointclass on X and Γ is closed under continuous preimages, i.e. f −1 (A) ∈ Γ(X) for every A ∈ Γ(Y ) and every continuous function f : X → Y .In particular, any pointclass Γ(X) in such a family is downward closed under the Wadge reducibility on X.\nRecall that A ⊆ X is Wadge reducible to B ⊆ Y if A = f −1 (B) for some continuous function f on X.\nTrivial examples of families of pointclasses are E, F, where E(X) = {∅} and F(X) = {X} for any space X.A basic example of a family of pointclasses is given by the family O = {τ X } of the topologies of all the spaces X.\n\nFinally, we define some operations on families of pointclasses which are relevant to hierarchy theory.First, the usual set-theoretic operations will be applied to the families of pointclasses pointwise: for example, the union i Γ i of the families of pointclasses Γ 0 , Γ 1 , . . . is defined by\n( i Γ i )(X) = i Γ i (X).\nSecond, a large class of such operations is induced by the set-theoretic operations of L.V. Kantorovich and E.M. Livenson (see e.g.[Se11] for the general definition).Among them are the operations Γ → Γ σ where Γ(X) σ is the set of all countable unions of sets in Γ(X), the operation Γ → Γ c where Γ(X) c is the set of all complements of sets in Γ(X), the operation Γ → Γ d where Γ(X) d is the set of all differences of sets in Γ(X), and the operation Γ → Γ p defined by\nΓ p (X) = {pr X (A) | A ∈ Γ(N × X)} where pr X (A) = {x ∈ X | ∃p ∈ N ((p, x) ∈ A)} is the projection of A ⊆ N × X along the axis N .\nThe next subsection contains some important examples of families of pointclasses from hierarchy theory.\n\n\nClassical Hierarchies on Topological Spaces\n\nLet us recall the definition of the Borel hierarchy on arbitrary spaces introduced in [Se04].\n\nDefinition 2.5 For α < ω 1 , define the family of pointclasses Σ 0 α = {Σ 0 α (X)} by induction on α as follows: Σ 0 0 (X) = {∅}, Σ 0 1 (X) = τ X is the collection of the open sets of X, Σ 0 2 (X) = ((Σ 0 1 (X)) d ) σ is the collection of all countable unions of differences of open sets, and Σ 0 α (X) = ( β<α (Σ 0 β (X)) c ) σ (for α > 2) is the class of countable unions of sets in β<α (Σ 0 β (X)) c .The sequence {Σ 0 α (X)} α<ω 1 is called the Borel hierarchy of X.We also let Π 0 β (X) = (Σ 0 β (X)) c and ∆ 0 α (X) = Σ 0 α (X) ∩ Π 0 α (X).The classes Σ 0 α (X), Π 0 α (X), ∆ 0 α (X) are called the levels of the Borel hierarchy of X.\n\nBy the definition and remarks at the end of the previous subsection, any of Σ 0 α , Π 0 α , ∆ 0 α is a family of pointclasses.It is straightforward to check by induction on α, β that using Definition 2.5 one has the following result.\n\nProposition 2.6 For every topological space X and for all α\n< β < ω 1 , Σ 0 α (X) ∪ Π 0 α (X) ⊆ ∆ 0 β (X).\nRemark.Definition 2.5 applies to all spaces X, and Proposition 2.6 holds true in the full generality.Note that Definition 2.5 differs from the classical definition for Polish spaces (see e.g.[Ke95, Section 11.B]) only for the level 2, and that for the case of Polish spaces our definition of the Borel hierarchy is equivalent to the classical one.Notice that the classical definition cannot be applied in general to non metrizable spaces (like e.g. the non discrete ω-algebraic domains) precisely because the inclusion Σ 0 1 ⊆ Σ 0 2 may fail.Let {Σ 1 n (X)} 1≤n<ω be Luzin's projective hierarchy in X (cf.[Br13]).Using the corresponding operation on families of pointclasses from the previous subsection we have Σ 1 1 (X) = (Π 0 2 (X)) p and Σ 1 n+1 (X) = (Π 1 n (X)) p for any n ≥ 1.Let also Σ 1 0 = Π 1 0 = ∆ 1 1 .The reason why the definition of the first level of the Luzin hierarchy is distinct from the classical definition Σ 1 1 (X) = (Π 0 1 (X)) p for Polish spaces is that the inclusion Σ 0 1 (X) ⊆ (Π 0 1 (X)) p may fail in general.Any level Σ 1 n , Π 1 n , ∆ 1 n of the Luzin hierarchy is a family of pointclasses, and we have the natural inclusions among them similar to the inclusions for levels of the Borel hierarchy.\n\nLevels of the introduced hierarchies in an arbitrary space have closure properties similar to those known for the classical hierarchies in Polish spaces, in particular: Proposition 2.7 Any non-zero Σ-level of the Borel hierarchy on an arbitrary topological space is closed under finite intersections, countable unions and binary products (for the case of binary product this means that\nA ∈ Σ 0 α (X) and B ∈ Σ 0 α (Y ) imply A × B ∈ Σ 0 α (X × Y )).\nAny non-zero Σ-level of the Luzin hierarchy in an arbitrary space is closed under countable unions, countable intersections, binary products and the projection along N -axis (for the case of projection this means that\nA ∈ Σ 1 n (N × X) implies pr X (A) ∈ Σ 1 n (X)).\nWe will often use the following straightforward result which also extends the corresponding well-known facts for the classical hierarchies.\n\nProposition 2.8 Let X be a subspace of a topological space Y , A ⊆ X, and let Γ be a Σ-or a Π-level of the introduced hierarchies.Then\nA ∈ Γ(X) iff A = X ∩ B for some B ∈ Γ(Y ). If in addition X ∈ Γ(Y ), then A ∈ Γ(X) iff A ∈ Γ(Y ).\nWe will often cite the following topological complexity of the equality test\nEQ X := {(x, x) | x ∈ X} on a topological space X. Proposition 2.9 If X is a CB 0 -space then EQ X ∈ Π 0 2 (X × X). If X is a Hausdorff space, then EQ X ∈ Π 0 1 (X × X).\nProof.The first statement has been shown in [Br13].The second (well-known) statement follows from the fact that in the Hausdorff case the complement of EQ X is equal to {U × V | U, V open and disjoint} and therefore open.✷ Let Γ be a family of pointclasses and X, Y be spaces.A function f :\nX → Y is called Γ- measurable if f −1 (A) ∈ Γ(X)\nfor each open set A ⊆ Y .Note that the continuous functions coincide with the Σ 0 1 -measurable functions.We will also use the following basic property of Γ-measurable functions.Note that addition + on ordinal numbers is associative, but not commutative.\nProposition 2.10 [BY09, Lemma 1] Let X, Y be CB 0 -spaces, let α, β < ω 1 be ordinals, let f : X → Y be a Σ 0 1+α -measurable function, and let A ∈ Σ 0 1+β (Y ) and B ∈ Π 0 1+β (Y ). Then f −1 (A) ∈ Σ 0 1+α+β (X) and f −1 (B) ∈ Π 0 1+α+β (X).\nNotice that the case (α, β) = (0, 0) denotes the fact that the preimage of an open (closed) set under a continuous function is open (resp.closed).\n\n\nPolish and quasi-Polish spaces\n\nRemember that a space X is Polish if it is countably based and metrisable with a metric d such that (X, d) is a complete metric space.Important examples of Polish spaces are the Baire space, the Cantor space, the space of reals R and its Cartesian powers R n (n < ω), the closed unit interval [0, 1], the Hilbert cube [0, 1] ω and the Hilbert space R ω .\n\nBelow we will cite the following result known as Lavrentyev's Theorem:\n\nProposition 2.11 [Ke95, Theorem 3.9] Let X, Y be Polish spaces, A ⊆ X, B ⊆ Y , and let f be a homeomorphism of A onto B (equipped with the subspace topologies induced from X and Y ).Then there exist\nA * ∈ Π 0 2 (X), B * ∈ Π 0 2 (Y ) and a homeomorphism f * of A * onto B * such that A ⊆ A * , B ⊆ B * , and f * | A = f .\nA natural variant of the class of Polish spaces has recently emerged.Given a set X, call a function d from X ×X to the nonnegative reals quasi-metric whenever x = y iff d(x, y) = d(y, x) = 0, and d(x, y) ≤ d(x, z)+d(z, y) (but we don't require d to be symmetric).In particular, every metric is a quasi-metric.Every quasi-metric on X canonically induces a topology on X which is denoted by τ d , where τ d is the topology generated by the open balls\nB d (x, ε) = {y ∈ x | d(x, y) < ε} for x ∈ X and 0 < ε ∈ R. A space X is called quasi-metrisable if there is a quasi-metric on X which generates its topology. If d is a quasi-metric on X, let d be the metric on X defined by d(x, y) = max{d(x, y), d(y, x)}. A sequence {x n } n<ω in X is called d-Cauchy sequence if for every ε > 0 there is n 0 ∈ ω such that d(x n , x m ) < ε for all n 0 ≤ n ≤ m.\nWe say that the quasi-metric d on X is complete if every d-Cauchy sequence converges with respect to d (notice that this definition is coherent with the notion of completeness for a metric d, as in this case d = d).\n\nA T 0 -space X is called quasi-Polish if it is countably based and there is a complete quasi-metric on X which generates its topology.In particular, every Polish space is quasi-Polish, but by [Br13,Corollary 45] also every ω-continuous domain is quasi-Polish.De Brecht [Br13] shows that there is a reasonable DST for the quasi-Polish spaces which extends the classical DST for Polish spaces [Ke95] and the DST for ω-continuous domains [Se06,Se08] in many directions.\n\nAn important example of a quasi-Polish space is the space Pω equipped with the following quasi-metric d: if A ⊆ B then d(A, B) = 0, otherwise d(A, B) = 1/2 a where a is the smallest number in A \\ B. Note that Proposition 2.2 implies that any CB 0 -space is quasi-metrisable.\n\nDe Brecht proved the following characterization of quasi-Polish spaces (cf.[Br13, Corollary 24]): Proposition 2.12 A topological space is quasi-Polish iff it is homeomorphic to a Π 0 2 -subset of Pω (endowed with the relative topology inherited from Pω).\n\n\nAdmissible Representations and QCB 0 -spaces\n\nA representation of a space X is a surjection of a subspace of the Baire space N onto X. Usually it is denoted as a partial function from N to X.The notion of admissible representation is basic in Computable Analysis (CA).A representation δ of X is admissible, if it is continuous and any continuous function ν : Z → X from a zero-dimensional CB 0 -space Z to X is continuously reducible to δ, i.e. ν = δg for some continuous function g : Z → N .A topological space is admissibly representable if it has an admissible representation.\n\nThe notion of admissibility was introduced in [KW85] for representations of countably based spaces (in a different but equivalent formulation) and was extensively studied by many authors.In [BH02] a close relation of admissible representations of countably based spaces to open continuous representations was established.In [Sch02,Sch03] the notion was extended to non-countably based spaces and a nice characterization of the admissibly represented spaces was achieved.Namely, the admissibly represented sequential topological spaces coincide with the QCB 0 -spaces, i.e., T 0 -spaces which are topological quotients of countably based spaces.\n\nThe category QCB 0 of QCB 0 -spaces as objects and continuous functions as morphisms is known to be cartesian closed (cf.[ELS04,Sch03]).The exponential Y X to QCB 0 -spaces X, Y has the set of continuous functions from X to Y as the underlying set, and its topology is the sequentialization of the compact-open topology on Y X .By the sequentialization of a topology τ we mean the family of all sequentially open sets pertaining to this topology.(Remember that sequentially open sets are defined to be the complements of the sets that are closed under forming limits of converging sequences.)The sequentialization of τ is finer than or equal to τ .The topology of the QCB 0 -product to X and Y , which we denote by X × Y , is the sequentialization of the classical Tychonoff topology on the cartesian product of the underlying sets of X and Y .So products and exponentials in QCB 0 are formed in the same way as in its supercategory Seq of sequential topological spaces.\n\nFrom [Br13] it follows that admissible total representations are closely related to quasi-Polish spaces.The following assertion is contained among the results in [Br13].\n\nProposition 2.13 For any CB 0 -space X the following statements are equivalent:\n\n(1) X is quasi-Polish.\n\n(2) X has an open continuous total representation.\n\n(3) X has an admissible total representation.\n\n(4) X has an admissible representation whose domain is a Polish space.\n\nWe will also cite the following facts from [Sch02,Sch03].\n\nProposition 2.14 Let δ and γ be admissible representations of QCB 0 -spaces X and Y , respectively.Then f : X → Y is continuous iff f δ = γg for some partial continuous function g on N .\n\nNote that if u :⊆ N 2 → N is continuous and p ∈ N then u p = λx.u(p,x) is a partial continuous function on N .\n\nProposition 2.15 There is a partial continuous function u : ⊆ N 2 → N such that dom(u) ∈ Π 0 2 (N 2 ) and for any partial continuous function g on N there is some p ∈ N such that u p is an extension of g.\n\nThe function u in Proposition 2.15 can be chosen as the application operator of the Second Kleene Algebra.We use it to describe the construction of admissible representations for function spaces formed in QCB 0 (cf.[Sch03,Wei00]).\n\nProposition 2.16 Let δ and γ be admissible representations for QCB 0 -spaces X and Y , respectively.Then admissible representations [δ × γ] for the QCB 0 -product X × Y and [δ → γ] for the QCB 0 -exponential Y X can be defined by:\n[δ × γ]( p, q ) = (x, y) iff δ(p) = x ∧ γ(q) = y and [δ → γ](p) = f iff f δ = γu p .\nfor p, q ∈ N , x ∈ X, y ∈ Y , and f : X → Y .\n\nDe Brecht and Yamamoto showed the following property of subsets of admissibly represented countably-based spaces.\n\nProposition 2.17 [BY09, Corollary 3].Let δ be an admissible representation of a countablybased T 0 -space X, let A ⊆ X and let\n1 ≤ α < ω 1 . Then A ∈ Σ 0 α (X) iff δ −1 (A) ∈ Σ 0 α (dom(δ)).\n\nMain Technical Facts\n\nIn this section we prove a couple of facts that serve as main technical tools in the sequel, but some of these facts might be also of independent interest.\n\nThe first result generalizes the well-known fact that Pω is an injective space (see Proposition 2.2), because the continuous functions coincide with the Σ 0 1 -measurable functions.\n\nTheorem 3.1 Let Y be a topological space, let X be a subspace of Y , and let\nΓ ∈ {Σ 0 α , Σ 1 n , Π 1 n | 1 ≤ α < ω 1 , 1 ≤ n < ω}. Then any Γ-measurable function f : X → Pω can be extended to a Γ-measurable function g : Y → Pω. Proof. For any n < ω, the set ↑{n} = {A ⊆ ω | n ∈ A} is open in Pω, hence f −1 (↑{n}) ∈ Γ(X), hence f −1 (↑{n}) = X ∩ A n for some A n ∈ Γ(Y )y ∈ A n ⇔ n ∈ g(y) ⇔ g(y) ∈ ↑{n} ⇔ y ∈ g −1 (↑{n}), hence g −1 (↑{n}) = A n ∈ Γ(Y ). Since {↑{n} | n ∈ ω}\nis a subbasis in Pω and Γ(Y ) is closed under finite intersection and countable union by Proposition 2.7, g is Γ-measurable.If y ∈ X then we have\nn ∈ f (y) ⇔ f (y) ∈ ↑{n} ⇔ y ∈ f −1 (↑{n}) ⇔ y ∈ A n ⇔ n ∈ g(y), hence g(y) = f (y). ✷\nThe second result is a remote relative of Lavrentyev's Theorem (cf.Proposition 2.11).\n\nTheorem 3.2 Let A, B ⊆ Pω be subspaces of Pω, α, β < ω 1 , and let f : A → B be a Σ 0 1+αmeasurable bijection such that its inverse g = f −1 is Σ 0 1+β -measurable.Then there exist A * ∈ Π 0 1+α+µ+1 (Pω), B * ∈ Π 0 1+β+µ+1 (Pω) (where µ = max{α, β}) and a Σ 0 1+α -measurable bijection\nf * : A * → B * such that g * = f * −1 is Σ 0 1+β -measurable, A ⊆ A * , B ⊆ B * , f * | A = f and g * | B = g.\nProof.By Theorem 3.1, there are a Σ 0 1+α -measurable extension f 1 : Pω → Pω of f and a Σ 0 1+β -measurable extension g 1 : Pω → Pω of g.Consider the set\nS = {(x, y) ∈ Pω × Pω | f 1 (x) = y ∧ x = g 1 (y)}. Clearly, S = S 1 ∩ S 2 where S 1 = {(x, y) | f 1 (x) = y} and S 2 = {(x, y) | x = g 1 (y)}. Since EQ Pω ∈ Π 0 2 (Pω × Pω)\nby Proposition 2.9 and S 1 is the preimage of EQ Pω under the Let us introduce one of the main notions of this paper.\nΣ 0 1+α - measurable function (x, y) → (f 1 (x), y), S 1 ∈ Π 0 1+α+1 (Pω × Pω)\nDefinition 3.4 Let Γ be a family of pointclasses.A topological space X is called a Γ-space if X is homeomorphic to a subspace A ⊆ Pω with A ∈ Γ(Pω).The class of all Γ-spaces is denoted CB 0 (Γ).\n\nThe third result of this section extends Corollary 24 in [Br13] (which is obtained when Γ = Π 0 2 ).\nProposition 3.5 Let Γ ∈ {Π 0 2 , Σ 0 α , Π 0 α , Σ 1 n , Π 1 n | 3 ≤ α < ω 1 , 1 ≤ n <\nω} and f :⊆ X → Y be a partial continuous function from a topological space X to a Γ-space Y .Then there is a continuous extension g :⊆ X → Y of f with dom(g) ∈ Γ(X).\n\nProof.Without loss of generality we assume Y ∈ Γ(Pω), so in particular f :⊆ X → Pω.By Proposition 2.3, there is a total continuous extension h : X → Pω of f .Since Y ∈ Γ(Pω), h −1 (Y ) ∈ Γ(X).Since dom(f ) ⊆ h −1 (Y ), we can take the restriction of h to h −1 (Y ) as the desired function g. ✷\n\nThe fourth result of this section is the following extension of Lavrentyev's Theorem (see Proposition 2.11).For Γ = Π 0 2 the result gives the extension of Lavrentyev's Theorem to quasi-Polish spaces (cf.[Br13]), and Lavrentyev's Theorem is obtained if we restrict the last fact to Polish spaces.\nTheorem 3.6 Let Γ ∈ {Π 0 2 , Σ 0 α , Π 0 α , Σ 1 n , Π 1 n | 3 ≤ α < ω 1 , 1 ≤ n < ω}, X, Y be Γ-spaces, A ⊆ X, B ⊆ Y ,* | A = f . Proof. Let g = f −1 . By Proposition 3.5 there exist A 1 ∈ Γ(X), B 1 ∈ Γ(Y ) and continuous functions f 1 : A 1 → Y , g 1 : B 1 → X such that A ⊆ A 1 , B ⊆ B 1 , f 1 | A = f and g 1 | B = g. Let S = {(x, y) ∈ A 1 × B 1 | f 1 (x) = y ∧ x = g 1 (y)}. Then graph(f ) ⊆ S and S ∈ Γ(X × Y ), because A 1 × B 1 ∈ Γ(X × Y ), EQ X ∈ Π 0 2 (X × X) and EQ Y ∈ Π 0 2 (Y × Y ) by Proposition 2.9, (x, y) → (f 1 (x), y)) is a continuous function from A 1 × B 1 to Y × Y and (x, y) → (x, g 1 (y)) is a continuous function from A 1 × B 1 to X × X.\n\nLet now\nA * = {x ∈ A 1 | ∃y((x, y) ∈ A 1 × B 1 )}, B * = {y ∈ B 1 | ∃x((x, y) ∈ A 1 × B 1 )}, f * = f 1 | A and g * = g 1 | B . Then A ⊆ A * , B ⊆ B * , f * | A = f , g * | B = g and g * = f * −1 (hence f * is a homeomorphism of A * onto B * ). Since A * = {x ∈ A 1 | (x, f 1 (x)) ∈ A 1 × B 1 } and B * = {y ∈ B 1 | (g 1 (y), y) ∈ A 1 × B 1 },\nA * ∈ Γ(A 1 ) and B * ∈ Γ(B 1 ).By Proposition 2.8 we have A * ∈ Γ(X) and B * ∈ Γ(Y ).✷ Finally, we give a natural version of the previous theorem related to retracts, although this version is not used in the sequel.\n\nProposition 3.7 Let Γ, X, Y, A, B be as in the previous theorem and let r : A → B and s : B → A be continuous functions such that rs = id B (hence, B is a retract of A).Then there exist A * ∈ Γ(X), B * ∈ Γ(Y ) and continuous functions r * :\nA * → B * , s * : B * → A * such that A ⊆ A * , B ⊆ B * , r * | A = r, s * | B = s and r * s * = id B * (hence, B * is a retract of A * ).\nProof.By Proposition 3.5 there exist A 1 ∈ Γ(X), B 1 ∈ Γ(Y ) and continuous functions r 1 :\nA 1 → Y , s 1 : B 1 → X such that A ⊆ A 1 , B ⊆ B 1 , r 1 | A = r and s 1 | B = s. Let S = {(x, y) ∈ A 1 × B 1 | r 1 (x) = y = r 1 s 1 (y)} and A * = {x ∈ A 1 | (x, r 1 (x)) ∈ S}, B * = {y ∈ B 1 | (s 1 (y), y) ∈ S}, r * = r 1 | A * and s * = s 1 | B * .\nSimilarly to the previous proof one checks that these objects have the desired properties.✷ 4 Hierarchies of CB 0 -Spaces\n\nHere we introduce and study natural hierarchies of CB 0 -spaces (see Definition 3.4) as well as the Borel hierarchy and the Luzin hierarchy on Pω discussed in Subsection 2.4.\n\nDefinition 4.1 (1) By the Borel hierarchy of CB 0 -spaces we mean the sequence of classes {CB 0 (Σ 0 α )} α<ω 1 .By levels of this hierarchy we mean the classes CB 0 (Σ 0 α ) as well as the classes CB 0 (Π 0 α ) and CB 0 (∆ 0 α ).\n\n(2) By the Luzin hierarchy of CB 0 -spaces we mean the sequence of classes {CB 0 (Σ 1 n )} n<ω .By levels of this hierarchy we mean the classes CB 0 (Σ 1 n ) as well as the classes CB 0 (Π 1 n ) and CB 0 (∆ 1 n ).\n\nNote that the Borel hierarchy of spaces includes some natural classes of spaces identified earlier.E.g., by Proposition 2.12 CB 0 (Π 0 2 ) coincides with the class of quasi-Polish spaces and, by Corollary 33 in [Br13], CB 0 (Π 0 3 ) coincides with the class of CB 0 -spaces that admit a compatible bicomplete quasi-metric in the sense of [JK98].\n\nObviously, for any families of pointclasses Γ, ∆ we have: if Γ ⊆ ∆ (i.e.Γ(X) ⊆ ∆(X) for each space X) then CB 0 (Γ) ⊆ CB 0 (∆).Therefore, we have the natural inclusions for levels of the introduced hierarchies, in particular CB 0 (∆ 0 α ) ⊆ CB 0 (Σ 0 α )∩CB 0 (Π 0 α ) and CB 0 (Σ 0 α )∪CB 0 (Π 0 α ) ⊆ CB 0 (∆ 0 β ) for all α < β < ω 1 .Below we establish some basic properties of the Borel and the Luzin hierarchy of spaces, but first we show that most of levels of the introduced hierarchies of spaces are closed under retracts (for Π 0 2 the result is already known from [Br13]).\nProposition 4.2 Let Γ ∈ {Π 0 2 , Σ 0 α , Π 0 α , Σ 1 n , Π 1 n | 3 ≤ α < ω 1 , 1 ≤ n < ω}.\nThen any retract of a Γ-space is a Γ-space.\n\nProof.Let Y be a retract of a Γ-space X via a section-retraction pair (s, r) of continuous functions; we have to show that Y is a Γ-space.We may assume without loss of generality that X ∈ Γ(Pω).Since s(Y ) = {x ∈ X | sr(x) = x}, s(Y ) is the preimage of EQ X under the continuous function x → (sr(x), x).Since EQ X ∈ Π 0 2 (X ×X) by Proposition 2.9, s(Y ) ∈ Π 0 2 (X) and thus s(Y ) ∈ Γ(X) by Proposition 2.6.Proposition 2.8 yields s(Y ) ∈ Γ(Pω).Since Y is homeomorphic to s(Y ), Y is a Γ-space.✷ Now we establish an interesting property of the introduced hierarchies of spaces which implies the non-collapse property.\n\n\nProposition 4.3 For any countable ordinal\nα ≥ 2, CB 0 (Σ 0 α ) ∩ CB 0 (Π 0 α ) = CB 0 (∆ 0 α ). For any positive integer n, CB 0 (Σ 1 n ) ∩ CB 0 (Π 1 n ) = CB 0 (∆ 1 n ).\nThis proposition is based on the following lemma.\nLemma 4.4 Let Γ ∈ {Π 0 2 , Σ 0 α , Π 0 α , Σ 1 n , Π 1 n | 3 ≤ α < ω 1 , 1 ≤ n < ω} and let X be a subspace of Pω. Then X ∈ CB 0 (Γ) iff X ∈ Γ(Pω).\nProof.If the underlying set of X is in Γ(Pω), then X is a Γ-space by Definition 3.4.Conversely, let X be a Γ-space.Then there is some B ∈ Γ(Pω) such that B (endowed with the subspace topology inherited from Pω) is homeomorphic to X. Let h : X → B be a homeomorphism.We apply Theorem 3.6 to extend h to a homeomorphism h * : X * → B * , where X * and B * are sets in Γ(Pω) with X ⊆ X * and B ⊆ B * .By Proposition 2.8 we have B ∈ Γ(B * ).Hence X ∈ Γ(X * ), because X = (h * ) −1 (B).Proposition 2.8 yields X ∈ Γ(Pω).✷ Now we are ready to give the proof of Proposition 4.3.\n\nProof.The inclusions from right to left are obvious.It remains to check CB 0 (Γ)\n∩ CB 0 (Γ c ) ⊆ CB 0 (Γ ∩ Γ c ) for each Γ ∈ {Π 0 α , Π 1 n | 2 ≤ α < ω 1 , 1 ≤ n < ω}. Let Z ∈ CB 0 (Γ) ∩ CB 0 (Γ c ).\nThen Z is homeomorphic to some subspaces A, B of Pω with A ∈ Γ(Pω) and B ∈ Γ c (Pω).Hence B is a Γ-space by being homeomorphic to A. Lemma 4.4 yields B ∈ Γ(Pω).Therefore B ∈ (Γ ∩ Γ c )(Pω) and hence Z ∈ CB 0 (Γ ∩ Γ c ). ✷ Corollary 4.5 The Borel hierarchy and the Luzin hierarchy of CB 0 -spaces do not collapse.More precisely CB 0 (Σ 0 α ) ⊆ CB 0 (Π 0 α ) for each countable ordinal α ≥ 2, and CB 0 (Σ 1 n ) ⊆ CB 0 (Π 1 n ) for each positive integer n.\n\nProof.Proofs for both hierarchies are similar, so consider only the Borel hierarchy.According to [Se05a], there is a set A in Σ 0 α (Pω) \\ Π 0 α (Pω).The space A (with the topology induced from Pω) is obviously a Σ 0 α -space.If it were a Π 0 α -space, then by Lemma 4.4 we would have A ∈ Π 0 α (Pω), a contradiction.✷ 5 Hierarchies of QCB 0 -Spaces\n\nAs shown in [Sch03], any QCB 0 -space has an admissible representation.Here we introduce and study natural hierarchies of QCB 0 -spaces induced by this fact.For any representation δ of a space X, let EQ(δ) := {(p, q) ∈ N 2 | p, q ∈ dom(δ) ∧ δ(p) = δ(q)}.\n\nDefinition 5.1 (1) Let Γ be a family of pointclasses.A topological space X is called Γrepresentable if X has an admissible representation δ with EQ(δ) ∈ Γ(N × N ).The class of all Γ-representable spaces is denoted QCB 0 (Γ).\n\n(2) By the Borel hierarchy of QCB 0 -spaces we mean the sequence {QCB 0 (Σ 0 α )} α<ω 1 .By levels of this hierarchy we mean the classes QCB 0 (Σ 0 α ) as well as the classes QCB 0 (Π 0 α ) and QCB 0 (∆ 0 α ).\n\n(3) By the Luzin hierarchy of QCB 0 -spaces we mean the sequence {QCB 0 (Σ 1 n )} n<ω .By levels of this hierarchy we mean the classes QCB 0 (Σ 1 n ) as well as the classes QCB 0 (Π 1 n ) and QCB 0 (∆ 1 n ).\n\nThe next assertion establishes an equivalent simpler definition of most levels for the case of CB 0 -spaces.\nProposition 5.2 (1) Let Γ ∈ {Π 0 2 , Σ 0 α , Π 0 α , Σ 1 n , Π 1 n | 3 ≤ α < ω 1 , 1 ≤ n < ω} and let X be a CB 0 -space. Then X is Γ-representable iff X has an admissible representation δ with dom(δ) ∈ Γ(N ). (2) Let Γ ∈ {Π 0 1 , Σ 0 α , Π 0 α , Σ 1 n , Π 1 n | 2 ≤ α < ω 1 , 1 ≤ n <\nω} and let X be a Hausdorff space.Then X is Γ-representable iff X has an admissible representation δ with dom(δ) ∈ Γ(N ).\n\nProof.The only-if-part of the first statement holds for any topological space X and any family of pointclasses Γ.Indeed, let δ be an admissible representation of X with EQ(δ) ∈ Γ(N ×N ).Since dom(δ) is the preimage of EQ(δ) under the continuous function x → (x, x), dom(δ) ∈ Γ(N ).\n\nConversely, let δ be an admissible representation of X with dom(δ) ∈ Γ(N ).By Proposition 2.9, EQ X ∈ Π 0 2 (X × X), so EQ(δ) is a Π 0 2 -set in dom(δ) × dom(δ) by the continuity of δ.The set dom(δ) × dom(δ) is a Γ-subset of N × N by being the intersection of the sets {(x, y)\n∈ N × N | x ∈ dom(δ)} ∈ Γ(N × N ) and {(x, y) ∈ N × N | y ∈ dom(δ)} ∈ Γ(N × N )\n. By Propositions 2.6 and 2.8 we obtain EQ(δ) ∈ Γ(N × N ).\n\nThe second statement follows similarly by taking into account that EQ X is closed in X × X by Proposition 2.9, if X is a Hausdorff space.✷\n\nThe main advantage of the hierarchies of this section over the hierarchies from the previous section is that they include many natural non-countably based spaces, in particular the spaces of Kleene-Kreisel continuous functionals, as we will see later.\n\nObviously, for any families of pointclasses Γ, ∆ we have: Γ ⊆ ∆ implies QCB 0 (Γ) ⊆ QCB 0 (∆), hence we have the natural inclusions for levels of the introduced hierarchies, in particular\nQCB 0 (∆ 0 α ) ⊆ QCB 0 (Σ 0 α ) ∩ QCB 0 (Π 0 α ) and QCB 0 (Σ 0 α ) ∪ QCB 0 (Π 0 α ) ⊆ QCB 0 (∆ 0 β\n) for all α < β < ω 1 .We will show that these hierarchies do not collapse, but first we establish the closure of the levels under retracts.\nProposition 5.3 Let Γ ∈ {Σ 0 α , Π 0 α , Σ 1 n , Π 1 n | 1 ≤ α < ω 1 , 1 ≤ n < ω}.\nThen any retract of a Γ-representable space is a Γ-representable space.\n\nProof.Let Y be a retract of a Γ-representable space X via a section-retraction pair (s, r) of continuous functions; we have to show that Y is a Γ-representable space.Let δ be an admissible representation for X with EQ(δ) ∈ Γ(N ×N ).Then rδ is clearly an admissible representation for Y , so it suffices to show that EQ(rδ) ∈ Γ(N ×N ).Since s is an injection, EQ(rδ) is the preimage of EQ(δ) under the continuous function (x, y) → (sr(x), sr(y)).Therefore, EQ(rδ) ∈ Γ(N × N ).✷\n\nThe following observation is well-known.Proof.Let Y be a retract of D via a section-retraction pair (s, r) of continuous functions.Then r :⊆ N → Y is admissible, because any continuous function ν : Z → Y defined on a zero-dimensional CB 0 -space Z is reducible to r via sν.\n\nConversely, assume that r ⊆ : N → Y is an admissible representation of Y .Then the identity function id Y is reducible to r via some continuous function s : Y → D. Then id Y = rs, hence Y is a retract of D via the continuous section s and the continuous retraction r. ✷ Finally, we establish the non-collapse property of the introduced hierarchies of spaces.\n\nTheorem 5.5 The Borel hierarchy and the Luzin hierarchy of QCB 0 -spaces do not collapse.More precisely, QCB 0 (Σ 0 α ) ⊆ QCB 0 (Π 0 α ) for each countable ordinal α ≥ 2, and QCB 0 (Σ 1 n ) ⊆ QCB 0 (Π 1 n ) for each positive integer n.\n\nProof.Proofs for both hierarchies are similar, so consider only the Borel hierarchy.As is well-known Since N is a Π 0 2 -space, hence a Γ-space, we can use Theorem 3.6 (applied to X ′ = Y ′ = N ) to extend the homeomorphism s : Y → S to a homeomorphism s * between larger sets Y * , S * ∈ Π 0 2 (N ).By Proposition 2.6 we have Y * , S * ∈ Π 0 α (N ).Since S ∈ Π 0 α (S * ) by Proposition 2.8,\n[Ke95], there is a set Y in Σ 0 α (N ) \\ Π 0 α (N ). Consider Y as a subspace of N . Since id Y is an admissible representation for Y , Y ∈ QCB 0 (Σ 0 α ). Suppose for a contradiction that Y ∈ QCB 0 (Π 0 α ),Y = s * −1 (S) ∈ Π 0 α (Y * ). Therefore Y ∈ Π 0 α (N ) by Proposition 2.8, a contradiction. ✷ Remark 5.6\nThe spaces Y witnessing the non-collapse property above are rather artificial.In Theorem 7.7 we will find very natural spaces witnessing the non-collapse of the Luzin hierarchy of QCB 0 -spaces.\n\n\nRelating the Hierarchies\n\nIn this section we establish close relationships of the hierarchies of CB 0 -spaces with the corresponding hierarchies of QCB 0 -spaces.The next result implies that any level of a hierarchy of QCB 0 -spaces extends the corresponding level in the corresponding hierarchy of CB 0 -spaces.\nProposition 6.1 Let Γ ∈ {Π 0 2 , Σ 0 α , Π 0 α , Σ 1 n , Π 1 n | 3 ≤ α < ω 1 , 1 ≤ n < ω}. Then CB 0 (Γ) ⊆ QCB 0 (Γ).\nProof.It suffices to show that any space X ∈ Γ(Pω) is a Γ-representable space.Let ρ : N → Pω be the total admissible representation of Pω defined by ρ(x) = {n | ∃i(x(i) = n + 1)} (see e.g.[Wei00,Br13] for details).Then the restriction of ρ to ρ −1 (X) ∈ Γ(N ) is an admissible representation of X.Since X is a CB 0 -space, X is a Γ-representable space by Proposition 5.2.✷\n\nIn particular, we have CB 0 (Γ) ⊆ QCB 0 (Γ) ∩ CB 0 for each level Γ of the Borel hierarchy or the Luzin hierarchy.The main question of this section is: for which levels Γ we have the equality CB 0 (Γ) = QCB 0 (Γ) ∩ CB 0 ?Proposition 2.12 implies that the equality holds for Γ = Π 0 2 .The next result implies that the equality holds for all zero-dimensional CB 0 -spaces.\nProposition 6.2 Let Γ ∈ {Π 0 2 , Σ 0 α , Π 0 α , Σ 1 n , Π 1 n | 3 ≤ α < ω 1 , 1 ≤ n < ω} and X be a zero- dimensional space in QCB 0 (Γ) ∩ CB 0 . Then X ∈ CB 0 (Γ).\nProof.Let r : D → X be an admissible representation of X with D ∈ Γ(N ).By Lemma 5.4 there is a continuous function s :\nX → D satisfying rs = id X . Then s(X) = {z ∈ D | sr(z) = z} ∈ Π 0 2 (D) by Proposition 2.9, hence s(X) ∈ Γ(N ) by Proposition 2.8. Since N is a Π 0 2 - space, there is a homeomorphism f of N onto a subspace Y of Pω with Y ∈ Π 0 2 (Pω).\nAs f s is a homeomorphism of X onto f s(X), we have f s(X) ∈ Γ(Y ).Propositions 2.6 and 2.8 yield f s(X) ∈ Γ(Pω).Therefore X ∈ CB 0 (Γ).✷ By the next theorem the equality CB 0 (Γ) = QCB 0 (Γ) ∩ CB 0 holds for almost all levels.For finite levels, this was pointed out to us by Matthew de Brecht.We thank him for giving the permission to use his proof of Theorem 6.3(2).Note that his proof may be used to obtain also a proof of Theorem 6.3(1) which is slightly different from ours.\nTheorem 6.3 (1) For any level Γ ∈ {Σ 0 α , Π 0 α , Σ 1 n , Π 1 n | ω ≤ α < ω 1 , 1 ≤ n < ω}, we have QCB 0 (Γ) ∩ CB 0 = CB 0 (Γ) .\n(2) For all natural numbers m ≥ 2 and n ≥ 3, we have\nQCB 0 (Π 0 m ) ∩ CB 0 = CB 0 (Π 0 m ) and QCB 0 (Σ 0 n ) ∩ CB 0 = CB 0 (Σ 0 n ) .\nProof.\n\n(\n) Let Γ ∈ {Σ 0 α , Π 0 α , Σ 1 n , Π 1 n | ω ≤ α < ω 1 , 1 ≤ n < ω}.1\nWe have already seen CB 0 (Γ) ⊆ QCB 0 (Γ) ∩ CB 0 .\n\nLet X ∈ QCB 0 (Γ) ∩ CB 0 .Then there is an admissible representation δ : D → X of X with D ∈ Γ(N ).By Proposition 2.1 we may assume w.l.o.g.X to be a subspace of Pω.We have to show X ∈ CB 0 (Γ).\n\nLet χ : Pω → C be the bijection between Pω and the Cantor space C = 2 ω that sends any A ⊆ ω to its characteristic function χ A .Obviously, χ is Σ 0 2 -measurable while its inverse χ −1 is continuous (i.e.Σ 0 1 -measurable).Let σ = χ| X be the restriction of χ to X ⊆ Pω.By Proposition 2.8, σ is a Σ 0 2 -measurable bijection between X and σ(X) ⊆ C such that σ −1 is Σ 0 1 -measurable.Since σ −1 is continuous as a function from the zero-dimensional CB 0 -space σ(X) to X and δ is admissible, there is a continuous function g : σ(X) → N with σ −1 = δg.Then id X = σ −1 σ = δgσ, hence the function h = gσ satisfies h(X) = {y ∈ D | hδ(y) = y}.Thus, h is a Σ 0 2 -measurable bijection between X and H = h(X), whereas its inverse h −1 = δ| h(X) is Σ 0 1 -measurable (i.e.continuous).By Proposition 6.2 we may w.l.o.g.assume that D is a subspace of Pω and D ∈ Γ(Pω).Since H is the preimage of EQ D ∈ Π 0 2 (D × D) under the Σ 0 2 -measurable function y → (hδ(y), y), we have H ∈ Π 0 3 (D) by Proposition 2.10 and hence H ∈ Γ(D) by Proposition 2.6.Proposition 2.8 yields H ∈ Γ(Pω).By Theorem 3.2 and Corollary 3.3, there exist H * ∈ Π 0 3 (Pω), X * ∈ Π 0 4 (Pω), a continuous bijection δ * : H * → X * and a Σ 0 2 -measurable bijection h * : X * → H * such that δ * is the inverse of h * , δ * is an extension of h −1 and h * is an extension of h.Since H ∈ Γ(H * ) by Proposition 2.8 and X = h * −1 (H), Proposition 2.10 yields us X ∈ Γ(X * ).By Propositions 2.6 and 2.8, we obtain X ∈ Γ(Pω), hence X ∈ CB 0 (Γ).This completes the proof.\n\n(2) Let δ be an admissible representation of a subspace X of Pω such that dom(δ) ∈ Γ(N ), where Γ ∈ {Π 0 m , Σ 0 n | m ≥ 2, n ≥ 3}.Let ρ be the total admissible representation of Pω defined in the proof of Proposition 6.1.By Proposition 2.17 it is enough to show that ρ −1 (X) is in Γ(N ).Since the corestriction of ρ to X is continuous, there is a continuous function h : ρ −1 (X) → N satisfying ρ(p) = δh(p) for all p ∈ ρ −1 (X).As Pω is an injective space and N is a Polish space, δ and h can be extended to continuous functions δ * : N → Pω and h * : G → N , where G is a Π 0 2 -subset of N , see Propositions 2.3 and 2.4.By Proposition 2.9,\nA := {p ∈ G | ρ(p) = δ * h * (p)} is a Π 0\n2 -subset of G and thus of N .It is easy to verify that ρ −1 (X) is the intersection of A with (h * ) −1 (dom(δ)).By Proposition 2.7 and 2.8 this implies that ρ −1 (X) is Γ-subset of N .Proposition 2.17 yields that X is a Γ-subset of Pω, i.e., X ∈ CB 0 (Γ).✷ required function g : N × ω Y → N in such a way that g maps (x, H) to x, if H is a witness for x.Otherwise g assigns to (x, H) some element of a countable dense subset of M .\n\nTo construct an appropriate dense subset of M , we choose for every finite sequence w in the set W := {x <k | x ∈ M, k ∈ ω} some α(w) ∈ M such that α(w) has w as an initial segment.Clearly, {α(w) | w ∈ W } is dense in M .\n\nBy being a QCB 0 -space, Y has a countable dense subset {β j | j ∈ ω} (see Proposition 3.3.1 in [Sch03]).We construct a sequence {C k } k of subsets of the QCB 0 -space N ×ω Y by C 0 := N ×ω Y and C k := (x, H) ∈ N × ω Y x <k ∈ W and (x, β j ) ∈ D H(β j ) for all j ≤ k for all k > 0. The set C k is clopen in the Tychonoff topology on N × ω Y and thus in the QCB 0 -topology, because C k is the union of the clopen sets\n(w • N ) × ω Y ∩ k j=0 {x ∈ N | (x, β j ) ∈ D a j } × {H ∈ ω Y | β j ∈ H −1 {a j }} ,\nwhere w ranges over the finite sequence of length k in W and a 0 , . . ., a k vary over the natural numbers.Note that {H ∈ ω Y | β j ∈ H −1 {a j }} is clopen even in the compact-open topology on ω Y and therefore in the QCB 0 -topology on ω Y , which is finer than the former.\n\nNext we define a function ℓ\n: N × ω Y → ω ∪ {∞} by ℓ(x, H) := ∞ if (x, H) ∈ k∈ω C k max{k ∈ ω | (x, H) ∈ C k } otherwise We use {C k } k and ℓ to define our function g : N × ω Y → N by g(x, H) := x if (x, H) ∈ k∈ω C k α(x <ℓ(x,H) ) otherwise Note that ∞ = i ≤ ℓ(x, H) implies x <i ∈ W and g(x, H) <i = x <i .\nFirst we show the continuity of g.Let O be an open set of the Baire space and let (x, H) ∈ g −1 (O).Then there is some m ∈ ω such that g(x, H) <m • N ⊆ O.\n\nCase 1: Assume ℓ(x, H) = ∞.Then (x, H) ∈ k∈ω C k .We define a set U by\nU := (x <m • N × ω Y ) ∩ m i=0 C i Then U is open and all (x ′ , H ′ ) ∈ U satisfy ℓ(x ′ , H ′ ) ≥ m and g(x ′ , H ′ ) <m = g(x, H) <m = x <m . Hence (x, H) ∈ U ⊆ g −1 (O).\nCase 2: Assume ℓ(x, H) < ∞.Let k := ℓ(x, H) and define a set U by\nU := (x <k • N × ω Y ) ∩ k i=0 C i \\ C k+1 Then U is open, (x, H) ∈ U and all (x ′ , H ′ ) ∈ U satisfy even g(x ′ , H ′ ) = g(x, H) = α(x <k ) ∈ U .\nWe conclude that in both cases g is continuous in the point (x, H).Therefore g is continuous, even with respect to the Tychonoff product on the set N × ω Y , which is coarser than the QCB 0 -topology.Now we show M ⊆ rng (g).Let x ∈ M .We define a function H x : Y → ω by\nH x (y) := min{k ∈ ω | (x, y) ∈ D k } .\nNote that H x is total, because we have f (y) = x and therefore (x, y) ∈ k∈ω D k for every y ∈ Y .Furthermore, H is continuous in every point y ∈ Y , because the set\nU := b ∈ Y (x, b) ∈ D Hx(y) \\ Hx(y)−1 i=0 D i satisfies y ∈ U ⊆ H −1 x {H x (Y . Since (x, H x ) ∈ k∈ω C k , we have g(x, H x ) = x.\nIt remains to show rng(g) ⊆ M .So let x ∈ rng (g).The only interesting case is x / ∈ {α(w) | w ∈ W }. Then there is some continuous function H : Y → ω such that g(x, H) = x and thus (x, H) ∈ k∈ω C k .Suppose for a contradiction that there is some y ∈ Y such that (x, y) / ∈ D H(y) .Then the set\nV := {b ∈ Y | (x, b) / ∈ D H(y) } ∩ H −1 {H(y)} is non-empty and open in Y , because D H(y) is closed and H is continuous. So there exists some k ∈ ω with β k ∈ V . The element β k satisfies (x, β k ) / ∈ D H(y) and H(β k ) = H(y). But since (x, H) ∈ C k , we have (x, β k ) ∈ D H(β k ) , a contradiction.\nWe conclude (x, y) ∈ D H(y) and thus f (y) = x for every y ∈ Y .Therefore x / ∈ rng (f ) and x ∈ M .So g satisfies the required properties.✷ Using the cartesian closedness of QCB 0 , we define a sequence of spaces {N k } k<ω by induction on k as follows: N 0 := ω and N k + 1 := ω N k , where ω denotes the space of natural numbers endowed with the discrete topology.Obviously N 1 is equal to the Baire space N .The space N k is referred to as the sequential space of (Kleene-Kreisel) continuous functionals of type k.For any k ≥ 2, the sequential topology on N k is strictly finer than the corresponding compactopen topology [Hy79].Furthermore it is neither zero-dimensional nor regular [Sch09].Any of these spaces has a natural admissible representation.From Theorem 7.1 we obtain:\nCorollary 7.3 For any integer k ≥ 1, N k ∈ QCB 0 (Π 1 k−1 ). Proof. Obviously, N 0 , N 1 ∈ QCB 0 (Π 0 0 ) ⊆ QCB 0 (∆ 1 1 ). By Theorem 7.1, N 2 ∈ QCB 0 (Π 1 1 ). Assuming by induction that N k ∈ QCB 0 (Π 1 k−1 ) for a given k ≥ 2, we obtain N k + 1 ∈ QCB 0 (Π 1\nk ) by Theorem 7.1.✷ We remark that Lemma 2.34 in [No80] implies that N k has an admissible representation such that its domain is even in the effective version Π 1 k−1 of Π 1 k−1 (N ).This follows from the fact that the function which maps \"associates\" for N k to the encoded functionals forms an admissible representation.\n\nNote that the continuous functionals (also known under the somewhat misleading name \"countable functionals\") were first defined in [Kl959,Kr59] independently by S. Kleene and G. Kreisel.Their definitions look different from each other, as well as from the definition above, although all Proof.We only have to check that QCB 0 (P) is closed under binary products and exponentiation formed in the supercategory QCB 0 .For exponentiation this follows from Theorem 7.1.It is an easy exercise to show that the product representation [δ × γ] constructed in Proposition 2.16 satisfies EQ([δ × γ]) ∈ Σ 1 n (N 2 ), whenever EQ(δ) and EQ(γ) are Σ 1 n -subsets of N 2 .✷ It turns out that QCB 0 (P) is in a sense the smallest cartesian closed subcategory of QCB 0 containing ω. Proposition 8.2 There is no full cartesian closed subcategory C of QCB 0 such that C inherits binary products from QCB 0 , contains the discrete space ω of natural numbers and is contained itself in QCB 0 (Σ 1 n ) for some 1 ≤ n < ω.\n\nProof.We show at first that any cartesian closed subcategory D of QCB 0 that contains ω and inherits binary products from QCB 0 has the property that exponentials formed in D are homeomorphic to the corresponding QCB 0 -exponentials.\n\nLet E be an exponential formed in D to spaces X, Y ∈ D. Remember that this means that there exists a continuous evaluation function eval : E × X → Y such that for every space Z ∈ D and every continuous function h : Z × X → Y there is a unique continuous function ĥ : Z → E such that h(z, x) = eval ĥ(z)(x) for all z ∈ Z and x ∈ X.\n\nFor every continuous function f : X → Y it follows from the uniqueness condition (applied to Z = ω and the continuous function (i, x) → f (x)) that there exists a unique element t(f ) ∈ E satisfying f (x) = eval (t(f ), x) for all x ∈ X.Conversely, since Y X is an exponential to X and Y in the supercategory QCB 0 , there is a continuous function Ev : E → Y X satisfying Ev (e)(x) = eval (e, x) for all e ∈ E and x ∈ X.Clearly, Ev (t(f )) = f for every f ∈ Y X .The uniqueness condition implies t(Ev (e)) = e for every e ∈ E. Hence Ev is a continuous bijection with t as its inverse.\n\nTo show that t is a continuous function from Y X to E, let {f i } i be a sequence that converges in Y X to some function f ∞ ∈ Y X .This can be reformulated by stating that the function F : N ∞ × X → Y defined by F (i, x) := f i (x) for i ∈ ω ∪ {∞} and x ∈ X is continuous, where N ∞ denotes the one-point compactification of the discrete natural number with ∞ as the infinity point.(This is a well-known property of exponentials in the category of sequential topological spaces and therefore in the category QCB 0 , see e.g.[ELS04,Sch03]).\n\nBelow we will show that N ∞ is a continuous retract of the exponential E N formed in D to X = Y = ω.So let (s, r) be a continuous section-retraction pair.We define G : E N × X → ω by G(p, x) := F (r(p), x).Clearly, G is continuous.Since E is an exponential in D, there is a continuous function G : E N → E satisfying eval ( G(p), x) = G(p, x) for all p ∈ E N and x ∈ X.\n\nThe uniqueness condition implies G(p) = t(f r(p) ).As {i} i converges to ∞ in N ∞ , { G(s(i))} i converges to G(s(∞)) in E. Clearly, we have G(s(i)) = t(f i ) for all i ∈ ω ∪ {∞}.Therefore {t(f i )} i converges to t(f ∞ ) in E.\n\nWe conclude that t is sequentially continuous.Since Y X is a sequential topological space, t is even continuous in the topological sense.This means that Y X is homeomorphic to E. Now we show that N ∞ is indeed a retract of E N .As in the general case, there is a continuous bijection Ev N : E N → N .Hence E N is Hausdorff, but not discrete by being an uncountable space with a dense countable subset (see Proposition 3.3.1 in [Sch03]).So there exists an injective sequence {z i } i converging in E N to some point z ∞ ∈ E N \\ {z i | i ∈ ω}.Since N is zero-dimensional and Ev N is injective, there is a sequence {C k } k of clopen sets in N satisfying Then both functions are continuous and satisfy rs = id N∞ .Hence N ∞ is a continuous retract of E N .\n\nFinally, suppose that C were a cartesian closed subcategory of QCB 0 with the desired properties.By the above statement, C contains a space homeomorphic to N k for all k ∈ ω.In particular, there is a space E ∈ C homeomorphic to N n + 1 .Hence N n + 1 has an admissible representation δ such that EQ(δ) ∈ Σ 1 n (N 2 ).This contradicts Theorem 7.7.✷\n\n\nRemark 8.3\n\nThe assumption ω ∈ C in the last proposition is essential, because several cartesian closed categories of ω-algebraic domains are known [Ju90], which are all QCB 0 (Π 0 2 )-spaces by being quasi-Polish.\n\n\nConclusion\n\nHopefully, the results of this paper show that the introduced hierarchies are natural and interesting for CA and DST.The study of these hierarchies is of course in the very beginning and many natural questions remain open.In particular, we would like to see more natural and important witnesses for the non-collapse property of the hierarchies.A systematic development of DST for the introduced classes of spaces (in particular, for the projective QCB 0 -spaces) seems also a natural direction of future research.\n\n\n\nby Proposition 2.8.Define the function g : Y → Pω by g(y) = {n | y ∈ A n }.Then for any y ∈ Y we have\n\n\n\n\nby Proposition 2.10.Similarly, S 2 ∈ Π 0 1+β+1 (Pω × Pω) and therefore S ∈ Π 0 1+µ+1 (Pω × Pω) by Propositions 2.6 and 2.7.Now let A * = {x ∈ Pω | (x, f 1 (x)) ∈ S} and B * = {y ∈ Pω | (g 1 (y), y) ∈ S}.Since the function x → (x, f 1 (x)) is Σ 0 1+α -measurable and A * is the preimage of S under this function, A * ∈ Π 0 1+α+µ+1 (Pω) by Proposition 2.10.Similarly, B * ∈ Π 0 1+β+µ+1 (Pω).Then the sets A * , B * and the functions f * = f 1 | A * , g * = g 1 | B * have the desired properties.✷ The next fact is a special case of the previous theorem.Corollary 3.3 If we take in the previous theorem α = 0 and β = 1 then we obtain A * ∈ Π 0 3 (Pω) and B * ∈ Π 0 4 (Pω).\n\n\n\n\nand let f be a homeomorphism of A onto B. Then there exist A * ∈ Γ(X), B * ∈ Γ(Y ) and a homeomorphism f * of A * onto B * such that A ⊆ A * , B ⊆ B * , and f\n\n\nLemma 5. 4\n4\nLet r : D → Y be a continuous function from a subspace D of N to a zerodimensional CB 0 -space Y .Then r (viewed as a partial function from N to Y ) is an admissible representation for Y iff there is a continuous function s : Y → D satisfying rs = id Y .\n\n\n\n\nso there is an admissible representation δ : D → Y for Y with D ∈ Π 0 α (N ).By Lemma 5.4 there is a continuous function s : Y → D satisfying δs = id Y .Since D is Hausdorff, S = s(Y ) is a closed subset of D, hence S ∈ Π 0 α (D) by Proposition 2.7.By Proposition 2.8 we obtain S ∈ Π 0 α (N ).\n\n\n\n\ny)} and is open by being the preimage of the open set D Hx(y) \\ Hx(y)−1 i=0 D i under the continuous map b → (x, b).Therefore H x is an element of the function space ω\n\n\n\n\nEv N (z k ) ∈ C k and {Ev N (z i ) | i ≤ ∞, i = k} ⊆ N \\ C k for every k ∈ ω.We define functions s : N ∞ → E N and r : E N → N ∞ by s(i) := z i and r(z) := min ∞, k ∈ ω z ∈ Ev −1 N (C k ) .\n\nAcknowledgementsWe thank the referees for valuable remarks.We are grateful to Matthew de Brecht for permission to include his proof of Theorem 6.3(2).* Victor Selivanov has been supported by a Marie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme.Both authors have been supported by DFG-RFBR Grant 436 RUS 113/1002/01.The Luzin Hierarchy and Continuous FunctionalsIn this section we establish close relations of the Luzin hierarchy to the continuous functionals of finite types.We start by relating the exponentiation operation on admissibly represented spaces (see Proposition 2.16) to the Luzin hierarchy.Theorem 7.1 Let k ∈ ω, X ∈ QCB 0 (Π 1 k ) and Y ∈ QCB 0 (Σ 1 k ).Then Y X ∈ QCB 0 (Π 1 k+1 ).Proof.Let δ and γ be admissible representations of X and Y respectively such that EQ(δ) ∈ Π 1 k (N 2 ) and EQ(γ) ∈ Σ 1 k (N 2 ).By Proposition 2.16 it suffices to show that EQ([δ → γ]) ∈ Π 1 k+1 (N 2 ).From the definition of [δ → γ] we obtain (p, q) ∈ EQ([δ → γ]) ⇐⇒ for all (x, y) ∈ EQ(δ) u p (x), u p (y) , u q (x), u q (y) , u p (x), u q (x) ∈ EQ(γ).So we havewhere M := (p, q, x, y) ∈ N 4 (x, y) / ∈ EQ(δ) or there is some (a, b, c, d) ∈ N 4 such that (p, x, a), (p, y, b), (q, x, c), (q, y, d) ∈ graph(u) and (a, b), (c, d), (a, c) ∈ EQ(γ) .Since the universal function u :⊆ N 2 → N is continuous and has a Π 0 2 -set as domain, graph(u) is a Π 0 2 (N 3 )-set.This implies that M is a Σ 1 n (N 4 )-set by Propositions 2.6 and 2.7.We conclude that EQ([δ → γ]) is a Π 1 n+1 (N 2 )-set.✷ The next result (which is known from[Kr59]for the particular case Y = N k ) is the key technical fact of this section.Theorem 7.2 Let Y be a QCB 0 -space and let f : Y → N be a continuous function with rng (f ) = N .Then there exists a continuous function g : N × ω Y → N with rng(g) = N \\ rng(f ).We remark that by N × ω Y we mean the product formed in the category of QCB 0 -spaces (see Subsection 2.6), the topology of which is finer than (or equal to) the Tychonoff topology.However, the function g constructed in the proof is even continuous w.r.t. the Tychonoff topology.where (j, a, b) varies over all triples of natural numbers with a = b.Let {D i } i denote a sequence consisting of these clopen sets.So we have NEQ = i∈ω D i and x ∈ M ⇐⇒ ∀y ∈ Y.∃i ∈ ω.(x, y) ∈ D i .Motivated by the above equivalence, we call a continuous function H : Y → ω a witness for an element x ∈ M , if (x, y) ∈ D H(y) holds for all y ∈ Y .The idea of the proof is to construct the three definitions are known to be equivalent.Another equivalent definition in terms of domains was given in[Er74].Additional information may be found in[No80,No99].The following observation follows from the cartesian closedness of QCB 0 .Lemma 7.4 For k ≥ 1, the QCB 0 -spaces N k − 1 and N × N k are continuous retracts of N k .Proof.As QCB 0 is a cartesian closed category, for all QCB 0 -spaces X, Y, Z such that X is a continuous retract of Y the function space Z X is a continuous retract of Z Y and the QCB 0product X ×Z is a continuous retract of Y ×Z.Clearly, ω is a continuous retract of N = N 1 .So the first statement follows by induction on k.A further induction establishes N as a continuous retract ofhence N k and N k × N k are isomorphic in QCB 0 and thus homeomorphic.We conclude that N × N k is a continuous retract of N k .✷ From Theorem 7.2 and Lemma 7.4 we can easily infer the following nice result.Proposition 7.5 For any positive integer k and for any non-empty setProof.We proceed by induction on k ≥ 1.For k = 1 the claim is well-known (cf.Section 14 in[Ke95]).Let M ∈ Σ 1 k+1 (N ).Then there is some setwhere •, • denotes a canonical homeomorphism from N 2 to N .We setThe induction hypothesis yields us a continuous function f B : N k → N such that rng (f B ) = B.By Theorem 7.2 there exists a continuous function g : N × N k + 1 → N such that rng(g) = A. By Lemma 7.4 there is a continuous retraction r : N k + 1 → N × N k + 1 .Since r is surjective, A is also the range of gr.Using the unique continuous function π2 (gr(z)).Then f is continuous and satisfiesHence M is the range of f .✷The last result can be even improved to the following characterization of levels of the Luzin hierarchy in terms of the Kleene-Kreisel continuous functionals.Theorem 7.6 Let k be a positive integer and B a non-empty subset of N .ThenProof.The only-if-part is given by Proposition 7.5.For the if-part let f : N k → N be a continuous function.According to Corollary 7.3, N k has an admissible representation δ such that its domain dom(δ) is a Π 1 k−1 -set.We define a partial function g : N → N by g(p) := f δ(p).Then g is continuous.So it can be extended to a continuous function g * : X → N such that X is a Π 0 2 -subset of N (cf.Proposition 2.15 or 3.5).The graph of g * is a closed set in X × N and therefore a Π 0 2 -set in N ×N .By Propositions 2.6 and 2.7, the set Finally, we relate continuous functionals to the Luzin hierarchy of QCB 0 -spaces (for a similar relationship see[No81]).The next result provides the exact estimation of the spaces of continuous functionals of finite types in the Luzin hierarchy of QCB 0 -spaces.On the other hand, the result provides \"natural\" witnesses for the non-collapse property of this hierarchy.Theorem 7.7 For any positive integer k,We prove the stronger assertion that N k + 1 has no continuous representation δ with dom(δ) ∈ Σ 1 k (N ).Suppose for a contradiction that δ is a continuous representation of N k + 1 with dom(δ) ∈ Σ 1 k (N ).Then there is a continuous function f : N k → N with rng(f ) = dom(δ) by Proposition 7.5.We define a function g : N k → ω by g(φ) := 1 + δ(f (φ))(φ) = 1 + eval δ(f (φ)), φ .Then g is continuous, because f , δ and the evaluation function eval : N k + 1 × N k → ω mapping (ψ, φ) to ψ(φ) are continuous (note that N k + 1 × N k carries the sequential QCB 0topology).So there is some p ∈ dom(δ) and some h ∈ N k such that δ(p) = g and f (h) = p.We obtain g(h) = 1 + δ(f (h))(h) = 1 + g(h) , a contradiction.We conclude N k + 1 / ∈ QCB 0 (Σ 1 k ).✷ 8 The category of projective QCB 0 -spacesIn this section we show that the Luzin hierarchy of QCB 0 -spaces gives rise to a nice cartesian closed category.This is the full subcategory of the category QCB 0 consisting of the n QCB 0 (Σ 1 n ) as objects and all continuous function between them as morphisms.We denote this category by QCB 0 (P) and call its objects projective qcb-spaces.Theorem 8.1 The category QCB 0 (P) of projective qcb-spaces is cartesian closed.\nTopological properties of real number representations. V Brattka, P Hertling, Theoretical Computer Science. 2842002\n\nQuasi-Polish spaces. M De Brecht, arXiv:1108.1445v1Annals of Pure and Applied Logic. 16432013\n\nM De Brecht, A Yamamoto, Sixth International Conference on Computability and Complexity in Analysis. A Bauer, P Hertling, -I Ker, Ko, 2009. 2009OASICS 11 Schloss Dagstuhl -Leibniz-Zentrum für Informatik\n\nR Engelking, General Topology. Heldermann. Berlin1989\n\nMaximal and everywhere defined functionals. Algebra and Logic. Yu Ershov, 197413\n\nComparing Cartesian Closed Categories of (Core) Compactly Generated Spaces. M Escard, J Lawson, A Simpson, Topology and its Applications. 1432004\n\nG Gierz, K H Hoffmann, K Keimel, J D Lawson, M W Mislove, D S Scott, A compendium of Continuous Lattices. BerlinSpringer1980\n\nFilter spaces and continuous functionals. J M L Hyland, Annals of Mathematical Logic. 161979\n\nCharacterizations of absolute F σδ -sets. H Junnila, H.-P Künzi, Chechoslovak Mathematical Journal. 481998\n\nCartesian closed categories of algebraic CPO's. Theoretical Computer Science. A Jung, 199070\n\nClassical Descriptive Set Theory. A S Kechris, 1995SpringerNew York\n\nS C Kleene, Countable functionals. Constructivity in Mathematics. A Heyting, North Holland, Amsterdam1959\n\nInterpretation of analysis by means of constructive functionals of finite types. G Kreisel, North Holland, Amsterdam. Constructivity in Mathematics. A Heyting, 1959\n\nC Kreitz, K Weihrauch, Theory of representations. Theoretical Computer Science. 198538\n\nRecursion on the Countable Functionals. D Normann, Lecture Notes in Mathematics. 8111980\n\nCountable functionals and the projective hierarchy. D Normann, Journal of Symbolic Logic. 461981\n\nThe continuous functionals. Handbook of Computability Theory. D Normann, E.R. Griffor Ed1999\n\nExtended Admissibility. M Schröder, Theoretical Computer Science. 2842002\n\nAdmissible Representations for Continuous Computations. M Schröder, 2003PhD Thesist Hagen\n\nThe sequential topology on N N N is not regular. M Schröder, Mathematical Structures in Computer Science. 192009\n\nDifference hierarchy in ϕ-spaces. Algebra and Logic. V L Selivanov, 200443\n\nVariations on the Wadge reducibility. V L Selivanov, Siberian Advances in Mathematics. 200515\n\nV L Selivanov, Hierarchies in ϕ-spaces and applications. Mathematical Logic Quarterly. 200551\n\nTowards a descriptive set theory for domain-like structures. V L Selivanov, Theoretical Computer Science. 3652006\n\nOn the difference hierarchy in countably based T 0 -spaces. Electronic Notes in Theoretical. V L Selivanov, Computer Science. 2212008\n\nV L Selivanov, arXiv:1304.1239Total representations. Accepted for publication by Logical Methods in Computer Science. 2011\n\nK Weihrauch, Computable Analysis. Springer2000\n"},"corpus_id":{"kind":"number","value":15509555,"string":"15,509,555"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1304.1647\",\n \"mag\": 2963651973,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"journals/mscs/SchroderS15\",\n \"doi\": \"10.1017/s0960129513000376\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1304.1647v1.pdf\"\n ],\n \"pdf_sha\": \"5085c86d9fc3ec91522cb1a42f26919dacad5597\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1304.1647\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Some Hierarchies of QCB 0 -Spaces"},"authors":{"kind":"list like","value":["Matthias Schröder","Victor Selivanov","A P Ershov","University of Bundeswehr-Munich\nNeubiebergGermany","Institute of Informatics Systems SB RAS Novisibirsk\nRussia"],"string":"[\n \"Matthias Schröder\",\n \"Victor Selivanov\",\n \"A P Ershov\",\n \"University of Bundeswehr-Munich\\nNeubiebergGermany\",\n \"Institute of Informatics Systems SB RAS Novisibirsk\\nRussia\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We define and study hierarchies of topological spaces induced by the classical Borel and Luzin hierarchies of sets.Our hierarchies are divided into two classes: hierarchies of countably based spaces induced by their embeddings into Pω, and hierarchies of spaces (not necessarily countably based) induced by their admissible representations.We concentrate on the non-collapse property of the hierarchies and on the relationships between hierarchies in the two classes."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"A basic notion of Computable Analysis (CA) [Wei00] is the notion of an admissible representation of a topological space X.This is a partial continuous surjection δ from the Baire space N onto X satisfying a certain universality property (see Subsection 2.6 for some more details).Such a representation of X induces a reasonable computability theory on X, and the class of admissibly represented spaces is wide enough to include most spaces of interest for Analysis or Numerical Mathematics.As shown by the first author [Sch03], this class coincides with the class of the so-called QCB 0 -spaces, i.e.T 0 -spaces which are quotients of countably based spaces, and it forms a cartesian closed category (with the continuous functions as morphisms).Thus, among QCB 0 -spaces one meets many important function spaces including the Kleene-Kreisel continuous functionals [Kl959,Kr59] interesting for several branches of logic and computability theory.\nAlong with the mentioned nice properties of QCB 0 -spaces, this class seems to be too broad to admit a deep understanding.Hence, it makes sense to search for natural subclasses of this class which still include \"practically\" important spaces but are (hopefully) easier to study.Interesting examples of such subclasses are obtained if we consider, for each level Γ of the classical Borel or Luzin (projective) hierarchies of Descriptive Set Theory (DST) [Ke95], the class of spaces which have an admissible representation of the complexity Γ (below we make this precise).The study of the resulting Borel and Luzin hierarchies of QCB 0 -spaces is one of the aims of this paper.\nAlong with the hierarchies of QCB 0 -spaces, we will consider Borel and Luzin hierarchies of countably based T 0 -spaces (CB 0 -spaces for short) which are induced by the well-known fact that any CB 0 -space may be embedded in the algebraic domain Pω of all subsets of ω.The precise definition depends crucially on the possibility to find the \"right\" extensions of the classical hierarchies from the class of Polish spaces to the arbitrary spaces (including Pω).Such extensions were introduced by the second author in [Se04] and were studied in [Se05,Se05a,Se06,Se08,BY09,Br13].In particular, it was shown that many properties of these hierarchies in ω-continuous domains resemble the properties of classical hierarchies in Polish spaces [Ke95].\nHierarchies of spaces obtained in this way turn out to be closely related to the corresponding hierarchies of QCB 0 -spaces.Moreover, among the first levels of the Borel hierarchy of CB 0spaces we meet some classes of spaces which attracted attention of several researches in the field of quasi-metric spaces, in particular the class of quasi-Polish spaces.The class of quasi-Polish spaces identified and studied in [Br13] is a good solution to the problem from [Se08] of finding a natural class of spaces that includes the Polish spaces and the ω-continuous domains and has a reasonable DST.\nIn this paper we establish some basic properties of the above-mentioned hierarchies of spaces.In particular, we show that the hierarchies of spaces do not collapse, and that any level of any hierarchy is closed under retracts.As main technical tools to prove these results we use suitable generalizations of some classical facts (e.g. of the injectivity property of Pω and of Lavrentyev's Theorem on extending partial homeomorphisms in Polish spaces).Some of those generalizations might be also interesting in their own right.We also show that the class of all spaces in our hierarchies forms in a sense the smallest cartesian closed category of QCB 0 -spaces containing the discrete space ω of natural numbers, and establish the close relationship of the Luzin hierarchy of QCB 0 -spaces to the continuous functionals of finite type.Hence the class of all spaces in the Luzin hierarchy of QCB 0 -spaces seems to be a reasonable subclass of QCB 0 -spaces that contains most of spaces of interest for CA, including the Kleene-Kreisel continuous functionals.\nAfter recalling some notions and known facts in the next section, we establish the main technical facts in Section 3. In Sections 4 and 5 we introduce and study the mentioned hierarchies of CB 0spaces and of QCB 0 -spaces.In Section 6 we establish close relationships between hierarchies of CB 0 -spaces to those of QCB 0 -spaces (namely, any level of the CB 0 -hierarchies coincides with the class of CB 0 -spaces in the corresponding level of the corresponding QCB 0 -hierarchy).In Section 7 we relate the Luzin hierarchy of QCB 0 -spaces to the Kleene-Kreisel continuous functionals.In Section 8 we discuss the cartesian closedness of the corresponding categories, and we conclude in Section 9."},{"title":"Notation and Preliminaries","content":""},{"title":"Notation","content":"We freely use the standard set-theoretic notation like dom(f ), rng (f ) and graph(f ) for the domain, range and graph of a function f , respectively, X × Y for the Cartesian product, X ⊔ Y for the disjoint union of sets X and Y , Y X for the set of functions f : X → Y (but in the case when X, Y are QCB 0 -spaces we use the same notation to denote the space of continuous functions from X to Y ), and P (X) for the set of all subsets of X.For A ⊆ X, A denotes the complement X \\ A of A in X.We identify the set of natural numbers with the first infinite ordinal ω.The first uncountable ordinal is denoted by ω 1 .The notation f : X → Y (resp.f :⊆ X → Y ) means that f is a total (resp.a partial) function from a set X to a set Y ."},{"title":"Topological Spaces","content":"We assume the reader to be familiar with the basic notions of topology.The collection of all open subsets of a topological space X (i.e. the topology of X) is denoted by τ X ; for the underlying set of X we will write X in abuse of notation.We will usually abbreviate \"topological space\" to \"space\".Remember that a space is zero-dimensional if it has a basis of clopen sets.\nA space Y is called a (continuous) retract of a space X if there are continuous functions s : Y → X and r : X → Y such that composition rs coincides with the identity function id Y on Y .Such a pair of functions (s, r) is called a section-retraction pair.Note that the section s is a homeomorphism between Y and the subspace s(Y ) = {x ∈ X | sr(x) = x} of X, and\nLet ω be the space of non-negative integers with the discrete topology.Of course, the spaces ω × ω = ω 2 , and ω ⊔ ω are homeomorphic to ω, the first homeomorphism is realized by the Cantor pairing function •, • .\nLet N = ω ω be the set of all infinite sequences of natural numbers (i.e., of all functions ξ : ω → ω).Let ω * be the set of finite sequences of elements of ω, including the empty sequence.For σ ∈ ω * and ξ ∈ N , we write σ ⊑ ξ to denote that σ is an initial segment of the sequence ξ.By σξ = σ • ξ we denote the concatenation of σ and ξ, and by σ • N the set of all extensions of σ in N .For x ∈ N , we can write x = x(0)x(1) . . .where x(i) ∈ ω for each i < ω.For x ∈ N and n < ω, let x <n = x(0) . . .x(n − 1) denote the initial segment of x of length n.Notations in the style of regular expressions like 0 ω , 0 * 1 or 0 m 1 n have the obvious standard meaning.\nBy endowing N with the product of the discrete topologies on ω, we obtain the so-called Baire space.The product topology coincides with the topology generated by the collection of sets of the form σ • N for σ ∈ ω * .The Baire space is of primary importance for DST and CA.The importance stems from the fact that many countable objects are coded straightforwardly by elements of N , and it has very specific topological properties.In particular, it is a perfect zero-dimensional space and the spaces N 2 , N ω , ω ×N = N ⊔N ⊔• • • (endowed with the product topology) are all homeomorphic to N .Let (x, y) → x, y be a homeomorphism between N 2 and N .The Baire space N has the following universality property for zero-dimensional CB 0 -spaces:\nThe subspace C = 2 ω of N formed by the infinite binary strings (endowed with the relative topology inherited from N ) is known as the Cantor space.Along with N and C, the space Pω is of principal importance for this paper.This is the space of subsets of the natural numbers with the Scott topology on (P (ω); ⊆).The basic open sets of this topology are of the form {A ⊆ ω | F ⊆ A}, where F ranges over the finite subsets of ω.\nThe importance of Pω for this paper is explained by its following well-known properties.First, Pω is universal for CB 0 -spaces.\nProof.Since Pω is a CB 0 -space, any space X homeomorphic to a subspace Pω is a CB 0 -space as well.Conversely, if {β i | i ∈ ω} is a base for a countably-based T 0 -space X, then the function e : X → Pω mapping\nThe second property shows that Pω is an injective object in the category of all topological spaces.\nProposition 2.3 [G+80, Proposition 3.5] Let Y be topological space and X be a subspace of Y .Then any continuous function f : X → Pω can be extended to a continuous function g : Y → Pω.\nCompletely metrisable spaces satisfy the following extension theorem by Kuratowski.\nProposition 2.4 [Ke95, Theorem 3.8] Let X be a metrisable space and Y be a completely metrisable space.Then any continuous function f : A → Y defined on a subset A of X can be extended to a continuous function g :"},{"title":"Families of Pointclasses","content":"Here we recall a useful technical notion of a family of pointclasses introduced in [Se11].\nA pointclass on X is simply a collection Γ(X) of subsets of X.We need the following \"parameterized\" version of the notion of pointclass.A family of pointclasses is a family Γ = {Γ(X)} indexed by arbitrary topological spaces X such that each Γ(X) is a pointclass on X and Γ is closed under continuous preimages, i.e. f −1 (A) ∈ Γ(X) for every A ∈ Γ(Y ) and every continuous function f : X → Y .In particular, any pointclass Γ(X) in such a family is downward closed under the Wadge reducibility on X.\nTrivial examples of families of pointclasses are E, F, where E(X) = {∅} and F(X) = {X} for any space X.A basic example of a family of pointclasses is given by the family O = {τ X } of the topologies of all the spaces X.\nFinally, we define some operations on families of pointclasses which are relevant to hierarchy theory.First, the usual set-theoretic operations will be applied to the families of pointclasses pointwise: for example, the union i Γ i of the families of pointclasses Γ 0 , Γ 1 , . . . is defined by\nSecond, a large class of such operations is induced by the set-theoretic operations of L.V. Kantorovich and E.M. Livenson (see e.g.[Se11] for the general definition).Among them are the operations Γ → Γ σ where Γ(X) σ is the set of all countable unions of sets in Γ(X), the operation Γ → Γ c where Γ(X) c is the set of all complements of sets in Γ(X), the operation Γ → Γ d where Γ(X) d is the set of all differences of sets in Γ(X), and the operation Γ → Γ p defined by\nThe next subsection contains some important examples of families of pointclasses from hierarchy theory."},{"title":"Classical Hierarchies on Topological Spaces","content":"Let us recall the definition of the Borel hierarchy on arbitrary spaces introduced in [Se04].\nDefinition 2.5 For α < ω 1 , define the family of pointclasses Σ 0 α = {Σ 0 α (X)} by induction on α as follows: Σ 0 0 (X) = {∅}, Σ 0 1 (X) = τ X is the collection of the open sets of X, Σ 0 2 (X) = ((Σ 0 1 (X)) d ) σ is the collection of all countable unions of differences of open sets, and Σ 0 α (X) = ( β<α (Σ 0 β (X)) c ) σ (for α > 2) is the class of countable unions of sets in β<α (Σ 0 β (X)) c .The sequence {Σ 0 α (X)} α<ω 1 is called the Borel hierarchy of X.We also let Π 0 β (X) = (Σ 0 β (X)) c and ∆ 0 α (X) = Σ 0 α (X) ∩ Π 0 α (X).The classes Σ 0 α (X), Π 0 α (X), ∆ 0 α (X) are called the levels of the Borel hierarchy of X.\nBy the definition and remarks at the end of the previous subsection, any of Σ 0 α , Π 0 α , ∆ 0 α is a family of pointclasses.It is straightforward to check by induction on α, β that using Definition 2.5 one has the following result.\nProposition 2.6 For every topological space X and for all α\nRemark.Definition 2.5 applies to all spaces X, and Proposition 2.6 holds true in the full generality.Note that Definition 2.5 differs from the classical definition for Polish spaces (see e.g.[Ke95, Section 11.B]) only for the level 2, and that for the case of Polish spaces our definition of the Borel hierarchy is equivalent to the classical one.Notice that the classical definition cannot be applied in general to non metrizable spaces (like e.g. the non discrete ω-algebraic domains) precisely because the inclusion Σ 0 1 ⊆ Σ 0 2 may fail.Let {Σ 1 n (X)} 1≤n<ω be Luzin's projective hierarchy in X (cf.[Br13]).Using the corresponding operation on families of pointclasses from the previous subsection we have Σ 1 1 (X) = (Π 0 2 (X)) p and Σ 1 n+1 (X) = (Π 1 n (X)) p for any n ≥ 1.Let also Σ 1 0 = Π 1 0 = ∆ 1 1 .The reason why the definition of the first level of the Luzin hierarchy is distinct from the classical definition Σ 1 1 (X) = (Π 0 1 (X)) p for Polish spaces is that the inclusion Σ 0 1 (X) ⊆ (Π 0 1 (X)) p may fail in general.Any level Σ 1 n , Π 1 n , ∆ 1 n of the Luzin hierarchy is a family of pointclasses, and we have the natural inclusions among them similar to the inclusions for levels of the Borel hierarchy.\nLevels of the introduced hierarchies in an arbitrary space have closure properties similar to those known for the classical hierarchies in Polish spaces, in particular: Proposition 2.7 Any non-zero Σ-level of the Borel hierarchy on an arbitrary topological space is closed under finite intersections, countable unions and binary products (for the case of binary product this means that\nAny non-zero Σ-level of the Luzin hierarchy in an arbitrary space is closed under countable unions, countable intersections, binary products and the projection along N -axis (for the case of projection this means that\nWe will often use the following straightforward result which also extends the corresponding well-known facts for the classical hierarchies.\nProposition 2.8 Let X be a subspace of a topological space Y , A ⊆ X, and let Γ be a Σ-or a Π-level of the introduced hierarchies.Then\nWe will often cite the following topological complexity of the equality test\nProof.The first statement has been shown in [Br13].The second (well-known) statement follows from the fact that in the Hausdorff case the complement of EQ X is equal to {U × V | U, V open and disjoint} and therefore open.✷ Let Γ be a family of pointclasses and X, Y be spaces.A function f :\nfor each open set A ⊆ Y .Note that the continuous functions coincide with the Σ 0 1 -measurable functions.We will also use the following basic property of Γ-measurable functions.Note that addition + on ordinal numbers is associative, but not commutative.\nNotice that the case (α, β) = (0, 0) denotes the fact that the preimage of an open (closed) set under a continuous function is open (resp.closed)."},{"title":"Polish and quasi-Polish spaces","content":"Remember that a space X is Polish if it is countably based and metrisable with a metric d such that (X, d) is a complete metric space.Important examples of Polish spaces are the Baire space, the Cantor space, the space of reals R and its Cartesian powers R n (n < ω), the closed unit interval [0, 1], the Hilbert cube [0, 1] ω and the Hilbert space R ω .\nBelow we will cite the following result known as Lavrentyev's Theorem:\nProposition 2.11 [Ke95, Theorem 3.9] Let X, Y be Polish spaces, A ⊆ X, B ⊆ Y , and let f be a homeomorphism of A onto B (equipped with the subspace topologies induced from X and Y ).Then there exist\nA natural variant of the class of Polish spaces has recently emerged.Given a set X, call a function d from X ×X to the nonnegative reals quasi-metric whenever x = y iff d(x, y) = d(y, x) = 0, and d(x, y) ≤ d(x, z)+d(z, y) (but we don't require d to be symmetric).In particular, every metric is a quasi-metric.Every quasi-metric on X canonically induces a topology on X which is denoted by τ d , where τ d is the topology generated by the open balls\nWe say that the quasi-metric d on X is complete if every d-Cauchy sequence converges with respect to d (notice that this definition is coherent with the notion of completeness for a metric d, as in this case d = d).\nA T 0 -space X is called quasi-Polish if it is countably based and there is a complete quasi-metric on X which generates its topology.In particular, every Polish space is quasi-Polish, but by [Br13,Corollary 45] also every ω-continuous domain is quasi-Polish.De Brecht [Br13] shows that there is a reasonable DST for the quasi-Polish spaces which extends the classical DST for Polish spaces [Ke95] and the DST for ω-continuous domains [Se06,Se08] in many directions.\nAn important example of a quasi-Polish space is the space Pω equipped with the following quasi-metric d: if A ⊆ B then d(A, B) = 0, otherwise d(A, B) = 1/2 a where a is the smallest number in A \\ B. Note that Proposition 2.2 implies that any CB 0 -space is quasi-metrisable.\nDe Brecht proved the following characterization of quasi-Polish spaces (cf.[Br13, Corollary 24]): Proposition 2.12 A topological space is quasi-Polish iff it is homeomorphic to a Π 0 2 -subset of Pω (endowed with the relative topology inherited from Pω)."},{"title":"Admissible Representations and QCB 0 -spaces","content":"A representation of a space X is a surjection of a subspace of the Baire space N onto X. Usually it is denoted as a partial function from N to X.The notion of admissible representation is basic in Computable Analysis (CA).A representation δ of X is admissible, if it is continuous and any continuous function ν : Z → X from a zero-dimensional CB 0 -space Z to X is continuously reducible to δ, i.e. ν = δg for some continuous function g : Z → N .A topological space is admissibly representable if it has an admissible representation.\nThe notion of admissibility was introduced in [KW85] for representations of countably based spaces (in a different but equivalent formulation) and was extensively studied by many authors.In [BH02] a close relation of admissible representations of countably based spaces to open continuous representations was established.In [Sch02,Sch03] the notion was extended to non-countably based spaces and a nice characterization of the admissibly represented spaces was achieved.Namely, the admissibly represented sequential topological spaces coincide with the QCB 0 -spaces, i.e., T 0 -spaces which are topological quotients of countably based spaces.\nThe category QCB 0 of QCB 0 -spaces as objects and continuous functions as morphisms is known to be cartesian closed (cf.[ELS04,Sch03]).The exponential Y X to QCB 0 -spaces X, Y has the set of continuous functions from X to Y as the underlying set, and its topology is the sequentialization of the compact-open topology on Y X .By the sequentialization of a topology τ we mean the family of all sequentially open sets pertaining to this topology.(Remember that sequentially open sets are defined to be the complements of the sets that are closed under forming limits of converging sequences.)The sequentialization of τ is finer than or equal to τ .The topology of the QCB 0 -product to X and Y , which we denote by X × Y , is the sequentialization of the classical Tychonoff topology on the cartesian product of the underlying sets of X and Y .So products and exponentials in QCB 0 are formed in the same way as in its supercategory Seq of sequential topological spaces.\nFrom [Br13] it follows that admissible total representations are closely related to quasi-Polish spaces.The following assertion is contained among the results in [Br13].\nProposition 2.13 For any CB 0 -space X the following statements are equivalent:\n(1) X is quasi-Polish.\n(2) X has an open continuous total representation.\n(3) X has an admissible total representation.\n(4) X has an admissible representation whose domain is a Polish space.\nWe will also cite the following facts from [Sch02,Sch03].\nProposition 2.14 Let δ and γ be admissible representations of QCB 0 -spaces X and Y , respectively.Then f : X → Y is continuous iff f δ = γg for some partial continuous function g on N .\nNote that if u :⊆ N 2 → N is continuous and p ∈ N then u p = λx.u(p,x) is a partial continuous function on N .\nProposition 2.15 There is a partial continuous function u : ⊆ N 2 → N such that dom(u) ∈ Π 0 2 (N 2 ) and for any partial continuous function g on N there is some p ∈ N such that u p is an extension of g.\nThe function u in Proposition 2.15 can be chosen as the application operator of the Second Kleene Algebra.We use it to describe the construction of admissible representations for function spaces formed in QCB 0 (cf.[Sch03,Wei00]).\nProposition 2.16 Let δ and γ be admissible representations for QCB 0 -spaces X and Y , respectively.Then admissible representations [δ × γ] for the QCB 0 -product X × Y and [δ → γ] for the QCB 0 -exponential Y X can be defined by:\nfor p, q ∈ N , x ∈ X, y ∈ Y , and f : X → Y .\nDe Brecht and Yamamoto showed the following property of subsets of admissibly represented countably-based spaces.\nProposition 2.17 [BY09, Corollary 3].Let δ be an admissible representation of a countablybased T 0 -space X, let A ⊆ X and let"},{"title":"Main Technical Facts","content":"In this section we prove a couple of facts that serve as main technical tools in the sequel, but some of these facts might be also of independent interest.\nThe first result generalizes the well-known fact that Pω is an injective space (see Proposition 2.2), because the continuous functions coincide with the Σ 0 1 -measurable functions.\nTheorem 3.1 Let Y be a topological space, let X be a subspace of Y , and let\nis a subbasis in Pω and Γ(Y ) is closed under finite intersection and countable union by Proposition 2.7, g is Γ-measurable.If y ∈ X then we have\nThe second result is a remote relative of Lavrentyev's Theorem (cf.Proposition 2.11).\nTheorem 3.2 Let A, B ⊆ Pω be subspaces of Pω, α, β < ω 1 , and let f : A → B be a Σ 0 1+αmeasurable bijection such that its inverse g = f −1 is Σ 0 1+β -measurable.Then there exist A * ∈ Π 0 1+α+µ+1 (Pω), B * ∈ Π 0 1+β+µ+1 (Pω) (where µ = max{α, β}) and a Σ 0 1+α -measurable bijection\nProof.By Theorem 3.1, there are a Σ 0 1+α -measurable extension f 1 : Pω → Pω of f and a Σ 0 1+β -measurable extension g 1 : Pω → Pω of g.Consider the set\nby Proposition 2.9 and S 1 is the preimage of EQ Pω under the Let us introduce one of the main notions of this paper.\nDefinition 3.4 Let Γ be a family of pointclasses.A topological space X is called a Γ-space if X is homeomorphic to a subspace A ⊆ Pω with A ∈ Γ(Pω).The class of all Γ-spaces is denoted CB 0 (Γ).\nThe third result of this section extends Corollary 24 in [Br13] (which is obtained when Γ = Π 0 2 ).\nω} and f :⊆ X → Y be a partial continuous function from a topological space X to a Γ-space Y .Then there is a continuous extension g :⊆ X → Y of f with dom(g) ∈ Γ(X).\nProof.Without loss of generality we assume Y ∈ Γ(Pω), so in particular f :⊆ X → Pω.By Proposition 2.3, there is a total continuous extension h : X → Pω of f .Since Y ∈ Γ(Pω), h −1 (Y ) ∈ Γ(X).Since dom(f ) ⊆ h −1 (Y ), we can take the restriction of h to h −1 (Y ) as the desired function g. ✷\nThe fourth result of this section is the following extension of Lavrentyev's Theorem (see Proposition 2.11).For Γ = Π 0 2 the result gives the extension of Lavrentyev's Theorem to quasi-Polish spaces (cf.[Br13]), and Lavrentyev's Theorem is obtained if we restrict the last fact to Polish spaces."},{"title":"Let now","content":"A * ∈ Γ(A 1 ) and B * ∈ Γ(B 1 ).By Proposition 2.8 we have A * ∈ Γ(X) and B * ∈ Γ(Y ).✷ Finally, we give a natural version of the previous theorem related to retracts, although this version is not used in the sequel.\nProposition 3.7 Let Γ, X, Y, A, B be as in the previous theorem and let r : A → B and s : B → A be continuous functions such that rs = id B (hence, B is a retract of A).Then there exist A * ∈ Γ(X), B * ∈ Γ(Y ) and continuous functions r * :\nProof.By Proposition 3.5 there exist A 1 ∈ Γ(X), B 1 ∈ Γ(Y ) and continuous functions r 1 :\nSimilarly to the previous proof one checks that these objects have the desired properties.✷ 4 Hierarchies of CB 0 -Spaces\nHere we introduce and study natural hierarchies of CB 0 -spaces (see Definition 3.4) as well as the Borel hierarchy and the Luzin hierarchy on Pω discussed in Subsection 2.4.\nDefinition 4.1 (1) By the Borel hierarchy of CB 0 -spaces we mean the sequence of classes {CB 0 (Σ 0 α )} α<ω 1 .By levels of this hierarchy we mean the classes CB 0 (Σ 0 α ) as well as the classes CB 0 (Π 0 α ) and CB 0 (∆ 0 α ).\n(2) By the Luzin hierarchy of CB 0 -spaces we mean the sequence of classes {CB 0 (Σ 1 n )} n<ω .By levels of this hierarchy we mean the classes CB 0 (Σ 1 n ) as well as the classes CB 0 (Π 1 n ) and CB 0 (∆ 1 n ).\nNote that the Borel hierarchy of spaces includes some natural classes of spaces identified earlier.E.g., by Proposition 2.12 CB 0 (Π 0 2 ) coincides with the class of quasi-Polish spaces and, by Corollary 33 in [Br13], CB 0 (Π 0 3 ) coincides with the class of CB 0 -spaces that admit a compatible bicomplete quasi-metric in the sense of [JK98].\nObviously, for any families of pointclasses Γ, ∆ we have: if Γ ⊆ ∆ (i.e.Γ(X) ⊆ ∆(X) for each space X) then CB 0 (Γ) ⊆ CB 0 (∆).Therefore, we have the natural inclusions for levels of the introduced hierarchies, in particular CB 0 (∆ 0 α ) ⊆ CB 0 (Σ 0 α )∩CB 0 (Π 0 α ) and CB 0 (Σ 0 α )∪CB 0 (Π 0 α ) ⊆ CB 0 (∆ 0 β ) for all α < β < ω 1 .Below we establish some basic properties of the Borel and the Luzin hierarchy of spaces, but first we show that most of levels of the introduced hierarchies of spaces are closed under retracts (for Π 0 2 the result is already known from [Br13]).\nThen any retract of a Γ-space is a Γ-space.\nProof.Let Y be a retract of a Γ-space X via a section-retraction pair (s, r) of continuous functions; we have to show that Y is a Γ-space.We may assume without loss of generality that X ∈ Γ(Pω).Since s(Y ) = {x ∈ X | sr(x) = x}, s(Y ) is the preimage of EQ X under the continuous function x → (sr(x), x).Since EQ X ∈ Π 0 2 (X ×X) by Proposition 2.9, s(Y ) ∈ Π 0 2 (X) and thus s(Y ) ∈ Γ(X) by Proposition 2.6.Proposition 2.8 yields s(Y ) ∈ Γ(Pω).Since Y is homeomorphic to s(Y ), Y is a Γ-space.✷ Now we establish an interesting property of the introduced hierarchies of spaces which implies the non-collapse property."},{"title":"Proposition 4.3 For any countable ordinal","content":"This proposition is based on the following lemma.\nProof.If the underlying set of X is in Γ(Pω), then X is a Γ-space by Definition 3.4.Conversely, let X be a Γ-space.Then there is some B ∈ Γ(Pω) such that B (endowed with the subspace topology inherited from Pω) is homeomorphic to X. Let h : X → B be a homeomorphism.We apply Theorem 3.6 to extend h to a homeomorphism h * : X * → B * , where X * and B * are sets in Γ(Pω) with X ⊆ X * and B ⊆ B * .By Proposition 2.8 we have B ∈ Γ(B * ).Hence X ∈ Γ(X * ), because X = (h * ) −1 (B).Proposition 2.8 yields X ∈ Γ(Pω).✷ Now we are ready to give the proof of Proposition 4.3.\nProof.The inclusions from right to left are obvious.It remains to check CB 0 (Γ)\nThen Z is homeomorphic to some subspaces A, B of Pω with A ∈ Γ(Pω) and B ∈ Γ c (Pω).Hence B is a Γ-space by being homeomorphic to A. Lemma 4.4 yields B ∈ Γ(Pω).Therefore B ∈ (Γ ∩ Γ c )(Pω) and hence Z ∈ CB 0 (Γ ∩ Γ c ). ✷ Corollary 4.5 The Borel hierarchy and the Luzin hierarchy of CB 0 -spaces do not collapse.More precisely CB 0 (Σ 0 α ) ⊆ CB 0 (Π 0 α ) for each countable ordinal α ≥ 2, and CB 0 (Σ 1 n ) ⊆ CB 0 (Π 1 n ) for each positive integer n.\nProof.Proofs for both hierarchies are similar, so consider only the Borel hierarchy.According to [Se05a], there is a set A in Σ 0 α (Pω) \\ Π 0 α (Pω).The space A (with the topology induced from Pω) is obviously a Σ 0 α -space.If it were a Π 0 α -space, then by Lemma 4.4 we would have A ∈ Π 0 α (Pω), a contradiction.✷ 5 Hierarchies of QCB 0 -Spaces\nAs shown in [Sch03], any QCB 0 -space has an admissible representation.Here we introduce and study natural hierarchies of QCB 0 -spaces induced by this fact.For any representation δ of a space X, let EQ(δ) := {(p, q) ∈ N 2 | p, q ∈ dom(δ) ∧ δ(p) = δ(q)}.\nDefinition 5.1 (1) Let Γ be a family of pointclasses.A topological space X is called Γrepresentable if X has an admissible representation δ with EQ(δ) ∈ Γ(N × N ).The class of all Γ-representable spaces is denoted QCB 0 (Γ).\n(2) By the Borel hierarchy of QCB 0 -spaces we mean the sequence {QCB 0 (Σ 0 α )} α<ω 1 .By levels of this hierarchy we mean the classes QCB 0 (Σ 0 α ) as well as the classes QCB 0 (Π 0 α ) and QCB 0 (∆ 0 α ).\n(3) By the Luzin hierarchy of QCB 0 -spaces we mean the sequence {QCB 0 (Σ 1 n )} n<ω .By levels of this hierarchy we mean the classes QCB 0 (Σ 1 n ) as well as the classes QCB 0 (Π 1 n ) and QCB 0 (∆ 1 n ).\nThe next assertion establishes an equivalent simpler definition of most levels for the case of CB 0 -spaces.\nω} and let X be a Hausdorff space.Then X is Γ-representable iff X has an admissible representation δ with dom(δ) ∈ Γ(N ).\nProof.The only-if-part of the first statement holds for any topological space X and any family of pointclasses Γ.Indeed, let δ be an admissible representation of X with EQ(δ) ∈ Γ(N ×N ).Since dom(δ) is the preimage of EQ(δ) under the continuous function x → (x, x), dom(δ) ∈ Γ(N ).\nConversely, let δ be an admissible representation of X with dom(δ) ∈ Γ(N ).By Proposition 2.9, EQ X ∈ Π 0 2 (X × X), so EQ(δ) is a Π 0 2 -set in dom(δ) × dom(δ) by the continuity of δ.The set dom(δ) × dom(δ) is a Γ-subset of N × N by being the intersection of the sets {(x, y)\n. By Propositions 2.6 and 2.8 we obtain EQ(δ) ∈ Γ(N × N ).\nThe second statement follows similarly by taking into account that EQ X is closed in X × X by Proposition 2.9, if X is a Hausdorff space.✷\nThe main advantage of the hierarchies of this section over the hierarchies from the previous section is that they include many natural non-countably based spaces, in particular the spaces of Kleene-Kreisel continuous functionals, as we will see later.\nObviously, for any families of pointclasses Γ, ∆ we have: Γ ⊆ ∆ implies QCB 0 (Γ) ⊆ QCB 0 (∆), hence we have the natural inclusions for levels of the introduced hierarchies, in particular\n) for all α < β < ω 1 .We will show that these hierarchies do not collapse, but first we establish the closure of the levels under retracts.\nThen any retract of a Γ-representable space is a Γ-representable space.\nProof.Let Y be a retract of a Γ-representable space X via a section-retraction pair (s, r) of continuous functions; we have to show that Y is a Γ-representable space.Let δ be an admissible representation for X with EQ(δ) ∈ Γ(N ×N ).Then rδ is clearly an admissible representation for Y , so it suffices to show that EQ(rδ) ∈ Γ(N ×N ).Since s is an injection, EQ(rδ) is the preimage of EQ(δ) under the continuous function (x, y) → (sr(x), sr(y)).Therefore, EQ(rδ) ∈ Γ(N × N ).✷\nThe following observation is well-known.Proof.Let Y be a retract of D via a section-retraction pair (s, r) of continuous functions.Then r :⊆ N → Y is admissible, because any continuous function ν : Z → Y defined on a zero-dimensional CB 0 -space Z is reducible to r via sν.\nConversely, assume that r ⊆ : N → Y is an admissible representation of Y .Then the identity function id Y is reducible to r via some continuous function s : Y → D. Then id Y = rs, hence Y is a retract of D via the continuous section s and the continuous retraction r. ✷ Finally, we establish the non-collapse property of the introduced hierarchies of spaces.\nTheorem 5.5 The Borel hierarchy and the Luzin hierarchy of QCB 0 -spaces do not collapse.More precisely, QCB 0 (Σ 0 α ) ⊆ QCB 0 (Π 0 α ) for each countable ordinal α ≥ 2, and QCB 0 (Σ 1 n ) ⊆ QCB 0 (Π 1 n ) for each positive integer n.\nProof.Proofs for both hierarchies are similar, so consider only the Borel hierarchy.As is well-known Since N is a Π 0 2 -space, hence a Γ-space, we can use Theorem 3.6 (applied to X ′ = Y ′ = N ) to extend the homeomorphism s : Y → S to a homeomorphism s * between larger sets Y * , S * ∈ Π 0 2 (N ).By Proposition 2.6 we have Y * , S * ∈ Π 0 α (N ).Since S ∈ Π 0 α (S * ) by Proposition 2.8,\nThe spaces Y witnessing the non-collapse property above are rather artificial.In Theorem 7.7 we will find very natural spaces witnessing the non-collapse of the Luzin hierarchy of QCB 0 -spaces."},{"title":"Relating the Hierarchies","content":"In this section we establish close relationships of the hierarchies of CB 0 -spaces with the corresponding hierarchies of QCB 0 -spaces.The next result implies that any level of a hierarchy of QCB 0 -spaces extends the corresponding level in the corresponding hierarchy of CB 0 -spaces.\nProof.It suffices to show that any space X ∈ Γ(Pω) is a Γ-representable space.Let ρ : N → Pω be the total admissible representation of Pω defined by ρ(x) = {n | ∃i(x(i) = n + 1)} (see e.g.[Wei00,Br13] for details).Then the restriction of ρ to ρ −1 (X) ∈ Γ(N ) is an admissible representation of X.Since X is a CB 0 -space, X is a Γ-representable space by Proposition 5.2.✷\nIn particular, we have CB 0 (Γ) ⊆ QCB 0 (Γ) ∩ CB 0 for each level Γ of the Borel hierarchy or the Luzin hierarchy.The main question of this section is: for which levels Γ we have the equality CB 0 (Γ) = QCB 0 (Γ) ∩ CB 0 ?Proposition 2.12 implies that the equality holds for Γ = Π 0 2 .The next result implies that the equality holds for all zero-dimensional CB 0 -spaces.\nProof.Let r : D → X be an admissible representation of X with D ∈ Γ(N ).By Lemma 5.4 there is a continuous function s :\nAs f s is a homeomorphism of X onto f s(X), we have f s(X) ∈ Γ(Y ).Propositions 2.6 and 2.8 yield f s(X) ∈ Γ(Pω).Therefore X ∈ CB 0 (Γ).✷ By the next theorem the equality CB 0 (Γ) = QCB 0 (Γ) ∩ CB 0 holds for almost all levels.For finite levels, this was pointed out to us by Matthew de Brecht.We thank him for giving the permission to use his proof of Theorem 6.3(2).Note that his proof may be used to obtain also a proof of Theorem 6.3(1) which is slightly different from ours.\n(2) For all natural numbers m ≥ 2 and n ≥ 3, we have\nProof.\n(\nWe have already seen CB 0 (Γ) ⊆ QCB 0 (Γ) ∩ CB 0 .\nLet X ∈ QCB 0 (Γ) ∩ CB 0 .Then there is an admissible representation δ : D → X of X with D ∈ Γ(N ).By Proposition 2.1 we may assume w.l.o.g.X to be a subspace of Pω.We have to show X ∈ CB 0 (Γ).\nLet χ : Pω → C be the bijection between Pω and the Cantor space C = 2 ω that sends any A ⊆ ω to its characteristic function χ A .Obviously, χ is Σ 0 2 -measurable while its inverse χ −1 is continuous (i.e.Σ 0 1 -measurable).Let σ = χ| X be the restriction of χ to X ⊆ Pω.By Proposition 2.8, σ is a Σ 0 2 -measurable bijection between X and σ(X) ⊆ C such that σ −1 is Σ 0 1 -measurable.Since σ −1 is continuous as a function from the zero-dimensional CB 0 -space σ(X) to X and δ is admissible, there is a continuous function g : σ(X) → N with σ −1 = δg.Then id X = σ −1 σ = δgσ, hence the function h = gσ satisfies h(X) = {y ∈ D | hδ(y) = y}.Thus, h is a Σ 0 2 -measurable bijection between X and H = h(X), whereas its inverse h −1 = δ| h(X) is Σ 0 1 -measurable (i.e.continuous).By Proposition 6.2 we may w.l.o.g.assume that D is a subspace of Pω and D ∈ Γ(Pω).Since H is the preimage of EQ D ∈ Π 0 2 (D × D) under the Σ 0 2 -measurable function y → (hδ(y), y), we have H ∈ Π 0 3 (D) by Proposition 2.10 and hence H ∈ Γ(D) by Proposition 2.6.Proposition 2.8 yields H ∈ Γ(Pω).By Theorem 3.2 and Corollary 3.3, there exist H * ∈ Π 0 3 (Pω), X * ∈ Π 0 4 (Pω), a continuous bijection δ * : H * → X * and a Σ 0 2 -measurable bijection h * : X * → H * such that δ * is the inverse of h * , δ * is an extension of h −1 and h * is an extension of h.Since H ∈ Γ(H * ) by Proposition 2.8 and X = h * −1 (H), Proposition 2.10 yields us X ∈ Γ(X * ).By Propositions 2.6 and 2.8, we obtain X ∈ Γ(Pω), hence X ∈ CB 0 (Γ).This completes the proof.\n(2) Let δ be an admissible representation of a subspace X of Pω such that dom(δ) ∈ Γ(N ), where Γ ∈ {Π 0 m , Σ 0 n | m ≥ 2, n ≥ 3}.Let ρ be the total admissible representation of Pω defined in the proof of Proposition 6.1.By Proposition 2.17 it is enough to show that ρ −1 (X) is in Γ(N ).Since the corestriction of ρ to X is continuous, there is a continuous function h : ρ −1 (X) → N satisfying ρ(p) = δh(p) for all p ∈ ρ −1 (X).As Pω is an injective space and N is a Polish space, δ and h can be extended to continuous functions δ * : N → Pω and h * : G → N , where G is a Π 0 2 -subset of N , see Propositions 2.3 and 2.4.By Proposition 2.9,\n2 -subset of G and thus of N .It is easy to verify that ρ −1 (X) is the intersection of A with (h * ) −1 (dom(δ)).By Proposition 2.7 and 2.8 this implies that ρ −1 (X) is Γ-subset of N .Proposition 2.17 yields that X is a Γ-subset of Pω, i.e., X ∈ CB 0 (Γ).✷ required function g : N × ω Y → N in such a way that g maps (x, H) to x, if H is a witness for x.Otherwise g assigns to (x, H) some element of a countable dense subset of M .\nTo construct an appropriate dense subset of M , we choose for every finite sequence w in the set W := {x <k | x ∈ M, k ∈ ω} some α(w) ∈ M such that α(w) has w as an initial segment.Clearly, {α(w) | w ∈ W } is dense in M .\nBy being a QCB 0 -space, Y has a countable dense subset {β j | j ∈ ω} (see Proposition 3.3.1 in [Sch03]).We construct a sequence {C k } k of subsets of the QCB 0 -space N ×ω Y by C 0 := N ×ω Y and C k := (x, H) ∈ N × ω Y x <k ∈ W and (x, β j ) ∈ D H(β j ) for all j ≤ k for all k > 0. The set C k is clopen in the Tychonoff topology on N × ω Y and thus in the QCB 0 -topology, because C k is the union of the clopen sets\nwhere w ranges over the finite sequence of length k in W and a 0 , . . ., a k vary over the natural numbers.Note that {H ∈ ω Y | β j ∈ H −1 {a j }} is clopen even in the compact-open topology on ω Y and therefore in the QCB 0 -topology on ω Y , which is finer than the former.\nNext we define a function ℓ\nFirst we show the continuity of g.Let O be an open set of the Baire space and let (x, H) ∈ g −1 (O).Then there is some m ∈ ω such that g(x, H) <m • N ⊆ O.\nCase 1: Assume ℓ(x, H) = ∞.Then (x, H) ∈ k∈ω C k .We define a set U by\nCase 2: Assume ℓ(x, H) < ∞.Let k := ℓ(x, H) and define a set U by\nWe conclude that in both cases g is continuous in the point (x, H).Therefore g is continuous, even with respect to the Tychonoff product on the set N × ω Y , which is coarser than the QCB 0 -topology.Now we show M ⊆ rng (g).Let x ∈ M .We define a function H x : Y → ω by\nNote that H x is total, because we have f (y) = x and therefore (x, y) ∈ k∈ω D k for every y ∈ Y .Furthermore, H is continuous in every point y ∈ Y , because the set\nIt remains to show rng(g) ⊆ M .So let x ∈ rng (g).The only interesting case is x / ∈ {α(w) | w ∈ W }. Then there is some continuous function H : Y → ω such that g(x, H) = x and thus (x, H) ∈ k∈ω C k .Suppose for a contradiction that there is some y ∈ Y such that (x, y) / ∈ D H(y) .Then the set\nWe conclude (x, y) ∈ D H(y) and thus f (y) = x for every y ∈ Y .Therefore x / ∈ rng (f ) and x ∈ M .So g satisfies the required properties.✷ Using the cartesian closedness of QCB 0 , we define a sequence of spaces {N k } k<ω by induction on k as follows: N 0 := ω and N k + 1 := ω N k , where ω denotes the space of natural numbers endowed with the discrete topology.Obviously N 1 is equal to the Baire space N .The space N k is referred to as the sequential space of (Kleene-Kreisel) continuous functionals of type k.For any k ≥ 2, the sequential topology on N k is strictly finer than the corresponding compactopen topology [Hy79].Furthermore it is neither zero-dimensional nor regular [Sch09].Any of these spaces has a natural admissible representation.From Theorem 7.1 we obtain:\nk ) by Theorem 7.1.✷ We remark that Lemma 2.34 in [No80] implies that N k has an admissible representation such that its domain is even in the effective version Π 1 k−1 of Π 1 k−1 (N ).This follows from the fact that the function which maps \"associates\" for N k to the encoded functionals forms an admissible representation.\nNote that the continuous functionals (also known under the somewhat misleading name \"countable functionals\") were first defined in [Kl959,Kr59] independently by S. Kleene and G. Kreisel.Their definitions look different from each other, as well as from the definition above, although all Proof.We only have to check that QCB 0 (P) is closed under binary products and exponentiation formed in the supercategory QCB 0 .For exponentiation this follows from Theorem 7.1.It is an easy exercise to show that the product representation [δ × γ] constructed in Proposition 2.16 satisfies EQ([δ × γ]) ∈ Σ 1 n (N 2 ), whenever EQ(δ) and EQ(γ) are Σ 1 n -subsets of N 2 .✷ It turns out that QCB 0 (P) is in a sense the smallest cartesian closed subcategory of QCB 0 containing ω. Proposition 8.2 There is no full cartesian closed subcategory C of QCB 0 such that C inherits binary products from QCB 0 , contains the discrete space ω of natural numbers and is contained itself in QCB 0 (Σ 1 n ) for some 1 ≤ n < ω.\nProof.We show at first that any cartesian closed subcategory D of QCB 0 that contains ω and inherits binary products from QCB 0 has the property that exponentials formed in D are homeomorphic to the corresponding QCB 0 -exponentials.\nLet E be an exponential formed in D to spaces X, Y ∈ D. Remember that this means that there exists a continuous evaluation function eval : E × X → Y such that for every space Z ∈ D and every continuous function h : Z × X → Y there is a unique continuous function ĥ : Z → E such that h(z, x) = eval ĥ(z)(x) for all z ∈ Z and x ∈ X.\nFor every continuous function f : X → Y it follows from the uniqueness condition (applied to Z = ω and the continuous function (i, x) → f (x)) that there exists a unique element t(f ) ∈ E satisfying f (x) = eval (t(f ), x) for all x ∈ X.Conversely, since Y X is an exponential to X and Y in the supercategory QCB 0 , there is a continuous function Ev : E → Y X satisfying Ev (e)(x) = eval (e, x) for all e ∈ E and x ∈ X.Clearly, Ev (t(f )) = f for every f ∈ Y X .The uniqueness condition implies t(Ev (e)) = e for every e ∈ E. Hence Ev is a continuous bijection with t as its inverse.\nTo show that t is a continuous function from Y X to E, let {f i } i be a sequence that converges in Y X to some function f ∞ ∈ Y X .This can be reformulated by stating that the function F : N ∞ × X → Y defined by F (i, x) := f i (x) for i ∈ ω ∪ {∞} and x ∈ X is continuous, where N ∞ denotes the one-point compactification of the discrete natural number with ∞ as the infinity point.(This is a well-known property of exponentials in the category of sequential topological spaces and therefore in the category QCB 0 , see e.g.[ELS04,Sch03]).\nBelow we will show that N ∞ is a continuous retract of the exponential E N formed in D to X = Y = ω.So let (s, r) be a continuous section-retraction pair.We define G : E N × X → ω by G(p, x) := F (r(p), x).Clearly, G is continuous.Since E is an exponential in D, there is a continuous function G : E N → E satisfying eval ( G(p), x) = G(p, x) for all p ∈ E N and x ∈ X.\nThe uniqueness condition implies G(p) = t(f r(p) ).As {i} i converges to ∞ in N ∞ , { G(s(i))} i converges to G(s(∞)) in E. Clearly, we have G(s(i)) = t(f i ) for all i ∈ ω ∪ {∞}.Therefore {t(f i )} i converges to t(f ∞ ) in E.\nWe conclude that t is sequentially continuous.Since Y X is a sequential topological space, t is even continuous in the topological sense.This means that Y X is homeomorphic to E. Now we show that N ∞ is indeed a retract of E N .As in the general case, there is a continuous bijection Ev N : E N → N .Hence E N is Hausdorff, but not discrete by being an uncountable space with a dense countable subset (see Proposition 3.3.1 in [Sch03]).So there exists an injective sequence {z i } i converging in E N to some point z ∞ ∈ E N \\ {z i | i ∈ ω}.Since N is zero-dimensional and Ev N is injective, there is a sequence {C k } k of clopen sets in N satisfying Then both functions are continuous and satisfy rs = id N∞ .Hence N ∞ is a continuous retract of E N .\nFinally, suppose that C were a cartesian closed subcategory of QCB 0 with the desired properties.By the above statement, C contains a space homeomorphic to N k for all k ∈ ω.In particular, there is a space E ∈ C homeomorphic to N n + 1 .Hence N n + 1 has an admissible representation δ such that EQ(δ) ∈ Σ 1 n (N 2 ).This contradicts Theorem 7.7.✷"},{"title":"Remark 8.3","content":"The assumption ω ∈ C in the last proposition is essential, because several cartesian closed categories of ω-algebraic domains are known [Ju90], which are all QCB 0 (Π 0 2 )-spaces by being quasi-Polish."},{"title":"Conclusion","content":"Hopefully, the results of this paper show that the introduced hierarchies are natural and interesting for CA and DST.The study of these hierarchies is of course in the very beginning and many natural questions remain open.In particular, we would like to see more natural and important witnesses for the non-collapse property of the hierarchies.A systematic development of DST for the introduced classes of spaces (in particular, for the projective QCB 0 -spaces) seems also a natural direction of future research.\nby Proposition 2.8.Define the function g : Y → Pω by g(y) = {n | y ∈ A n }.Then for any y ∈ Y we have\nby Proposition 2.10.Similarly, S 2 ∈ Π 0 1+β+1 (Pω × Pω) and therefore S ∈ Π 0 1+µ+1 (Pω × Pω) by Propositions 2.6 and 2.7.Now let A * = {x ∈ Pω | (x, f 1 (x)) ∈ S} and B * = {y ∈ Pω | (g 1 (y), y) ∈ S}.Since the function x → (x, f 1 (x)) is Σ 0 1+α -measurable and A * is the preimage of S under this function, A * ∈ Π 0 1+α+µ+1 (Pω) by Proposition 2.10.Similarly, B * ∈ Π 0 1+β+µ+1 (Pω).Then the sets A * , B * and the functions f * = f 1 | A * , g * = g 1 | B * have the desired properties.✷ The next fact is a special case of the previous theorem.Corollary 3.3 If we take in the previous theorem α = 0 and β = 1 then we obtain A * ∈ Π 0 3 (Pω) and B * ∈ Π 0 4 (Pω).\nand let f be a homeomorphism of A onto B. Then there exist A * ∈ Γ(X), B * ∈ Γ(Y ) and a homeomorphism f * of A * onto B * such that A ⊆ A * , B ⊆ B * , and f"},{"title":"Lemma 5. 4","content":"Let r : D → Y be a continuous function from a subspace D of N to a zerodimensional CB 0 -space Y .Then r (viewed as a partial function from N to Y ) is an admissible representation for Y iff there is a continuous function s : Y → D satisfying rs = id Y .\nso there is an admissible representation δ : D → Y for Y with D ∈ Π 0 α (N ).By Lemma 5.4 there is a continuous function s : Y → D satisfying δs = id Y .Since D is Hausdorff, S = s(Y ) is a closed subset of D, hence S ∈ Π 0 α (D) by Proposition 2.7.By Proposition 2.8 we obtain S ∈ Π 0 α (N ).\ny)} and is open by being the preimage of the open set D Hx(y) \\ Hx(y)−1 i=0 D i under the continuous map b → (x, b).Therefore H x is an element of the function space ω\nEv N (z k ) ∈ C k and {Ev N (z i ) | i ≤ ∞, i = k} ⊆ N \\ C k for every k ∈ ω.We define functions s : N ∞ → E N and r : E N → N ∞ by s(i) := z i and r(z) := min ∞, k ∈ ω z ∈ Ev −1 N (C k ) ."}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"A basic notion of Computable Analysis (CA) [Wei00] is the notion of an admissible representation of a topological space X.This is a partial continuous surjection δ from the Baire space N onto X satisfying a certain universality property (see Subsection 2.6 for some more details).Such a representation of X induces a reasonable computability theory on X, and the class of admissibly represented spaces is wide enough to include most spaces of interest for Analysis or Numerical Mathematics.As shown by the first author [Sch03], this class coincides with the class of the so-called QCB 0 -spaces, i.e.T 0 -spaces which are quotients of countably based spaces, and it forms a cartesian closed category (with the continuous functions as morphisms).Thus, among QCB 0 -spaces one meets many important function spaces including the Kleene-Kreisel continuous functionals [Kl959,Kr59] interesting for several branches of logic and computability theory.\\nAlong with the mentioned nice properties of QCB 0 -spaces, this class seems to be too broad to admit a deep understanding.Hence, it makes sense to search for natural subclasses of this class which still include \\\"practically\\\" important spaces but are (hopefully) easier to study.Interesting examples of such subclasses are obtained if we consider, for each level Γ of the classical Borel or Luzin (projective) hierarchies of Descriptive Set Theory (DST) [Ke95], the class of spaces which have an admissible representation of the complexity Γ (below we make this precise).The study of the resulting Borel and Luzin hierarchies of QCB 0 -spaces is one of the aims of this paper.\\nAlong with the hierarchies of QCB 0 -spaces, we will consider Borel and Luzin hierarchies of countably based T 0 -spaces (CB 0 -spaces for short) which are induced by the well-known fact that any CB 0 -space may be embedded in the algebraic domain Pω of all subsets of ω.The precise definition depends crucially on the possibility to find the \\\"right\\\" extensions of the classical hierarchies from the class of Polish spaces to the arbitrary spaces (including Pω).Such extensions were introduced by the second author in [Se04] and were studied in [Se05,Se05a,Se06,Se08,BY09,Br13].In particular, it was shown that many properties of these hierarchies in ω-continuous domains resemble the properties of classical hierarchies in Polish spaces [Ke95].\\nHierarchies of spaces obtained in this way turn out to be closely related to the corresponding hierarchies of QCB 0 -spaces.Moreover, among the first levels of the Borel hierarchy of CB 0spaces we meet some classes of spaces which attracted attention of several researches in the field of quasi-metric spaces, in particular the class of quasi-Polish spaces.The class of quasi-Polish spaces identified and studied in [Br13] is a good solution to the problem from [Se08] of finding a natural class of spaces that includes the Polish spaces and the ω-continuous domains and has a reasonable DST.\\nIn this paper we establish some basic properties of the above-mentioned hierarchies of spaces.In particular, we show that the hierarchies of spaces do not collapse, and that any level of any hierarchy is closed under retracts.As main technical tools to prove these results we use suitable generalizations of some classical facts (e.g. of the injectivity property of Pω and of Lavrentyev's Theorem on extending partial homeomorphisms in Polish spaces).Some of those generalizations might be also interesting in their own right.We also show that the class of all spaces in our hierarchies forms in a sense the smallest cartesian closed category of QCB 0 -spaces containing the discrete space ω of natural numbers, and establish the close relationship of the Luzin hierarchy of QCB 0 -spaces to the continuous functionals of finite type.Hence the class of all spaces in the Luzin hierarchy of QCB 0 -spaces seems to be a reasonable subclass of QCB 0 -spaces that contains most of spaces of interest for CA, including the Kleene-Kreisel continuous functionals.\\nAfter recalling some notions and known facts in the next section, we establish the main technical facts in Section 3. In Sections 4 and 5 we introduce and study the mentioned hierarchies of CB 0spaces and of QCB 0 -spaces.In Section 6 we establish close relationships between hierarchies of CB 0 -spaces to those of QCB 0 -spaces (namely, any level of the CB 0 -hierarchies coincides with the class of CB 0 -spaces in the corresponding level of the corresponding QCB 0 -hierarchy).In Section 7 we relate the Luzin hierarchy of QCB 0 -spaces to the Kleene-Kreisel continuous functionals.In Section 8 we discuss the cartesian closedness of the corresponding categories, and we conclude in Section 9.\"\n },\n {\n \"title\": \"Notation and Preliminaries\",\n \"content\": \"\"\n },\n {\n \"title\": \"Notation\",\n \"content\": \"We freely use the standard set-theoretic notation like dom(f ), rng (f ) and graph(f ) for the domain, range and graph of a function f , respectively, X × Y for the Cartesian product, X ⊔ Y for the disjoint union of sets X and Y , Y X for the set of functions f : X → Y (but in the case when X, Y are QCB 0 -spaces we use the same notation to denote the space of continuous functions from X to Y ), and P (X) for the set of all subsets of X.For A ⊆ X, A denotes the complement X \\\\ A of A in X.We identify the set of natural numbers with the first infinite ordinal ω.The first uncountable ordinal is denoted by ω 1 .The notation f : X → Y (resp.f :⊆ X → Y ) means that f is a total (resp.a partial) function from a set X to a set Y .\"\n },\n {\n \"title\": \"Topological Spaces\",\n \"content\": \"We assume the reader to be familiar with the basic notions of topology.The collection of all open subsets of a topological space X (i.e. the topology of X) is denoted by τ X ; for the underlying set of X we will write X in abuse of notation.We will usually abbreviate \\\"topological space\\\" to \\\"space\\\".Remember that a space is zero-dimensional if it has a basis of clopen sets.\\nA space Y is called a (continuous) retract of a space X if there are continuous functions s : Y → X and r : X → Y such that composition rs coincides with the identity function id Y on Y .Such a pair of functions (s, r) is called a section-retraction pair.Note that the section s is a homeomorphism between Y and the subspace s(Y ) = {x ∈ X | sr(x) = x} of X, and\\nLet ω be the space of non-negative integers with the discrete topology.Of course, the spaces ω × ω = ω 2 , and ω ⊔ ω are homeomorphic to ω, the first homeomorphism is realized by the Cantor pairing function •, • .\\nLet N = ω ω be the set of all infinite sequences of natural numbers (i.e., of all functions ξ : ω → ω).Let ω * be the set of finite sequences of elements of ω, including the empty sequence.For σ ∈ ω * and ξ ∈ N , we write σ ⊑ ξ to denote that σ is an initial segment of the sequence ξ.By σξ = σ • ξ we denote the concatenation of σ and ξ, and by σ • N the set of all extensions of σ in N .For x ∈ N , we can write x = x(0)x(1) . . .where x(i) ∈ ω for each i < ω.For x ∈ N and n < ω, let x <n = x(0) . . .x(n − 1) denote the initial segment of x of length n.Notations in the style of regular expressions like 0 ω , 0 * 1 or 0 m 1 n have the obvious standard meaning.\\nBy endowing N with the product of the discrete topologies on ω, we obtain the so-called Baire space.The product topology coincides with the topology generated by the collection of sets of the form σ • N for σ ∈ ω * .The Baire space is of primary importance for DST and CA.The importance stems from the fact that many countable objects are coded straightforwardly by elements of N , and it has very specific topological properties.In particular, it is a perfect zero-dimensional space and the spaces N 2 , N ω , ω ×N = N ⊔N ⊔• • • (endowed with the product topology) are all homeomorphic to N .Let (x, y) → x, y be a homeomorphism between N 2 and N .The Baire space N has the following universality property for zero-dimensional CB 0 -spaces:\\nThe subspace C = 2 ω of N formed by the infinite binary strings (endowed with the relative topology inherited from N ) is known as the Cantor space.Along with N and C, the space Pω is of principal importance for this paper.This is the space of subsets of the natural numbers with the Scott topology on (P (ω); ⊆).The basic open sets of this topology are of the form {A ⊆ ω | F ⊆ A}, where F ranges over the finite subsets of ω.\\nThe importance of Pω for this paper is explained by its following well-known properties.First, Pω is universal for CB 0 -spaces.\\nProof.Since Pω is a CB 0 -space, any space X homeomorphic to a subspace Pω is a CB 0 -space as well.Conversely, if {β i | i ∈ ω} is a base for a countably-based T 0 -space X, then the function e : X → Pω mapping\\nThe second property shows that Pω is an injective object in the category of all topological spaces.\\nProposition 2.3 [G+80, Proposition 3.5] Let Y be topological space and X be a subspace of Y .Then any continuous function f : X → Pω can be extended to a continuous function g : Y → Pω.\\nCompletely metrisable spaces satisfy the following extension theorem by Kuratowski.\\nProposition 2.4 [Ke95, Theorem 3.8] Let X be a metrisable space and Y be a completely metrisable space.Then any continuous function f : A → Y defined on a subset A of X can be extended to a continuous function g :\"\n },\n {\n \"title\": \"Families of Pointclasses\",\n \"content\": \"Here we recall a useful technical notion of a family of pointclasses introduced in [Se11].\\nA pointclass on X is simply a collection Γ(X) of subsets of X.We need the following \\\"parameterized\\\" version of the notion of pointclass.A family of pointclasses is a family Γ = {Γ(X)} indexed by arbitrary topological spaces X such that each Γ(X) is a pointclass on X and Γ is closed under continuous preimages, i.e. f −1 (A) ∈ Γ(X) for every A ∈ Γ(Y ) and every continuous function f : X → Y .In particular, any pointclass Γ(X) in such a family is downward closed under the Wadge reducibility on X.\\nTrivial examples of families of pointclasses are E, F, where E(X) = {∅} and F(X) = {X} for any space X.A basic example of a family of pointclasses is given by the family O = {τ X } of the topologies of all the spaces X.\\nFinally, we define some operations on families of pointclasses which are relevant to hierarchy theory.First, the usual set-theoretic operations will be applied to the families of pointclasses pointwise: for example, the union i Γ i of the families of pointclasses Γ 0 , Γ 1 , . . . is defined by\\nSecond, a large class of such operations is induced by the set-theoretic operations of L.V. Kantorovich and E.M. Livenson (see e.g.[Se11] for the general definition).Among them are the operations Γ → Γ σ where Γ(X) σ is the set of all countable unions of sets in Γ(X), the operation Γ → Γ c where Γ(X) c is the set of all complements of sets in Γ(X), the operation Γ → Γ d where Γ(X) d is the set of all differences of sets in Γ(X), and the operation Γ → Γ p defined by\\nThe next subsection contains some important examples of families of pointclasses from hierarchy theory.\"\n },\n {\n \"title\": \"Classical Hierarchies on Topological Spaces\",\n \"content\": \"Let us recall the definition of the Borel hierarchy on arbitrary spaces introduced in [Se04].\\nDefinition 2.5 For α < ω 1 , define the family of pointclasses Σ 0 α = {Σ 0 α (X)} by induction on α as follows: Σ 0 0 (X) = {∅}, Σ 0 1 (X) = τ X is the collection of the open sets of X, Σ 0 2 (X) = ((Σ 0 1 (X)) d ) σ is the collection of all countable unions of differences of open sets, and Σ 0 α (X) = ( β<α (Σ 0 β (X)) c ) σ (for α > 2) is the class of countable unions of sets in β<α (Σ 0 β (X)) c .The sequence {Σ 0 α (X)} α<ω 1 is called the Borel hierarchy of X.We also let Π 0 β (X) = (Σ 0 β (X)) c and ∆ 0 α (X) = Σ 0 α (X) ∩ Π 0 α (X).The classes Σ 0 α (X), Π 0 α (X), ∆ 0 α (X) are called the levels of the Borel hierarchy of X.\\nBy the definition and remarks at the end of the previous subsection, any of Σ 0 α , Π 0 α , ∆ 0 α is a family of pointclasses.It is straightforward to check by induction on α, β that using Definition 2.5 one has the following result.\\nProposition 2.6 For every topological space X and for all α\\nRemark.Definition 2.5 applies to all spaces X, and Proposition 2.6 holds true in the full generality.Note that Definition 2.5 differs from the classical definition for Polish spaces (see e.g.[Ke95, Section 11.B]) only for the level 2, and that for the case of Polish spaces our definition of the Borel hierarchy is equivalent to the classical one.Notice that the classical definition cannot be applied in general to non metrizable spaces (like e.g. the non discrete ω-algebraic domains) precisely because the inclusion Σ 0 1 ⊆ Σ 0 2 may fail.Let {Σ 1 n (X)} 1≤n<ω be Luzin's projective hierarchy in X (cf.[Br13]).Using the corresponding operation on families of pointclasses from the previous subsection we have Σ 1 1 (X) = (Π 0 2 (X)) p and Σ 1 n+1 (X) = (Π 1 n (X)) p for any n ≥ 1.Let also Σ 1 0 = Π 1 0 = ∆ 1 1 .The reason why the definition of the first level of the Luzin hierarchy is distinct from the classical definition Σ 1 1 (X) = (Π 0 1 (X)) p for Polish spaces is that the inclusion Σ 0 1 (X) ⊆ (Π 0 1 (X)) p may fail in general.Any level Σ 1 n , Π 1 n , ∆ 1 n of the Luzin hierarchy is a family of pointclasses, and we have the natural inclusions among them similar to the inclusions for levels of the Borel hierarchy.\\nLevels of the introduced hierarchies in an arbitrary space have closure properties similar to those known for the classical hierarchies in Polish spaces, in particular: Proposition 2.7 Any non-zero Σ-level of the Borel hierarchy on an arbitrary topological space is closed under finite intersections, countable unions and binary products (for the case of binary product this means that\\nAny non-zero Σ-level of the Luzin hierarchy in an arbitrary space is closed under countable unions, countable intersections, binary products and the projection along N -axis (for the case of projection this means that\\nWe will often use the following straightforward result which also extends the corresponding well-known facts for the classical hierarchies.\\nProposition 2.8 Let X be a subspace of a topological space Y , A ⊆ X, and let Γ be a Σ-or a Π-level of the introduced hierarchies.Then\\nWe will often cite the following topological complexity of the equality test\\nProof.The first statement has been shown in [Br13].The second (well-known) statement follows from the fact that in the Hausdorff case the complement of EQ X is equal to {U × V | U, V open and disjoint} and therefore open.✷ Let Γ be a family of pointclasses and X, Y be spaces.A function f :\\nfor each open set A ⊆ Y .Note that the continuous functions coincide with the Σ 0 1 -measurable functions.We will also use the following basic property of Γ-measurable functions.Note that addition + on ordinal numbers is associative, but not commutative.\\nNotice that the case (α, β) = (0, 0) denotes the fact that the preimage of an open (closed) set under a continuous function is open (resp.closed).\"\n },\n {\n \"title\": \"Polish and quasi-Polish spaces\",\n \"content\": \"Remember that a space X is Polish if it is countably based and metrisable with a metric d such that (X, d) is a complete metric space.Important examples of Polish spaces are the Baire space, the Cantor space, the space of reals R and its Cartesian powers R n (n < ω), the closed unit interval [0, 1], the Hilbert cube [0, 1] ω and the Hilbert space R ω .\\nBelow we will cite the following result known as Lavrentyev's Theorem:\\nProposition 2.11 [Ke95, Theorem 3.9] Let X, Y be Polish spaces, A ⊆ X, B ⊆ Y , and let f be a homeomorphism of A onto B (equipped with the subspace topologies induced from X and Y ).Then there exist\\nA natural variant of the class of Polish spaces has recently emerged.Given a set X, call a function d from X ×X to the nonnegative reals quasi-metric whenever x = y iff d(x, y) = d(y, x) = 0, and d(x, y) ≤ d(x, z)+d(z, y) (but we don't require d to be symmetric).In particular, every metric is a quasi-metric.Every quasi-metric on X canonically induces a topology on X which is denoted by τ d , where τ d is the topology generated by the open balls\\nWe say that the quasi-metric d on X is complete if every d-Cauchy sequence converges with respect to d (notice that this definition is coherent with the notion of completeness for a metric d, as in this case d = d).\\nA T 0 -space X is called quasi-Polish if it is countably based and there is a complete quasi-metric on X which generates its topology.In particular, every Polish space is quasi-Polish, but by [Br13,Corollary 45] also every ω-continuous domain is quasi-Polish.De Brecht [Br13] shows that there is a reasonable DST for the quasi-Polish spaces which extends the classical DST for Polish spaces [Ke95] and the DST for ω-continuous domains [Se06,Se08] in many directions.\\nAn important example of a quasi-Polish space is the space Pω equipped with the following quasi-metric d: if A ⊆ B then d(A, B) = 0, otherwise d(A, B) = 1/2 a where a is the smallest number in A \\\\ B. Note that Proposition 2.2 implies that any CB 0 -space is quasi-metrisable.\\nDe Brecht proved the following characterization of quasi-Polish spaces (cf.[Br13, Corollary 24]): Proposition 2.12 A topological space is quasi-Polish iff it is homeomorphic to a Π 0 2 -subset of Pω (endowed with the relative topology inherited from Pω).\"\n },\n {\n \"title\": \"Admissible Representations and QCB 0 -spaces\",\n \"content\": \"A representation of a space X is a surjection of a subspace of the Baire space N onto X. Usually it is denoted as a partial function from N to X.The notion of admissible representation is basic in Computable Analysis (CA).A representation δ of X is admissible, if it is continuous and any continuous function ν : Z → X from a zero-dimensional CB 0 -space Z to X is continuously reducible to δ, i.e. ν = δg for some continuous function g : Z → N .A topological space is admissibly representable if it has an admissible representation.\\nThe notion of admissibility was introduced in [KW85] for representations of countably based spaces (in a different but equivalent formulation) and was extensively studied by many authors.In [BH02] a close relation of admissible representations of countably based spaces to open continuous representations was established.In [Sch02,Sch03] the notion was extended to non-countably based spaces and a nice characterization of the admissibly represented spaces was achieved.Namely, the admissibly represented sequential topological spaces coincide with the QCB 0 -spaces, i.e., T 0 -spaces which are topological quotients of countably based spaces.\\nThe category QCB 0 of QCB 0 -spaces as objects and continuous functions as morphisms is known to be cartesian closed (cf.[ELS04,Sch03]).The exponential Y X to QCB 0 -spaces X, Y has the set of continuous functions from X to Y as the underlying set, and its topology is the sequentialization of the compact-open topology on Y X .By the sequentialization of a topology τ we mean the family of all sequentially open sets pertaining to this topology.(Remember that sequentially open sets are defined to be the complements of the sets that are closed under forming limits of converging sequences.)The sequentialization of τ is finer than or equal to τ .The topology of the QCB 0 -product to X and Y , which we denote by X × Y , is the sequentialization of the classical Tychonoff topology on the cartesian product of the underlying sets of X and Y .So products and exponentials in QCB 0 are formed in the same way as in its supercategory Seq of sequential topological spaces.\\nFrom [Br13] it follows that admissible total representations are closely related to quasi-Polish spaces.The following assertion is contained among the results in [Br13].\\nProposition 2.13 For any CB 0 -space X the following statements are equivalent:\\n(1) X is quasi-Polish.\\n(2) X has an open continuous total representation.\\n(3) X has an admissible total representation.\\n(4) X has an admissible representation whose domain is a Polish space.\\nWe will also cite the following facts from [Sch02,Sch03].\\nProposition 2.14 Let δ and γ be admissible representations of QCB 0 -spaces X and Y , respectively.Then f : X → Y is continuous iff f δ = γg for some partial continuous function g on N .\\nNote that if u :⊆ N 2 → N is continuous and p ∈ N then u p = λx.u(p,x) is a partial continuous function on N .\\nProposition 2.15 There is a partial continuous function u : ⊆ N 2 → N such that dom(u) ∈ Π 0 2 (N 2 ) and for any partial continuous function g on N there is some p ∈ N such that u p is an extension of g.\\nThe function u in Proposition 2.15 can be chosen as the application operator of the Second Kleene Algebra.We use it to describe the construction of admissible representations for function spaces formed in QCB 0 (cf.[Sch03,Wei00]).\\nProposition 2.16 Let δ and γ be admissible representations for QCB 0 -spaces X and Y , respectively.Then admissible representations [δ × γ] for the QCB 0 -product X × Y and [δ → γ] for the QCB 0 -exponential Y X can be defined by:\\nfor p, q ∈ N , x ∈ X, y ∈ Y , and f : X → Y .\\nDe Brecht and Yamamoto showed the following property of subsets of admissibly represented countably-based spaces.\\nProposition 2.17 [BY09, Corollary 3].Let δ be an admissible representation of a countablybased T 0 -space X, let A ⊆ X and let\"\n },\n {\n \"title\": \"Main Technical Facts\",\n \"content\": \"In this section we prove a couple of facts that serve as main technical tools in the sequel, but some of these facts might be also of independent interest.\\nThe first result generalizes the well-known fact that Pω is an injective space (see Proposition 2.2), because the continuous functions coincide with the Σ 0 1 -measurable functions.\\nTheorem 3.1 Let Y be a topological space, let X be a subspace of Y , and let\\nis a subbasis in Pω and Γ(Y ) is closed under finite intersection and countable union by Proposition 2.7, g is Γ-measurable.If y ∈ X then we have\\nThe second result is a remote relative of Lavrentyev's Theorem (cf.Proposition 2.11).\\nTheorem 3.2 Let A, B ⊆ Pω be subspaces of Pω, α, β < ω 1 , and let f : A → B be a Σ 0 1+αmeasurable bijection such that its inverse g = f −1 is Σ 0 1+β -measurable.Then there exist A * ∈ Π 0 1+α+µ+1 (Pω), B * ∈ Π 0 1+β+µ+1 (Pω) (where µ = max{α, β}) and a Σ 0 1+α -measurable bijection\\nProof.By Theorem 3.1, there are a Σ 0 1+α -measurable extension f 1 : Pω → Pω of f and a Σ 0 1+β -measurable extension g 1 : Pω → Pω of g.Consider the set\\nby Proposition 2.9 and S 1 is the preimage of EQ Pω under the Let us introduce one of the main notions of this paper.\\nDefinition 3.4 Let Γ be a family of pointclasses.A topological space X is called a Γ-space if X is homeomorphic to a subspace A ⊆ Pω with A ∈ Γ(Pω).The class of all Γ-spaces is denoted CB 0 (Γ).\\nThe third result of this section extends Corollary 24 in [Br13] (which is obtained when Γ = Π 0 2 ).\\nω} and f :⊆ X → Y be a partial continuous function from a topological space X to a Γ-space Y .Then there is a continuous extension g :⊆ X → Y of f with dom(g) ∈ Γ(X).\\nProof.Without loss of generality we assume Y ∈ Γ(Pω), so in particular f :⊆ X → Pω.By Proposition 2.3, there is a total continuous extension h : X → Pω of f .Since Y ∈ Γ(Pω), h −1 (Y ) ∈ Γ(X).Since dom(f ) ⊆ h −1 (Y ), we can take the restriction of h to h −1 (Y ) as the desired function g. ✷\\nThe fourth result of this section is the following extension of Lavrentyev's Theorem (see Proposition 2.11).For Γ = Π 0 2 the result gives the extension of Lavrentyev's Theorem to quasi-Polish spaces (cf.[Br13]), and Lavrentyev's Theorem is obtained if we restrict the last fact to Polish spaces.\"\n },\n {\n \"title\": \"Let now\",\n \"content\": \"A * ∈ Γ(A 1 ) and B * ∈ Γ(B 1 ).By Proposition 2.8 we have A * ∈ Γ(X) and B * ∈ Γ(Y ).✷ Finally, we give a natural version of the previous theorem related to retracts, although this version is not used in the sequel.\\nProposition 3.7 Let Γ, X, Y, A, B be as in the previous theorem and let r : A → B and s : B → A be continuous functions such that rs = id B (hence, B is a retract of A).Then there exist A * ∈ Γ(X), B * ∈ Γ(Y ) and continuous functions r * :\\nProof.By Proposition 3.5 there exist A 1 ∈ Γ(X), B 1 ∈ Γ(Y ) and continuous functions r 1 :\\nSimilarly to the previous proof one checks that these objects have the desired properties.✷ 4 Hierarchies of CB 0 -Spaces\\nHere we introduce and study natural hierarchies of CB 0 -spaces (see Definition 3.4) as well as the Borel hierarchy and the Luzin hierarchy on Pω discussed in Subsection 2.4.\\nDefinition 4.1 (1) By the Borel hierarchy of CB 0 -spaces we mean the sequence of classes {CB 0 (Σ 0 α )} α<ω 1 .By levels of this hierarchy we mean the classes CB 0 (Σ 0 α ) as well as the classes CB 0 (Π 0 α ) and CB 0 (∆ 0 α ).\\n(2) By the Luzin hierarchy of CB 0 -spaces we mean the sequence of classes {CB 0 (Σ 1 n )} n<ω .By levels of this hierarchy we mean the classes CB 0 (Σ 1 n ) as well as the classes CB 0 (Π 1 n ) and CB 0 (∆ 1 n ).\\nNote that the Borel hierarchy of spaces includes some natural classes of spaces identified earlier.E.g., by Proposition 2.12 CB 0 (Π 0 2 ) coincides with the class of quasi-Polish spaces and, by Corollary 33 in [Br13], CB 0 (Π 0 3 ) coincides with the class of CB 0 -spaces that admit a compatible bicomplete quasi-metric in the sense of [JK98].\\nObviously, for any families of pointclasses Γ, ∆ we have: if Γ ⊆ ∆ (i.e.Γ(X) ⊆ ∆(X) for each space X) then CB 0 (Γ) ⊆ CB 0 (∆).Therefore, we have the natural inclusions for levels of the introduced hierarchies, in particular CB 0 (∆ 0 α ) ⊆ CB 0 (Σ 0 α )∩CB 0 (Π 0 α ) and CB 0 (Σ 0 α )∪CB 0 (Π 0 α ) ⊆ CB 0 (∆ 0 β ) for all α < β < ω 1 .Below we establish some basic properties of the Borel and the Luzin hierarchy of spaces, but first we show that most of levels of the introduced hierarchies of spaces are closed under retracts (for Π 0 2 the result is already known from [Br13]).\\nThen any retract of a Γ-space is a Γ-space.\\nProof.Let Y be a retract of a Γ-space X via a section-retraction pair (s, r) of continuous functions; we have to show that Y is a Γ-space.We may assume without loss of generality that X ∈ Γ(Pω).Since s(Y ) = {x ∈ X | sr(x) = x}, s(Y ) is the preimage of EQ X under the continuous function x → (sr(x), x).Since EQ X ∈ Π 0 2 (X ×X) by Proposition 2.9, s(Y ) ∈ Π 0 2 (X) and thus s(Y ) ∈ Γ(X) by Proposition 2.6.Proposition 2.8 yields s(Y ) ∈ Γ(Pω).Since Y is homeomorphic to s(Y ), Y is a Γ-space.✷ Now we establish an interesting property of the introduced hierarchies of spaces which implies the non-collapse property.\"\n },\n {\n \"title\": \"Proposition 4.3 For any countable ordinal\",\n \"content\": \"This proposition is based on the following lemma.\\nProof.If the underlying set of X is in Γ(Pω), then X is a Γ-space by Definition 3.4.Conversely, let X be a Γ-space.Then there is some B ∈ Γ(Pω) such that B (endowed with the subspace topology inherited from Pω) is homeomorphic to X. Let h : X → B be a homeomorphism.We apply Theorem 3.6 to extend h to a homeomorphism h * : X * → B * , where X * and B * are sets in Γ(Pω) with X ⊆ X * and B ⊆ B * .By Proposition 2.8 we have B ∈ Γ(B * ).Hence X ∈ Γ(X * ), because X = (h * ) −1 (B).Proposition 2.8 yields X ∈ Γ(Pω).✷ Now we are ready to give the proof of Proposition 4.3.\\nProof.The inclusions from right to left are obvious.It remains to check CB 0 (Γ)\\nThen Z is homeomorphic to some subspaces A, B of Pω with A ∈ Γ(Pω) and B ∈ Γ c (Pω).Hence B is a Γ-space by being homeomorphic to A. Lemma 4.4 yields B ∈ Γ(Pω).Therefore B ∈ (Γ ∩ Γ c )(Pω) and hence Z ∈ CB 0 (Γ ∩ Γ c ). ✷ Corollary 4.5 The Borel hierarchy and the Luzin hierarchy of CB 0 -spaces do not collapse.More precisely CB 0 (Σ 0 α ) ⊆ CB 0 (Π 0 α ) for each countable ordinal α ≥ 2, and CB 0 (Σ 1 n ) ⊆ CB 0 (Π 1 n ) for each positive integer n.\\nProof.Proofs for both hierarchies are similar, so consider only the Borel hierarchy.According to [Se05a], there is a set A in Σ 0 α (Pω) \\\\ Π 0 α (Pω).The space A (with the topology induced from Pω) is obviously a Σ 0 α -space.If it were a Π 0 α -space, then by Lemma 4.4 we would have A ∈ Π 0 α (Pω), a contradiction.✷ 5 Hierarchies of QCB 0 -Spaces\\nAs shown in [Sch03], any QCB 0 -space has an admissible representation.Here we introduce and study natural hierarchies of QCB 0 -spaces induced by this fact.For any representation δ of a space X, let EQ(δ) := {(p, q) ∈ N 2 | p, q ∈ dom(δ) ∧ δ(p) = δ(q)}.\\nDefinition 5.1 (1) Let Γ be a family of pointclasses.A topological space X is called Γrepresentable if X has an admissible representation δ with EQ(δ) ∈ Γ(N × N ).The class of all Γ-representable spaces is denoted QCB 0 (Γ).\\n(2) By the Borel hierarchy of QCB 0 -spaces we mean the sequence {QCB 0 (Σ 0 α )} α<ω 1 .By levels of this hierarchy we mean the classes QCB 0 (Σ 0 α ) as well as the classes QCB 0 (Π 0 α ) and QCB 0 (∆ 0 α ).\\n(3) By the Luzin hierarchy of QCB 0 -spaces we mean the sequence {QCB 0 (Σ 1 n )} n<ω .By levels of this hierarchy we mean the classes QCB 0 (Σ 1 n ) as well as the classes QCB 0 (Π 1 n ) and QCB 0 (∆ 1 n ).\\nThe next assertion establishes an equivalent simpler definition of most levels for the case of CB 0 -spaces.\\nω} and let X be a Hausdorff space.Then X is Γ-representable iff X has an admissible representation δ with dom(δ) ∈ Γ(N ).\\nProof.The only-if-part of the first statement holds for any topological space X and any family of pointclasses Γ.Indeed, let δ be an admissible representation of X with EQ(δ) ∈ Γ(N ×N ).Since dom(δ) is the preimage of EQ(δ) under the continuous function x → (x, x), dom(δ) ∈ Γ(N ).\\nConversely, let δ be an admissible representation of X with dom(δ) ∈ Γ(N ).By Proposition 2.9, EQ X ∈ Π 0 2 (X × X), so EQ(δ) is a Π 0 2 -set in dom(δ) × dom(δ) by the continuity of δ.The set dom(δ) × dom(δ) is a Γ-subset of N × N by being the intersection of the sets {(x, y)\\n. By Propositions 2.6 and 2.8 we obtain EQ(δ) ∈ Γ(N × N ).\\nThe second statement follows similarly by taking into account that EQ X is closed in X × X by Proposition 2.9, if X is a Hausdorff space.✷\\nThe main advantage of the hierarchies of this section over the hierarchies from the previous section is that they include many natural non-countably based spaces, in particular the spaces of Kleene-Kreisel continuous functionals, as we will see later.\\nObviously, for any families of pointclasses Γ, ∆ we have: Γ ⊆ ∆ implies QCB 0 (Γ) ⊆ QCB 0 (∆), hence we have the natural inclusions for levels of the introduced hierarchies, in particular\\n) for all α < β < ω 1 .We will show that these hierarchies do not collapse, but first we establish the closure of the levels under retracts.\\nThen any retract of a Γ-representable space is a Γ-representable space.\\nProof.Let Y be a retract of a Γ-representable space X via a section-retraction pair (s, r) of continuous functions; we have to show that Y is a Γ-representable space.Let δ be an admissible representation for X with EQ(δ) ∈ Γ(N ×N ).Then rδ is clearly an admissible representation for Y , so it suffices to show that EQ(rδ) ∈ Γ(N ×N ).Since s is an injection, EQ(rδ) is the preimage of EQ(δ) under the continuous function (x, y) → (sr(x), sr(y)).Therefore, EQ(rδ) ∈ Γ(N × N ).✷\\nThe following observation is well-known.Proof.Let Y be a retract of D via a section-retraction pair (s, r) of continuous functions.Then r :⊆ N → Y is admissible, because any continuous function ν : Z → Y defined on a zero-dimensional CB 0 -space Z is reducible to r via sν.\\nConversely, assume that r ⊆ : N → Y is an admissible representation of Y .Then the identity function id Y is reducible to r via some continuous function s : Y → D. Then id Y = rs, hence Y is a retract of D via the continuous section s and the continuous retraction r. ✷ Finally, we establish the non-collapse property of the introduced hierarchies of spaces.\\nTheorem 5.5 The Borel hierarchy and the Luzin hierarchy of QCB 0 -spaces do not collapse.More precisely, QCB 0 (Σ 0 α ) ⊆ QCB 0 (Π 0 α ) for each countable ordinal α ≥ 2, and QCB 0 (Σ 1 n ) ⊆ QCB 0 (Π 1 n ) for each positive integer n.\\nProof.Proofs for both hierarchies are similar, so consider only the Borel hierarchy.As is well-known Since N is a Π 0 2 -space, hence a Γ-space, we can use Theorem 3.6 (applied to X ′ = Y ′ = N ) to extend the homeomorphism s : Y → S to a homeomorphism s * between larger sets Y * , S * ∈ Π 0 2 (N ).By Proposition 2.6 we have Y * , S * ∈ Π 0 α (N ).Since S ∈ Π 0 α (S * ) by Proposition 2.8,\\nThe spaces Y witnessing the non-collapse property above are rather artificial.In Theorem 7.7 we will find very natural spaces witnessing the non-collapse of the Luzin hierarchy of QCB 0 -spaces.\"\n },\n {\n \"title\": \"Relating the Hierarchies\",\n \"content\": \"In this section we establish close relationships of the hierarchies of CB 0 -spaces with the corresponding hierarchies of QCB 0 -spaces.The next result implies that any level of a hierarchy of QCB 0 -spaces extends the corresponding level in the corresponding hierarchy of CB 0 -spaces.\\nProof.It suffices to show that any space X ∈ Γ(Pω) is a Γ-representable space.Let ρ : N → Pω be the total admissible representation of Pω defined by ρ(x) = {n | ∃i(x(i) = n + 1)} (see e.g.[Wei00,Br13] for details).Then the restriction of ρ to ρ −1 (X) ∈ Γ(N ) is an admissible representation of X.Since X is a CB 0 -space, X is a Γ-representable space by Proposition 5.2.✷\\nIn particular, we have CB 0 (Γ) ⊆ QCB 0 (Γ) ∩ CB 0 for each level Γ of the Borel hierarchy or the Luzin hierarchy.The main question of this section is: for which levels Γ we have the equality CB 0 (Γ) = QCB 0 (Γ) ∩ CB 0 ?Proposition 2.12 implies that the equality holds for Γ = Π 0 2 .The next result implies that the equality holds for all zero-dimensional CB 0 -spaces.\\nProof.Let r : D → X be an admissible representation of X with D ∈ Γ(N ).By Lemma 5.4 there is a continuous function s :\\nAs f s is a homeomorphism of X onto f s(X), we have f s(X) ∈ Γ(Y ).Propositions 2.6 and 2.8 yield f s(X) ∈ Γ(Pω).Therefore X ∈ CB 0 (Γ).✷ By the next theorem the equality CB 0 (Γ) = QCB 0 (Γ) ∩ CB 0 holds for almost all levels.For finite levels, this was pointed out to us by Matthew de Brecht.We thank him for giving the permission to use his proof of Theorem 6.3(2).Note that his proof may be used to obtain also a proof of Theorem 6.3(1) which is slightly different from ours.\\n(2) For all natural numbers m ≥ 2 and n ≥ 3, we have\\nProof.\\n(\\nWe have already seen CB 0 (Γ) ⊆ QCB 0 (Γ) ∩ CB 0 .\\nLet X ∈ QCB 0 (Γ) ∩ CB 0 .Then there is an admissible representation δ : D → X of X with D ∈ Γ(N ).By Proposition 2.1 we may assume w.l.o.g.X to be a subspace of Pω.We have to show X ∈ CB 0 (Γ).\\nLet χ : Pω → C be the bijection between Pω and the Cantor space C = 2 ω that sends any A ⊆ ω to its characteristic function χ A .Obviously, χ is Σ 0 2 -measurable while its inverse χ −1 is continuous (i.e.Σ 0 1 -measurable).Let σ = χ| X be the restriction of χ to X ⊆ Pω.By Proposition 2.8, σ is a Σ 0 2 -measurable bijection between X and σ(X) ⊆ C such that σ −1 is Σ 0 1 -measurable.Since σ −1 is continuous as a function from the zero-dimensional CB 0 -space σ(X) to X and δ is admissible, there is a continuous function g : σ(X) → N with σ −1 = δg.Then id X = σ −1 σ = δgσ, hence the function h = gσ satisfies h(X) = {y ∈ D | hδ(y) = y}.Thus, h is a Σ 0 2 -measurable bijection between X and H = h(X), whereas its inverse h −1 = δ| h(X) is Σ 0 1 -measurable (i.e.continuous).By Proposition 6.2 we may w.l.o.g.assume that D is a subspace of Pω and D ∈ Γ(Pω).Since H is the preimage of EQ D ∈ Π 0 2 (D × D) under the Σ 0 2 -measurable function y → (hδ(y), y), we have H ∈ Π 0 3 (D) by Proposition 2.10 and hence H ∈ Γ(D) by Proposition 2.6.Proposition 2.8 yields H ∈ Γ(Pω).By Theorem 3.2 and Corollary 3.3, there exist H * ∈ Π 0 3 (Pω), X * ∈ Π 0 4 (Pω), a continuous bijection δ * : H * → X * and a Σ 0 2 -measurable bijection h * : X * → H * such that δ * is the inverse of h * , δ * is an extension of h −1 and h * is an extension of h.Since H ∈ Γ(H * ) by Proposition 2.8 and X = h * −1 (H), Proposition 2.10 yields us X ∈ Γ(X * ).By Propositions 2.6 and 2.8, we obtain X ∈ Γ(Pω), hence X ∈ CB 0 (Γ).This completes the proof.\\n(2) Let δ be an admissible representation of a subspace X of Pω such that dom(δ) ∈ Γ(N ), where Γ ∈ {Π 0 m , Σ 0 n | m ≥ 2, n ≥ 3}.Let ρ be the total admissible representation of Pω defined in the proof of Proposition 6.1.By Proposition 2.17 it is enough to show that ρ −1 (X) is in Γ(N ).Since the corestriction of ρ to X is continuous, there is a continuous function h : ρ −1 (X) → N satisfying ρ(p) = δh(p) for all p ∈ ρ −1 (X).As Pω is an injective space and N is a Polish space, δ and h can be extended to continuous functions δ * : N → Pω and h * : G → N , where G is a Π 0 2 -subset of N , see Propositions 2.3 and 2.4.By Proposition 2.9,\\n2 -subset of G and thus of N .It is easy to verify that ρ −1 (X) is the intersection of A with (h * ) −1 (dom(δ)).By Proposition 2.7 and 2.8 this implies that ρ −1 (X) is Γ-subset of N .Proposition 2.17 yields that X is a Γ-subset of Pω, i.e., X ∈ CB 0 (Γ).✷ required function g : N × ω Y → N in such a way that g maps (x, H) to x, if H is a witness for x.Otherwise g assigns to (x, H) some element of a countable dense subset of M .\\nTo construct an appropriate dense subset of M , we choose for every finite sequence w in the set W := {x <k | x ∈ M, k ∈ ω} some α(w) ∈ M such that α(w) has w as an initial segment.Clearly, {α(w) | w ∈ W } is dense in M .\\nBy being a QCB 0 -space, Y has a countable dense subset {β j | j ∈ ω} (see Proposition 3.3.1 in [Sch03]).We construct a sequence {C k } k of subsets of the QCB 0 -space N ×ω Y by C 0 := N ×ω Y and C k := (x, H) ∈ N × ω Y x <k ∈ W and (x, β j ) ∈ D H(β j ) for all j ≤ k for all k > 0. The set C k is clopen in the Tychonoff topology on N × ω Y and thus in the QCB 0 -topology, because C k is the union of the clopen sets\\nwhere w ranges over the finite sequence of length k in W and a 0 , . . ., a k vary over the natural numbers.Note that {H ∈ ω Y | β j ∈ H −1 {a j }} is clopen even in the compact-open topology on ω Y and therefore in the QCB 0 -topology on ω Y , which is finer than the former.\\nNext we define a function ℓ\\nFirst we show the continuity of g.Let O be an open set of the Baire space and let (x, H) ∈ g −1 (O).Then there is some m ∈ ω such that g(x, H) <m • N ⊆ O.\\nCase 1: Assume ℓ(x, H) = ∞.Then (x, H) ∈ k∈ω C k .We define a set U by\\nCase 2: Assume ℓ(x, H) < ∞.Let k := ℓ(x, H) and define a set U by\\nWe conclude that in both cases g is continuous in the point (x, H).Therefore g is continuous, even with respect to the Tychonoff product on the set N × ω Y , which is coarser than the QCB 0 -topology.Now we show M ⊆ rng (g).Let x ∈ M .We define a function H x : Y → ω by\\nNote that H x is total, because we have f (y) = x and therefore (x, y) ∈ k∈ω D k for every y ∈ Y .Furthermore, H is continuous in every point y ∈ Y , because the set\\nIt remains to show rng(g) ⊆ M .So let x ∈ rng (g).The only interesting case is x / ∈ {α(w) | w ∈ W }. Then there is some continuous function H : Y → ω such that g(x, H) = x and thus (x, H) ∈ k∈ω C k .Suppose for a contradiction that there is some y ∈ Y such that (x, y) / ∈ D H(y) .Then the set\\nWe conclude (x, y) ∈ D H(y) and thus f (y) = x for every y ∈ Y .Therefore x / ∈ rng (f ) and x ∈ M .So g satisfies the required properties.✷ Using the cartesian closedness of QCB 0 , we define a sequence of spaces {N k } k<ω by induction on k as follows: N 0 := ω and N k + 1 := ω N k , where ω denotes the space of natural numbers endowed with the discrete topology.Obviously N 1 is equal to the Baire space N .The space N k is referred to as the sequential space of (Kleene-Kreisel) continuous functionals of type k.For any k ≥ 2, the sequential topology on N k is strictly finer than the corresponding compactopen topology [Hy79].Furthermore it is neither zero-dimensional nor regular [Sch09].Any of these spaces has a natural admissible representation.From Theorem 7.1 we obtain:\\nk ) by Theorem 7.1.✷ We remark that Lemma 2.34 in [No80] implies that N k has an admissible representation such that its domain is even in the effective version Π 1 k−1 of Π 1 k−1 (N ).This follows from the fact that the function which maps \\\"associates\\\" for N k to the encoded functionals forms an admissible representation.\\nNote that the continuous functionals (also known under the somewhat misleading name \\\"countable functionals\\\") were first defined in [Kl959,Kr59] independently by S. Kleene and G. Kreisel.Their definitions look different from each other, as well as from the definition above, although all Proof.We only have to check that QCB 0 (P) is closed under binary products and exponentiation formed in the supercategory QCB 0 .For exponentiation this follows from Theorem 7.1.It is an easy exercise to show that the product representation [δ × γ] constructed in Proposition 2.16 satisfies EQ([δ × γ]) ∈ Σ 1 n (N 2 ), whenever EQ(δ) and EQ(γ) are Σ 1 n -subsets of N 2 .✷ It turns out that QCB 0 (P) is in a sense the smallest cartesian closed subcategory of QCB 0 containing ω. Proposition 8.2 There is no full cartesian closed subcategory C of QCB 0 such that C inherits binary products from QCB 0 , contains the discrete space ω of natural numbers and is contained itself in QCB 0 (Σ 1 n ) for some 1 ≤ n < ω.\\nProof.We show at first that any cartesian closed subcategory D of QCB 0 that contains ω and inherits binary products from QCB 0 has the property that exponentials formed in D are homeomorphic to the corresponding QCB 0 -exponentials.\\nLet E be an exponential formed in D to spaces X, Y ∈ D. Remember that this means that there exists a continuous evaluation function eval : E × X → Y such that for every space Z ∈ D and every continuous function h : Z × X → Y there is a unique continuous function ĥ : Z → E such that h(z, x) = eval ĥ(z)(x) for all z ∈ Z and x ∈ X.\\nFor every continuous function f : X → Y it follows from the uniqueness condition (applied to Z = ω and the continuous function (i, x) → f (x)) that there exists a unique element t(f ) ∈ E satisfying f (x) = eval (t(f ), x) for all x ∈ X.Conversely, since Y X is an exponential to X and Y in the supercategory QCB 0 , there is a continuous function Ev : E → Y X satisfying Ev (e)(x) = eval (e, x) for all e ∈ E and x ∈ X.Clearly, Ev (t(f )) = f for every f ∈ Y X .The uniqueness condition implies t(Ev (e)) = e for every e ∈ E. Hence Ev is a continuous bijection with t as its inverse.\\nTo show that t is a continuous function from Y X to E, let {f i } i be a sequence that converges in Y X to some function f ∞ ∈ Y X .This can be reformulated by stating that the function F : N ∞ × X → Y defined by F (i, x) := f i (x) for i ∈ ω ∪ {∞} and x ∈ X is continuous, where N ∞ denotes the one-point compactification of the discrete natural number with ∞ as the infinity point.(This is a well-known property of exponentials in the category of sequential topological spaces and therefore in the category QCB 0 , see e.g.[ELS04,Sch03]).\\nBelow we will show that N ∞ is a continuous retract of the exponential E N formed in D to X = Y = ω.So let (s, r) be a continuous section-retraction pair.We define G : E N × X → ω by G(p, x) := F (r(p), x).Clearly, G is continuous.Since E is an exponential in D, there is a continuous function G : E N → E satisfying eval ( G(p), x) = G(p, x) for all p ∈ E N and x ∈ X.\\nThe uniqueness condition implies G(p) = t(f r(p) ).As {i} i converges to ∞ in N ∞ , { G(s(i))} i converges to G(s(∞)) in E. Clearly, we have G(s(i)) = t(f i ) for all i ∈ ω ∪ {∞}.Therefore {t(f i )} i converges to t(f ∞ ) in E.\\nWe conclude that t is sequentially continuous.Since Y X is a sequential topological space, t is even continuous in the topological sense.This means that Y X is homeomorphic to E. Now we show that N ∞ is indeed a retract of E N .As in the general case, there is a continuous bijection Ev N : E N → N .Hence E N is Hausdorff, but not discrete by being an uncountable space with a dense countable subset (see Proposition 3.3.1 in [Sch03]).So there exists an injective sequence {z i } i converging in E N to some point z ∞ ∈ E N \\\\ {z i | i ∈ ω}.Since N is zero-dimensional and Ev N is injective, there is a sequence {C k } k of clopen sets in N satisfying Then both functions are continuous and satisfy rs = id N∞ .Hence N ∞ is a continuous retract of E N .\\nFinally, suppose that C were a cartesian closed subcategory of QCB 0 with the desired properties.By the above statement, C contains a space homeomorphic to N k for all k ∈ ω.In particular, there is a space E ∈ C homeomorphic to N n + 1 .Hence N n + 1 has an admissible representation δ such that EQ(δ) ∈ Σ 1 n (N 2 ).This contradicts Theorem 7.7.✷\"\n },\n {\n \"title\": \"Remark 8.3\",\n \"content\": \"The assumption ω ∈ C in the last proposition is essential, because several cartesian closed categories of ω-algebraic domains are known [Ju90], which are all QCB 0 (Π 0 2 )-spaces by being quasi-Polish.\"\n },\n {\n \"title\": \"Conclusion\",\n \"content\": \"Hopefully, the results of this paper show that the introduced hierarchies are natural and interesting for CA and DST.The study of these hierarchies is of course in the very beginning and many natural questions remain open.In particular, we would like to see more natural and important witnesses for the non-collapse property of the hierarchies.A systematic development of DST for the introduced classes of spaces (in particular, for the projective QCB 0 -spaces) seems also a natural direction of future research.\\nby Proposition 2.8.Define the function g : Y → Pω by g(y) = {n | y ∈ A n }.Then for any y ∈ Y we have\\nby Proposition 2.10.Similarly, S 2 ∈ Π 0 1+β+1 (Pω × Pω) and therefore S ∈ Π 0 1+µ+1 (Pω × Pω) by Propositions 2.6 and 2.7.Now let A * = {x ∈ Pω | (x, f 1 (x)) ∈ S} and B * = {y ∈ Pω | (g 1 (y), y) ∈ S}.Since the function x → (x, f 1 (x)) is Σ 0 1+α -measurable and A * is the preimage of S under this function, A * ∈ Π 0 1+α+µ+1 (Pω) by Proposition 2.10.Similarly, B * ∈ Π 0 1+β+µ+1 (Pω).Then the sets A * , B * and the functions f * = f 1 | A * , g * = g 1 | B * have the desired properties.✷ The next fact is a special case of the previous theorem.Corollary 3.3 If we take in the previous theorem α = 0 and β = 1 then we obtain A * ∈ Π 0 3 (Pω) and B * ∈ Π 0 4 (Pω).\\nand let f be a homeomorphism of A onto B. Then there exist A * ∈ Γ(X), B * ∈ Γ(Y ) and a homeomorphism f * of A * onto B * such that A ⊆ A * , B ⊆ B * , and f\"\n },\n {\n \"title\": \"Lemma 5. 4\",\n \"content\": \"Let r : D → Y be a continuous function from a subspace D of N to a zerodimensional CB 0 -space Y .Then r (viewed as a partial function from N to Y ) is an admissible representation for Y iff there is a continuous function s : Y → D satisfying rs = id Y .\\nso there is an admissible representation δ : D → Y for Y with D ∈ Π 0 α (N ).By Lemma 5.4 there is a continuous function s : Y → D satisfying δs = id Y .Since D is Hausdorff, S = s(Y ) is a closed subset of D, hence S ∈ Π 0 α (D) by Proposition 2.7.By Proposition 2.8 we obtain S ∈ Π 0 α (N ).\\ny)} and is open by being the preimage of the open set D Hx(y) \\\\ Hx(y)−1 i=0 D i under the continuous map b → (x, b).Therefore H x is an element of the function space ω\\nEv N (z k ) ∈ C k and {Ev N (z i ) | i ≤ ∞, i = k} ⊆ N \\\\ C k for every k ∈ ω.We define functions s : N ∞ → E N and r : E N → N ∞ by s(i) := z i and r(z) := min ∞, k ∈ ω z ∈ Ev −1 N (C k ) .\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"by Proposition 2.8.Define the function g : Y → Pω by g(y) = {n | y ∈ A n }.Then for any y ∈ Y we have","content":"by Proposition 2.8.Define the function g : Y → Pω by g(y) = {n | y ∈ A n }.Then for any y ∈ Y we have"},{"id":"","caption":"by Proposition 2.10.Similarly, S 2 ∈ Π 0 1+β+1 (Pω × Pω) and therefore S ∈ Π 0 1+µ+1 (Pω × Pω) by Propositions 2.6 and 2.7.Now let A * = {x ∈ Pω | (x, f 1 (x)) ∈ S} and B * = {y ∈ Pω | (g 1 (y), y) ∈ S}.Since the function x → (x, f 1 (x)) is Σ 0 1+α -measurable and A * is the preimage of S under this function, A * ∈ Π 0 1+α+µ+1 (Pω) by Proposition 2.10.Similarly, B * ∈ Π 0 1+β+µ+1 (Pω).Then the sets A * , B * and the functions f * = f 1 | A * , g * = g 1 | B * have the desired properties.✷ The next fact is a special case of the previous theorem.Corollary 3.3 If we take in the previous theorem α = 0 and β = 1 then we obtain A * ∈ Π 0 3 (Pω) and B * ∈ Π 0 4 (Pω).","content":"by Proposition 2.10.Similarly, S 2 ∈ Π 0 1+β+1 (Pω × Pω) and therefore S ∈ Π 0 1+µ+1 (Pω × Pω) by Propositions 2.6 and 2.7.Now let A * = {x ∈ Pω | (x, f 1 (x)) ∈ S} and B * = {y ∈ Pω | (g 1 (y), y) ∈ S}.Since the function x → (x, f 1 (x)) is Σ 0 1+α -measurable and A * is the preimage of S under this function, A * ∈ Π 0 1+α+µ+1 (Pω) by Proposition 2.10.Similarly, B * ∈ Π 0 1+β+µ+1 (Pω).Then the sets A * , B * and the functions f * = f 1 | A * , g * = g 1 | B * have the desired properties.✷ The next fact is a special case of the previous theorem.Corollary 3.3 If we take in the previous theorem α = 0 and β = 1 then we obtain A * ∈ Π 0 3 (Pω) and B * ∈ Π 0 4 (Pω)."},{"id":"","caption":"and let f be a homeomorphism of A onto B. Then there exist A * ∈ Γ(X), B * ∈ Γ(Y ) and a homeomorphism f * of A * onto B * such that A ⊆ A * , B ⊆ B * , and f","content":"and let f be a homeomorphism of A onto B. Then there exist A * ∈ Γ(X), B * ∈ Γ(Y ) and a homeomorphism f * of A * onto B * such that A ⊆ A * , B ⊆ B * , and f"},{"id":"","caption":"Let r : D → Y be a continuous function from a subspace D of N to a zerodimensional CB 0 -space Y .Then r (viewed as a partial function from N to Y ) is an admissible representation for Y iff there is a continuous function s : Y → D satisfying rs = id Y .","content":"Lemma 5. 4\n4\nLet r : D → Y be a continuous function from a subspace D of N to a zerodimensional CB 0 -space Y .Then r (viewed as a partial function from N to Y ) is an admissible representation for Y iff there is a continuous function s : Y → D satisfying rs = id Y ."},{"id":"","caption":"so there is an admissible representation δ : D → Y for Y with D ∈ Π 0 α (N ).By Lemma 5.4 there is a continuous function s : Y → D satisfying δs = id Y .Since D is Hausdorff, S = s(Y ) is a closed subset of D, hence S ∈ Π 0 α (D) by Proposition 2.7.By Proposition 2.8 we obtain S ∈ Π 0 α (N ).","content":"so there is an admissible representation δ : D → Y for Y with D ∈ Π 0 α (N ).By Lemma 5.4 there is a continuous function s : Y → D satisfying δs = id Y .Since D is Hausdorff, S = s(Y ) is a closed subset of D, hence S ∈ Π 0 α (D) by Proposition 2.7.By Proposition 2.8 we obtain S ∈ Π 0 α (N )."},{"id":"","caption":"y)} and is open by being the preimage of the open set D Hx(y) \\ Hx(y)−1 i=0 D i under the continuous map b → (x, b).Therefore H x is an element of the function space ω","content":"y)} and is open by being the preimage of the open set D Hx(y) \\ Hx(y)−1 i=0 D i under the continuous map b → (x, b).Therefore H x is an element of the function space ω"},{"id":"","caption":"Ev N (z k ) ∈ C k and {Ev N (z i ) | i ≤ ∞, i = k} ⊆ N \\ C k for every k ∈ ω.We define functions s : N ∞ → E N and r : E N → N ∞ by s(i) := z i and r(z) := min ∞, k ∈ ω z ∈ Ev −1 N (C k ) .","content":"Ev N (z k ) ∈ C k and {Ev N (z i ) | i ≤ ∞, i = k} ⊆ N \\ C k for every k ∈ ω.We define functions s : N ∞ → E N and r : E N → N ∞ by s(i) := z i and r(z) := min ∞, k ∈ ω z ∈ Ev −1 N (C k ) ."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"by Proposition 2.8.Define the function g : Y → Pω by g(y) = {n | y ∈ A n }.Then for any y ∈ Y we have\",\n \"content\": \"by Proposition 2.8.Define the function g : Y → Pω by g(y) = {n | y ∈ A n }.Then for any y ∈ Y we have\"\n },\n {\n \"id\": \"\",\n \"caption\": \"by Proposition 2.10.Similarly, S 2 ∈ Π 0 1+β+1 (Pω × Pω) and therefore S ∈ Π 0 1+µ+1 (Pω × Pω) by Propositions 2.6 and 2.7.Now let A * = {x ∈ Pω | (x, f 1 (x)) ∈ S} and B * = {y ∈ Pω | (g 1 (y), y) ∈ S}.Since the function x → (x, f 1 (x)) is Σ 0 1+α -measurable and A * is the preimage of S under this function, A * ∈ Π 0 1+α+µ+1 (Pω) by Proposition 2.10.Similarly, B * ∈ Π 0 1+β+µ+1 (Pω).Then the sets A * , B * and the functions f * = f 1 | A * , g * = g 1 | B * have the desired properties.✷ The next fact is a special case of the previous theorem.Corollary 3.3 If we take in the previous theorem α = 0 and β = 1 then we obtain A * ∈ Π 0 3 (Pω) and B * ∈ Π 0 4 (Pω).\",\n \"content\": \"by Proposition 2.10.Similarly, S 2 ∈ Π 0 1+β+1 (Pω × Pω) and therefore S ∈ Π 0 1+µ+1 (Pω × Pω) by Propositions 2.6 and 2.7.Now let A * = {x ∈ Pω | (x, f 1 (x)) ∈ S} and B * = {y ∈ Pω | (g 1 (y), y) ∈ S}.Since the function x → (x, f 1 (x)) is Σ 0 1+α -measurable and A * is the preimage of S under this function, A * ∈ Π 0 1+α+µ+1 (Pω) by Proposition 2.10.Similarly, B * ∈ Π 0 1+β+µ+1 (Pω).Then the sets A * , B * and the functions f * = f 1 | A * , g * = g 1 | B * have the desired properties.✷ The next fact is a special case of the previous theorem.Corollary 3.3 If we take in the previous theorem α = 0 and β = 1 then we obtain A * ∈ Π 0 3 (Pω) and B * ∈ Π 0 4 (Pω).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"and let f be a homeomorphism of A onto B. Then there exist A * ∈ Γ(X), B * ∈ Γ(Y ) and a homeomorphism f * of A * onto B * such that A ⊆ A * , B ⊆ B * , and f\",\n \"content\": \"and let f be a homeomorphism of A onto B. Then there exist A * ∈ Γ(X), B * ∈ Γ(Y ) and a homeomorphism f * of A * onto B * such that A ⊆ A * , B ⊆ B * , and f\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let r : D → Y be a continuous function from a subspace D of N to a zerodimensional CB 0 -space Y .Then r (viewed as a partial function from N to Y ) is an admissible representation for Y iff there is a continuous function s : Y → D satisfying rs = id Y .\",\n \"content\": \"Lemma 5. 4\\n4\\nLet r : D → Y be a continuous function from a subspace D of N to a zerodimensional CB 0 -space Y .Then r (viewed as a partial function from N to Y ) is an admissible representation for Y iff there is a continuous function s : Y → D satisfying rs = id Y .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"so there is an admissible representation δ : D → Y for Y with D ∈ Π 0 α (N ).By Lemma 5.4 there is a continuous function s : Y → D satisfying δs = id Y .Since D is Hausdorff, S = s(Y ) is a closed subset of D, hence S ∈ Π 0 α (D) by Proposition 2.7.By Proposition 2.8 we obtain S ∈ Π 0 α (N ).\",\n \"content\": \"so there is an admissible representation δ : D → Y for Y with D ∈ Π 0 α (N ).By Lemma 5.4 there is a continuous function s : Y → D satisfying δs = id Y .Since D is Hausdorff, S = s(Y ) is a closed subset of D, hence S ∈ Π 0 α (D) by Proposition 2.7.By Proposition 2.8 we obtain S ∈ Π 0 α (N ).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"y)} and is open by being the preimage of the open set D Hx(y) \\\\ Hx(y)−1 i=0 D i under the continuous map b → (x, b).Therefore H x is an element of the function space ω\",\n \"content\": \"y)} and is open by being the preimage of the open set D Hx(y) \\\\ Hx(y)−1 i=0 D i under the continuous map b → (x, b).Therefore H x is an element of the function space ω\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Ev N (z k ) ∈ C k and {Ev N (z i ) | i ≤ ∞, i = k} ⊆ N \\\\ C k for every k ∈ ω.We define functions s : N ∞ → E N and r : E N → N ∞ by s(i) := z i and r(z) := min ∞, k ∈ ω z ∈ Ev −1 N (C k ) .\",\n \"content\": \"Ev N (z k ) ∈ C k and {Ev N (z i ) | i ≤ ∞, i = k} ⊆ N \\\\ C k for every k ∈ ω.We define functions s : N ∞ → E N and r : E N → N ∞ by s(i) := z i and r(z) := min ∞, k ∈ ω z ∈ Ev −1 N (C k ) .\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Topological properties of real number representations. V Brattka, P Hertling, Theoretical Computer Science. 2842002","doi":""},{"text":"Quasi-Polish spaces. M De Brecht, arXiv:1108.1445v1Annals of Pure and Applied Logic. 16432013","doi":""},{"text":"M De Brecht, A Yamamoto, Sixth International Conference on Computability and Complexity in Analysis. A Bauer, P Hertling, -I Ker, Ko, 2009. 2009OASICS 11 Schloss Dagstuhl -Leibniz-Zentrum für Informatik","doi":""},{"text":"R Engelking, General Topology. Heldermann. Berlin1989","doi":""},{"text":"Maximal and everywhere defined functionals. Algebra and Logic. Yu Ershov, 197413","doi":""},{"text":"Comparing Cartesian Closed Categories of (Core) Compactly Generated Spaces. M Escard, J Lawson, A Simpson, Topology and its Applications. 1432004","doi":""},{"text":"G Gierz, K H Hoffmann, K Keimel, J D Lawson, M W Mislove, D S Scott, A compendium of Continuous Lattices. BerlinSpringer1980","doi":""},{"text":"Filter spaces and continuous functionals. J M L Hyland, Annals of Mathematical Logic. 161979","doi":""},{"text":"Characterizations of absolute F σδ -sets. H Junnila, H.-P Künzi, Chechoslovak Mathematical Journal. 481998","doi":""},{"text":"Cartesian closed categories of algebraic CPO's. Theoretical Computer Science. A Jung, 199070","doi":""},{"text":"Classical Descriptive Set Theory. A S Kechris, 1995SpringerNew York","doi":""},{"text":"S C Kleene, Countable functionals. Constructivity in Mathematics. A Heyting, North Holland, Amsterdam1959","doi":""},{"text":"Interpretation of analysis by means of constructive functionals of finite types. G Kreisel, North Holland, Amsterdam. Constructivity in Mathematics. A Heyting, 1959","doi":""},{"text":"C Kreitz, K Weihrauch, Theory of representations. Theoretical Computer Science. 198538","doi":""},{"text":"Recursion on the Countable Functionals. D Normann, Lecture Notes in Mathematics. 8111980","doi":""},{"text":"Countable functionals and the projective hierarchy. D Normann, Journal of Symbolic Logic. 461981","doi":""},{"text":"The continuous functionals. Handbook of Computability Theory. D Normann, E.R. Griffor Ed1999","doi":""},{"text":"Extended Admissibility. M Schröder, Theoretical Computer Science. 2842002","doi":""},{"text":"Admissible Representations for Continuous Computations. M Schröder, 2003PhD Thesist Hagen","doi":""},{"text":"The sequential topology on N N N is not regular. M Schröder, Mathematical Structures in Computer Science. 192009","doi":""},{"text":"Difference hierarchy in ϕ-spaces. Algebra and Logic. V L Selivanov, 200443","doi":""},{"text":"Variations on the Wadge reducibility. V L Selivanov, Siberian Advances in Mathematics. 200515","doi":""},{"text":"V L Selivanov, Hierarchies in ϕ-spaces and applications. Mathematical Logic Quarterly. 200551","doi":""},{"text":"Towards a descriptive set theory for domain-like structures. V L Selivanov, Theoretical Computer Science. 3652006","doi":""},{"text":"On the difference hierarchy in countably based T 0 -spaces. Electronic Notes in Theoretical. V L Selivanov, Computer Science. 2212008","doi":""},{"text":"V L Selivanov, arXiv:1304.1239Total representations. Accepted for publication by Logical Methods in Computer Science. 2011","doi":""},{"text":"K Weihrauch, Computable Analysis. Springer2000","doi":""}],"string":"[\n {\n \"text\": \"Topological properties of real number representations. V Brattka, P Hertling, Theoretical Computer Science. 2842002\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quasi-Polish spaces. M De Brecht, arXiv:1108.1445v1Annals of Pure and Applied Logic. 16432013\",\n \"doi\": \"\"\n },\n {\n \"text\": \"M De Brecht, A Yamamoto, Sixth International Conference on Computability and Complexity in Analysis. A Bauer, P Hertling, -I Ker, Ko, 2009. 2009OASICS 11 Schloss Dagstuhl -Leibniz-Zentrum für Informatik\",\n \"doi\": \"\"\n },\n {\n \"text\": \"R Engelking, General Topology. Heldermann. Berlin1989\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Maximal and everywhere defined functionals. Algebra and Logic. Yu Ershov, 197413\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Comparing Cartesian Closed Categories of (Core) Compactly Generated Spaces. M Escard, J Lawson, A Simpson, Topology and its Applications. 1432004\",\n \"doi\": \"\"\n },\n {\n \"text\": \"G Gierz, K H Hoffmann, K Keimel, J D Lawson, M W Mislove, D S Scott, A compendium of Continuous Lattices. BerlinSpringer1980\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Filter spaces and continuous functionals. J M L Hyland, Annals of Mathematical Logic. 161979\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Characterizations of absolute F σδ -sets. H Junnila, H.-P Künzi, Chechoslovak Mathematical Journal. 481998\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cartesian closed categories of algebraic CPO's. Theoretical Computer Science. A Jung, 199070\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Classical Descriptive Set Theory. A S Kechris, 1995SpringerNew York\",\n \"doi\": \"\"\n },\n {\n \"text\": \"S C Kleene, Countable functionals. Constructivity in Mathematics. A Heyting, North Holland, Amsterdam1959\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Interpretation of analysis by means of constructive functionals of finite types. G Kreisel, North Holland, Amsterdam. Constructivity in Mathematics. A Heyting, 1959\",\n \"doi\": \"\"\n },\n {\n \"text\": \"C Kreitz, K Weihrauch, Theory of representations. Theoretical Computer Science. 198538\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Recursion on the Countable Functionals. D Normann, Lecture Notes in Mathematics. 8111980\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Countable functionals and the projective hierarchy. D Normann, Journal of Symbolic Logic. 461981\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The continuous functionals. Handbook of Computability Theory. D Normann, E.R. Griffor Ed1999\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Extended Admissibility. M Schröder, Theoretical Computer Science. 2842002\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Admissible Representations for Continuous Computations. M Schröder, 2003PhD Thesist Hagen\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The sequential topology on N N N is not regular. M Schröder, Mathematical Structures in Computer Science. 192009\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Difference hierarchy in ϕ-spaces. Algebra and Logic. V L Selivanov, 200443\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Variations on the Wadge reducibility. V L Selivanov, Siberian Advances in Mathematics. 200515\",\n \"doi\": \"\"\n },\n {\n \"text\": \"V L Selivanov, Hierarchies in ϕ-spaces and applications. Mathematical Logic Quarterly. 200551\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Towards a descriptive set theory for domain-like structures. V L Selivanov, Theoretical Computer Science. 3652006\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the difference hierarchy in countably based T 0 -spaces. Electronic Notes in Theoretical. V L Selivanov, Computer Science. 2212008\",\n \"doi\": \"\"\n },\n {\n \"text\": \"V L Selivanov, arXiv:1304.1239Total representations. Accepted for publication by Logical Methods in Computer Science. 2011\",\n \"doi\": \"\"\n },\n {\n \"text\": \"K Weihrauch, Computable Analysis. Springer2000\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":491,"cells":{"text":{"kind":"string","value":"\nHidden AGNs in Early-Type Galaxies\nversion April 8, 2016\n\nAlessandro Paggi apaggi@cfa.harvard.edu \nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA\n\nGiuseppina Fabbiano \nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA\n\nFrancesca Civano \nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA\n\nDepartment of Physics and Yale Center for Astronomy and Astrophysics\nYale University\nP.O. Box 20812106520-8121New HavenCT\n\nSilvia Pellegrini \nDepartment of Astronomy\nUniversity of Bologna\nvia Ranzani 140127BolognaItaly\n\nMartin Elvis \nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA\n\nDong-Woo Kim \nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA\n\nHidden AGNs in Early-Type Galaxies\nversion April 8, 2016Subject headings: galaxies: elliptical -surveys -X-rays: galaxies\nWe present a stacking analysis of the complete sample of Early Type Galaxies (ETGs) in the Chandra COSMOS (C-COSMOS) survey, to explore the nature of the X-ray luminosity in the redshift and stellar luminosity ranges 0 < z < 1.5 and 10 9 < L K /L < 10 13 . Using established scaling relations, we subtract the contribution of X-ray binary populations, to estimate the combined emission of hot ISM and AGN. To discriminate between the relative importance of these two components, we (1) compare our results with the relation observed in the local universe L X,gas ∝ L 4.5 K for hot gaseous halos emission in ETGs, and (2) evaluate the spectral signature of each stacked bin. We find two regimes where the non-stellar X-ray emission is hard, consistent with AGN emission. First, there is evidence of hard, absorbed X-ray emission in stacked bins including relatively high z (∼ 1.2) ETGs with average high X-ray luminosity (L X−LMXB 6 × 10 42 erg/s). These luminosities are consistent with the presence of highly absorbed \"hidden\" AGNs in these ETGs, which are not visible in their optical-IR spectra and spectral energy distributions. Second, confirming the early indication from our C-COSMOS study of X-ray detected ETGs, we find significantly enhanced X-ray luminoaity in lower stellar mass ETGs (L K 10 11 L ), relative to the local L X,gas ∝ L 4.5 K relation. The stacked spectra of these ETGs also suggest X-ray emission harder than expected from gaseous hot halos. This emission is consistent with inefficient accretion 10 −5 − 10 −4Ṁ Edd onto M BH ∼ 10 6 − 10 8 M .\n\nIntroduction\n\nDiffuse X-ray emission from normal early type galaxies (ETG) was first detected in the Virgo cluster with the Einstein X-ray Observatory (Forman et al. 1979). Although in a few cases (e.g., M86) the displacement of the X-ray emission from the stellar body of the galaxy pointed to hot gaseous halos, the ∼ arcminute resolution of the Einstein IPC data could not disentangle observationally hot halos from the populations of low-mass X-ray binaries (LMXBs), which was likely to dominate the X-ray emission in a large number of the observed ETGs (Trinchieri & Fabbiano 1985). Nonetheless, several authors discussed how the X-ray emission and the inferred hot gas contribution could be used to constrain both the binding mass of the galaxies and the physical evolution of the hot halos, including feedback from SNIa and AGNs, and interaction with intracluster and group media (see e.g., Fabbiano 1989;Canizares et al. 1987;Ciotti et al. 1991;David et al. 1991;White & Sarazin 1991;Mathews & Brighenti 2003;Ciotti & Ostriker 2007).\n\nThanks to the high spatial resolution of Chandra and imaging/spectral capabilities of the ACIS detector (Garmire et al. 2003), it has become possible to directly detect and characterize the different contributions to the X-ray emission of ETGs in the near universe. With Chandra, LMXB populations -both associated with Globular Clusters (GC) and in the stellar field of the ETG -have been studied to distances of several tens of megaparsecs (e.g., Sarazin et al. 2000;Kundu et al. 2002; see review by Fabbiano 2006). The integrated LMXB luminosity correlates well with the integrated ETG stellar luminosity (e.g., Kim & Fabbiano 2004;Gilfanov 2004). The scatter in this relation is a factor of 2-3, and it depends on both GC content (Kim & Fabbiano 2004), and the age of the stellar population (Kim et al. 2009;Kim & Fabbiano 2010;Boroson et al. 2011;Lehmer et al. 2014). These detailed high resolution studies allowed Boroson et al. (2011, BKF11) to accurately estimate the gaseous contribution to the X-ray luminosity L X,gas of a sample of 30 ETGs within 32 Mpc. After subtracting LMXB, stellar, and nuclear contributions to the X-ray luminosity, BKF11 found steep scaling relations between L X,gas and both the stellar L K luminosity and the hot gas temperature T . Subsequent work by Kim & Fabbiano (2013, KF13;2015, KF15) suggests that for L X,gas > 10 40 erg s −1 the hot gas around ETGs is virialized, giving a new way to measure the total mass of these galaxies.\n\nBeyond the local universe, Chandra Deep Field studies (both detections and stacking of almost 1,000 ETGs) found that the mean L X of L B = 10 10 L ETGs remains equal or mildly increases over z = 0 − 0.7 (Lehmer et al. 2007;Tzanavaris & Georgantopoulos 2008) and that the ratio of the soft X-ray luminosity to the B-band luminosity evolves mildly as L (0.5−2 keV) /L B ∝ (1 + z) 1.1±0.7 since z ≈ 1.2 (Danielson et al. 2012), suggesting some heating mechanism preventing the hot gas from cooling, consistent with mechanical heating from radio AGNs. The evolution of the X-ray binary contribution to the X-ray emission (both of LMXBs and high mass binaries) across cosmic time, modeled by Fragos et al. (2013a), led Fragos et al. (2013b) to conclude that energy feedback from X-ray binaries is also important in the early universe.\n\nThe COSMOS survey, covering ∼ 2 square degrees of sky with multi-wavelength coverage, which includes Chandra X-ray coverage (Scoville et al. 2007;Elvis et al. 2009), has increased the available sample of ETGs to over 6,600 galaxies Civano et al. (2014, C14). This survey provides a good z coverage out to z ∼ 2.5 (and a few galaxies are detected out to z ∼ 5). C14 studied the sample of 69 X-ray detected ETGs in the Chandra-COSMOS survey (C-COSMOS, Elvis et al. 2009), covering a redshift range 0 < z < 1.5, to explore redshift dependencies in the BKF11, KF13 scaling relations. In particular, C14 focused on the relation between L X,gas and the K-band stellar luminosity (L K ) of ETGs which is a good proxy of stellar mass for old stellar systems. The latter has been evaluated in the local Universe by BKF11 as L X,gas ∝ L K 2.6±0.4 for sources with L X,gas ≤ 2 × 10 41 erg s −1 , while including sources with L X,gas up to 9 × 10 42 erg s −1 KF13 found a steeper L X,gas ∝ L K 4.5±0.8 relation. C14 found that this latter relation also holds for X-ray detected ETGs with L X,gas ≤ ×10 42 erg s −1 , suggesting that the hot gas in these more distant galaxies may be virialized as well. However, more distant (z > 0.9) and younger stellar age galaxies generally fall outside the relation observed in the local sample, and are X-ray overluminous with respect to their L K . For these sources C14 suggested that the higher X-ray luminosity may be due to the presence of hidden AGNs (as suggested by observed hardness ratios), merging phenomena, which are expected to increase the halo X-ray luminosity, or nuclear accretion (that could also induce AGN activity during the merging phase, Cox et al. 2006).\n\nIn this work we perform a stacking analysis of the C-COSMOS ETG data in order to investigate the nature of fainter, X-ray undetected ETGs. We compare their properties with those of the sample of local ETGs investigated by BKF11 and KF13, and with the sample of ETGs detected in the C-COSMOS survey studied by C14. This stacking study also allow us to further explore redshift dependencies. Our aim is to explore the properties of the non-stellar Xray emission, i.e. the X-ray emission resulting from the subtraction of the expected X-ray binary contribution (taking into account the cosmic evolution of the X-ray binary population Fragos et al. 2013b). The resulting emission will probe both hot X-ray halos and the possible emission of active nuclei in these galaxies. In particular, the large number of low-stellar-mass ETGs in the sample will provide a unique probe of the existence of relatively low-mass massive black holes in their nuclei. These black holes should produce low-luminosity nuclear X-ray sources, if they are fueled by the ETG stellar outgassing (Volonteri et al. 2011).\n\nThe paper is organized as follows: Section 2 describes the sample selection and the data sets used in this work, Section 3.3 is dedicated to the description of the adopted stacking procedure, in Section 4 we discuss our results, and Section 5 is dedicated to our conclusions.In the following, we adopt the standard flat cosmology with Ω Λ = 0.73 and H 0 = 70 km s −1 Mpc −1 .\n\n\nSample Selection\n\nThe COSMOS survey (Scoville et al. 2007) provides a multi-wavelength characterization of over 1.5 million galaxies at redshift extending out to 5 in 2 deg 2 of the sky. The Chandra X-ray coverage (C-COSMOS Elvis et al. 2009) over 0.9 deg 2 , gives us the means to study X-ray emission of ETGs, extending local studies to higher redshifts, and explore the redshift evolution of hot halos with large samples of galaxies.\n\nOur ETG sample is selected from the most recent version of the COSMOS photometric catalog by Ilbert et al. (2009). The spectral energy distribution (SED) identification performed by Ilbert et al. (2009) makes use of 7 elliptical galaxy templates with ages from 2 to 13 Gyr (Polletta et al. 2007). The SED templates were generated with the stellar population synthesis package developed by Bruzual & Charlot (2003), assuming an initial mass function from Chabrier (2003) and an exponentially declining star formation history. We also collected additional galaxy properties (stellar mass, age and star formation rate) from Ilbert et al. (2010), and we made use of the most recent near-infrared photometry from the Ultra Deep Survey with the VISTA telescope (Ultra-VISTA McCracken et al. 2012) 1 .\n\nWe selected galaxies classified as E or S0 and that have a reliable photometric redshift (Ilbert et al. 2009). In addition we selected only sources with Ultra-VISTA detection both in J and K bands, since these are needed to evaluate the rest frame L K (see below). This narrows our sample to 10,808 (the COSMOS ETG sample considered in C14). We then selected only the 6436 ETGs falling in the C-COSMOS field, and then excluded the 48 X-ray detected ETGs in common with the C-COSMOS identification catalog (Civano et al. 2012). This leads us to our final sample of 6388 C-COSMOS ETGs -not detected in X-rays -which constitutes the starting point of the stacking analysis described in this paper. Note that we selected sources with photometric redshift estimate in the catalog by Ilbert et al. (2009), while C14 selected their sources to have a spectroscopic or photometric redshift estimates in the C-COSMOS identification catalog, so the sample of ETGs selected by the latter authors (∼ 6600 sources) is not fully contained in our sample.\n\nFollowing C14 we evaluated the K band luminosity of our sources from the Ultra-VISTA K-band aperture magnitude of the COSMOS photometric catalog. To calculate the rest frame K band luminosities we assumed a power law spectrum (as appropriate over this small frequency range) f ν ∝ ν −α with α = (J − K)/2.5 log (ν J /ν K ) (where J and K are the AB magnitudes from the COSMOS photometric catalog and ν J and ν K are the corresponding frequencies), and express the rest frame luminosities in solar luminosity as L K = 10 −(K+ext−5.19)/2.5 × (1 + z) α−1 × (D L /10) 2 , where z is the photometric redshift from the COSMOS catalog, D L is the luminosity distance expressed in parsecs, and ext is the aperture correction parameter computed from the FWHM measured for all the COSMOS extended sources in the Hubble ACS data.\n\nWe compared the distributions of the physical parameters of the 6388 C-COSMOS ETGs with the entire 10,808 1 Note that stellar mass reported in COSMOS photometric catalog, and obtained from the Bruzual & Charlot (2003) best-fit templates, do not incorporate the Ultra-VISTA K-band NIR data. Although this is only an issue if there is any residual star formation in the ETGs, L K and M * may not necessarily be completely equivalent.\n\nCOSMOS ETG sample finding no significant difference, demonstrating as expected that the sample of ETGs not detected in X-ray is representative of COSMOS ETGs overall. Note that the luminosity threshold is a function of redshift because of the uniform magnitude limit of the Ultra-VISTA COSMOS survey (K s ∼ 24, McCracken et al. 2012). The specific star formation rate for the majority of the ETGs is in the regime where Ilbert et al. (2010) defines galaxies as quiescent, although higher star formation rate is observed in young ETGs. The properties (luminosity, mass, star formation rate and age) of the COSMOS ETGs are in agreement with those reported by (Moresco et al. 2013) for a sample of ETGs selected using multiple criteria (including color-color diagrams, spectra, star formation rate and SED classification). Masses and luminosities are in agreement with those of the K-selected elliptical galaxies of Jones et al. (2014), extracted from a small area of the COSMOS field.\n\n\nAnalysis and Results\n\nBelow we describe the procedures followed in our stacking analysis. The results are summarized in Tables 1 and 2.\n\n\nCount Extraction at ETG Positions\n\nWe used all 49 C-COSMOS observations to evaluate source and background counts for our sample of ETGs, excluding data in the ACIS-I chip gaps. We also excluded data from X-ray detected sources removing counts within a 10\" radius of all sources in the C-COSMOS source identification catalog (Civano et al. 2012). We replaced the excluded counts by sampling the Poisson distribution of the pixel values nearby background regions with dmfilth CIAO task 2 . When a source lay in the field of view of more than one observation, we combined the data from the different observations using the prescriptions from Puccetti et al. (2009), excluding observations where the source was more than 10 acrmin off-axis from the aim point.\n\nDue to the variation of the Chandra Point Spread Function (PSF) radius R PS F 3 , we used an extraction radius r ext that was determined by the source position in each observation in order to extract an optimal fraction of X-ray photons from the diffuse halo. For sources with morphological information available in the HST-ACS catalog (Leauthaud et al. 2007) we adopted the source semi-major axes R as a measure of its physical extent. For sources with R > R PS F we adopted values of r ext that was the convolution of R and R PS F (R 2 + R 2 PS F ); otherwise we adopted r ext = R PS F as we did for sources without morphological information in HST-ACS catalog. We notice however that sources with R > R PS F represent only ∼ 1% of our sample, with extraction radii typically ranging between 1\" and 7\". The average exposure time for each source within the extraction circle, E, was then evaluated. The histogram of extracted ETG position counts is shown in Figure 1 (blue line). The small fraction of extraction regions with counts above ∼ 5 represent off-axis sources with large extraction radii.\n\n\nBackground Evaluation\n\nTo evaluate the background we extracted counts from an annulus centered on the source coordinates, with innerradius 2r ext and outer radius 4r ext . Due to the very low signal, the background represents the largest source of uncertainty in this kind of analysis. To estimate the systematic uncertainties introduced by the evaluation of the background, we compared the results of three different methods, obtaining consistent results. First we simply evaluated the background contribution to the total source counts from the mean count value of the pixels in the whole background annulus. Second, we adopted the clipping procedure of Willott (2011, method b), which evaluates the mean count value µ -and the standard deviation σ -of all pixels within a circle of radius 4r ext around each source position and then sets all pixels with count value larger than µ + 3σ to µ. This method suppresses the noise in the background annulus, but also removes signal from the source 4 . Lastly, we divided the background annulus into eight 45 degree sectors, and excluded all the sectors in which the mean count value was larger than µ + 3σ, thus excluding regions of higher background without removing signal from the source. The net counts N obtained with all three methods are consistent within statistics, so in the following we adopted the clipping procedure of Willott (2011, method b).\n\nTo test the robustness of our results we also extracted counts from positions on the sky offset from the ETG coordinates by random distances between 15\" and 30\" both in right ascension and declination. The net counts of the non-source positions are shown in Figure 1. They follow an approximatively Gaussian distribution centered on zero, as expected from background fluctuations, while the ETG counts show an excess of positive signal. To test if our results were dominated by a sub-population of ETGs we repeated the stacking procedure on sub-samples randomly drawn from the complete sample, progressively halving the subsample, obtaining results compatible with the full sample.\n\n\nStacking Procedure\n\nTo evaluate the count rate C for each stacked bin we stacked the observed source net counts in 0.5-7 keV range N and exposure E to obtain C = i N i / i E i . From these counts we subtracted the expected contributions from stellar sources (LMXBs and Ultra Luminous X-ray sources, ULXs), and evaluated the average bin rest-frame X-ray luminosity as explained below. We corrected L K to take into account the evolution of the stellar population (i.e., the fading effect, see C14). Note that, at a given redshift, younger galaxies have larger fading correction than older ones. To compare our results with those of KF13 and C14 we considered sources with 9 < log (L K ) < 13.\n\nWe constructed the stacks by binning in both L K and z, to ensure that the binning scheme did not produce any significant bias on the results. For the L K binning, we first divided our sample in four log (L K ) bins, 9 − 10, 10 − 11, 11 − 12 and 12 − 13. We then ordered the sources in each bin in increasing redshift, and progressively stacked them until we reached a minimum signal to noise ratio of 3; if this 3σ threshold could not be reached we stacked the signal from all the remaining sources in the bin. Examples of 0.5-7 keV images for these stacking bins are presented in Figure 2. For the z binning, we followed the same procedure, dividing the sample into three bins 0 − 0.5, 0.5 − 1 and 1 − 1.5, and then ordered the sources in each bin in increasing L K .\n\n\nEvaluation of Non-stellar L X for Each Stacked Bin\n\nTo subtract the LMXB contribution from the total X-ray emission, we evaluated the average LMXB contribution to each bin using the 0.3-8 keV relation from Fragos et al. (2013b), as in C14:\nlog (L LMXB /M * ) = 30.259 − 1.505 × log age − 0.421 × log 2 age + 0.425 × log 3 age + 0.135 × log 4 age , (1)\nwhere L X is expressed in erg s −1 , M * is the average stellar mass of the stacking bin expressed in solar masses and age is the average stellar age of the stacking bin expressed in Gyr. 5 We also evaluated the contribution to X-ray luminosity from ULXs that could be present in the galaxies using the scaling relations from Calzetti et al. (2007) and Mineo et al. (2012) between the ULX luminosity and the galaxy star formation rate. ULX contribution is more than an order of magnitude smaller than L X−LMXB even for the least luminous L K bins.\n\nWe converted the average 0.3 − 8 keV LMXB luminosities in each bin to count rates in the 0.5 − 7 keV interval using a power law model with index Γ = 1.8 (consistent with the typical spectrum of LMXBs) and Galactic absorption N H = 2.6 × 10 20 cm −2 . For these parameters the conversion factor used from Chandra count rates to fluxes is 1.04 × 10 −11 erg cm −2 s −1 cts −16 . We then subtracted these count rates from the stacked signals and converted back the resulting count rates into luminosities using a model comprising Galactic N H plus a thermal APEC component with temperature depending on the source total X-ray luminosity. We adopted kT = 0.7, 1 and 2 keV for X-ray luminosities of L X < 10 41 , 10 41 < L X < 10 42 and L X > 10 42 erg s −1 , respectively, following BKF11 and Dai et al. (2007). Finally, we K-corrected the luminosity to the 0.3 − 8 keV rest frame band. These LMXB-subtracted luminosities, that we indicate as L X−LMXB , are expected to include the emission from the hot gaseous halos, but may also include a possible contribution from any AGN harbored within these ETGs. Figure 3 shows the results in the L X−LMXB − L K plane for both L K and z binning (Left and Right panels, respectively), where on the upper axes we plot M * as taken from COSMOS catalog. In both binning schemes the overall scatter plots are consistent, showing that we do not suffer from strong binning biases. By construction, the L K -first binning provides information on the extreme redshift bins (both high and low), while the z-first binning covers more fully the L K range. The bins are numbered in Figure 3, following Tables 1 and 2. We use different colors to visually divide the stacking bins into four groups: red, for bins X-ray over-luminous with respect to the local L X−LMXB and L K relation (BFK11, KF13, see C14); yellow, for X-ray under-luminous bins; and grey and green for bins following the local relation with L X−LMXB lower and higher than 10 41 erg s −1 , respectively.\n\nAs noted in Section 1, Eq. 1 has an observed spread of a factor 2-3, which can be related to both the presence of GC LMXB in the ETGs and to the spread in age of the stellar population (Kim & Fabbiano 2004BKF11;Lehmer et al. 2014). Given the parameter space of Figure 3, this spread will not change our conclusions.\n\n\nHardness Ratios\n\nHardness ratios (HRs) were evaluated to characterize the ETG average spectral properties. HR = (H −S )/(H +S ), where S is the soft 0.5 − 2 keV counts and H the hard 2 − 7 keV counts. These HRs are those of the LMXB-subtracted emission. The contributions of these sources were evaluated and subtracted by applying the procedure in Section 3.4 to each energy band. We evaluated the errors on HRs as due to both uncertainties on counts and to uncertainties on estimate of LMXB contribution due to spread of stellar age and mass in the bins.\n\nIn Figure 4 we show the HRs for the stacking bins using the same color code as in Figure 3. We plot HR versus z in the Left panel, using the L K -first binning, which gives a better z sampling. We plot HR versus L K in the Right panel, using the z-first binning, to explore the L K behavior. The left panel of Figure 4 shows a distinct clustering in HR-z space of bins from separate loci of the L X−LMXB − L K plane, as shown by the clusters of different color points. The Right panel shows that ETGs with larger stellar mass (L K ) tend to have lower HR. Given the scaling relations of ETGs (BKF11; KF13; KF15), this could just reflect a relation with total binding mass, with more massive galaxies being more efficient at retaining large hot gaseous halos.\n\n\nDiscussion\n\nFigure 5 summarizes our results in the L X−LMXB − L K plane and compares them with the COSMOS ETG X-ray detections (C14) and with the near universe ETG data (BKF11, KF13). We show the local KF13 relation with a solid line, and the scatter in this relation with the two dashed lines. These lines are chosen to pass through the points for the most L X,gas luminous source (M87) and the most L K luminous source (NGC1316), respectively. The COSMOS stacked-ETG results are overall consistent with the C14 COSMOS detections, but significantly increase the coverage at lower total stellar luminosities / masses. While most of the detected signals are compatible with the local KF13 relation, there are several detections that lie both to the left and to the right side of it, that is, bins that appear to be respectively over and under-luminous in X-ray with respect to their L K or M * .\n\nIn Figure 5 we also show the hardness ratios for each stacked bin with lines with slopes proportional to HR. We find positive HRs (i.e., harder spectra) in stacked bins outside the dashed line on the left. These are the lower L K galaxies with significant X-ray excess (the red points in Figure 4, see also C14). We also find hard HR > 0 in some points with L X−LMXB > 10 41 erg s −1 and z > 1 (the three top green points in Figure 4), to the left of the solid line.\n\nInstead, lower HR ratios (i.e., softer spectra) are found in bins following the local relation (between the two dashed lines). This is especially so for those to the right of the solid line. This suggests that they have a larger than average hot gas content up to z ∼ 1. Figure 6 compares the HR-z scatter diagram with a set of spectral models. On the Left, we show the expected HRs for thermal APEC (dashed lines; from bottom to top, kT = 0.7, 1, 2 and 3 keV) and power-law models (solid lines; fixed slope Γ = 2 and increasing intrinsic absorption from bottom to top, N H = 0, 10 21 , 10 22 and 10 23 cm −2 ). The APEC models are the expected emission of thermal gaseous emission, while the power-law models are typical AGN spectra. To illustrate cases of mixed emission from a hot gas component and a heavily obscured AGN, on the right panel of Figure 6 we plot composite models consisting of kT = 1 keV APEC emission with line of sight N H plus power-law models with slope Γ = 2 and intrinsic absorption N H = 10 23 cm −2 . From the bottom to the top the relative normalization of the two models varies, to reflect an increasing contribution of the power-law model to the total flux, ranging from ∼ 0.2% to ∼ 100%. The X-ray over-luminous bins (red points) have generally high HRs consistent with being dominated by AGN emission, either with \"normal\" AGNs with absorbing columns 10 22 cm −2 , or in the case of the composite gas + AGN emission even highly obscured Compton Thick AGNs (N H > 10 24 cm −2 , Risaliti et al. 1999;Levenson et al. 2006;Lehmer et al. 2010).\n\nThe bins following the local relation with L X−LMXB < 10 41 erg s −1 (grey points) show HR ∼ (−0.5) − 0.3, lower than those of the over-luminous X-ray bins. A soft HR first suggests emission from hot gas. However, Figure 6 shows that pure APEC thermal models with solar metallicity always have HR < −0.6. Hence these objects have HR values inconsistent with pure gaseous emission.\n\nThese HRs can instead be explained with combinations of thermal and AGN emission. Figure 6 Right shows models with varied ratios of APEC and power-law emission. In this model the observed HRs imply AGN components accounting for 15% − 60% of the X-ray luminosity. Given the X-ray luminosities of this sub-sample (Figures 3, 5), these ratios then imply AGN luminosities ranging from ∼ 5 × 10 39 erg s −1 to ∼ 5 × 10 40 erg s −1 . This range is in agreement with the low-luminosity Chandra X-ray sources in the near universe sample (BKF11, Pellegrini 2005), which have luminosities ranging from ∼ 10 38 erg s −1 to ∼ 5 × 10 41 erg s −1 .\n\nThe bins with L X−LMXB > 10 41 erg s −1 (green points) have hard HR > 0 incompatible with a low absorption power-law model (e.g., bins 21, 22 and 25). The three highest redshift bins in this group (e.g., bins 28, 29 and 30) have even harder HR > 0.5 that are compatible only with absorbing columns larger than 10 23 cm −2 . This suggests that these galaxies harbor (on average) highly obscured Compton Thick AGNs that dominate their X-ray luminosity.\n\nIn summary, our stacking analysis suggests that on average AGN emission is likely to be present in ETGs. These AGNs may be faint and hidden in hot gas dominated ETGs, which are consistent with the near Universe scaling relation of virialized halos (KF13, KF15, see C14). In contrast, AGN emission appears to dominate the non-stellar X-ray emission in two classes of ETGs: (1) low-stellar mass ETGs (log (L K /L ) 10.5), and (2) higher redshift ETGs (z > 0.5, > 1) especially those overluminous in X-rays relative to the near universe L X−LMXB − L K scaling relation. The latter group has HR values suggesting highly absorbed Compton Thick AGNs. We note that these AGNs were not visible in the COSMOS colors and spectra, since AGNs were excluded in our ETG sample selection process (C14). (2007), obtaining black hole masses ∼ 0.5 dex larger than the former. Uncertainties in these estimate are typically of a factor of ∼ 2 − 3 (Tremaine et al. 2002;Graham 2007) and evolution to z ∼ 1 − 2 may provide a systematic black hole mass increase of a similar factor (Merloni et al. 2010). These effects will not change our conclusions in the 3-dex wide range covered by Figure 7.\n\n\nAccretion Regimes for the AGNs\n\nOn the right hand axes we plot L bol = 10 L X−LMXB , an appropriate bolometric correction for low-luminosity AGNs that radiate at a few 10s % of L Edd (Kollmeier et al. 2006;Steinhardt & Elvis 2010). Given this nominal L bol we can plot diagonal lines of constant L bol /L Edd . These lines range from 10 −2 to 10 −6 L bol /L Edd .\n\nWe see that the massive high L X−LMXB galaxies (green points) and the low-mass X-ray overluminous (red points) galaxies have implied Eddington ratios in the 10 −3 to 10 −5 range. RIAFs are generally believed to apply to all black hole accretion below 10 −2 L/L Edd , so we would expect all these sources to be RIAFs. If the bolometric correction for RIAFs is smaller than that considered here, as may well be the case (Narayan & Yi 1995), the L bol values on the y-axis should be correspondingly lower (as the Eddington ratios on the currently plotted dashed lines).\n\nAlternatively the true L bol of the three high z bins of massive high L X−LMXB galaxies (green points 28, 29, 30) could be greatly underestimated. The HRs of these bins also imply highly absorbed emission (N H > 10 23 cm −2 , Figure 6). If these are then Compton Thick AGN they may have much larger intrinsic luminosities, with factors of 100 being quite plausible (e.g., Risaliti et al. 1999;Levenson et al. 2006). They would have intrinsic luminosities L 0.5−10 keV ∼ 10 45 erg s −1 and L bol /L Edd in the ∼ 10 −2 range, so that they could lie within the normal AGN regime.\n\nTo investigate this possibility, we built SEDs of ETGs in our sample using optical-infrared magnitudes reported in COSMOS photometric catalog, since a \"hidden\" AGN may give rise to an IR excess with respect to an average ETG SED. We then fitted these SEDs with the same elliptical galaxy template used by Ilbert et al. (2009) without finding any significant deviation. To confirm this, we evaluated the L 2−10−LMXB luminosity of the stacking bins assuming a power-law spectrum (as appropriate for AGN spectra) and compared that with their IR luminosity. In all cases the Xray luminosities are at least 3 order of magnitudes lower than the IR luminosities, even assuming absorbing columns ∼ 10 23 cm −2 , indicating that at IR wavelength a putative AGN component is expected to be significantly smaller than the galactic emission.\n\nThe galaxies that are consistent with the local L K − L X−LMXB gaseous relation (grey points), with X-ray colors suggesting thermal emission plus a possible AGN component, have smaller Eddington ratios ∼ 10 −5 to 10 −6 (these ratios of course assume that the entire emission is due to an AGN; as shown in Figure 6, the AGN component may range from ∼ 15 − 60%).\n\nWe also plot in Figure 7 the predictions from Volonteri et al. (2011) of L X,AGN for three values of M * for both efficient and radiatively inefficient accretion flows (RIAFs Narayan & Yi 1995). These authors' models use the stellar density profile scaling with stellar mass, and predict the accretion rate from the normal stellar mass loss due to stellar evolution, and the consequent massive black hole luminosity for various radiative efficiencies 7 . The X-ray luminosity coming from accretion is computed as L X,AGN /L Edd = η X ηṁ, where η X = 0.1 is the X-ray fraction of the bolometric luminosity, andṁ = 0.1Ṁ/Ṁ Edd is the accretion rate in Eddington units with a 10% accretion efficiency. Radiatively efficient accretion flows have η = 1, while inefficient ones have η =ṁ/ṁ cr , and for themṁ <ṁ cr = 3 × 10 −2 .\n\nIn order to compare our results with Volonteri et al. (2011) models we converted the 0.5−7 keV LMXB subtracted counts into 0.5 − 10 keV band luminosities, L 0.5−10 keV assuming a pow-law spectrum as appropriate for AGN spectra. Based on these models, inefficient accretion into the nuclei could explain even all the L X−LMXB of the massive, high L X−LMXB ETGs (green points), except for the most luminous ones, which are likely to be Compton thick AGNs (see above). The low-mass X-ray overluminous ETGs (red points) are also consistent with the Volonteri et al. model. Considering that a large fraction of the X-ray emission is due to hot gas (see above), the grey points show instead a possible AGN emission even lower than predicted for radiatively inefficient accretion. In these ETGs, the L X−LMXB is consistent with the local L X,gas − L K relation. This result can have the following possible explanations, that are not mutually exclusive.\n\n1. The Volonteri et al.'s models assume that the whole of their estimated mass accretion rate (coming from the instantaneous mass loss of all stars, and obtained by integrating over the whole stellar density profile) indeed reaches the massive black hole, while this represents an upper limit on the mass accretion rate of the massive black hole (some gas may be outflowing, or may reach the massive black hole with a delay, or may possess angular momentum and form a disk, etc.).\n\n2. The accreting mass, that has already entered the accretion radius, is reduced on its way to the massive black hole, due to a wind from a RIAF (Narayan & Yi 1995;Blandford & Begelman 1999); similarly, close to the nucleus, there may be a heating mechanism for the accreting gas, as a nuclear jet, again impeding or lowering the accretion rate (e.g. Pellegrini et al. 2012;Yuan & Li 2011).\n\n3. Accretion onto the black hole is (temporarily) not taking place, as may be the case during feedback-regulated activity cycles. Numerical simulations show that, during these cycles, AGN outbursts are followed by major degassing of the circumnuclear region, with a precipitous drop of the nuclear accretion rate; the outbursts are 7 We note that in Volonteri et al. (2011) models assume for ETGs a stellar age of 12 Gyr. We corrected for the average age 10 9.5 Gyr of C-COSMOS ETGs, which yields slightly smaller emitted X-ray luminosities (see Eq. 9 in Volonteri et al. 2011 ). separated by long time intervals during which the galaxy is replenished again by gas from the stellar mass losses, until a new nuclear outburst takes place (Ciotti et al. 2010).\n\n4. η X might be lower than 1 as assumed here (see also Volonteri et al. 2011), or that η at lowṁ decreases faster than adopted in the Volonteri et al.'s models, so that lower L X,AGN should be predicted (e.g., Yuan & Narayan 2014).\n\nOur results extend to higher X-ray luminosities a recent analysis of ETGs in the AMUSE survey of Virgo and field galaxies with Chandra (Miller et al. 2015). AMUSE results indicate a significant presence of supermassive black holes in low stellar mass galaxies. Due to the lower redshifts of the AMUSE ETGs, that survey is sensitive to X-ray luminosities (∼ 1.3×10 38 erg/s) one order of magnitude fainter are than our faintest bin 1 (∼ 2.4×10 39 erg/s). For these dim sources the stellar wind accretion proposed by Volonteri et al. (2011) would predict higher X-ray luminosities than those observed, Miller et al. conclude that these black holes are probably powered by inefficient advection dominated accretion flows or, alternatively, by an outflow/jet component. We found that low-luminosity, X-ray \"overluminous\" ETGs can be adequately described by the RIAF fueled by stellar winds as proposed by Volonteri et al. (2011). Instead, ETGs with L X−LMXB > 10 42 erg s −1 may host Compton Thick AGNs. Similarly, when comparing with the results of Lehmer et al. (2007) and Danielson et al. (2012) on Chandra Deep Fields, we extend to higher X-ray luminosities and to lower optical luminosities (that is, lower stellar masses), finding sources that significantly exceed the expected emission from hot-gas dominated ETGs, with log (L 0.5−2 keV /L B )/(erg/s/L ) > 30 (see Figure 7 in Danielson et al. 2012).\n\nWe also note that, as reported by Comastri et al. (2015), the revision of black hole scaling relations by Kormendy & Ho (2013) indicate that the local mass density in black holes should be five times higher than previous estimates. These authors advance the possibility that a sizeable population of highly obscured AGNs with covering fraction of obscuring material ∼ 4π would explain the revised upward value of the SMBH local mass density without exceeding the limits imposed by the X-ray and IR background. An example of these obscured AGNs can be that observed in the ETG ESO565-G019 with Suzaku and Swift/BAT (Gandhi et al. 2013).\n\nFinally, we note that the hard X-ray slopes of the more luminous ETG bins could be due to contamination from X-ray jets. However, the combined 1.4GHz VLA COSMOS Large and Deep Project catalogs (Schinnerer et al. 2010) offer radio counterparts only to ∼ 2% of our sample of ETGs. The presence of radio emitters among non X-ray detected ETGs will be addressed after the release of the JVLA-COSMOS Project data.\n\n\nSummary and Conclusions\n\nWe have followed up our previous work (C14) investigating the X-ray / K-band properties of the 69 ETGs detected in the Chandra COSMOS survey (Elvis et al. 2009), with a luminosity and hardness ratio stacking analysis of the Chandra data for the sample of 6388 early type galaxies (ETGs) not detected in X-rays. The stacking analysis allows the investigation of the average features of individually X-ray undetected ETGs. Our ETG sample was selected as representative of normal galaxies, with no evidence of AGN emission in their spectra and multi-band photometry (C14, Section 2). Our purpose was to investigate the properties of the X-ray emission from hot gas and from possible \"hidden\" AGN components that may be revealed by their X-ray emission. To this purpose, the expected contribution of LMXBs was subtracted from the X-ray luminosity of each stacked bin, using established scaling relations with the K-band luminosity (BKF11). All luminosities were corrected to the local rest frame; X-ray hardness ratios were based on the observed counts (after subtraction of the LMXB contribution), and were compared to emission models for the appropriate average bin redshift. This analysis has led to the following conclusions:\n\n1. On average AGN emission may be present in all ETGs.\n\n2. AGNs may be faint and hidden in hot gas dominated ETGs, which are consistent with the local universe scaling relation of virialized halos (KF13, KF15, see C14).\n\n3. AGN emission appears to dominate the non-stellar X-ray emission in low-stellar mass ETGs (log (L K ) 10.5 in solar units), which show marked X-ray luminosity excesses relative to their K-band luminosity (a proxy of stellar mass), when compared with the local ETG L X,Gas L K relation of the local universe (BKF11). This result confirms the conclusions of C14 and extends them to lower stellar mass galaxies.\n\n4. AGN emission is also prominent in higher redshift ETGs (z > 0.5, > 1) especially those with X-ray luminosity 10 42 erg s −1 . The latter group have HR values suggesting highly absorbed Compton Thick AGNs. We note that these AGNs were not visible in the COSMOS colors and spectra, since AGNs were excluded by our ETG sample (C14). 5. Given the nuclear BH mass predicted by the M BH − L K relation (Kormendy & Ho 2013), the hidden AGNs in the X-ray over-luminous low stellar mass ETGs are consistent with the presence of radiatively inefficient accretion fueled by stellar outgassing (Volonteri et al. 2011). The highly absorbed higher L X−LMXB AGNs hidden in massive, higher z ETGs instead suggest efficient accretion in highly obscured nuclei.\n\nThis work was partially supported by NASA contract NAS8-03060 (CXC), and NASA Chandra grant G01-12110X. SP acknowledges financial support from MIUR grant PRIN 2010-2011, project 'The Chemical and Dynamical Evolution of the Milky Way and Local Group Galaxies', prot. 2010LY5N2T. FC acknowledges financial support by the NASA contract 11-ADAP11-0218. AP thanks Andy Goulding for useful discussion and suggestions. ME and GF thank the Aspen Center for Physics and the NSF Grant #1066293 for hospitality during the completion of this paper. This research has made use of software provided by the Chandra X-ray Center (CXC) in the application packages CIAO, ChIPS, and Sherpa. This research is based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere, Chile, program 179.A-2005 (UltraVISTA survey). (2) number of stacked sources; (3) average rest frame L K in the bin (with corresponding standard deviation); (4) average redshift in the bin (with corresponding standard deviation); (5) average stellar age in the bin (with corresponding standard deviation); (6) average stellar mass in the bin (with corresponding standard deviation); (7) signal to noise ration of the stacked signal; (8) average 0.3-8 Xray luminosity after LMXB subtraction estimated from a thermal model. The first uncertainty reported in parenthesis is due to standard deviation of counts, stellar age and stellar mass in the bin, while the second one takes into account the additional standard deviation of redshift.; (9) hardness ratio (with error due to standard deviation of counts, stellar age and stellar mass in the bin)\n\n(1) bin number (2) number of sources (3) < log(LK/L ) > (4) < z > (5) < log(age/yr) > (6) < log(M * /M ) > (7) S /N (8) < LX−LMXB, 0.3−8/(10 40 erg s −1 ) > (1) bin number (2) 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 12 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Table 1 (for the left panel) and Table 2 (for the right panel): red, X-ray over-luminous with respect to the local L X,gas − L K relation; yellow, X-ray under-luminous bins; grey and green bins following the local relation with L X−LMXB lower and higher than 10 41 erg s −1 , respectively. Triangles pointing down represent upper limits. with increasing temperatures from bottom to top 0.7, 1, 2 and 3 keV. The full lines represent simulated HR from powerlaw models with fixed slope Γ = 2 and increasing intrinsic absorption form bottom to top 0, 10 21 , 10 22 and 10 23 cm −2 . Right: Same as left panel, but for models comprising a power-law PL with fixed slope Γ = 2 and intrinsic absorption of 10 23 cm −2 , plus a thermal model with temperature of 1 keV (APEC). The relative normalization of the two models is, from bottom to top, 10 −3 , 10 −2 , 10 −1 , 1, 10, 10 2 and 10 3 , corresponding to a contribution of the power-law component to the total flux of ∼ 0.2%, ∼ 2%, ∼ 15%, ∼ 62%, ∼ 94%, ∼ 99% and 100%, respectively. 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 \n\nFigure 7\n7shows the results of the L K -first stacking, with L X−LMXB plotted against stellar mass M * and the mass of the nuclear black hole, M BH , obtained from the M BH − M * relation of Kormendy & Ho (2013). Evaluating M BH from the M BH − M * relation McConnell & Ma (2013) would yield black hole masses between 0.1 and 0.3 dex smaller than the former. Similarly, one could derive M BH from the M BH − L K relation from Graham\n\nFig. 1 .Fig. 2 .\n12-Distribution of net counts for randomly distributed extraction regions (red line) and for the selected sample of sources (blue line). -Examples of 0.5-7 keV images of the L K selected stacking bins.\n\n\nnumber of sources (3) < log(LK/L ) > (4) < z > (5) < log(age/yr) > (6) < log(M * /M ) > (7) S /N (8) < LX−LMXB, 0.3−8/(10 40 erg s −1 ) >\n\nFig. 3 .\n3-X-ray luminosity (after LMXBs subtraction) versus K-band luminosity for the stacking bins selected in L K (Left Panel) and redshift (Right Panel). The horizontal and vertical black error bars indicate the statistical uncertainties on L K and L X−LMXB , respectively, while the vertical green error bars show the uncertainty on L X−LMXB due to redshift spread in each bin. The stellar mass M * from COSMOS catalog is plotted on the top axis. The local relation by KF13 is represented with a full black line. The dashed lines are drawn to include the most L X,gas luminous source ETG (M87) and the most L K luminous ETG (NGC1316) in the Local Sample (BKF11, KF13), respectively. Bins are labelled as in\n\nFigFig. 5 .Fig. 6 .\n56. 4.-Left: HRs of the L K -first stacking bins versus redshift. Right: HRs of the z-first stacking bins versus L K . Triangles pointing up and down represent lower and upper limits on HR, respectively, while the box for bin 28 represents unconstrained HR. -Same as Figure 3, but with bins labelled according to their average redshift, and HRs indicated by the slope of the orange segments. Black error bars reflect statistical uncertainties, vertical green bars show the uncertainty on L X−LMXB due to redshift spread in each bin. Red boxes show the BKF11 and KF13 samples, blue circles show the C14 sample of C-COSMOS X-ray detected ETGs. Triangles pointing down represent upper limits. -Left: HR-z scatter diagram of L K -first bin. The vertical error bars indicate uncertainties on L X−LMXB due to counts and to LMXB subtraction. Triangles pointing up and down represent lower and upper limits on HR, respectively. Squares represent unconstrained HRs. Dashed lines show HRs expected from thermal models (APEC)\n\nFig. 7 .\n7-X-ray luminosity (after LMXBs subtraction) versus black hole mass M BH (evaluated from M * according to theKormendy & Ho (2013) relation) for the L K -first stacking bins. The stellar mass M * from COSMOS catalog is plotted on the top axis. Lines of constant L bol /L Edd , given M BH and L bol , are shown as diagonals (from bottom to top 10 −6 , 10 −5 , 10 −4 , 10 −3 and 10 −2 ). The bins are color-labeled as inFigure 3. Light blue boxes represent the predictions of the Volonteri et al. (2011) models for M * = 10 9 M , 10 10 M and 10 11 M , where the top of each box refers to radiatively efficient accretion flows and the bottom refers to RIAFs.\n\nTable 1 :\n1Main features of the L K selected stacking bins: (1) bin number; \n\nTable 2 :\n2Same as Table 1 but for redshift selected stacking bins.\nhttp://cxc.harvard.edu/ciao/ahelp/dmfilth.html 3 In the following we consider the Chandra-ACIS radius enclosing 90% of the energy.\nFor the rare cases where µ + 3σ < 1, only pixels with two or more counts were clipped.\nThese average quantities, as well as their standard deviation in each bin is reported in Tables 1 and 2. 6 http://cxc.harvard.edu/toolkit/pimms.jsp\n1 1 16 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10\n. R D Blandford, M C Begelman, MNRAS. 3031Blandford, R. D., & Begelman, M. C. 1999, MNRAS, 303, L1\n\n. B Boroson, D.-W Kim, G Fabbiano, ApJ. 729BKF1112Boroson, B., Kim, D.-W., & Fabbiano, G. 2011, ApJ, 729, 12 (BKF11)\n\n. G Bruzual, S Charlot, MNRAS. 3441000Bruzual, G., & Charlot, S. 2003, MNRAS, 344, 1000\n\n. D Calzetti, R C Kennicutt, C W Engelbracht, ApJ. 666870Calzetti, D., Kennicutt, R. C., Engelbracht, C. W., et al. 2007, ApJ, 666, 870\n\n. C R Canizares, G Fabbiano, G Trinchieri, ApJ. 312503Canizares, C. R., Fabbiano, G., & Trinchieri, G. 1987, ApJ, 312, 503\n\n. G Chabrier, PASP. 115763Chabrier, G. 2003, PASP, 115, 763\n\n. L Ciotti, A D'ercole, S Pellegrini, A Renzini, ApJ. 376380Ciotti, L., D'Ercole, A., Pellegrini, S., & Renzini, A. 1991, ApJ, 376, 380\n\n. L Ciotti, J P Ostriker, ApJ. 6651038Ciotti, L., & Ostriker, J. P. 2007, ApJ, 665, 1038\n\n. L Ciotti, J P Ostriker, D Proga, ApJ. 717708Ciotti, L., Ostriker, J. P., & Proga, D. 2010, ApJ, 717, 708\n\n. F Civano, M Elvis, M Brusa, ApJS. 20130Civano, F., Elvis, M., Brusa, M., et al. 2012, ApJS, 201, 30\n\n. F Civano, G Fabbiano, S Pellegrini, ApJ. 790C1416Civano, F., Fabbiano, G., Pellegrini, S., et al. 2014, ApJ, 790, 16 (C14)\n\n. A Comastri, R Gilli, A Marconi, G Risaliti, M Salvati, A&A. 57410Comastri, A., Gilli, R., Marconi, A., Risaliti, G., & Salvati, M. 2015, A&A, 574, L10\n\n. T J Cox, T Di Matteo, L Hernquist, ApJ. 643692Cox, T. J., Di Matteo, T., Hernquist, L., et al. 2006, ApJ, 643, 692\n\n. X Dai, C S Kochanek, N D Morgan, ApJ. 658917Dai, X., Kochanek, C. S., & Morgan, N. D. 2007, ApJ, 658, 917\n\n. A L R Danielson, B D Lehmer, D M Alexander, MNRAS. 422494Danielson, A. L. R., Lehmer, B. D., Alexander, D. M., et al. 2012, MNRAS, 422, 494\n\n. L P David, W Forman, C Jones, ApJ. 369121David, L. P., Forman, W., & Jones, C. 1991, ApJ, 369, 121\n\n. M Elvis, F Civano, C Vignali, ApJS. 184158Elvis, M., Civano, F., Vignali, C., et al. 2009, ApJS, 184, 158\n\n. G Fabbiano, ARA&A. 2787Fabbiano, G. 1989, ARA&A, 27, 87\n\n. G Fabbiano, ARA&A. 44323Fabbiano, G. 2006, ARA&A, 44, 323\n\n. W Forman, J Schwarz, C Jones, W Liller, A C Fabian, ApJ. 23427Forman, W., Schwarz, J., Jones, C., Liller, W., & Fabian, A. C. 1979, ApJ, 234, L27\n\n. W Forman, C Jones, W Tucker, ApJ. 293102Forman, W., Jones, C., & Tucker, W. 1985, ApJ, 293, 102\n\n. T Fragos, B Lehmer, M Tremmel, ApJ. 76441Fragos, T., Lehmer, B., Tremmel, M., et al. 2013, ApJ, 764, 41\n\n. T Fragos, B . Lehmer, S Naoz, A Zezas, A Basu-Zych, ApJ. 77631Fragos, T., Lehmer, B. ., Naoz, S., Zezas, A., & Basu-Zych, A. 2013, ApJ, 776, L31\n\n. P Gandhi, Y Terashima, S Yamada, ApJ. 77351Gandhi, P., Terashima, Y., Yamada, S., et al. 2013, ApJ, 773, 51\n\nG P Garmire, M W Bautz, P G Ford, J A Nousek, G R Ricker, Jr, Proc. SPIE. SPIE485128Garmire, G. P., Bautz, M. W., Ford, P. G., Nousek, J. A., & Ricker, G. R., Jr. 2003, Proc. SPIE, 4851, 28\n\n. M Gilfanov, MNRAS. 349146Gilfanov, M. 2004, MNRAS, 349, 146\n\n. A W Graham, MNRAS. 379711Graham, A. W. 2007, MNRAS, 379, 711\n\n. O Ilbert, P Capak, M Salvato, ApJ. 6901236Ilbert, O., Capak, P., Salvato, M., et al. 2009, ApJ, 690, 1236\n\n. O Ilbert, M Salvato, E Le Floc'h, ApJ. 709644Ilbert, O., Salvato, M., Le Floc'h, E., et al. 2010, ApJ, 709, 644\n\nB T Jannuzi, A Dey, Photometric Redshifts and the Detection of High Redshift Galaxies. 191111Jannuzi, B. T., & Dey, A. 1999, Photometric Redshifts and the Detection of High Redshift Galaxies, 191, 111\n\n. T M Jones, M Kriek, P G Van Dokkum, ApJ. 78325Jones, T. M., Kriek, M., van Dokkum, P. G., et al. 2014, ApJ, 783, 25\n\n. D.-W Kim, G Fabbiano, ApJ. 611846Kim, D.-W., & Fabbiano, G. 2004, ApJ, 611, 846\n\n. D.-W Kim, G Fabbiano, N J Brassington, ApJ. 703829Kim, D.-W., Fabbiano, G., Brassington, N. J., et al. 2009, ApJ, 703, 829\n\n. D.-W Kim, G Fabbiano, ApJ. 7211523Kim, D.-W., & Fabbiano, G. 2010, ApJ, 721, 1523\n\n. D.-W Kim, G Fabbiano, ApJ. 776KF13116Kim, D.-W., & Fabbiano, G. 2013, ApJ, 776, 116 (KF13)\n\n. D.-W Kim, G Fabbiano, arXiv:1504.00899Kim, D.-W., & Fabbiano, G. 2015, arXiv:1504.00899\n\n. C S Kochanek, D J Eisenstein, R J Cool, ApJS. 8Kochanek, C. S., Eisenstein, D. J., Cool, R. J., et al. 2012, ApJS, 200, 8\n\n. J A Kollmeier, C A Onken, C S Kochanek, ApJ. 648128Kollmeier, J. A., Onken, C. A., Kochanek, C. S., et al. 2006, ApJ, 648, 128\n\n. J Kormendy, L C Ho, ARA&A. 51511Kormendy, J., & Ho, L. C. 2013, ARA&A, 51, 511\n\n. A Kundu, T J Maccarone, S E Zepf, ApJ. 5745Kundu, A., Maccarone, T. J., & Zepf, S. E. 2002, ApJ, 574, L5\n\n. A Leauthaud, R Massey, J.-P Kneib, ApJS. 172219Leauthaud, A., Massey, R., Kneib, J.-P., et al. 2007, ApJS, 172, 219\n\n. B D Lehmer, W N Brandt, D M Alexander, ApJ. 657681Lehmer, B. D., Brandt, W. N., Alexander, D. M., et al. 2007, ApJ, 657, 681\n\n. B D Lehmer, D M Alexander, F E Bauer, ApJ. 724559Lehmer, B. D., Alexander, D. M., Bauer, F. E., et al. 2010, ApJ, 724, 559\n\n. B D Lehmer, M Berkeley, A Zezas, ApJ. 78952Lehmer, B. D., Berkeley, M., Zezas, A., et al. 2014, ApJ, 789, 52\n\n. N A Levenson, T M Heckman, J H Krolik, K A Weaver, P T Życki, ApJ. 648111Levenson N. A., Heckman T. M., Krolik J. H., Weaver K. A.,Życki P. T., 2006, ApJ, 648, 111\n\n. W G Mathews, F Brighenti, ARA&A. 41191Mathews, W. G., & Brighenti, F. 2003, ARA&A, 41, 191\n\n. N J Mcconnell, C.-P Ma, ApJ. 764184McConnell, N. J., & Ma, C.-P. 2013, ApJ, 764, 184\n\n. H J Mccracken, B Milvang-Jensen, J Dunlop, A&A. 544156McCracken, H. J., Milvang-Jensen, B., Dunlop,J., et al. 2012, A&A, 544, A156\n\n. A Merloni, A Bongiorno, M Bolzonella, ApJ. 708137Merloni, A., Bongiorno, A., Bolzonella, M., et al. 2010, ApJ, 708, 137\n\n. B P Miller, E Gallo, J E Greene, ApJ. 79998Miller, B. P., Gallo, E., Greene, J. E., et al. 2015, ApJ, 799, 98\n\n. S Mineo, M Gilfanov, R Sunyaev, MNRAS. 4192095Mineo, S., Gilfanov, M., & Sunyaev, R. 2012, MNRAS, 419, 2095\n\n. M Moresco, L Pozzetti, A Cimatti, A&A. 55861Moresco, M., Pozzetti, L., Cimatti, A., et al. 2013, A&A, 558, A61\n\n. S S Murray, A Kenter, W R Forman, ApJS. 1611Murray, S. S., Kenter, A., Forman, W. R., et al. 2005, ApJS, 161, 1\n\n. R Narayan, I Yi, ApJ. 452710Narayan, R., & Yi, I. 1995, ApJ, 452, 710\n\n. S Pellegrini, ApJ. 624155Pellegrini, S. 2005, ApJ, 624, 155\n\n. S Pellegrini, J Wang, G Fabbiano, ApJ. 75894Pellegrini, S., Wang, J., Fabbiano, G., et al. 2012, ApJ, 758, 94\n\n. M Polletta, M Tajer, L Maraschi, ApJ. 66381Polletta, M., Tajer, M., Maraschi, L., et al. 2007, ApJ, 663, 81\n\n. S Puccetti, C Vignali, N Cappelluti, ApJS. 185586Puccetti, S., Vignali, C., Cappelluti, N., et al. 2009, ApJS, 185, 586\n\n. G Risaliti, R Maiolino, M Salvati, ApJ. 522157Risaliti, G., Maiolino, R., & Salvati, M. 1999, ApJ, 522, 157\n\n. C L Sarazin, J A Irwin, J N Bregman, ApJ. 544101Sarazin, C. L., Irwin, J. A., & Bregman, J. N. 2000, ApJ, 544, L101\n\n. N Scoville, R G Abraham, H Aussel, ApJS. 17238Scoville, N., Abraham, R. G., Aussel, H., et al. 2007, ApJS, 172, 38\n\n. E Schinnerer, M T Sargent, M Bondi, ApJS. 188384Schinnerer, E., Sargent, M. T., Bondi, M., et al. 2010, ApJS, 188, 384\n\n. C L Steinhardt, M Elvis, MNRAS. 4022637Steinhardt, C. L., & Elvis, M. 2010, MNRAS, 402, 2637\n\n. S Tremaine, K Gebhardt, R Bender, ApJ. 574740Tremaine, S., Gebhardt, K., Bender, R., et al. 2002, ApJ, 574, 740\n\n. G Trinchieri, G Fabbiano, ApJ. 296447Trinchieri, G., & Fabbiano, G. 1985, ApJ, 296, 447\n\n. P Tzanavaris, I Georgantopoulos, A&A. 480Tzanavaris, P., & Georgantopoulos, I. 2008, A&A, 480, 663 -14 -\n\n. M Volonteri, M Dotti, D Campbell, M Mateo, ApJ. 730145Volonteri, M., Dotti, M., Campbell, D., & Mateo, M. 2011, ApJ, 730, 145\n\n. C R Watson, C S Kochanek, W R Forman, ApJ. 6962206Watson, C. R., Kochanek, C. S., Forman, W. R., et al. 2009, ApJ, 696, 2206\n\n. R E White, Iii, C L Sarazin, ApJ. 367476White, R. E., III, & Sarazin, C. L. 1991, ApJ, 367, 476\n\n. C J Willott, ApJ. 7428Willott, C. J. 2011, ApJ, 742, L8\n\n. F Yuan, M Li, ApJ. 73723Yuan, F., & Li, M. 2011, ApJ, 737, 23\n\n. F Yuan, R Narayan, ARA&A. 52529Yuan, F., & Narayan, R. 2014, ARA&A, 52, 529\n"},"corpus_id":{"kind":"number","value":118373611,"string":"118,373,611"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1507.03170\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.3847/0004-637x/823/2/112\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1507.03170v2.pdf\"\n ],\n \"pdf_sha\": \"841b18653dc96f857f4f5621a35dc255a2c6bd18\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1507.03170\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Hidden AGNs in Early-Type Galaxies"},"authors":{"kind":"list like","value":["Alessandro Paggi apaggi@cfa.harvard.edu \nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA","Giuseppina Fabbiano \nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA","Francesca Civano \nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA\n\nDepartment of Physics and Yale Center for Astronomy and Astrophysics\nYale University\nP.O. Box 20812106520-8121New HavenCT","Silvia Pellegrini \nDepartment of Astronomy\nUniversity of Bologna\nvia Ranzani 140127BolognaItaly","Martin Elvis \nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA","Dong-Woo Kim \nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA"],"string":"[\n \"Alessandro Paggi apaggi@cfa.harvard.edu \\nHarvard-Smithsonian Center for Astrophysics\\n60 Garden St02138CambridgeMAUSA\",\n \"Giuseppina Fabbiano \\nHarvard-Smithsonian Center for Astrophysics\\n60 Garden St02138CambridgeMAUSA\",\n \"Francesca Civano \\nHarvard-Smithsonian Center for Astrophysics\\n60 Garden St02138CambridgeMAUSA\\n\\nDepartment of Physics and Yale Center for Astronomy and Astrophysics\\nYale University\\nP.O. Box 20812106520-8121New HavenCT\",\n \"Silvia Pellegrini \\nDepartment of Astronomy\\nUniversity of Bologna\\nvia Ranzani 140127BolognaItaly\",\n \"Martin Elvis \\nHarvard-Smithsonian Center for Astrophysics\\n60 Garden St02138CambridgeMAUSA\",\n \"Dong-Woo Kim \\nHarvard-Smithsonian Center for Astrophysics\\n60 Garden St02138CambridgeMAUSA\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We present a stacking analysis of the complete sample of Early Type Galaxies (ETGs) in the Chandra COSMOS (C-COSMOS) survey, to explore the nature of the X-ray luminosity in the redshift and stellar luminosity ranges 0 < z < 1.5 and 10 9 < L K /L < 10 13 . Using established scaling relations, we subtract the contribution of X-ray binary populations, to estimate the combined emission of hot ISM and AGN. To discriminate between the relative importance of these two components, we (1) compare our results with the relation observed in the local universe L X,gas ∝ L 4.5 K for hot gaseous halos emission in ETGs, and (2) evaluate the spectral signature of each stacked bin. We find two regimes where the non-stellar X-ray emission is hard, consistent with AGN emission. First, there is evidence of hard, absorbed X-ray emission in stacked bins including relatively high z (∼ 1.2) ETGs with average high X-ray luminosity (L X−LMXB 6 × 10 42 erg/s). These luminosities are consistent with the presence of highly absorbed \"hidden\" AGNs in these ETGs, which are not visible in their optical-IR spectra and spectral energy distributions. Second, confirming the early indication from our C-COSMOS study of X-ray detected ETGs, we find significantly enhanced X-ray luminoaity in lower stellar mass ETGs (L K 10 11 L ), relative to the local L X,gas ∝ L 4.5 K relation. The stacked spectra of these ETGs also suggest X-ray emission harder than expected from gaseous hot halos. This emission is consistent with inefficient accretion 10 −5 − 10 −4Ṁ Edd onto M BH ∼ 10 6 − 10 8 M ."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Diffuse X-ray emission from normal early type galaxies (ETG) was first detected in the Virgo cluster with the Einstein X-ray Observatory (Forman et al. 1979). Although in a few cases (e.g., M86) the displacement of the X-ray emission from the stellar body of the galaxy pointed to hot gaseous halos, the ∼ arcminute resolution of the Einstein IPC data could not disentangle observationally hot halos from the populations of low-mass X-ray binaries (LMXBs), which was likely to dominate the X-ray emission in a large number of the observed ETGs (Trinchieri & Fabbiano 1985). Nonetheless, several authors discussed how the X-ray emission and the inferred hot gas contribution could be used to constrain both the binding mass of the galaxies and the physical evolution of the hot halos, including feedback from SNIa and AGNs, and interaction with intracluster and group media (see e.g., Fabbiano 1989;Canizares et al. 1987;Ciotti et al. 1991;David et al. 1991;White & Sarazin 1991;Mathews & Brighenti 2003;Ciotti & Ostriker 2007).\nThanks to the high spatial resolution of Chandra and imaging/spectral capabilities of the ACIS detector (Garmire et al. 2003), it has become possible to directly detect and characterize the different contributions to the X-ray emission of ETGs in the near universe. With Chandra, LMXB populations -both associated with Globular Clusters (GC) and in the stellar field of the ETG -have been studied to distances of several tens of megaparsecs (e.g., Sarazin et al. 2000;Kundu et al. 2002; see review by Fabbiano 2006). The integrated LMXB luminosity correlates well with the integrated ETG stellar luminosity (e.g., Kim & Fabbiano 2004;Gilfanov 2004). The scatter in this relation is a factor of 2-3, and it depends on both GC content (Kim & Fabbiano 2004), and the age of the stellar population (Kim et al. 2009;Kim & Fabbiano 2010;Boroson et al. 2011;Lehmer et al. 2014). These detailed high resolution studies allowed Boroson et al. (2011, BKF11) to accurately estimate the gaseous contribution to the X-ray luminosity L X,gas of a sample of 30 ETGs within 32 Mpc. After subtracting LMXB, stellar, and nuclear contributions to the X-ray luminosity, BKF11 found steep scaling relations between L X,gas and both the stellar L K luminosity and the hot gas temperature T . Subsequent work by Kim & Fabbiano (2013, KF13;2015, KF15) suggests that for L X,gas > 10 40 erg s −1 the hot gas around ETGs is virialized, giving a new way to measure the total mass of these galaxies.\nBeyond the local universe, Chandra Deep Field studies (both detections and stacking of almost 1,000 ETGs) found that the mean L X of L B = 10 10 L ETGs remains equal or mildly increases over z = 0 − 0.7 (Lehmer et al. 2007;Tzanavaris & Georgantopoulos 2008) and that the ratio of the soft X-ray luminosity to the B-band luminosity evolves mildly as L (0.5−2 keV) /L B ∝ (1 + z) 1.1±0.7 since z ≈ 1.2 (Danielson et al. 2012), suggesting some heating mechanism preventing the hot gas from cooling, consistent with mechanical heating from radio AGNs. The evolution of the X-ray binary contribution to the X-ray emission (both of LMXBs and high mass binaries) across cosmic time, modeled by Fragos et al. (2013a), led Fragos et al. (2013b) to conclude that energy feedback from X-ray binaries is also important in the early universe.\nThe COSMOS survey, covering ∼ 2 square degrees of sky with multi-wavelength coverage, which includes Chandra X-ray coverage (Scoville et al. 2007;Elvis et al. 2009), has increased the available sample of ETGs to over 6,600 galaxies Civano et al. (2014, C14). This survey provides a good z coverage out to z ∼ 2.5 (and a few galaxies are detected out to z ∼ 5). C14 studied the sample of 69 X-ray detected ETGs in the Chandra-COSMOS survey (C-COSMOS, Elvis et al. 2009), covering a redshift range 0 < z < 1.5, to explore redshift dependencies in the BKF11, KF13 scaling relations. In particular, C14 focused on the relation between L X,gas and the K-band stellar luminosity (L K ) of ETGs which is a good proxy of stellar mass for old stellar systems. The latter has been evaluated in the local Universe by BKF11 as L X,gas ∝ L K 2.6±0.4 for sources with L X,gas ≤ 2 × 10 41 erg s −1 , while including sources with L X,gas up to 9 × 10 42 erg s −1 KF13 found a steeper L X,gas ∝ L K 4.5±0.8 relation. C14 found that this latter relation also holds for X-ray detected ETGs with L X,gas ≤ ×10 42 erg s −1 , suggesting that the hot gas in these more distant galaxies may be virialized as well. However, more distant (z > 0.9) and younger stellar age galaxies generally fall outside the relation observed in the local sample, and are X-ray overluminous with respect to their L K . For these sources C14 suggested that the higher X-ray luminosity may be due to the presence of hidden AGNs (as suggested by observed hardness ratios), merging phenomena, which are expected to increase the halo X-ray luminosity, or nuclear accretion (that could also induce AGN activity during the merging phase, Cox et al. 2006).\nIn this work we perform a stacking analysis of the C-COSMOS ETG data in order to investigate the nature of fainter, X-ray undetected ETGs. We compare their properties with those of the sample of local ETGs investigated by BKF11 and KF13, and with the sample of ETGs detected in the C-COSMOS survey studied by C14. This stacking study also allow us to further explore redshift dependencies. Our aim is to explore the properties of the non-stellar Xray emission, i.e. the X-ray emission resulting from the subtraction of the expected X-ray binary contribution (taking into account the cosmic evolution of the X-ray binary population Fragos et al. 2013b). The resulting emission will probe both hot X-ray halos and the possible emission of active nuclei in these galaxies. In particular, the large number of low-stellar-mass ETGs in the sample will provide a unique probe of the existence of relatively low-mass massive black holes in their nuclei. These black holes should produce low-luminosity nuclear X-ray sources, if they are fueled by the ETG stellar outgassing (Volonteri et al. 2011).\nThe paper is organized as follows: Section 2 describes the sample selection and the data sets used in this work, Section 3.3 is dedicated to the description of the adopted stacking procedure, in Section 4 we discuss our results, and Section 5 is dedicated to our conclusions.In the following, we adopt the standard flat cosmology with Ω Λ = 0.73 and H 0 = 70 km s −1 Mpc −1 ."},{"title":"Sample Selection","content":"The COSMOS survey (Scoville et al. 2007) provides a multi-wavelength characterization of over 1.5 million galaxies at redshift extending out to 5 in 2 deg 2 of the sky. The Chandra X-ray coverage (C-COSMOS Elvis et al. 2009) over 0.9 deg 2 , gives us the means to study X-ray emission of ETGs, extending local studies to higher redshifts, and explore the redshift evolution of hot halos with large samples of galaxies.\nOur ETG sample is selected from the most recent version of the COSMOS photometric catalog by Ilbert et al. (2009). The spectral energy distribution (SED) identification performed by Ilbert et al. (2009) makes use of 7 elliptical galaxy templates with ages from 2 to 13 Gyr (Polletta et al. 2007). The SED templates were generated with the stellar population synthesis package developed by Bruzual & Charlot (2003), assuming an initial mass function from Chabrier (2003) and an exponentially declining star formation history. We also collected additional galaxy properties (stellar mass, age and star formation rate) from Ilbert et al. (2010), and we made use of the most recent near-infrared photometry from the Ultra Deep Survey with the VISTA telescope (Ultra-VISTA McCracken et al. 2012) 1 .\nWe selected galaxies classified as E or S0 and that have a reliable photometric redshift (Ilbert et al. 2009). In addition we selected only sources with Ultra-VISTA detection both in J and K bands, since these are needed to evaluate the rest frame L K (see below). This narrows our sample to 10,808 (the COSMOS ETG sample considered in C14). We then selected only the 6436 ETGs falling in the C-COSMOS field, and then excluded the 48 X-ray detected ETGs in common with the C-COSMOS identification catalog (Civano et al. 2012). This leads us to our final sample of 6388 C-COSMOS ETGs -not detected in X-rays -which constitutes the starting point of the stacking analysis described in this paper. Note that we selected sources with photometric redshift estimate in the catalog by Ilbert et al. (2009), while C14 selected their sources to have a spectroscopic or photometric redshift estimates in the C-COSMOS identification catalog, so the sample of ETGs selected by the latter authors (∼ 6600 sources) is not fully contained in our sample.\nFollowing C14 we evaluated the K band luminosity of our sources from the Ultra-VISTA K-band aperture magnitude of the COSMOS photometric catalog. To calculate the rest frame K band luminosities we assumed a power law spectrum (as appropriate over this small frequency range) f ν ∝ ν −α with α = (J − K)/2.5 log (ν J /ν K ) (where J and K are the AB magnitudes from the COSMOS photometric catalog and ν J and ν K are the corresponding frequencies), and express the rest frame luminosities in solar luminosity as L K = 10 −(K+ext−5.19)/2.5 × (1 + z) α−1 × (D L /10) 2 , where z is the photometric redshift from the COSMOS catalog, D L is the luminosity distance expressed in parsecs, and ext is the aperture correction parameter computed from the FWHM measured for all the COSMOS extended sources in the Hubble ACS data.\nWe compared the distributions of the physical parameters of the 6388 C-COSMOS ETGs with the entire 10,808 1 Note that stellar mass reported in COSMOS photometric catalog, and obtained from the Bruzual & Charlot (2003) best-fit templates, do not incorporate the Ultra-VISTA K-band NIR data. Although this is only an issue if there is any residual star formation in the ETGs, L K and M * may not necessarily be completely equivalent.\nCOSMOS ETG sample finding no significant difference, demonstrating as expected that the sample of ETGs not detected in X-ray is representative of COSMOS ETGs overall. Note that the luminosity threshold is a function of redshift because of the uniform magnitude limit of the Ultra-VISTA COSMOS survey (K s ∼ 24, McCracken et al. 2012). The specific star formation rate for the majority of the ETGs is in the regime where Ilbert et al. (2010) defines galaxies as quiescent, although higher star formation rate is observed in young ETGs. The properties (luminosity, mass, star formation rate and age) of the COSMOS ETGs are in agreement with those reported by (Moresco et al. 2013) for a sample of ETGs selected using multiple criteria (including color-color diagrams, spectra, star formation rate and SED classification). Masses and luminosities are in agreement with those of the K-selected elliptical galaxies of Jones et al. (2014), extracted from a small area of the COSMOS field."},{"title":"Analysis and Results","content":"Below we describe the procedures followed in our stacking analysis. The results are summarized in Tables 1 and 2."},{"title":"Count Extraction at ETG Positions","content":"We used all 49 C-COSMOS observations to evaluate source and background counts for our sample of ETGs, excluding data in the ACIS-I chip gaps. We also excluded data from X-ray detected sources removing counts within a 10\" radius of all sources in the C-COSMOS source identification catalog (Civano et al. 2012). We replaced the excluded counts by sampling the Poisson distribution of the pixel values nearby background regions with dmfilth CIAO task 2 . When a source lay in the field of view of more than one observation, we combined the data from the different observations using the prescriptions from Puccetti et al. (2009), excluding observations where the source was more than 10 acrmin off-axis from the aim point.\nDue to the variation of the Chandra Point Spread Function (PSF) radius R PS F 3 , we used an extraction radius r ext that was determined by the source position in each observation in order to extract an optimal fraction of X-ray photons from the diffuse halo. For sources with morphological information available in the HST-ACS catalog (Leauthaud et al. 2007) we adopted the source semi-major axes R as a measure of its physical extent. For sources with R > R PS F we adopted values of r ext that was the convolution of R and R PS F (R 2 + R 2 PS F ); otherwise we adopted r ext = R PS F as we did for sources without morphological information in HST-ACS catalog. We notice however that sources with R > R PS F represent only ∼ 1% of our sample, with extraction radii typically ranging between 1\" and 7\". The average exposure time for each source within the extraction circle, E, was then evaluated. The histogram of extracted ETG position counts is shown in Figure 1 (blue line). The small fraction of extraction regions with counts above ∼ 5 represent off-axis sources with large extraction radii."},{"title":"Background Evaluation","content":"To evaluate the background we extracted counts from an annulus centered on the source coordinates, with innerradius 2r ext and outer radius 4r ext . Due to the very low signal, the background represents the largest source of uncertainty in this kind of analysis. To estimate the systematic uncertainties introduced by the evaluation of the background, we compared the results of three different methods, obtaining consistent results. First we simply evaluated the background contribution to the total source counts from the mean count value of the pixels in the whole background annulus. Second, we adopted the clipping procedure of Willott (2011, method b), which evaluates the mean count value µ -and the standard deviation σ -of all pixels within a circle of radius 4r ext around each source position and then sets all pixels with count value larger than µ + 3σ to µ. This method suppresses the noise in the background annulus, but also removes signal from the source 4 . Lastly, we divided the background annulus into eight 45 degree sectors, and excluded all the sectors in which the mean count value was larger than µ + 3σ, thus excluding regions of higher background without removing signal from the source. The net counts N obtained with all three methods are consistent within statistics, so in the following we adopted the clipping procedure of Willott (2011, method b).\nTo test the robustness of our results we also extracted counts from positions on the sky offset from the ETG coordinates by random distances between 15\" and 30\" both in right ascension and declination. The net counts of the non-source positions are shown in Figure 1. They follow an approximatively Gaussian distribution centered on zero, as expected from background fluctuations, while the ETG counts show an excess of positive signal. To test if our results were dominated by a sub-population of ETGs we repeated the stacking procedure on sub-samples randomly drawn from the complete sample, progressively halving the subsample, obtaining results compatible with the full sample."},{"title":"Stacking Procedure","content":"To evaluate the count rate C for each stacked bin we stacked the observed source net counts in 0.5-7 keV range N and exposure E to obtain C = i N i / i E i . From these counts we subtracted the expected contributions from stellar sources (LMXBs and Ultra Luminous X-ray sources, ULXs), and evaluated the average bin rest-frame X-ray luminosity as explained below. We corrected L K to take into account the evolution of the stellar population (i.e., the fading effect, see C14). Note that, at a given redshift, younger galaxies have larger fading correction than older ones. To compare our results with those of KF13 and C14 we considered sources with 9 < log (L K ) < 13.\nWe constructed the stacks by binning in both L K and z, to ensure that the binning scheme did not produce any significant bias on the results. For the L K binning, we first divided our sample in four log (L K ) bins, 9 − 10, 10 − 11, 11 − 12 and 12 − 13. We then ordered the sources in each bin in increasing redshift, and progressively stacked them until we reached a minimum signal to noise ratio of 3; if this 3σ threshold could not be reached we stacked the signal from all the remaining sources in the bin. Examples of 0.5-7 keV images for these stacking bins are presented in Figure 2. For the z binning, we followed the same procedure, dividing the sample into three bins 0 − 0.5, 0.5 − 1 and 1 − 1.5, and then ordered the sources in each bin in increasing L K ."},{"title":"Evaluation of Non-stellar L X for Each Stacked Bin","content":"To subtract the LMXB contribution from the total X-ray emission, we evaluated the average LMXB contribution to each bin using the 0.3-8 keV relation from Fragos et al. (2013b), as in C14:\nwhere L X is expressed in erg s −1 , M * is the average stellar mass of the stacking bin expressed in solar masses and age is the average stellar age of the stacking bin expressed in Gyr. 5 We also evaluated the contribution to X-ray luminosity from ULXs that could be present in the galaxies using the scaling relations from Calzetti et al. (2007) and Mineo et al. (2012) between the ULX luminosity and the galaxy star formation rate. ULX contribution is more than an order of magnitude smaller than L X−LMXB even for the least luminous L K bins.\nWe converted the average 0.3 − 8 keV LMXB luminosities in each bin to count rates in the 0.5 − 7 keV interval using a power law model with index Γ = 1.8 (consistent with the typical spectrum of LMXBs) and Galactic absorption N H = 2.6 × 10 20 cm −2 . For these parameters the conversion factor used from Chandra count rates to fluxes is 1.04 × 10 −11 erg cm −2 s −1 cts −16 . We then subtracted these count rates from the stacked signals and converted back the resulting count rates into luminosities using a model comprising Galactic N H plus a thermal APEC component with temperature depending on the source total X-ray luminosity. We adopted kT = 0.7, 1 and 2 keV for X-ray luminosities of L X < 10 41 , 10 41 < L X < 10 42 and L X > 10 42 erg s −1 , respectively, following BKF11 and Dai et al. (2007). Finally, we K-corrected the luminosity to the 0.3 − 8 keV rest frame band. These LMXB-subtracted luminosities, that we indicate as L X−LMXB , are expected to include the emission from the hot gaseous halos, but may also include a possible contribution from any AGN harbored within these ETGs. Figure 3 shows the results in the L X−LMXB − L K plane for both L K and z binning (Left and Right panels, respectively), where on the upper axes we plot M * as taken from COSMOS catalog. In both binning schemes the overall scatter plots are consistent, showing that we do not suffer from strong binning biases. By construction, the L K -first binning provides information on the extreme redshift bins (both high and low), while the z-first binning covers more fully the L K range. The bins are numbered in Figure 3, following Tables 1 and 2. We use different colors to visually divide the stacking bins into four groups: red, for bins X-ray over-luminous with respect to the local L X−LMXB and L K relation (BFK11, KF13, see C14); yellow, for X-ray under-luminous bins; and grey and green for bins following the local relation with L X−LMXB lower and higher than 10 41 erg s −1 , respectively.\nAs noted in Section 1, Eq. 1 has an observed spread of a factor 2-3, which can be related to both the presence of GC LMXB in the ETGs and to the spread in age of the stellar population (Kim & Fabbiano 2004BKF11;Lehmer et al. 2014). Given the parameter space of Figure 3, this spread will not change our conclusions."},{"title":"Hardness Ratios","content":"Hardness ratios (HRs) were evaluated to characterize the ETG average spectral properties. HR = (H −S )/(H +S ), where S is the soft 0.5 − 2 keV counts and H the hard 2 − 7 keV counts. These HRs are those of the LMXB-subtracted emission. The contributions of these sources were evaluated and subtracted by applying the procedure in Section 3.4 to each energy band. We evaluated the errors on HRs as due to both uncertainties on counts and to uncertainties on estimate of LMXB contribution due to spread of stellar age and mass in the bins.\nIn Figure 4 we show the HRs for the stacking bins using the same color code as in Figure 3. We plot HR versus z in the Left panel, using the L K -first binning, which gives a better z sampling. We plot HR versus L K in the Right panel, using the z-first binning, to explore the L K behavior. The left panel of Figure 4 shows a distinct clustering in HR-z space of bins from separate loci of the L X−LMXB − L K plane, as shown by the clusters of different color points. The Right panel shows that ETGs with larger stellar mass (L K ) tend to have lower HR. Given the scaling relations of ETGs (BKF11; KF13; KF15), this could just reflect a relation with total binding mass, with more massive galaxies being more efficient at retaining large hot gaseous halos."},{"title":"Discussion","content":"Figure 5 summarizes our results in the L X−LMXB − L K plane and compares them with the COSMOS ETG X-ray detections (C14) and with the near universe ETG data (BKF11, KF13). We show the local KF13 relation with a solid line, and the scatter in this relation with the two dashed lines. These lines are chosen to pass through the points for the most L X,gas luminous source (M87) and the most L K luminous source (NGC1316), respectively. The COSMOS stacked-ETG results are overall consistent with the C14 COSMOS detections, but significantly increase the coverage at lower total stellar luminosities / masses. While most of the detected signals are compatible with the local KF13 relation, there are several detections that lie both to the left and to the right side of it, that is, bins that appear to be respectively over and under-luminous in X-ray with respect to their L K or M * .\nIn Figure 5 we also show the hardness ratios for each stacked bin with lines with slopes proportional to HR. We find positive HRs (i.e., harder spectra) in stacked bins outside the dashed line on the left. These are the lower L K galaxies with significant X-ray excess (the red points in Figure 4, see also C14). We also find hard HR > 0 in some points with L X−LMXB > 10 41 erg s −1 and z > 1 (the three top green points in Figure 4), to the left of the solid line.\nInstead, lower HR ratios (i.e., softer spectra) are found in bins following the local relation (between the two dashed lines). This is especially so for those to the right of the solid line. This suggests that they have a larger than average hot gas content up to z ∼ 1. Figure 6 compares the HR-z scatter diagram with a set of spectral models. On the Left, we show the expected HRs for thermal APEC (dashed lines; from bottom to top, kT = 0.7, 1, 2 and 3 keV) and power-law models (solid lines; fixed slope Γ = 2 and increasing intrinsic absorption from bottom to top, N H = 0, 10 21 , 10 22 and 10 23 cm −2 ). The APEC models are the expected emission of thermal gaseous emission, while the power-law models are typical AGN spectra. To illustrate cases of mixed emission from a hot gas component and a heavily obscured AGN, on the right panel of Figure 6 we plot composite models consisting of kT = 1 keV APEC emission with line of sight N H plus power-law models with slope Γ = 2 and intrinsic absorption N H = 10 23 cm −2 . From the bottom to the top the relative normalization of the two models varies, to reflect an increasing contribution of the power-law model to the total flux, ranging from ∼ 0.2% to ∼ 100%. The X-ray over-luminous bins (red points) have generally high HRs consistent with being dominated by AGN emission, either with \"normal\" AGNs with absorbing columns 10 22 cm −2 , or in the case of the composite gas + AGN emission even highly obscured Compton Thick AGNs (N H > 10 24 cm −2 , Risaliti et al. 1999;Levenson et al. 2006;Lehmer et al. 2010).\nThe bins following the local relation with L X−LMXB < 10 41 erg s −1 (grey points) show HR ∼ (−0.5) − 0.3, lower than those of the over-luminous X-ray bins. A soft HR first suggests emission from hot gas. However, Figure 6 shows that pure APEC thermal models with solar metallicity always have HR < −0.6. Hence these objects have HR values inconsistent with pure gaseous emission.\nThese HRs can instead be explained with combinations of thermal and AGN emission. Figure 6 Right shows models with varied ratios of APEC and power-law emission. In this model the observed HRs imply AGN components accounting for 15% − 60% of the X-ray luminosity. Given the X-ray luminosities of this sub-sample (Figures 3, 5), these ratios then imply AGN luminosities ranging from ∼ 5 × 10 39 erg s −1 to ∼ 5 × 10 40 erg s −1 . This range is in agreement with the low-luminosity Chandra X-ray sources in the near universe sample (BKF11, Pellegrini 2005), which have luminosities ranging from ∼ 10 38 erg s −1 to ∼ 5 × 10 41 erg s −1 .\nThe bins with L X−LMXB > 10 41 erg s −1 (green points) have hard HR > 0 incompatible with a low absorption power-law model (e.g., bins 21, 22 and 25). The three highest redshift bins in this group (e.g., bins 28, 29 and 30) have even harder HR > 0.5 that are compatible only with absorbing columns larger than 10 23 cm −2 . This suggests that these galaxies harbor (on average) highly obscured Compton Thick AGNs that dominate their X-ray luminosity.\nIn summary, our stacking analysis suggests that on average AGN emission is likely to be present in ETGs. These AGNs may be faint and hidden in hot gas dominated ETGs, which are consistent with the near Universe scaling relation of virialized halos (KF13, KF15, see C14). In contrast, AGN emission appears to dominate the non-stellar X-ray emission in two classes of ETGs: (1) low-stellar mass ETGs (log (L K /L ) 10.5), and (2) higher redshift ETGs (z > 0.5, > 1) especially those overluminous in X-rays relative to the near universe L X−LMXB − L K scaling relation. The latter group has HR values suggesting highly absorbed Compton Thick AGNs. We note that these AGNs were not visible in the COSMOS colors and spectra, since AGNs were excluded in our ETG sample selection process (C14). (2007), obtaining black hole masses ∼ 0.5 dex larger than the former. Uncertainties in these estimate are typically of a factor of ∼ 2 − 3 (Tremaine et al. 2002;Graham 2007) and evolution to z ∼ 1 − 2 may provide a systematic black hole mass increase of a similar factor (Merloni et al. 2010). These effects will not change our conclusions in the 3-dex wide range covered by Figure 7."},{"title":"Accretion Regimes for the AGNs","content":"On the right hand axes we plot L bol = 10 L X−LMXB , an appropriate bolometric correction for low-luminosity AGNs that radiate at a few 10s % of L Edd (Kollmeier et al. 2006;Steinhardt & Elvis 2010). Given this nominal L bol we can plot diagonal lines of constant L bol /L Edd . These lines range from 10 −2 to 10 −6 L bol /L Edd .\nWe see that the massive high L X−LMXB galaxies (green points) and the low-mass X-ray overluminous (red points) galaxies have implied Eddington ratios in the 10 −3 to 10 −5 range. RIAFs are generally believed to apply to all black hole accretion below 10 −2 L/L Edd , so we would expect all these sources to be RIAFs. If the bolometric correction for RIAFs is smaller than that considered here, as may well be the case (Narayan & Yi 1995), the L bol values on the y-axis should be correspondingly lower (as the Eddington ratios on the currently plotted dashed lines).\nAlternatively the true L bol of the three high z bins of massive high L X−LMXB galaxies (green points 28, 29, 30) could be greatly underestimated. The HRs of these bins also imply highly absorbed emission (N H > 10 23 cm −2 , Figure 6). If these are then Compton Thick AGN they may have much larger intrinsic luminosities, with factors of 100 being quite plausible (e.g., Risaliti et al. 1999;Levenson et al. 2006). They would have intrinsic luminosities L 0.5−10 keV ∼ 10 45 erg s −1 and L bol /L Edd in the ∼ 10 −2 range, so that they could lie within the normal AGN regime.\nTo investigate this possibility, we built SEDs of ETGs in our sample using optical-infrared magnitudes reported in COSMOS photometric catalog, since a \"hidden\" AGN may give rise to an IR excess with respect to an average ETG SED. We then fitted these SEDs with the same elliptical galaxy template used by Ilbert et al. (2009) without finding any significant deviation. To confirm this, we evaluated the L 2−10−LMXB luminosity of the stacking bins assuming a power-law spectrum (as appropriate for AGN spectra) and compared that with their IR luminosity. In all cases the Xray luminosities are at least 3 order of magnitudes lower than the IR luminosities, even assuming absorbing columns ∼ 10 23 cm −2 , indicating that at IR wavelength a putative AGN component is expected to be significantly smaller than the galactic emission.\nThe galaxies that are consistent with the local L K − L X−LMXB gaseous relation (grey points), with X-ray colors suggesting thermal emission plus a possible AGN component, have smaller Eddington ratios ∼ 10 −5 to 10 −6 (these ratios of course assume that the entire emission is due to an AGN; as shown in Figure 6, the AGN component may range from ∼ 15 − 60%).\nWe also plot in Figure 7 the predictions from Volonteri et al. (2011) of L X,AGN for three values of M * for both efficient and radiatively inefficient accretion flows (RIAFs Narayan & Yi 1995). These authors' models use the stellar density profile scaling with stellar mass, and predict the accretion rate from the normal stellar mass loss due to stellar evolution, and the consequent massive black hole luminosity for various radiative efficiencies 7 . The X-ray luminosity coming from accretion is computed as L X,AGN /L Edd = η X ηṁ, where η X = 0.1 is the X-ray fraction of the bolometric luminosity, andṁ = 0.1Ṁ/Ṁ Edd is the accretion rate in Eddington units with a 10% accretion efficiency. Radiatively efficient accretion flows have η = 1, while inefficient ones have η =ṁ/ṁ cr , and for themṁ <ṁ cr = 3 × 10 −2 .\nIn order to compare our results with Volonteri et al. (2011) models we converted the 0.5−7 keV LMXB subtracted counts into 0.5 − 10 keV band luminosities, L 0.5−10 keV assuming a pow-law spectrum as appropriate for AGN spectra. Based on these models, inefficient accretion into the nuclei could explain even all the L X−LMXB of the massive, high L X−LMXB ETGs (green points), except for the most luminous ones, which are likely to be Compton thick AGNs (see above). The low-mass X-ray overluminous ETGs (red points) are also consistent with the Volonteri et al. model. Considering that a large fraction of the X-ray emission is due to hot gas (see above), the grey points show instead a possible AGN emission even lower than predicted for radiatively inefficient accretion. In these ETGs, the L X−LMXB is consistent with the local L X,gas − L K relation. This result can have the following possible explanations, that are not mutually exclusive.\n1. The Volonteri et al.'s models assume that the whole of their estimated mass accretion rate (coming from the instantaneous mass loss of all stars, and obtained by integrating over the whole stellar density profile) indeed reaches the massive black hole, while this represents an upper limit on the mass accretion rate of the massive black hole (some gas may be outflowing, or may reach the massive black hole with a delay, or may possess angular momentum and form a disk, etc.).\n2. The accreting mass, that has already entered the accretion radius, is reduced on its way to the massive black hole, due to a wind from a RIAF (Narayan & Yi 1995;Blandford & Begelman 1999); similarly, close to the nucleus, there may be a heating mechanism for the accreting gas, as a nuclear jet, again impeding or lowering the accretion rate (e.g. Pellegrini et al. 2012;Yuan & Li 2011).\n3. Accretion onto the black hole is (temporarily) not taking place, as may be the case during feedback-regulated activity cycles. Numerical simulations show that, during these cycles, AGN outbursts are followed by major degassing of the circumnuclear region, with a precipitous drop of the nuclear accretion rate; the outbursts are 7 We note that in Volonteri et al. (2011) models assume for ETGs a stellar age of 12 Gyr. We corrected for the average age 10 9.5 Gyr of C-COSMOS ETGs, which yields slightly smaller emitted X-ray luminosities (see Eq. 9 in Volonteri et al. 2011 ). separated by long time intervals during which the galaxy is replenished again by gas from the stellar mass losses, until a new nuclear outburst takes place (Ciotti et al. 2010).\n4. η X might be lower than 1 as assumed here (see also Volonteri et al. 2011), or that η at lowṁ decreases faster than adopted in the Volonteri et al.'s models, so that lower L X,AGN should be predicted (e.g., Yuan & Narayan 2014).\nOur results extend to higher X-ray luminosities a recent analysis of ETGs in the AMUSE survey of Virgo and field galaxies with Chandra (Miller et al. 2015). AMUSE results indicate a significant presence of supermassive black holes in low stellar mass galaxies. Due to the lower redshifts of the AMUSE ETGs, that survey is sensitive to X-ray luminosities (∼ 1.3×10 38 erg/s) one order of magnitude fainter are than our faintest bin 1 (∼ 2.4×10 39 erg/s). For these dim sources the stellar wind accretion proposed by Volonteri et al. (2011) would predict higher X-ray luminosities than those observed, Miller et al. conclude that these black holes are probably powered by inefficient advection dominated accretion flows or, alternatively, by an outflow/jet component. We found that low-luminosity, X-ray \"overluminous\" ETGs can be adequately described by the RIAF fueled by stellar winds as proposed by Volonteri et al. (2011). Instead, ETGs with L X−LMXB > 10 42 erg s −1 may host Compton Thick AGNs. Similarly, when comparing with the results of Lehmer et al. (2007) and Danielson et al. (2012) on Chandra Deep Fields, we extend to higher X-ray luminosities and to lower optical luminosities (that is, lower stellar masses), finding sources that significantly exceed the expected emission from hot-gas dominated ETGs, with log (L 0.5−2 keV /L B )/(erg/s/L ) > 30 (see Figure 7 in Danielson et al. 2012).\nWe also note that, as reported by Comastri et al. (2015), the revision of black hole scaling relations by Kormendy & Ho (2013) indicate that the local mass density in black holes should be five times higher than previous estimates. These authors advance the possibility that a sizeable population of highly obscured AGNs with covering fraction of obscuring material ∼ 4π would explain the revised upward value of the SMBH local mass density without exceeding the limits imposed by the X-ray and IR background. An example of these obscured AGNs can be that observed in the ETG ESO565-G019 with Suzaku and Swift/BAT (Gandhi et al. 2013).\nFinally, we note that the hard X-ray slopes of the more luminous ETG bins could be due to contamination from X-ray jets. However, the combined 1.4GHz VLA COSMOS Large and Deep Project catalogs (Schinnerer et al. 2010) offer radio counterparts only to ∼ 2% of our sample of ETGs. The presence of radio emitters among non X-ray detected ETGs will be addressed after the release of the JVLA-COSMOS Project data."},{"title":"Summary and Conclusions","content":"We have followed up our previous work (C14) investigating the X-ray / K-band properties of the 69 ETGs detected in the Chandra COSMOS survey (Elvis et al. 2009), with a luminosity and hardness ratio stacking analysis of the Chandra data for the sample of 6388 early type galaxies (ETGs) not detected in X-rays. The stacking analysis allows the investigation of the average features of individually X-ray undetected ETGs. Our ETG sample was selected as representative of normal galaxies, with no evidence of AGN emission in their spectra and multi-band photometry (C14, Section 2). Our purpose was to investigate the properties of the X-ray emission from hot gas and from possible \"hidden\" AGN components that may be revealed by their X-ray emission. To this purpose, the expected contribution of LMXBs was subtracted from the X-ray luminosity of each stacked bin, using established scaling relations with the K-band luminosity (BKF11). All luminosities were corrected to the local rest frame; X-ray hardness ratios were based on the observed counts (after subtraction of the LMXB contribution), and were compared to emission models for the appropriate average bin redshift. This analysis has led to the following conclusions:\n1. On average AGN emission may be present in all ETGs.\n2. AGNs may be faint and hidden in hot gas dominated ETGs, which are consistent with the local universe scaling relation of virialized halos (KF13, KF15, see C14).\n3. AGN emission appears to dominate the non-stellar X-ray emission in low-stellar mass ETGs (log (L K ) 10.5 in solar units), which show marked X-ray luminosity excesses relative to their K-band luminosity (a proxy of stellar mass), when compared with the local ETG L X,Gas L K relation of the local universe (BKF11). This result confirms the conclusions of C14 and extends them to lower stellar mass galaxies.\n4. AGN emission is also prominent in higher redshift ETGs (z > 0.5, > 1) especially those with X-ray luminosity 10 42 erg s −1 . The latter group have HR values suggesting highly absorbed Compton Thick AGNs. We note that these AGNs were not visible in the COSMOS colors and spectra, since AGNs were excluded by our ETG sample (C14). 5. Given the nuclear BH mass predicted by the M BH − L K relation (Kormendy & Ho 2013), the hidden AGNs in the X-ray over-luminous low stellar mass ETGs are consistent with the presence of radiatively inefficient accretion fueled by stellar outgassing (Volonteri et al. 2011). The highly absorbed higher L X−LMXB AGNs hidden in massive, higher z ETGs instead suggest efficient accretion in highly obscured nuclei.\nThis work was partially supported by NASA contract NAS8-03060 (CXC), and NASA Chandra grant G01-12110X. SP acknowledges financial support from MIUR grant PRIN 2010-2011, project 'The Chemical and Dynamical Evolution of the Milky Way and Local Group Galaxies', prot. 2010LY5N2T. FC acknowledges financial support by the NASA contract 11-ADAP11-0218. AP thanks Andy Goulding for useful discussion and suggestions. ME and GF thank the Aspen Center for Physics and the NSF Grant #1066293 for hospitality during the completion of this paper. This research has made use of software provided by the Chandra X-ray Center (CXC) in the application packages CIAO, ChIPS, and Sherpa. This research is based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere, Chile, program 179.A-2005 (UltraVISTA survey). (2) number of stacked sources; (3) average rest frame L K in the bin (with corresponding standard deviation); (4) average redshift in the bin (with corresponding standard deviation); (5) average stellar age in the bin (with corresponding standard deviation); (6) average stellar mass in the bin (with corresponding standard deviation); (7) signal to noise ration of the stacked signal; (8) average 0.3-8 Xray luminosity after LMXB subtraction estimated from a thermal model. The first uncertainty reported in parenthesis is due to standard deviation of counts, stellar age and stellar mass in the bin, while the second one takes into account the additional standard deviation of redshift.; (9) hardness ratio (with error due to standard deviation of counts, stellar age and stellar mass in the bin)\n(1) bin number (2) number of sources (3) < log(LK/L ) > (4) < z > (5) < log(age/yr) > (6) < log(M * /M ) > (7) S /N (8) < LX−LMXB, 0.3−8/(10 40 erg s −1 ) > (1) bin number (2) 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 12 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Table 1 (for the left panel) and Table 2 (for the right panel): red, X-ray over-luminous with respect to the local L X,gas − L K relation; yellow, X-ray under-luminous bins; grey and green bins following the local relation with L X−LMXB lower and higher than 10 41 erg s −1 , respectively. Triangles pointing down represent upper limits. with increasing temperatures from bottom to top 0.7, 1, 2 and 3 keV. The full lines represent simulated HR from powerlaw models with fixed slope Γ = 2 and increasing intrinsic absorption form bottom to top 0, 10 21 , 10 22 and 10 23 cm −2 . Right: Same as left panel, but for models comprising a power-law PL with fixed slope Γ = 2 and intrinsic absorption of 10 23 cm −2 , plus a thermal model with temperature of 1 keV (APEC). The relative normalization of the two models is, from bottom to top, 10 −3 , 10 −2 , 10 −1 , 1, 10, 10 2 and 10 3 , corresponding to a contribution of the power-law component to the total flux of ∼ 0.2%, ∼ 2%, ∼ 15%, ∼ 62%, ∼ 94%, ∼ 99% and 100%, respectively. 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10"},{"title":"Figure 7","content":""},{"title":"Fig. 1 .Fig. 2 .","content":""},{"title":"Fig. 3 .","content":""},{"title":"FigFig. 5 .Fig. 6 .","content":""},{"title":"Fig. 7 .","content":""},{"title":"Table 1 :","content":""},{"title":"Table 2 :","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Diffuse X-ray emission from normal early type galaxies (ETG) was first detected in the Virgo cluster with the Einstein X-ray Observatory (Forman et al. 1979). Although in a few cases (e.g., M86) the displacement of the X-ray emission from the stellar body of the galaxy pointed to hot gaseous halos, the ∼ arcminute resolution of the Einstein IPC data could not disentangle observationally hot halos from the populations of low-mass X-ray binaries (LMXBs), which was likely to dominate the X-ray emission in a large number of the observed ETGs (Trinchieri & Fabbiano 1985). Nonetheless, several authors discussed how the X-ray emission and the inferred hot gas contribution could be used to constrain both the binding mass of the galaxies and the physical evolution of the hot halos, including feedback from SNIa and AGNs, and interaction with intracluster and group media (see e.g., Fabbiano 1989;Canizares et al. 1987;Ciotti et al. 1991;David et al. 1991;White & Sarazin 1991;Mathews & Brighenti 2003;Ciotti & Ostriker 2007).\\nThanks to the high spatial resolution of Chandra and imaging/spectral capabilities of the ACIS detector (Garmire et al. 2003), it has become possible to directly detect and characterize the different contributions to the X-ray emission of ETGs in the near universe. With Chandra, LMXB populations -both associated with Globular Clusters (GC) and in the stellar field of the ETG -have been studied to distances of several tens of megaparsecs (e.g., Sarazin et al. 2000;Kundu et al. 2002; see review by Fabbiano 2006). The integrated LMXB luminosity correlates well with the integrated ETG stellar luminosity (e.g., Kim & Fabbiano 2004;Gilfanov 2004). The scatter in this relation is a factor of 2-3, and it depends on both GC content (Kim & Fabbiano 2004), and the age of the stellar population (Kim et al. 2009;Kim & Fabbiano 2010;Boroson et al. 2011;Lehmer et al. 2014). These detailed high resolution studies allowed Boroson et al. (2011, BKF11) to accurately estimate the gaseous contribution to the X-ray luminosity L X,gas of a sample of 30 ETGs within 32 Mpc. After subtracting LMXB, stellar, and nuclear contributions to the X-ray luminosity, BKF11 found steep scaling relations between L X,gas and both the stellar L K luminosity and the hot gas temperature T . Subsequent work by Kim & Fabbiano (2013, KF13;2015, KF15) suggests that for L X,gas > 10 40 erg s −1 the hot gas around ETGs is virialized, giving a new way to measure the total mass of these galaxies.\\nBeyond the local universe, Chandra Deep Field studies (both detections and stacking of almost 1,000 ETGs) found that the mean L X of L B = 10 10 L ETGs remains equal or mildly increases over z = 0 − 0.7 (Lehmer et al. 2007;Tzanavaris & Georgantopoulos 2008) and that the ratio of the soft X-ray luminosity to the B-band luminosity evolves mildly as L (0.5−2 keV) /L B ∝ (1 + z) 1.1±0.7 since z ≈ 1.2 (Danielson et al. 2012), suggesting some heating mechanism preventing the hot gas from cooling, consistent with mechanical heating from radio AGNs. The evolution of the X-ray binary contribution to the X-ray emission (both of LMXBs and high mass binaries) across cosmic time, modeled by Fragos et al. (2013a), led Fragos et al. (2013b) to conclude that energy feedback from X-ray binaries is also important in the early universe.\\nThe COSMOS survey, covering ∼ 2 square degrees of sky with multi-wavelength coverage, which includes Chandra X-ray coverage (Scoville et al. 2007;Elvis et al. 2009), has increased the available sample of ETGs to over 6,600 galaxies Civano et al. (2014, C14). This survey provides a good z coverage out to z ∼ 2.5 (and a few galaxies are detected out to z ∼ 5). C14 studied the sample of 69 X-ray detected ETGs in the Chandra-COSMOS survey (C-COSMOS, Elvis et al. 2009), covering a redshift range 0 < z < 1.5, to explore redshift dependencies in the BKF11, KF13 scaling relations. In particular, C14 focused on the relation between L X,gas and the K-band stellar luminosity (L K ) of ETGs which is a good proxy of stellar mass for old stellar systems. The latter has been evaluated in the local Universe by BKF11 as L X,gas ∝ L K 2.6±0.4 for sources with L X,gas ≤ 2 × 10 41 erg s −1 , while including sources with L X,gas up to 9 × 10 42 erg s −1 KF13 found a steeper L X,gas ∝ L K 4.5±0.8 relation. C14 found that this latter relation also holds for X-ray detected ETGs with L X,gas ≤ ×10 42 erg s −1 , suggesting that the hot gas in these more distant galaxies may be virialized as well. However, more distant (z > 0.9) and younger stellar age galaxies generally fall outside the relation observed in the local sample, and are X-ray overluminous with respect to their L K . For these sources C14 suggested that the higher X-ray luminosity may be due to the presence of hidden AGNs (as suggested by observed hardness ratios), merging phenomena, which are expected to increase the halo X-ray luminosity, or nuclear accretion (that could also induce AGN activity during the merging phase, Cox et al. 2006).\\nIn this work we perform a stacking analysis of the C-COSMOS ETG data in order to investigate the nature of fainter, X-ray undetected ETGs. We compare their properties with those of the sample of local ETGs investigated by BKF11 and KF13, and with the sample of ETGs detected in the C-COSMOS survey studied by C14. This stacking study also allow us to further explore redshift dependencies. Our aim is to explore the properties of the non-stellar Xray emission, i.e. the X-ray emission resulting from the subtraction of the expected X-ray binary contribution (taking into account the cosmic evolution of the X-ray binary population Fragos et al. 2013b). The resulting emission will probe both hot X-ray halos and the possible emission of active nuclei in these galaxies. In particular, the large number of low-stellar-mass ETGs in the sample will provide a unique probe of the existence of relatively low-mass massive black holes in their nuclei. These black holes should produce low-luminosity nuclear X-ray sources, if they are fueled by the ETG stellar outgassing (Volonteri et al. 2011).\\nThe paper is organized as follows: Section 2 describes the sample selection and the data sets used in this work, Section 3.3 is dedicated to the description of the adopted stacking procedure, in Section 4 we discuss our results, and Section 5 is dedicated to our conclusions.In the following, we adopt the standard flat cosmology with Ω Λ = 0.73 and H 0 = 70 km s −1 Mpc −1 .\"\n },\n {\n \"title\": \"Sample Selection\",\n \"content\": \"The COSMOS survey (Scoville et al. 2007) provides a multi-wavelength characterization of over 1.5 million galaxies at redshift extending out to 5 in 2 deg 2 of the sky. The Chandra X-ray coverage (C-COSMOS Elvis et al. 2009) over 0.9 deg 2 , gives us the means to study X-ray emission of ETGs, extending local studies to higher redshifts, and explore the redshift evolution of hot halos with large samples of galaxies.\\nOur ETG sample is selected from the most recent version of the COSMOS photometric catalog by Ilbert et al. (2009). The spectral energy distribution (SED) identification performed by Ilbert et al. (2009) makes use of 7 elliptical galaxy templates with ages from 2 to 13 Gyr (Polletta et al. 2007). The SED templates were generated with the stellar population synthesis package developed by Bruzual & Charlot (2003), assuming an initial mass function from Chabrier (2003) and an exponentially declining star formation history. We also collected additional galaxy properties (stellar mass, age and star formation rate) from Ilbert et al. (2010), and we made use of the most recent near-infrared photometry from the Ultra Deep Survey with the VISTA telescope (Ultra-VISTA McCracken et al. 2012) 1 .\\nWe selected galaxies classified as E or S0 and that have a reliable photometric redshift (Ilbert et al. 2009). In addition we selected only sources with Ultra-VISTA detection both in J and K bands, since these are needed to evaluate the rest frame L K (see below). This narrows our sample to 10,808 (the COSMOS ETG sample considered in C14). We then selected only the 6436 ETGs falling in the C-COSMOS field, and then excluded the 48 X-ray detected ETGs in common with the C-COSMOS identification catalog (Civano et al. 2012). This leads us to our final sample of 6388 C-COSMOS ETGs -not detected in X-rays -which constitutes the starting point of the stacking analysis described in this paper. Note that we selected sources with photometric redshift estimate in the catalog by Ilbert et al. (2009), while C14 selected their sources to have a spectroscopic or photometric redshift estimates in the C-COSMOS identification catalog, so the sample of ETGs selected by the latter authors (∼ 6600 sources) is not fully contained in our sample.\\nFollowing C14 we evaluated the K band luminosity of our sources from the Ultra-VISTA K-band aperture magnitude of the COSMOS photometric catalog. To calculate the rest frame K band luminosities we assumed a power law spectrum (as appropriate over this small frequency range) f ν ∝ ν −α with α = (J − K)/2.5 log (ν J /ν K ) (where J and K are the AB magnitudes from the COSMOS photometric catalog and ν J and ν K are the corresponding frequencies), and express the rest frame luminosities in solar luminosity as L K = 10 −(K+ext−5.19)/2.5 × (1 + z) α−1 × (D L /10) 2 , where z is the photometric redshift from the COSMOS catalog, D L is the luminosity distance expressed in parsecs, and ext is the aperture correction parameter computed from the FWHM measured for all the COSMOS extended sources in the Hubble ACS data.\\nWe compared the distributions of the physical parameters of the 6388 C-COSMOS ETGs with the entire 10,808 1 Note that stellar mass reported in COSMOS photometric catalog, and obtained from the Bruzual & Charlot (2003) best-fit templates, do not incorporate the Ultra-VISTA K-band NIR data. Although this is only an issue if there is any residual star formation in the ETGs, L K and M * may not necessarily be completely equivalent.\\nCOSMOS ETG sample finding no significant difference, demonstrating as expected that the sample of ETGs not detected in X-ray is representative of COSMOS ETGs overall. Note that the luminosity threshold is a function of redshift because of the uniform magnitude limit of the Ultra-VISTA COSMOS survey (K s ∼ 24, McCracken et al. 2012). The specific star formation rate for the majority of the ETGs is in the regime where Ilbert et al. (2010) defines galaxies as quiescent, although higher star formation rate is observed in young ETGs. The properties (luminosity, mass, star formation rate and age) of the COSMOS ETGs are in agreement with those reported by (Moresco et al. 2013) for a sample of ETGs selected using multiple criteria (including color-color diagrams, spectra, star formation rate and SED classification). Masses and luminosities are in agreement with those of the K-selected elliptical galaxies of Jones et al. (2014), extracted from a small area of the COSMOS field.\"\n },\n {\n \"title\": \"Analysis and Results\",\n \"content\": \"Below we describe the procedures followed in our stacking analysis. The results are summarized in Tables 1 and 2.\"\n },\n {\n \"title\": \"Count Extraction at ETG Positions\",\n \"content\": \"We used all 49 C-COSMOS observations to evaluate source and background counts for our sample of ETGs, excluding data in the ACIS-I chip gaps. We also excluded data from X-ray detected sources removing counts within a 10\\\" radius of all sources in the C-COSMOS source identification catalog (Civano et al. 2012). We replaced the excluded counts by sampling the Poisson distribution of the pixel values nearby background regions with dmfilth CIAO task 2 . When a source lay in the field of view of more than one observation, we combined the data from the different observations using the prescriptions from Puccetti et al. (2009), excluding observations where the source was more than 10 acrmin off-axis from the aim point.\\nDue to the variation of the Chandra Point Spread Function (PSF) radius R PS F 3 , we used an extraction radius r ext that was determined by the source position in each observation in order to extract an optimal fraction of X-ray photons from the diffuse halo. For sources with morphological information available in the HST-ACS catalog (Leauthaud et al. 2007) we adopted the source semi-major axes R as a measure of its physical extent. For sources with R > R PS F we adopted values of r ext that was the convolution of R and R PS F (R 2 + R 2 PS F ); otherwise we adopted r ext = R PS F as we did for sources without morphological information in HST-ACS catalog. We notice however that sources with R > R PS F represent only ∼ 1% of our sample, with extraction radii typically ranging between 1\\\" and 7\\\". The average exposure time for each source within the extraction circle, E, was then evaluated. The histogram of extracted ETG position counts is shown in Figure 1 (blue line). The small fraction of extraction regions with counts above ∼ 5 represent off-axis sources with large extraction radii.\"\n },\n {\n \"title\": \"Background Evaluation\",\n \"content\": \"To evaluate the background we extracted counts from an annulus centered on the source coordinates, with innerradius 2r ext and outer radius 4r ext . Due to the very low signal, the background represents the largest source of uncertainty in this kind of analysis. To estimate the systematic uncertainties introduced by the evaluation of the background, we compared the results of three different methods, obtaining consistent results. First we simply evaluated the background contribution to the total source counts from the mean count value of the pixels in the whole background annulus. Second, we adopted the clipping procedure of Willott (2011, method b), which evaluates the mean count value µ -and the standard deviation σ -of all pixels within a circle of radius 4r ext around each source position and then sets all pixels with count value larger than µ + 3σ to µ. This method suppresses the noise in the background annulus, but also removes signal from the source 4 . Lastly, we divided the background annulus into eight 45 degree sectors, and excluded all the sectors in which the mean count value was larger than µ + 3σ, thus excluding regions of higher background without removing signal from the source. The net counts N obtained with all three methods are consistent within statistics, so in the following we adopted the clipping procedure of Willott (2011, method b).\\nTo test the robustness of our results we also extracted counts from positions on the sky offset from the ETG coordinates by random distances between 15\\\" and 30\\\" both in right ascension and declination. The net counts of the non-source positions are shown in Figure 1. They follow an approximatively Gaussian distribution centered on zero, as expected from background fluctuations, while the ETG counts show an excess of positive signal. To test if our results were dominated by a sub-population of ETGs we repeated the stacking procedure on sub-samples randomly drawn from the complete sample, progressively halving the subsample, obtaining results compatible with the full sample.\"\n },\n {\n \"title\": \"Stacking Procedure\",\n \"content\": \"To evaluate the count rate C for each stacked bin we stacked the observed source net counts in 0.5-7 keV range N and exposure E to obtain C = i N i / i E i . From these counts we subtracted the expected contributions from stellar sources (LMXBs and Ultra Luminous X-ray sources, ULXs), and evaluated the average bin rest-frame X-ray luminosity as explained below. We corrected L K to take into account the evolution of the stellar population (i.e., the fading effect, see C14). Note that, at a given redshift, younger galaxies have larger fading correction than older ones. To compare our results with those of KF13 and C14 we considered sources with 9 < log (L K ) < 13.\\nWe constructed the stacks by binning in both L K and z, to ensure that the binning scheme did not produce any significant bias on the results. For the L K binning, we first divided our sample in four log (L K ) bins, 9 − 10, 10 − 11, 11 − 12 and 12 − 13. We then ordered the sources in each bin in increasing redshift, and progressively stacked them until we reached a minimum signal to noise ratio of 3; if this 3σ threshold could not be reached we stacked the signal from all the remaining sources in the bin. Examples of 0.5-7 keV images for these stacking bins are presented in Figure 2. For the z binning, we followed the same procedure, dividing the sample into three bins 0 − 0.5, 0.5 − 1 and 1 − 1.5, and then ordered the sources in each bin in increasing L K .\"\n },\n {\n \"title\": \"Evaluation of Non-stellar L X for Each Stacked Bin\",\n \"content\": \"To subtract the LMXB contribution from the total X-ray emission, we evaluated the average LMXB contribution to each bin using the 0.3-8 keV relation from Fragos et al. (2013b), as in C14:\\nwhere L X is expressed in erg s −1 , M * is the average stellar mass of the stacking bin expressed in solar masses and age is the average stellar age of the stacking bin expressed in Gyr. 5 We also evaluated the contribution to X-ray luminosity from ULXs that could be present in the galaxies using the scaling relations from Calzetti et al. (2007) and Mineo et al. (2012) between the ULX luminosity and the galaxy star formation rate. ULX contribution is more than an order of magnitude smaller than L X−LMXB even for the least luminous L K bins.\\nWe converted the average 0.3 − 8 keV LMXB luminosities in each bin to count rates in the 0.5 − 7 keV interval using a power law model with index Γ = 1.8 (consistent with the typical spectrum of LMXBs) and Galactic absorption N H = 2.6 × 10 20 cm −2 . For these parameters the conversion factor used from Chandra count rates to fluxes is 1.04 × 10 −11 erg cm −2 s −1 cts −16 . We then subtracted these count rates from the stacked signals and converted back the resulting count rates into luminosities using a model comprising Galactic N H plus a thermal APEC component with temperature depending on the source total X-ray luminosity. We adopted kT = 0.7, 1 and 2 keV for X-ray luminosities of L X < 10 41 , 10 41 < L X < 10 42 and L X > 10 42 erg s −1 , respectively, following BKF11 and Dai et al. (2007). Finally, we K-corrected the luminosity to the 0.3 − 8 keV rest frame band. These LMXB-subtracted luminosities, that we indicate as L X−LMXB , are expected to include the emission from the hot gaseous halos, but may also include a possible contribution from any AGN harbored within these ETGs. Figure 3 shows the results in the L X−LMXB − L K plane for both L K and z binning (Left and Right panels, respectively), where on the upper axes we plot M * as taken from COSMOS catalog. In both binning schemes the overall scatter plots are consistent, showing that we do not suffer from strong binning biases. By construction, the L K -first binning provides information on the extreme redshift bins (both high and low), while the z-first binning covers more fully the L K range. The bins are numbered in Figure 3, following Tables 1 and 2. We use different colors to visually divide the stacking bins into four groups: red, for bins X-ray over-luminous with respect to the local L X−LMXB and L K relation (BFK11, KF13, see C14); yellow, for X-ray under-luminous bins; and grey and green for bins following the local relation with L X−LMXB lower and higher than 10 41 erg s −1 , respectively.\\nAs noted in Section 1, Eq. 1 has an observed spread of a factor 2-3, which can be related to both the presence of GC LMXB in the ETGs and to the spread in age of the stellar population (Kim & Fabbiano 2004BKF11;Lehmer et al. 2014). Given the parameter space of Figure 3, this spread will not change our conclusions.\"\n },\n {\n \"title\": \"Hardness Ratios\",\n \"content\": \"Hardness ratios (HRs) were evaluated to characterize the ETG average spectral properties. HR = (H −S )/(H +S ), where S is the soft 0.5 − 2 keV counts and H the hard 2 − 7 keV counts. These HRs are those of the LMXB-subtracted emission. The contributions of these sources were evaluated and subtracted by applying the procedure in Section 3.4 to each energy band. We evaluated the errors on HRs as due to both uncertainties on counts and to uncertainties on estimate of LMXB contribution due to spread of stellar age and mass in the bins.\\nIn Figure 4 we show the HRs for the stacking bins using the same color code as in Figure 3. We plot HR versus z in the Left panel, using the L K -first binning, which gives a better z sampling. We plot HR versus L K in the Right panel, using the z-first binning, to explore the L K behavior. The left panel of Figure 4 shows a distinct clustering in HR-z space of bins from separate loci of the L X−LMXB − L K plane, as shown by the clusters of different color points. The Right panel shows that ETGs with larger stellar mass (L K ) tend to have lower HR. Given the scaling relations of ETGs (BKF11; KF13; KF15), this could just reflect a relation with total binding mass, with more massive galaxies being more efficient at retaining large hot gaseous halos.\"\n },\n {\n \"title\": \"Discussion\",\n \"content\": \"Figure 5 summarizes our results in the L X−LMXB − L K plane and compares them with the COSMOS ETG X-ray detections (C14) and with the near universe ETG data (BKF11, KF13). We show the local KF13 relation with a solid line, and the scatter in this relation with the two dashed lines. These lines are chosen to pass through the points for the most L X,gas luminous source (M87) and the most L K luminous source (NGC1316), respectively. The COSMOS stacked-ETG results are overall consistent with the C14 COSMOS detections, but significantly increase the coverage at lower total stellar luminosities / masses. While most of the detected signals are compatible with the local KF13 relation, there are several detections that lie both to the left and to the right side of it, that is, bins that appear to be respectively over and under-luminous in X-ray with respect to their L K or M * .\\nIn Figure 5 we also show the hardness ratios for each stacked bin with lines with slopes proportional to HR. We find positive HRs (i.e., harder spectra) in stacked bins outside the dashed line on the left. These are the lower L K galaxies with significant X-ray excess (the red points in Figure 4, see also C14). We also find hard HR > 0 in some points with L X−LMXB > 10 41 erg s −1 and z > 1 (the three top green points in Figure 4), to the left of the solid line.\\nInstead, lower HR ratios (i.e., softer spectra) are found in bins following the local relation (between the two dashed lines). This is especially so for those to the right of the solid line. This suggests that they have a larger than average hot gas content up to z ∼ 1. Figure 6 compares the HR-z scatter diagram with a set of spectral models. On the Left, we show the expected HRs for thermal APEC (dashed lines; from bottom to top, kT = 0.7, 1, 2 and 3 keV) and power-law models (solid lines; fixed slope Γ = 2 and increasing intrinsic absorption from bottom to top, N H = 0, 10 21 , 10 22 and 10 23 cm −2 ). The APEC models are the expected emission of thermal gaseous emission, while the power-law models are typical AGN spectra. To illustrate cases of mixed emission from a hot gas component and a heavily obscured AGN, on the right panel of Figure 6 we plot composite models consisting of kT = 1 keV APEC emission with line of sight N H plus power-law models with slope Γ = 2 and intrinsic absorption N H = 10 23 cm −2 . From the bottom to the top the relative normalization of the two models varies, to reflect an increasing contribution of the power-law model to the total flux, ranging from ∼ 0.2% to ∼ 100%. The X-ray over-luminous bins (red points) have generally high HRs consistent with being dominated by AGN emission, either with \\\"normal\\\" AGNs with absorbing columns 10 22 cm −2 , or in the case of the composite gas + AGN emission even highly obscured Compton Thick AGNs (N H > 10 24 cm −2 , Risaliti et al. 1999;Levenson et al. 2006;Lehmer et al. 2010).\\nThe bins following the local relation with L X−LMXB < 10 41 erg s −1 (grey points) show HR ∼ (−0.5) − 0.3, lower than those of the over-luminous X-ray bins. A soft HR first suggests emission from hot gas. However, Figure 6 shows that pure APEC thermal models with solar metallicity always have HR < −0.6. Hence these objects have HR values inconsistent with pure gaseous emission.\\nThese HRs can instead be explained with combinations of thermal and AGN emission. Figure 6 Right shows models with varied ratios of APEC and power-law emission. In this model the observed HRs imply AGN components accounting for 15% − 60% of the X-ray luminosity. Given the X-ray luminosities of this sub-sample (Figures 3, 5), these ratios then imply AGN luminosities ranging from ∼ 5 × 10 39 erg s −1 to ∼ 5 × 10 40 erg s −1 . This range is in agreement with the low-luminosity Chandra X-ray sources in the near universe sample (BKF11, Pellegrini 2005), which have luminosities ranging from ∼ 10 38 erg s −1 to ∼ 5 × 10 41 erg s −1 .\\nThe bins with L X−LMXB > 10 41 erg s −1 (green points) have hard HR > 0 incompatible with a low absorption power-law model (e.g., bins 21, 22 and 25). The three highest redshift bins in this group (e.g., bins 28, 29 and 30) have even harder HR > 0.5 that are compatible only with absorbing columns larger than 10 23 cm −2 . This suggests that these galaxies harbor (on average) highly obscured Compton Thick AGNs that dominate their X-ray luminosity.\\nIn summary, our stacking analysis suggests that on average AGN emission is likely to be present in ETGs. These AGNs may be faint and hidden in hot gas dominated ETGs, which are consistent with the near Universe scaling relation of virialized halos (KF13, KF15, see C14). In contrast, AGN emission appears to dominate the non-stellar X-ray emission in two classes of ETGs: (1) low-stellar mass ETGs (log (L K /L ) 10.5), and (2) higher redshift ETGs (z > 0.5, > 1) especially those overluminous in X-rays relative to the near universe L X−LMXB − L K scaling relation. The latter group has HR values suggesting highly absorbed Compton Thick AGNs. We note that these AGNs were not visible in the COSMOS colors and spectra, since AGNs were excluded in our ETG sample selection process (C14). (2007), obtaining black hole masses ∼ 0.5 dex larger than the former. Uncertainties in these estimate are typically of a factor of ∼ 2 − 3 (Tremaine et al. 2002;Graham 2007) and evolution to z ∼ 1 − 2 may provide a systematic black hole mass increase of a similar factor (Merloni et al. 2010). These effects will not change our conclusions in the 3-dex wide range covered by Figure 7.\"\n },\n {\n \"title\": \"Accretion Regimes for the AGNs\",\n \"content\": \"On the right hand axes we plot L bol = 10 L X−LMXB , an appropriate bolometric correction for low-luminosity AGNs that radiate at a few 10s % of L Edd (Kollmeier et al. 2006;Steinhardt & Elvis 2010). Given this nominal L bol we can plot diagonal lines of constant L bol /L Edd . These lines range from 10 −2 to 10 −6 L bol /L Edd .\\nWe see that the massive high L X−LMXB galaxies (green points) and the low-mass X-ray overluminous (red points) galaxies have implied Eddington ratios in the 10 −3 to 10 −5 range. RIAFs are generally believed to apply to all black hole accretion below 10 −2 L/L Edd , so we would expect all these sources to be RIAFs. If the bolometric correction for RIAFs is smaller than that considered here, as may well be the case (Narayan & Yi 1995), the L bol values on the y-axis should be correspondingly lower (as the Eddington ratios on the currently plotted dashed lines).\\nAlternatively the true L bol of the three high z bins of massive high L X−LMXB galaxies (green points 28, 29, 30) could be greatly underestimated. The HRs of these bins also imply highly absorbed emission (N H > 10 23 cm −2 , Figure 6). If these are then Compton Thick AGN they may have much larger intrinsic luminosities, with factors of 100 being quite plausible (e.g., Risaliti et al. 1999;Levenson et al. 2006). They would have intrinsic luminosities L 0.5−10 keV ∼ 10 45 erg s −1 and L bol /L Edd in the ∼ 10 −2 range, so that they could lie within the normal AGN regime.\\nTo investigate this possibility, we built SEDs of ETGs in our sample using optical-infrared magnitudes reported in COSMOS photometric catalog, since a \\\"hidden\\\" AGN may give rise to an IR excess with respect to an average ETG SED. We then fitted these SEDs with the same elliptical galaxy template used by Ilbert et al. (2009) without finding any significant deviation. To confirm this, we evaluated the L 2−10−LMXB luminosity of the stacking bins assuming a power-law spectrum (as appropriate for AGN spectra) and compared that with their IR luminosity. In all cases the Xray luminosities are at least 3 order of magnitudes lower than the IR luminosities, even assuming absorbing columns ∼ 10 23 cm −2 , indicating that at IR wavelength a putative AGN component is expected to be significantly smaller than the galactic emission.\\nThe galaxies that are consistent with the local L K − L X−LMXB gaseous relation (grey points), with X-ray colors suggesting thermal emission plus a possible AGN component, have smaller Eddington ratios ∼ 10 −5 to 10 −6 (these ratios of course assume that the entire emission is due to an AGN; as shown in Figure 6, the AGN component may range from ∼ 15 − 60%).\\nWe also plot in Figure 7 the predictions from Volonteri et al. (2011) of L X,AGN for three values of M * for both efficient and radiatively inefficient accretion flows (RIAFs Narayan & Yi 1995). These authors' models use the stellar density profile scaling with stellar mass, and predict the accretion rate from the normal stellar mass loss due to stellar evolution, and the consequent massive black hole luminosity for various radiative efficiencies 7 . The X-ray luminosity coming from accretion is computed as L X,AGN /L Edd = η X ηṁ, where η X = 0.1 is the X-ray fraction of the bolometric luminosity, andṁ = 0.1Ṁ/Ṁ Edd is the accretion rate in Eddington units with a 10% accretion efficiency. Radiatively efficient accretion flows have η = 1, while inefficient ones have η =ṁ/ṁ cr , and for themṁ <ṁ cr = 3 × 10 −2 .\\nIn order to compare our results with Volonteri et al. (2011) models we converted the 0.5−7 keV LMXB subtracted counts into 0.5 − 10 keV band luminosities, L 0.5−10 keV assuming a pow-law spectrum as appropriate for AGN spectra. Based on these models, inefficient accretion into the nuclei could explain even all the L X−LMXB of the massive, high L X−LMXB ETGs (green points), except for the most luminous ones, which are likely to be Compton thick AGNs (see above). The low-mass X-ray overluminous ETGs (red points) are also consistent with the Volonteri et al. model. Considering that a large fraction of the X-ray emission is due to hot gas (see above), the grey points show instead a possible AGN emission even lower than predicted for radiatively inefficient accretion. In these ETGs, the L X−LMXB is consistent with the local L X,gas − L K relation. This result can have the following possible explanations, that are not mutually exclusive.\\n1. The Volonteri et al.'s models assume that the whole of their estimated mass accretion rate (coming from the instantaneous mass loss of all stars, and obtained by integrating over the whole stellar density profile) indeed reaches the massive black hole, while this represents an upper limit on the mass accretion rate of the massive black hole (some gas may be outflowing, or may reach the massive black hole with a delay, or may possess angular momentum and form a disk, etc.).\\n2. The accreting mass, that has already entered the accretion radius, is reduced on its way to the massive black hole, due to a wind from a RIAF (Narayan & Yi 1995;Blandford & Begelman 1999); similarly, close to the nucleus, there may be a heating mechanism for the accreting gas, as a nuclear jet, again impeding or lowering the accretion rate (e.g. Pellegrini et al. 2012;Yuan & Li 2011).\\n3. Accretion onto the black hole is (temporarily) not taking place, as may be the case during feedback-regulated activity cycles. Numerical simulations show that, during these cycles, AGN outbursts are followed by major degassing of the circumnuclear region, with a precipitous drop of the nuclear accretion rate; the outbursts are 7 We note that in Volonteri et al. (2011) models assume for ETGs a stellar age of 12 Gyr. We corrected for the average age 10 9.5 Gyr of C-COSMOS ETGs, which yields slightly smaller emitted X-ray luminosities (see Eq. 9 in Volonteri et al. 2011 ). separated by long time intervals during which the galaxy is replenished again by gas from the stellar mass losses, until a new nuclear outburst takes place (Ciotti et al. 2010).\\n4. η X might be lower than 1 as assumed here (see also Volonteri et al. 2011), or that η at lowṁ decreases faster than adopted in the Volonteri et al.'s models, so that lower L X,AGN should be predicted (e.g., Yuan & Narayan 2014).\\nOur results extend to higher X-ray luminosities a recent analysis of ETGs in the AMUSE survey of Virgo and field galaxies with Chandra (Miller et al. 2015). AMUSE results indicate a significant presence of supermassive black holes in low stellar mass galaxies. Due to the lower redshifts of the AMUSE ETGs, that survey is sensitive to X-ray luminosities (∼ 1.3×10 38 erg/s) one order of magnitude fainter are than our faintest bin 1 (∼ 2.4×10 39 erg/s). For these dim sources the stellar wind accretion proposed by Volonteri et al. (2011) would predict higher X-ray luminosities than those observed, Miller et al. conclude that these black holes are probably powered by inefficient advection dominated accretion flows or, alternatively, by an outflow/jet component. We found that low-luminosity, X-ray \\\"overluminous\\\" ETGs can be adequately described by the RIAF fueled by stellar winds as proposed by Volonteri et al. (2011). Instead, ETGs with L X−LMXB > 10 42 erg s −1 may host Compton Thick AGNs. Similarly, when comparing with the results of Lehmer et al. (2007) and Danielson et al. (2012) on Chandra Deep Fields, we extend to higher X-ray luminosities and to lower optical luminosities (that is, lower stellar masses), finding sources that significantly exceed the expected emission from hot-gas dominated ETGs, with log (L 0.5−2 keV /L B )/(erg/s/L ) > 30 (see Figure 7 in Danielson et al. 2012).\\nWe also note that, as reported by Comastri et al. (2015), the revision of black hole scaling relations by Kormendy & Ho (2013) indicate that the local mass density in black holes should be five times higher than previous estimates. These authors advance the possibility that a sizeable population of highly obscured AGNs with covering fraction of obscuring material ∼ 4π would explain the revised upward value of the SMBH local mass density without exceeding the limits imposed by the X-ray and IR background. An example of these obscured AGNs can be that observed in the ETG ESO565-G019 with Suzaku and Swift/BAT (Gandhi et al. 2013).\\nFinally, we note that the hard X-ray slopes of the more luminous ETG bins could be due to contamination from X-ray jets. However, the combined 1.4GHz VLA COSMOS Large and Deep Project catalogs (Schinnerer et al. 2010) offer radio counterparts only to ∼ 2% of our sample of ETGs. The presence of radio emitters among non X-ray detected ETGs will be addressed after the release of the JVLA-COSMOS Project data.\"\n },\n {\n \"title\": \"Summary and Conclusions\",\n \"content\": \"We have followed up our previous work (C14) investigating the X-ray / K-band properties of the 69 ETGs detected in the Chandra COSMOS survey (Elvis et al. 2009), with a luminosity and hardness ratio stacking analysis of the Chandra data for the sample of 6388 early type galaxies (ETGs) not detected in X-rays. The stacking analysis allows the investigation of the average features of individually X-ray undetected ETGs. Our ETG sample was selected as representative of normal galaxies, with no evidence of AGN emission in their spectra and multi-band photometry (C14, Section 2). Our purpose was to investigate the properties of the X-ray emission from hot gas and from possible \\\"hidden\\\" AGN components that may be revealed by their X-ray emission. To this purpose, the expected contribution of LMXBs was subtracted from the X-ray luminosity of each stacked bin, using established scaling relations with the K-band luminosity (BKF11). All luminosities were corrected to the local rest frame; X-ray hardness ratios were based on the observed counts (after subtraction of the LMXB contribution), and were compared to emission models for the appropriate average bin redshift. This analysis has led to the following conclusions:\\n1. On average AGN emission may be present in all ETGs.\\n2. AGNs may be faint and hidden in hot gas dominated ETGs, which are consistent with the local universe scaling relation of virialized halos (KF13, KF15, see C14).\\n3. AGN emission appears to dominate the non-stellar X-ray emission in low-stellar mass ETGs (log (L K ) 10.5 in solar units), which show marked X-ray luminosity excesses relative to their K-band luminosity (a proxy of stellar mass), when compared with the local ETG L X,Gas L K relation of the local universe (BKF11). This result confirms the conclusions of C14 and extends them to lower stellar mass galaxies.\\n4. AGN emission is also prominent in higher redshift ETGs (z > 0.5, > 1) especially those with X-ray luminosity 10 42 erg s −1 . The latter group have HR values suggesting highly absorbed Compton Thick AGNs. We note that these AGNs were not visible in the COSMOS colors and spectra, since AGNs were excluded by our ETG sample (C14). 5. Given the nuclear BH mass predicted by the M BH − L K relation (Kormendy & Ho 2013), the hidden AGNs in the X-ray over-luminous low stellar mass ETGs are consistent with the presence of radiatively inefficient accretion fueled by stellar outgassing (Volonteri et al. 2011). The highly absorbed higher L X−LMXB AGNs hidden in massive, higher z ETGs instead suggest efficient accretion in highly obscured nuclei.\\nThis work was partially supported by NASA contract NAS8-03060 (CXC), and NASA Chandra grant G01-12110X. SP acknowledges financial support from MIUR grant PRIN 2010-2011, project 'The Chemical and Dynamical Evolution of the Milky Way and Local Group Galaxies', prot. 2010LY5N2T. FC acknowledges financial support by the NASA contract 11-ADAP11-0218. AP thanks Andy Goulding for useful discussion and suggestions. ME and GF thank the Aspen Center for Physics and the NSF Grant #1066293 for hospitality during the completion of this paper. This research has made use of software provided by the Chandra X-ray Center (CXC) in the application packages CIAO, ChIPS, and Sherpa. This research is based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere, Chile, program 179.A-2005 (UltraVISTA survey). (2) number of stacked sources; (3) average rest frame L K in the bin (with corresponding standard deviation); (4) average redshift in the bin (with corresponding standard deviation); (5) average stellar age in the bin (with corresponding standard deviation); (6) average stellar mass in the bin (with corresponding standard deviation); (7) signal to noise ration of the stacked signal; (8) average 0.3-8 Xray luminosity after LMXB subtraction estimated from a thermal model. The first uncertainty reported in parenthesis is due to standard deviation of counts, stellar age and stellar mass in the bin, while the second one takes into account the additional standard deviation of redshift.; (9) hardness ratio (with error due to standard deviation of counts, stellar age and stellar mass in the bin)\\n(1) bin number (2) number of sources (3) < log(LK/L ) > (4) < z > (5) < log(age/yr) > (6) < log(M * /M ) > (7) S /N (8) < LX−LMXB, 0.3−8/(10 40 erg s −1 ) > (1) bin number (2) 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 12 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Table 1 (for the left panel) and Table 2 (for the right panel): red, X-ray over-luminous with respect to the local L X,gas − L K relation; yellow, X-ray under-luminous bins; grey and green bins following the local relation with L X−LMXB lower and higher than 10 41 erg s −1 , respectively. Triangles pointing down represent upper limits. with increasing temperatures from bottom to top 0.7, 1, 2 and 3 keV. The full lines represent simulated HR from powerlaw models with fixed slope Γ = 2 and increasing intrinsic absorption form bottom to top 0, 10 21 , 10 22 and 10 23 cm −2 . Right: Same as left panel, but for models comprising a power-law PL with fixed slope Γ = 2 and intrinsic absorption of 10 23 cm −2 , plus a thermal model with temperature of 1 keV (APEC). The relative normalization of the two models is, from bottom to top, 10 −3 , 10 −2 , 10 −1 , 1, 10, 10 2 and 10 3 , corresponding to a contribution of the power-law component to the total flux of ∼ 0.2%, ∼ 2%, ∼ 15%, ∼ 62%, ∼ 94%, ∼ 99% and 100%, respectively. 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10\"\n },\n {\n \"title\": \"Figure 7\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 1 .Fig. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FigFig. 5 .Fig. 6 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 7 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 1 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 2 :\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"shows the results of the L K -first stacking, with L X−LMXB plotted against stellar mass M * and the mass of the nuclear black hole, M BH , obtained from the M BH − M * relation of Kormendy & Ho (2013). Evaluating M BH from the M BH − M * relation McConnell & Ma (2013) would yield black hole masses between 0.1 and 0.3 dex smaller than the former. Similarly, one could derive M BH from the M BH − L K relation from Graham","content":"Figure 7\n7shows the results of the L K -first stacking, with L X−LMXB plotted against stellar mass M * and the mass of the nuclear black hole, M BH , obtained from the M BH − M * relation of Kormendy & Ho (2013). Evaluating M BH from the M BH − M * relation McConnell & Ma (2013) would yield black hole masses between 0.1 and 0.3 dex smaller than the former. Similarly, one could derive M BH from the M BH − L K relation from Graham"},{"id":"","caption":"-Distribution of net counts for randomly distributed extraction regions (red line) and for the selected sample of sources (blue line). -Examples of 0.5-7 keV images of the L K selected stacking bins.","content":"Fig. 1 .Fig. 2 .\n12-Distribution of net counts for randomly distributed extraction regions (red line) and for the selected sample of sources (blue line). -Examples of 0.5-7 keV images of the L K selected stacking bins."},{"id":"","caption":"number of sources (3) < log(LK/L ) > (4) < z > (5) < log(age/yr) > (6) < log(M * /M ) > (7) S /N (8) < LX−LMXB, 0.3−8/(10 40 erg s −1 ) >","content":"number of sources (3) < log(LK/L ) > (4) < z > (5) < log(age/yr) > (6) < log(M * /M ) > (7) S /N (8) < LX−LMXB, 0.3−8/(10 40 erg s −1 ) >"},{"id":"","caption":"-X-ray luminosity (after LMXBs subtraction) versus K-band luminosity for the stacking bins selected in L K (Left Panel) and redshift (Right Panel). The horizontal and vertical black error bars indicate the statistical uncertainties on L K and L X−LMXB , respectively, while the vertical green error bars show the uncertainty on L X−LMXB due to redshift spread in each bin. The stellar mass M * from COSMOS catalog is plotted on the top axis. The local relation by KF13 is represented with a full black line. The dashed lines are drawn to include the most L X,gas luminous source ETG (M87) and the most L K luminous ETG (NGC1316) in the Local Sample (BKF11, KF13), respectively. Bins are labelled as in","content":"Fig. 3 .\n3-X-ray luminosity (after LMXBs subtraction) versus K-band luminosity for the stacking bins selected in L K (Left Panel) and redshift (Right Panel). The horizontal and vertical black error bars indicate the statistical uncertainties on L K and L X−LMXB , respectively, while the vertical green error bars show the uncertainty on L X−LMXB due to redshift spread in each bin. The stellar mass M * from COSMOS catalog is plotted on the top axis. The local relation by KF13 is represented with a full black line. The dashed lines are drawn to include the most L X,gas luminous source ETG (M87) and the most L K luminous ETG (NGC1316) in the Local Sample (BKF11, KF13), respectively. Bins are labelled as in"},{"id":"","caption":". 4.-Left: HRs of the L K -first stacking bins versus redshift. Right: HRs of the z-first stacking bins versus L K . Triangles pointing up and down represent lower and upper limits on HR, respectively, while the box for bin 28 represents unconstrained HR. -Same as Figure 3, but with bins labelled according to their average redshift, and HRs indicated by the slope of the orange segments. Black error bars reflect statistical uncertainties, vertical green bars show the uncertainty on L X−LMXB due to redshift spread in each bin. Red boxes show the BKF11 and KF13 samples, blue circles show the C14 sample of C-COSMOS X-ray detected ETGs. Triangles pointing down represent upper limits. -Left: HR-z scatter diagram of L K -first bin. The vertical error bars indicate uncertainties on L X−LMXB due to counts and to LMXB subtraction. Triangles pointing up and down represent lower and upper limits on HR, respectively. Squares represent unconstrained HRs. Dashed lines show HRs expected from thermal models (APEC)","content":"FigFig. 5 .Fig. 6 .\n56. 4.-Left: HRs of the L K -first stacking bins versus redshift. Right: HRs of the z-first stacking bins versus L K . Triangles pointing up and down represent lower and upper limits on HR, respectively, while the box for bin 28 represents unconstrained HR. -Same as Figure 3, but with bins labelled according to their average redshift, and HRs indicated by the slope of the orange segments. Black error bars reflect statistical uncertainties, vertical green bars show the uncertainty on L X−LMXB due to redshift spread in each bin. Red boxes show the BKF11 and KF13 samples, blue circles show the C14 sample of C-COSMOS X-ray detected ETGs. Triangles pointing down represent upper limits. -Left: HR-z scatter diagram of L K -first bin. The vertical error bars indicate uncertainties on L X−LMXB due to counts and to LMXB subtraction. Triangles pointing up and down represent lower and upper limits on HR, respectively. Squares represent unconstrained HRs. Dashed lines show HRs expected from thermal models (APEC)"},{"id":"","caption":"-X-ray luminosity (after LMXBs subtraction) versus black hole mass M BH (evaluated from M * according to theKormendy & Ho (2013) relation) for the L K -first stacking bins. The stellar mass M * from COSMOS catalog is plotted on the top axis. Lines of constant L bol /L Edd , given M BH and L bol , are shown as diagonals (from bottom to top 10 −6 , 10 −5 , 10 −4 , 10 −3 and 10 −2 ). The bins are color-labeled as inFigure 3. Light blue boxes represent the predictions of the Volonteri et al. (2011) models for M * = 10 9 M , 10 10 M and 10 11 M , where the top of each box refers to radiatively efficient accretion flows and the bottom refers to RIAFs.","content":"Fig. 7 .\n7-X-ray luminosity (after LMXBs subtraction) versus black hole mass M BH (evaluated from M * according to theKormendy & Ho (2013) relation) for the L K -first stacking bins. The stellar mass M * from COSMOS catalog is plotted on the top axis. Lines of constant L bol /L Edd , given M BH and L bol , are shown as diagonals (from bottom to top 10 −6 , 10 −5 , 10 −4 , 10 −3 and 10 −2 ). The bins are color-labeled as inFigure 3. Light blue boxes represent the predictions of the Volonteri et al. (2011) models for M * = 10 9 M , 10 10 M and 10 11 M , where the top of each box refers to radiatively efficient accretion flows and the bottom refers to RIAFs."},{"id":"","caption":"Same as Table 1 but for redshift selected stacking bins.","content":"Table 1 :\n1Main features of the L K selected stacking bins: (1) bin number;"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"shows the results of the L K -first stacking, with L X−LMXB plotted against stellar mass M * and the mass of the nuclear black hole, M BH , obtained from the M BH − M * relation of Kormendy & Ho (2013). Evaluating M BH from the M BH − M * relation McConnell & Ma (2013) would yield black hole masses between 0.1 and 0.3 dex smaller than the former. Similarly, one could derive M BH from the M BH − L K relation from Graham\",\n \"content\": \"Figure 7\\n7shows the results of the L K -first stacking, with L X−LMXB plotted against stellar mass M * and the mass of the nuclear black hole, M BH , obtained from the M BH − M * relation of Kormendy & Ho (2013). Evaluating M BH from the M BH − M * relation McConnell & Ma (2013) would yield black hole masses between 0.1 and 0.3 dex smaller than the former. Similarly, one could derive M BH from the M BH − L K relation from Graham\"\n },\n {\n \"id\": \"\",\n \"caption\": \"-Distribution of net counts for randomly distributed extraction regions (red line) and for the selected sample of sources (blue line). -Examples of 0.5-7 keV images of the L K selected stacking bins.\",\n \"content\": \"Fig. 1 .Fig. 2 .\\n12-Distribution of net counts for randomly distributed extraction regions (red line) and for the selected sample of sources (blue line). -Examples of 0.5-7 keV images of the L K selected stacking bins.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"number of sources (3) < log(LK/L ) > (4) < z > (5) < log(age/yr) > (6) < log(M * /M ) > (7) S /N (8) < LX−LMXB, 0.3−8/(10 40 erg s −1 ) >\",\n \"content\": \"number of sources (3) < log(LK/L ) > (4) < z > (5) < log(age/yr) > (6) < log(M * /M ) > (7) S /N (8) < LX−LMXB, 0.3−8/(10 40 erg s −1 ) >\"\n },\n {\n \"id\": \"\",\n \"caption\": \"-X-ray luminosity (after LMXBs subtraction) versus K-band luminosity for the stacking bins selected in L K (Left Panel) and redshift (Right Panel). The horizontal and vertical black error bars indicate the statistical uncertainties on L K and L X−LMXB , respectively, while the vertical green error bars show the uncertainty on L X−LMXB due to redshift spread in each bin. The stellar mass M * from COSMOS catalog is plotted on the top axis. The local relation by KF13 is represented with a full black line. The dashed lines are drawn to include the most L X,gas luminous source ETG (M87) and the most L K luminous ETG (NGC1316) in the Local Sample (BKF11, KF13), respectively. Bins are labelled as in\",\n \"content\": \"Fig. 3 .\\n3-X-ray luminosity (after LMXBs subtraction) versus K-band luminosity for the stacking bins selected in L K (Left Panel) and redshift (Right Panel). The horizontal and vertical black error bars indicate the statistical uncertainties on L K and L X−LMXB , respectively, while the vertical green error bars show the uncertainty on L X−LMXB due to redshift spread in each bin. The stellar mass M * from COSMOS catalog is plotted on the top axis. The local relation by KF13 is represented with a full black line. The dashed lines are drawn to include the most L X,gas luminous source ETG (M87) and the most L K luminous ETG (NGC1316) in the Local Sample (BKF11, KF13), respectively. Bins are labelled as in\"\n },\n {\n \"id\": \"\",\n \"caption\": \". 4.-Left: HRs of the L K -first stacking bins versus redshift. Right: HRs of the z-first stacking bins versus L K . Triangles pointing up and down represent lower and upper limits on HR, respectively, while the box for bin 28 represents unconstrained HR. -Same as Figure 3, but with bins labelled according to their average redshift, and HRs indicated by the slope of the orange segments. Black error bars reflect statistical uncertainties, vertical green bars show the uncertainty on L X−LMXB due to redshift spread in each bin. Red boxes show the BKF11 and KF13 samples, blue circles show the C14 sample of C-COSMOS X-ray detected ETGs. Triangles pointing down represent upper limits. -Left: HR-z scatter diagram of L K -first bin. The vertical error bars indicate uncertainties on L X−LMXB due to counts and to LMXB subtraction. Triangles pointing up and down represent lower and upper limits on HR, respectively. Squares represent unconstrained HRs. Dashed lines show HRs expected from thermal models (APEC)\",\n \"content\": \"FigFig. 5 .Fig. 6 .\\n56. 4.-Left: HRs of the L K -first stacking bins versus redshift. Right: HRs of the z-first stacking bins versus L K . Triangles pointing up and down represent lower and upper limits on HR, respectively, while the box for bin 28 represents unconstrained HR. -Same as Figure 3, but with bins labelled according to their average redshift, and HRs indicated by the slope of the orange segments. Black error bars reflect statistical uncertainties, vertical green bars show the uncertainty on L X−LMXB due to redshift spread in each bin. Red boxes show the BKF11 and KF13 samples, blue circles show the C14 sample of C-COSMOS X-ray detected ETGs. Triangles pointing down represent upper limits. -Left: HR-z scatter diagram of L K -first bin. The vertical error bars indicate uncertainties on L X−LMXB due to counts and to LMXB subtraction. Triangles pointing up and down represent lower and upper limits on HR, respectively. Squares represent unconstrained HRs. Dashed lines show HRs expected from thermal models (APEC)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"-X-ray luminosity (after LMXBs subtraction) versus black hole mass M BH (evaluated from M * according to theKormendy & Ho (2013) relation) for the L K -first stacking bins. The stellar mass M * from COSMOS catalog is plotted on the top axis. Lines of constant L bol /L Edd , given M BH and L bol , are shown as diagonals (from bottom to top 10 −6 , 10 −5 , 10 −4 , 10 −3 and 10 −2 ). The bins are color-labeled as inFigure 3. Light blue boxes represent the predictions of the Volonteri et al. (2011) models for M * = 10 9 M , 10 10 M and 10 11 M , where the top of each box refers to radiatively efficient accretion flows and the bottom refers to RIAFs.\",\n \"content\": \"Fig. 7 .\\n7-X-ray luminosity (after LMXBs subtraction) versus black hole mass M BH (evaluated from M * according to theKormendy & Ho (2013) relation) for the L K -first stacking bins. The stellar mass M * from COSMOS catalog is plotted on the top axis. Lines of constant L bol /L Edd , given M BH and L bol , are shown as diagonals (from bottom to top 10 −6 , 10 −5 , 10 −4 , 10 −3 and 10 −2 ). The bins are color-labeled as inFigure 3. Light blue boxes represent the predictions of the Volonteri et al. (2011) models for M * = 10 9 M , 10 10 M and 10 11 M , where the top of each box refers to radiatively efficient accretion flows and the bottom refers to RIAFs.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Same as Table 1 but for redshift selected stacking bins.\",\n \"content\": \"Table 1 :\\n1Main features of the L K selected stacking bins: (1) bin number;\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". R D Blandford, M C Begelman, MNRAS. 3031Blandford, R. D., & Begelman, M. C. 1999, MNRAS, 303, L1","doi":""},{"text":". B Boroson, D.-W Kim, G Fabbiano, ApJ. 729BKF1112Boroson, B., Kim, D.-W., & Fabbiano, G. 2011, ApJ, 729, 12 (BKF11)","doi":""},{"text":". G Bruzual, S Charlot, MNRAS. 3441000Bruzual, G., & Charlot, S. 2003, MNRAS, 344, 1000","doi":""},{"text":". D Calzetti, R C Kennicutt, C W Engelbracht, ApJ. 666870Calzetti, D., Kennicutt, R. C., Engelbracht, C. W., et al. 2007, ApJ, 666, 870","doi":""},{"text":". C R Canizares, G Fabbiano, G Trinchieri, ApJ. 312503Canizares, C. R., Fabbiano, G., & Trinchieri, G. 1987, ApJ, 312, 503","doi":""},{"text":". G Chabrier, PASP. 115763Chabrier, G. 2003, PASP, 115, 763","doi":""},{"text":". L Ciotti, A D'ercole, S Pellegrini, A Renzini, ApJ. 376380Ciotti, L., D'Ercole, A., Pellegrini, S., & Renzini, A. 1991, ApJ, 376, 380","doi":""},{"text":". L Ciotti, J P Ostriker, ApJ. 6651038Ciotti, L., & Ostriker, J. P. 2007, ApJ, 665, 1038","doi":""},{"text":". L Ciotti, J P Ostriker, D Proga, ApJ. 717708Ciotti, L., Ostriker, J. P., & Proga, D. 2010, ApJ, 717, 708","doi":""},{"text":". F Civano, M Elvis, M Brusa, ApJS. 20130Civano, F., Elvis, M., Brusa, M., et al. 2012, ApJS, 201, 30","doi":""},{"text":". F Civano, G Fabbiano, S Pellegrini, ApJ. 790C1416Civano, F., Fabbiano, G., Pellegrini, S., et al. 2014, ApJ, 790, 16 (C14)","doi":""},{"text":". A Comastri, R Gilli, A Marconi, G Risaliti, M Salvati, A&A. 57410Comastri, A., Gilli, R., Marconi, A., Risaliti, G., & Salvati, M. 2015, A&A, 574, L10","doi":""},{"text":". T J Cox, T Di Matteo, L Hernquist, ApJ. 643692Cox, T. J., Di Matteo, T., Hernquist, L., et al. 2006, ApJ, 643, 692","doi":""},{"text":". X Dai, C S Kochanek, N D Morgan, ApJ. 658917Dai, X., Kochanek, C. S., & Morgan, N. D. 2007, ApJ, 658, 917","doi":""},{"text":". A L R Danielson, B D Lehmer, D M Alexander, MNRAS. 422494Danielson, A. L. R., Lehmer, B. D., Alexander, D. M., et al. 2012, MNRAS, 422, 494","doi":""},{"text":". L P David, W Forman, C Jones, ApJ. 369121David, L. P., Forman, W., & Jones, C. 1991, ApJ, 369, 121","doi":""},{"text":". M Elvis, F Civano, C Vignali, ApJS. 184158Elvis, M., Civano, F., Vignali, C., et al. 2009, ApJS, 184, 158","doi":""},{"text":". G Fabbiano, ARA&A. 2787Fabbiano, G. 1989, ARA&A, 27, 87","doi":""},{"text":". G Fabbiano, ARA&A. 44323Fabbiano, G. 2006, ARA&A, 44, 323","doi":""},{"text":". W Forman, J Schwarz, C Jones, W Liller, A C Fabian, ApJ. 23427Forman, W., Schwarz, J., Jones, C., Liller, W., & Fabian, A. C. 1979, ApJ, 234, L27","doi":""},{"text":". W Forman, C Jones, W Tucker, ApJ. 293102Forman, W., Jones, C., & Tucker, W. 1985, ApJ, 293, 102","doi":""},{"text":". T Fragos, B Lehmer, M Tremmel, ApJ. 76441Fragos, T., Lehmer, B., Tremmel, M., et al. 2013, ApJ, 764, 41","doi":""},{"text":". T Fragos, B . Lehmer, S Naoz, A Zezas, A Basu-Zych, ApJ. 77631Fragos, T., Lehmer, B. ., Naoz, S., Zezas, A., & Basu-Zych, A. 2013, ApJ, 776, L31","doi":""},{"text":". P Gandhi, Y Terashima, S Yamada, ApJ. 77351Gandhi, P., Terashima, Y., Yamada, S., et al. 2013, ApJ, 773, 51","doi":""},{"text":"G P Garmire, M W Bautz, P G Ford, J A Nousek, G R Ricker, Jr, Proc. SPIE. SPIE485128Garmire, G. P., Bautz, M. W., Ford, P. G., Nousek, J. A., & Ricker, G. R., Jr. 2003, Proc. SPIE, 4851, 28","doi":""},{"text":". M Gilfanov, MNRAS. 349146Gilfanov, M. 2004, MNRAS, 349, 146","doi":""},{"text":". A W Graham, MNRAS. 379711Graham, A. W. 2007, MNRAS, 379, 711","doi":""},{"text":". O Ilbert, P Capak, M Salvato, ApJ. 6901236Ilbert, O., Capak, P., Salvato, M., et al. 2009, ApJ, 690, 1236","doi":""},{"text":". O Ilbert, M Salvato, E Le Floc'h, ApJ. 709644Ilbert, O., Salvato, M., Le Floc'h, E., et al. 2010, ApJ, 709, 644","doi":""},{"text":"B T Jannuzi, A Dey, Photometric Redshifts and the Detection of High Redshift Galaxies. 191111Jannuzi, B. T., & Dey, A. 1999, Photometric Redshifts and the Detection of High Redshift Galaxies, 191, 111","doi":""},{"text":". T M Jones, M Kriek, P G Van Dokkum, ApJ. 78325Jones, T. M., Kriek, M., van Dokkum, P. G., et al. 2014, ApJ, 783, 25","doi":""},{"text":". D.-W Kim, G Fabbiano, ApJ. 611846Kim, D.-W., & Fabbiano, G. 2004, ApJ, 611, 846","doi":""},{"text":". D.-W Kim, G Fabbiano, N J Brassington, ApJ. 703829Kim, D.-W., Fabbiano, G., Brassington, N. J., et al. 2009, ApJ, 703, 829","doi":""},{"text":". D.-W Kim, G Fabbiano, ApJ. 7211523Kim, D.-W., & Fabbiano, G. 2010, ApJ, 721, 1523","doi":""},{"text":". D.-W Kim, G Fabbiano, ApJ. 776KF13116Kim, D.-W., & Fabbiano, G. 2013, ApJ, 776, 116 (KF13)","doi":""},{"text":". D.-W Kim, G Fabbiano, arXiv:1504.00899Kim, D.-W., & Fabbiano, G. 2015, arXiv:1504.00899","doi":""},{"text":". C S Kochanek, D J Eisenstein, R J Cool, ApJS. 8Kochanek, C. S., Eisenstein, D. J., Cool, R. J., et al. 2012, ApJS, 200, 8","doi":""},{"text":". J A Kollmeier, C A Onken, C S Kochanek, ApJ. 648128Kollmeier, J. A., Onken, C. A., Kochanek, C. S., et al. 2006, ApJ, 648, 128","doi":""},{"text":". J Kormendy, L C Ho, ARA&A. 51511Kormendy, J., & Ho, L. C. 2013, ARA&A, 51, 511","doi":""},{"text":". A Kundu, T J Maccarone, S E Zepf, ApJ. 5745Kundu, A., Maccarone, T. J., & Zepf, S. E. 2002, ApJ, 574, L5","doi":""},{"text":". A Leauthaud, R Massey, J.-P Kneib, ApJS. 172219Leauthaud, A., Massey, R., Kneib, J.-P., et al. 2007, ApJS, 172, 219","doi":""},{"text":". B D Lehmer, W N Brandt, D M Alexander, ApJ. 657681Lehmer, B. D., Brandt, W. N., Alexander, D. M., et al. 2007, ApJ, 657, 681","doi":""},{"text":". B D Lehmer, D M Alexander, F E Bauer, ApJ. 724559Lehmer, B. D., Alexander, D. M., Bauer, F. E., et al. 2010, ApJ, 724, 559","doi":""},{"text":". B D Lehmer, M Berkeley, A Zezas, ApJ. 78952Lehmer, B. D., Berkeley, M., Zezas, A., et al. 2014, ApJ, 789, 52","doi":""},{"text":". N A Levenson, T M Heckman, J H Krolik, K A Weaver, P T Życki, ApJ. 648111Levenson N. A., Heckman T. M., Krolik J. H., Weaver K. A.,Życki P. T., 2006, ApJ, 648, 111","doi":""},{"text":". W G Mathews, F Brighenti, ARA&A. 41191Mathews, W. G., & Brighenti, F. 2003, ARA&A, 41, 191","doi":""},{"text":". N J Mcconnell, C.-P Ma, ApJ. 764184McConnell, N. J., & Ma, C.-P. 2013, ApJ, 764, 184","doi":""},{"text":". H J Mccracken, B Milvang-Jensen, J Dunlop, A&A. 544156McCracken, H. J., Milvang-Jensen, B., Dunlop,J., et al. 2012, A&A, 544, A156","doi":""},{"text":". A Merloni, A Bongiorno, M Bolzonella, ApJ. 708137Merloni, A., Bongiorno, A., Bolzonella, M., et al. 2010, ApJ, 708, 137","doi":""},{"text":". B P Miller, E Gallo, J E Greene, ApJ. 79998Miller, B. P., Gallo, E., Greene, J. E., et al. 2015, ApJ, 799, 98","doi":""},{"text":". S Mineo, M Gilfanov, R Sunyaev, MNRAS. 4192095Mineo, S., Gilfanov, M., & Sunyaev, R. 2012, MNRAS, 419, 2095","doi":""},{"text":". M Moresco, L Pozzetti, A Cimatti, A&A. 55861Moresco, M., Pozzetti, L., Cimatti, A., et al. 2013, A&A, 558, A61","doi":""},{"text":". S S Murray, A Kenter, W R Forman, ApJS. 1611Murray, S. S., Kenter, A., Forman, W. R., et al. 2005, ApJS, 161, 1","doi":""},{"text":". R Narayan, I Yi, ApJ. 452710Narayan, R., & Yi, I. 1995, ApJ, 452, 710","doi":""},{"text":". S Pellegrini, ApJ. 624155Pellegrini, S. 2005, ApJ, 624, 155","doi":""},{"text":". S Pellegrini, J Wang, G Fabbiano, ApJ. 75894Pellegrini, S., Wang, J., Fabbiano, G., et al. 2012, ApJ, 758, 94","doi":""},{"text":". M Polletta, M Tajer, L Maraschi, ApJ. 66381Polletta, M., Tajer, M., Maraschi, L., et al. 2007, ApJ, 663, 81","doi":""},{"text":". S Puccetti, C Vignali, N Cappelluti, ApJS. 185586Puccetti, S., Vignali, C., Cappelluti, N., et al. 2009, ApJS, 185, 586","doi":""},{"text":". G Risaliti, R Maiolino, M Salvati, ApJ. 522157Risaliti, G., Maiolino, R., & Salvati, M. 1999, ApJ, 522, 157","doi":""},{"text":". C L Sarazin, J A Irwin, J N Bregman, ApJ. 544101Sarazin, C. L., Irwin, J. A., & Bregman, J. N. 2000, ApJ, 544, L101","doi":""},{"text":". N Scoville, R G Abraham, H Aussel, ApJS. 17238Scoville, N., Abraham, R. G., Aussel, H., et al. 2007, ApJS, 172, 38","doi":""},{"text":". E Schinnerer, M T Sargent, M Bondi, ApJS. 188384Schinnerer, E., Sargent, M. T., Bondi, M., et al. 2010, ApJS, 188, 384","doi":""},{"text":". C L Steinhardt, M Elvis, MNRAS. 4022637Steinhardt, C. L., & Elvis, M. 2010, MNRAS, 402, 2637","doi":""},{"text":". S Tremaine, K Gebhardt, R Bender, ApJ. 574740Tremaine, S., Gebhardt, K., Bender, R., et al. 2002, ApJ, 574, 740","doi":""},{"text":". G Trinchieri, G Fabbiano, ApJ. 296447Trinchieri, G., & Fabbiano, G. 1985, ApJ, 296, 447","doi":""},{"text":". P Tzanavaris, I Georgantopoulos, A&A. 480Tzanavaris, P., & Georgantopoulos, I. 2008, A&A, 480, 663 -14 -","doi":""},{"text":". M Volonteri, M Dotti, D Campbell, M Mateo, ApJ. 730145Volonteri, M., Dotti, M., Campbell, D., & Mateo, M. 2011, ApJ, 730, 145","doi":""},{"text":". C R Watson, C S Kochanek, W R Forman, ApJ. 6962206Watson, C. R., Kochanek, C. S., Forman, W. R., et al. 2009, ApJ, 696, 2206","doi":""},{"text":". R E White, Iii, C L Sarazin, ApJ. 367476White, R. E., III, & Sarazin, C. L. 1991, ApJ, 367, 476","doi":""},{"text":". C J Willott, ApJ. 7428Willott, C. J. 2011, ApJ, 742, L8","doi":""},{"text":". F Yuan, M Li, ApJ. 73723Yuan, F., & Li, M. 2011, ApJ, 737, 23","doi":""},{"text":". F Yuan, R Narayan, ARA&A. 52529Yuan, F., & Narayan, R. 2014, ARA&A, 52, 529","doi":""}],"string":"[\n {\n \"text\": \". R D Blandford, M C Begelman, MNRAS. 3031Blandford, R. D., & Begelman, M. C. 1999, MNRAS, 303, L1\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Boroson, D.-W Kim, G Fabbiano, ApJ. 729BKF1112Boroson, B., Kim, D.-W., & Fabbiano, G. 2011, ApJ, 729, 12 (BKF11)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Bruzual, S Charlot, MNRAS. 3441000Bruzual, G., & Charlot, S. 2003, MNRAS, 344, 1000\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Calzetti, R C Kennicutt, C W Engelbracht, ApJ. 666870Calzetti, D., Kennicutt, R. C., Engelbracht, C. W., et al. 2007, ApJ, 666, 870\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C R Canizares, G Fabbiano, G Trinchieri, ApJ. 312503Canizares, C. R., Fabbiano, G., & Trinchieri, G. 1987, ApJ, 312, 503\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Chabrier, PASP. 115763Chabrier, G. 2003, PASP, 115, 763\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Ciotti, A D'ercole, S Pellegrini, A Renzini, ApJ. 376380Ciotti, L., D'Ercole, A., Pellegrini, S., & Renzini, A. 1991, ApJ, 376, 380\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Ciotti, J P Ostriker, ApJ. 6651038Ciotti, L., & Ostriker, J. P. 2007, ApJ, 665, 1038\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Ciotti, J P Ostriker, D Proga, ApJ. 717708Ciotti, L., Ostriker, J. P., & Proga, D. 2010, ApJ, 717, 708\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Civano, M Elvis, M Brusa, ApJS. 20130Civano, F., Elvis, M., Brusa, M., et al. 2012, ApJS, 201, 30\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Civano, G Fabbiano, S Pellegrini, ApJ. 790C1416Civano, F., Fabbiano, G., Pellegrini, S., et al. 2014, ApJ, 790, 16 (C14)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Comastri, R Gilli, A Marconi, G Risaliti, M Salvati, A&A. 57410Comastri, A., Gilli, R., Marconi, A., Risaliti, G., & Salvati, M. 2015, A&A, 574, L10\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T J Cox, T Di Matteo, L Hernquist, ApJ. 643692Cox, T. J., Di Matteo, T., Hernquist, L., et al. 2006, ApJ, 643, 692\",\n \"doi\": \"\"\n },\n {\n \"text\": \". X Dai, C S Kochanek, N D Morgan, ApJ. 658917Dai, X., Kochanek, C. S., & Morgan, N. D. 2007, ApJ, 658, 917\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A L R Danielson, B D Lehmer, D M Alexander, MNRAS. 422494Danielson, A. L. R., Lehmer, B. D., Alexander, D. M., et al. 2012, MNRAS, 422, 494\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L P David, W Forman, C Jones, ApJ. 369121David, L. P., Forman, W., & Jones, C. 1991, ApJ, 369, 121\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Elvis, F Civano, C Vignali, ApJS. 184158Elvis, M., Civano, F., Vignali, C., et al. 2009, ApJS, 184, 158\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Fabbiano, ARA&A. 2787Fabbiano, G. 1989, ARA&A, 27, 87\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Fabbiano, ARA&A. 44323Fabbiano, G. 2006, ARA&A, 44, 323\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W Forman, J Schwarz, C Jones, W Liller, A C Fabian, ApJ. 23427Forman, W., Schwarz, J., Jones, C., Liller, W., & Fabian, A. C. 1979, ApJ, 234, L27\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W Forman, C Jones, W Tucker, ApJ. 293102Forman, W., Jones, C., & Tucker, W. 1985, ApJ, 293, 102\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Fragos, B Lehmer, M Tremmel, ApJ. 76441Fragos, T., Lehmer, B., Tremmel, M., et al. 2013, ApJ, 764, 41\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Fragos, B . Lehmer, S Naoz, A Zezas, A Basu-Zych, ApJ. 77631Fragos, T., Lehmer, B. ., Naoz, S., Zezas, A., & Basu-Zych, A. 2013, ApJ, 776, L31\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Gandhi, Y Terashima, S Yamada, ApJ. 77351Gandhi, P., Terashima, Y., Yamada, S., et al. 2013, ApJ, 773, 51\",\n \"doi\": \"\"\n },\n {\n \"text\": \"G P Garmire, M W Bautz, P G Ford, J A Nousek, G R Ricker, Jr, Proc. SPIE. SPIE485128Garmire, G. P., Bautz, M. W., Ford, P. G., Nousek, J. A., & Ricker, G. R., Jr. 2003, Proc. SPIE, 4851, 28\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Gilfanov, MNRAS. 349146Gilfanov, M. 2004, MNRAS, 349, 146\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A W Graham, MNRAS. 379711Graham, A. W. 2007, MNRAS, 379, 711\",\n \"doi\": \"\"\n },\n {\n \"text\": \". O Ilbert, P Capak, M Salvato, ApJ. 6901236Ilbert, O., Capak, P., Salvato, M., et al. 2009, ApJ, 690, 1236\",\n \"doi\": \"\"\n },\n {\n \"text\": \". O Ilbert, M Salvato, E Le Floc'h, ApJ. 709644Ilbert, O., Salvato, M., Le Floc'h, E., et al. 2010, ApJ, 709, 644\",\n \"doi\": \"\"\n },\n {\n \"text\": \"B T Jannuzi, A Dey, Photometric Redshifts and the Detection of High Redshift Galaxies. 191111Jannuzi, B. T., & Dey, A. 1999, Photometric Redshifts and the Detection of High Redshift Galaxies, 191, 111\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T M Jones, M Kriek, P G Van Dokkum, ApJ. 78325Jones, T. M., Kriek, M., van Dokkum, P. G., et al. 2014, ApJ, 783, 25\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D.-W Kim, G Fabbiano, ApJ. 611846Kim, D.-W., & Fabbiano, G. 2004, ApJ, 611, 846\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D.-W Kim, G Fabbiano, N J Brassington, ApJ. 703829Kim, D.-W., Fabbiano, G., Brassington, N. J., et al. 2009, ApJ, 703, 829\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D.-W Kim, G Fabbiano, ApJ. 7211523Kim, D.-W., & Fabbiano, G. 2010, ApJ, 721, 1523\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D.-W Kim, G Fabbiano, ApJ. 776KF13116Kim, D.-W., & Fabbiano, G. 2013, ApJ, 776, 116 (KF13)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D.-W Kim, G Fabbiano, arXiv:1504.00899Kim, D.-W., & Fabbiano, G. 2015, arXiv:1504.00899\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C S Kochanek, D J Eisenstein, R J Cool, ApJS. 8Kochanek, C. S., Eisenstein, D. J., Cool, R. J., et al. 2012, ApJS, 200, 8\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J A Kollmeier, C A Onken, C S Kochanek, ApJ. 648128Kollmeier, J. A., Onken, C. A., Kochanek, C. S., et al. 2006, ApJ, 648, 128\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Kormendy, L C Ho, ARA&A. 51511Kormendy, J., & Ho, L. C. 2013, ARA&A, 51, 511\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Kundu, T J Maccarone, S E Zepf, ApJ. 5745Kundu, A., Maccarone, T. J., & Zepf, S. E. 2002, ApJ, 574, L5\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Leauthaud, R Massey, J.-P Kneib, ApJS. 172219Leauthaud, A., Massey, R., Kneib, J.-P., et al. 2007, ApJS, 172, 219\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B D Lehmer, W N Brandt, D M Alexander, ApJ. 657681Lehmer, B. D., Brandt, W. N., Alexander, D. M., et al. 2007, ApJ, 657, 681\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B D Lehmer, D M Alexander, F E Bauer, ApJ. 724559Lehmer, B. D., Alexander, D. M., Bauer, F. E., et al. 2010, ApJ, 724, 559\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B D Lehmer, M Berkeley, A Zezas, ApJ. 78952Lehmer, B. D., Berkeley, M., Zezas, A., et al. 2014, ApJ, 789, 52\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N A Levenson, T M Heckman, J H Krolik, K A Weaver, P T Życki, ApJ. 648111Levenson N. A., Heckman T. M., Krolik J. H., Weaver K. A.,Życki P. T., 2006, ApJ, 648, 111\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W G Mathews, F Brighenti, ARA&A. 41191Mathews, W. G., & Brighenti, F. 2003, ARA&A, 41, 191\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N J Mcconnell, C.-P Ma, ApJ. 764184McConnell, N. J., & Ma, C.-P. 2013, ApJ, 764, 184\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H J Mccracken, B Milvang-Jensen, J Dunlop, A&A. 544156McCracken, H. J., Milvang-Jensen, B., Dunlop,J., et al. 2012, A&A, 544, A156\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Merloni, A Bongiorno, M Bolzonella, ApJ. 708137Merloni, A., Bongiorno, A., Bolzonella, M., et al. 2010, ApJ, 708, 137\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B P Miller, E Gallo, J E Greene, ApJ. 79998Miller, B. P., Gallo, E., Greene, J. E., et al. 2015, ApJ, 799, 98\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Mineo, M Gilfanov, R Sunyaev, MNRAS. 4192095Mineo, S., Gilfanov, M., & Sunyaev, R. 2012, MNRAS, 419, 2095\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Moresco, L Pozzetti, A Cimatti, A&A. 55861Moresco, M., Pozzetti, L., Cimatti, A., et al. 2013, A&A, 558, A61\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S S Murray, A Kenter, W R Forman, ApJS. 1611Murray, S. S., Kenter, A., Forman, W. R., et al. 2005, ApJS, 161, 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Narayan, I Yi, ApJ. 452710Narayan, R., & Yi, I. 1995, ApJ, 452, 710\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Pellegrini, ApJ. 624155Pellegrini, S. 2005, ApJ, 624, 155\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Pellegrini, J Wang, G Fabbiano, ApJ. 75894Pellegrini, S., Wang, J., Fabbiano, G., et al. 2012, ApJ, 758, 94\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Polletta, M Tajer, L Maraschi, ApJ. 66381Polletta, M., Tajer, M., Maraschi, L., et al. 2007, ApJ, 663, 81\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Puccetti, C Vignali, N Cappelluti, ApJS. 185586Puccetti, S., Vignali, C., Cappelluti, N., et al. 2009, ApJS, 185, 586\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Risaliti, R Maiolino, M Salvati, ApJ. 522157Risaliti, G., Maiolino, R., & Salvati, M. 1999, ApJ, 522, 157\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C L Sarazin, J A Irwin, J N Bregman, ApJ. 544101Sarazin, C. L., Irwin, J. A., & Bregman, J. N. 2000, ApJ, 544, L101\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Scoville, R G Abraham, H Aussel, ApJS. 17238Scoville, N., Abraham, R. G., Aussel, H., et al. 2007, ApJS, 172, 38\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Schinnerer, M T Sargent, M Bondi, ApJS. 188384Schinnerer, E., Sargent, M. T., Bondi, M., et al. 2010, ApJS, 188, 384\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C L Steinhardt, M Elvis, MNRAS. 4022637Steinhardt, C. L., & Elvis, M. 2010, MNRAS, 402, 2637\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Tremaine, K Gebhardt, R Bender, ApJ. 574740Tremaine, S., Gebhardt, K., Bender, R., et al. 2002, ApJ, 574, 740\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Trinchieri, G Fabbiano, ApJ. 296447Trinchieri, G., & Fabbiano, G. 1985, ApJ, 296, 447\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Tzanavaris, I Georgantopoulos, A&A. 480Tzanavaris, P., & Georgantopoulos, I. 2008, A&A, 480, 663 -14 -\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Volonteri, M Dotti, D Campbell, M Mateo, ApJ. 730145Volonteri, M., Dotti, M., Campbell, D., & Mateo, M. 2011, ApJ, 730, 145\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C R Watson, C S Kochanek, W R Forman, ApJ. 6962206Watson, C. R., Kochanek, C. S., Forman, W. R., et al. 2009, ApJ, 696, 2206\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R E White, Iii, C L Sarazin, ApJ. 367476White, R. E., III, & Sarazin, C. L. 1991, ApJ, 367, 476\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C J Willott, ApJ. 7428Willott, C. J. 2011, ApJ, 742, L8\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Yuan, M Li, ApJ. 73723Yuan, F., & Li, M. 2011, ApJ, 737, 23\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Yuan, R Narayan, ARA&A. 52529Yuan, F., & Narayan, R. 2014, ARA&A, 52, 529\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":492,"cells":{"text":{"kind":"string","value":"\nPrecise radial velocities of giant stars. IV. ⋆ A correlation between surface gravity and radial velocity variation and a statistical investigation of companion properties\n4 Jan 2008 February 2, 2008\n\nS Hekker \nLeiden Observatory\nLeiden University\nP.O. Box 95132300 RALeidenThe Netherlands\n\nRoyal Observatory of Belgium\nRinglaan 31180BrusselsBelgium\n\nI A G Snellen \nLeiden Observatory\nLeiden University\nP.O. Box 95132300 RALeidenThe Netherlands\n\nC Aerts \nInstituut voor Sterrenkunde\nKatholieke Universiteit Leuven\nCelestijnenlaan 200 D3001LeuvenBelgium\n\nDepartment of Astrophysics\nUniversity of Nijmegen\nP.O. Box 90106500 GLNijmegenThe Netherlands\n\nA Quirrenbach \nLeiden Observatory\nLeiden University\nP.O. Box 95132300 RALeidenThe Netherlands\n\nZAH\nLandessternwarte Heidelberg, Königstuhl 12D-69117HeidelbergGermany\n\nS Reffert \nZAH\nLandessternwarte Heidelberg, Königstuhl 12D-69117HeidelbergGermany\n\nD S Mitchell \nCalifornia Polytechnic State University\n93407San Luis ObispoCAUSA\n\nPrecise radial velocities of giant stars. IV. ⋆ A correlation between surface gravity and radial velocity variation and a statistical investigation of companion properties\n4 Jan 2008 February 2, 2008Received ¡date¿; accepted ¡date¿Astronomy & Astrophysics manuscript no. 8321astrophstars: variables -techniques: radial velocities\nContext. Since 1999, we have been conducting a radial velocity survey of 179 K giants using the Coudé Auxiliary Telescope at UCO/Lick observatory. At present ∼20−100 measurements have been collected per star with a precision of 5 to 8 m s −1 . Of the stars monitored, 145 (80%) show radial velocity (RV) variations at a level >20 m s −1 , of which 43 exhibit significant periodicities. Aims. Our aim is to investigate possible mechanism(s) that cause these observed RV variations. We intend to test whether these variations are intrinsic in nature, or possibly induced by companions, or both. In addition, we aim to characterise the parameters of these companions. Methods. A relation between log g and the amplitude of the RV variations is investigated for all stars in the sample. Furthermore, the hypothesis that all periodic RV variations are caused by companions is investigated by comparing their inferred orbital statistics with the statistics of companions around main sequence F, G, and K dwarfs. Results. A strong relation is found between the amplitude of the RV variations and log g in K giant stars, as suggested earlier byHatzes & Cochran (1998). However, most of the stars exhibiting periodic variations are located above this relation. These RV variations can be split in a periodic component which is not correlated with log g and a random residual part which does correlate with log g. Compared to main-sequence dwarf stars, K giants frequently exhibit periodic RV variations. Interpreting these RV variations as being caused by companions, the orbital parameters are different from the companions orbiting dwarfs. Conclusions. Intrinsic mechanisms play an important role in producing RV variations in K giants stars, as suggested by their dependence on log g. However, it appears that periodic RV variations are additional to these intrinsic variations, consistent with them being caused by companions. If indeed the majority of the periodic RV variations in K giants is interpreted as due to substellar companions, then massive planets are significantly more common around K giants than around F, G, K main-sequence stars.\n\nIntroduction\n\nFor more than a decade, radial velocity observations with accuracies of order m s −1 have been within reach (see for instance Marcy & Butler (2000) and Queloz et al. (2001)). Even accuracies of less than 1 m s −1 (Pepe et al. 2003) are possible now. With these observations, more than 200 sub-stellar companions have been discovered by measuring the reflex motions of their parent stars. Most of these sub-stellar companions have been detected around F, G and K main sequence Send offprint requests to: S. Hekker, email: saskia@oma.be ⋆ Based on data obtained at UCO/Lick Observatory, USA stars, but detections around an A star (Galland et al. 2006) and several subgiants (Johnson et al. (2006), Johnson et al. (2007)) have also been reported recently. Moreover, 10 giant stars were reported to have sub-stellar companions (ι Draconis (K2III) Frink et al. (2002), HD104985 (G9III) Sato et al. (2003), HD47526 (K1III) Setiawan et al. (2003), HD13189 (K2II-III) Hatzes et al. (2005), HD11977 (G5III) Setiawan et al. (2005), Pollux (K0III) Hatzes et al. (2006), Reffert et al. (2006), 4UMa (K1III) Döllinger et al. (2007), NGC2423 No3 and NGC4349 No127 Lovis & Mayor (2007), and recently HD17092 (K0III) Niedzielski et al. (2007)) 1 . In addition to searches for extra-solar companions, radial velocity observations prove to be very useful for detecting solarlike oscillations in stars with turbulent atmospheres, such as the dwarf α Cen A (e.g. Bedding et al. 2006), the subgiant Procyon (e.g. Eggenberger et al. 2004: Martić et al. 2004) and the giant ǫ Ophiuchi (e.g. De Ridder et al. 2006). With techniques for accurate radial velocity observations at hand, a survey was started in 1999 to verify whether K giants are stable enough to be used as astrometric reference stars for SIM/PlanetQuest (Space Interferometry Mission) (Frink et al. 2001). This survey contains 179 stars and uses the Coudé Auxiliary Telescope (CAT) at University of California Observatories / Lick Observatory, in conjunction with the Hamilton Echelle Spectrograph. The survey has recently been expanded to about 380 giants and is still ongoing. For the analysis described in the present paper only data from the initial 179 stars are used.\n\nFrom this survey, companions have been announced for ι Draconis (Frink et al. 2002) and Pollux . Stars with radial velocity variations of less than 20 m s −1 have been presented as stable stars by Hekker et al. (2006). In addition, some binaries discovered with this survey, as well as an extensive overview of the sample, will be presented in forthcoming papers.\n\nAs almost all of the stars show significant radial velocity variations, we investigate here which mechanism causes these variations. Non-periodic radial velocity variations, of the order of the investigated timescales, are most likely caused by some intrinsic mechanism, while the periodic variability can also be caused by companions. We also investigate the characteristics of these companions.\n\nIn Sect. 2, the radial velocity observations are described in detail. In Sect. 3, the relation between the observed radial velocity amplitude and surface gravity is investigated. In Sect. 4, we explore the hypothesis that all periodic radial velocity variations are caused by sub-stellar companions, and we compare the inferred orbital parameters with those obtained for substellar companions orbiting main sequence stars. Our conclusions are presented in Sect. 5.\n\n\nRadial velocity variations\n\nThe initial 179 stars selected for the radial velocity survey are used in the present work. These stars have been selected from the Hipparcos catalogue (Perryman & ESA 1997), based on the criteria described by Frink et al. (2001). The selected stars are all brighter than 6 mag, are presumably single and have photometric variations < 0.06 mag in V. The survey started in 1999 at Lick Observatory using the Coudé Auxiliary Telescope (CAT) in conjunction with the Hamilton Echelle Spectrograph (R=60 000). The system with an iodine cell in the light path has been described by Marcy & Butler (1992) and Valenti et al. (1995). With integration times of up to thirty minutes for the faintest stars (m v = 6 mag) we reach a signal to noise ratio of about 80 − 100 at λ = 5500 Å, yielding a radial velocity precision of 5 − 8 m s −1 . As we are looking for radial velocity variations of order 10 to 100 m s −1 , this is adequate and hence no attempt has been made to reach the 3 m s −1 accuracy which is in principle possible with this setup (Butler et al. 1996). For the determination of the radial velocities the pipeline described by Butler et al. (1996) is used. In this pipeline, a template iodine spectrum and a template spectrum of the target star obtained without an iodine cell in the lightpath are used to model the stellar observations with a superposed iodine spectrum. The Doppler shift is a free parameter in this model. Note that with this method the absolute radial velocity is not measured, but the radial velocity relative to the stellar template is obtained.\n\nAll 179 stars are subjected to a period search. The periodicity of the radial velocity variations is determined first of all from a classical Lomb-Scargle (LS) periodogram (Scargle 1982). The significance threshold is set to 6σ, where the noise level is determined from the average power of the residual Scargle periodogram for frequencies between 0 and 0.03 cycles per day (c/d) (0.35 µHz) and a frequency step of 0.00001 c/d (0.12 · 10 −3 µHz). We adopted the conventional method of iterative sinewave fitting ('prewhitening') to search for subsequent frequencies (Kuschnig et al. 1997). In Figs 1 and 2, the radial velocity variation as a function of phase is shown for two stars. The period of the star in Fig. 1 is highly significant, while the one in Fig. 2 is close to the significance threshold. Periodograms are shown in the bottom panels of these figures. As properly emphasized by Cumming et al. (1999), such a classical period search may not be appropriate for unevenly spaced sparse data, even though we set the significance level at a conservatively high level. In order to check this, we have done an additional LS analysis after prewhitening the original data by a linear polynomial. This led almost always to the same frequencies. We only accepted a frequency when it was found to meet the significance criterion for both these analyses. The significant frequencies are listed in Table 1. Hatzes & Cochran (1998) already investigated the origin of the observed radial velocities in K giant stars. Although their sample contained only 9 stars, they suggested that the amplitude of the radial velocity increases with decreasing surface gravity (log g). In lower surface gravity it takes longer to decrease the velocity of a moving parcel which results in larger amplitudes and the relation suggested by Hatzes & Cochran (1998) would therefore be evidence for pulsations or rotational modulation as the mechanism for these long period radial velocity variations. For the present sample, log g values were determined spectroscopically by Hekker & Meléndez (2007), by imposing excitation and ionisation equilibrium of iron lines through stellar models. The equivalent width of about two dozen carefully selected iron lines were used for a spectroscopic LTE analysis based on the 2002 version of MOOG (Sneden 1973) and Kurucz model atmospheres which include overshooting (Castelli et al. 1997). These authors estimated the error on log g to be 0.22 dex from the scatter found in a comparison with literature values. A detailed description of the stellar parameters for individual stars and a comparison with literature values are available in Hekker & Meléndez (2007) and is therefore omitted here. In Fig. 3, we show half of the peak-to-peak value of the observed radial velocity variations as a function of log g for K giants in our sample. A clear trend is visible between increasing radial velocity variations in single stars, and decreasing log g, which provides a strong indication that, at least for a large fraction of stars in our sample, the observed radial velocity vari-\n\n\nRadial velocity amplitude -surface gravity relation\n\n\nFig. 3.\n\nHalf of the peak-to-peak variation of the radial velocity as a function of surface gravity (log g). The blue • indicate the single stars with periodic radial velocity variations (see text for periodicity criteria), the red indicate single stars with random radial velocity variations, and stellar binaries (companion mass > 100 M Jup ) are indicated with black ⋄ symbols. (Colours are only visible in the online version of the paper.) The solid line is the best fit through the random single stars, the dotted line indicates the 1σ interval around the best fit and the dashed line indicates the 3.5σ interval. Single stars with a higher radial velocity amplitude than expected based on their log g value (more than 3.5σ above the best fit) are indicated by arrows. Six of the 8 stars with periodic radial velocity variations and log g < 1.6 are classified bright giants or supergiants (Perryman & ESA 1997).\n\nations are induced by a mechanism intrinsic to the star. This trend is present for stars with random as well as stars with periodic radial velocity variations. Also, nearly all stars with periodic radial velocity variations and log g ≥ 1.6 are located above the fit in Fig. 3. Seven single stars have a higher radial velocity variation than expected based on their log g value, i.e. they are situated more than 3.5σ above the best fit for the relation obtained for single stars. These stars are indicated with arrows in Fig. 3. The radial velocity variation observed for HIP53229 may be due to a stellar companion in a wide orbit, with a period much longer than the observation time span. Due to this long period the companion mass, and, therefore, the (sub-)stellar nature, is still very uncertain. HIP33152 is classified as a supergiant. The observed radial velocity variations for HIP80693, HIP36616, and HIP88048 can be fitted with two Keplerian orbits, while HIP75458 can be explained by an eccentric substellar companion (Frink et al. 2002) and an additional linear trend, indicating a companion in a wide orbit. HIP34693 can be fitted very accurately with a Keplerian orbit of a single nearly sinusoidal sub-stellar companion.\n\nIn order to investigate the simultaneous occurrence of substellar companions and oscillations in giants, the best Keplerian fits are subtracted from the observed radial velocity variations of both the binaries and the single stars with significant periodic radial velocity variations. The half peak-to-peak values of these residuals are plotted as a function of surface gravity in Fig. 4, also showing the linear fit through the non-periodic stars, and the amplitudes of the subtracted periodic signals. There are several interesting points to make about this graph. Firstly, almost all periodic stars show larger radial velocity variations than predicted by the relation found for non-periodic stars, but when the periodicities are removed, their residual radial velocity variations follow the same relation as found for the non-periodic stars. This could be interpreted as evidence for both intrinsic (non-periodic radial velocity variation) and extrinsic (periodic variations) mechanisms playing a role in these stars. Secondly, there is no correlation between the amplitude of the subtracted periodic signal and log g, which provides additional evidence for the presence of companions. Thirdly, almost all (8 out of 9) stars with log g ≤ 1.6 exhibit periodic variations. If these indeed have an extrinsic mechanism, it would mean that ∼ 90% of these stars have sub-stellar companions. However, from an astrophysical point of view, stars with such low surface gravities are already very high on the giant branch or even on the asymptotic giant branch. At these low gravities, stars cannot be constant anymore, as the outer layers are so diluted that instabilities occur easily, either periodic or random. Stated differently, these stars are very close to the semi-regulars, which are on their way to become Mira variables. Hence, these periodic variations could well be intrinsic.\n\n\nCompanion Interpretation\n\nFrom the analysis of the correlation between radial velocity amplitude and surface gravity we have evidence for the presence of both intrinsic variability and companions in at least a fraction of the K giant stars with periodic radial velocity variations. In order to study the characteristics of these companions, we take the hypothesis that the periodic radial velocity variations detected in 43 of our K giant stars, excluding binaries, are caused by sub-stellar companions. Under this hypothesis, we investigate the statistical properties of the orbital parameters of the sample and compare these with the statistical properties of companions orbiting F, G and K dwarfs.\n\nAccording to our analysis, 55 stars in the sample would have a single companion, and 11 stars multiple companions.\n\nTwenty-three (22 single and 1 in a multiple system) of these companions have m sin i larger than 100 M Jup and should be interpreted as stellar binaries. By advancing the multiple substellar systems forward in time via a Runga-Kutta integration, we investigated the stability of the systems, taking into account the mutual interactions of the companions. With the orbital parameters that minimise χ 2 taken at face value, we found that most of the inferred sub-stellar multiple systems would be 'likely unstable', with a change in semi-major axis > 1% and < 10%, or 'unstable' with a change in semi-major axis > 10% on a time scale of 100 years due to companion-companion interaction.\n\nHowever, the inferred stability depends on the starting epoch of the computations, as well as on the orbital parameters, which might change with an increasing number of observations. Furthermore, there is no guarantee that the obtained χ 2 minimum is a global minimum. Therefore, stars with multiple inferred sub-stellar companions that seem to be unstable, might also have stable solutions. One could also use the equations for dynamical stability described by Gladman (1993) and Marcy et al. (2001). Gladman (1993) also notes that the Hill stability criteria for companions in initially eccentric orbits may not be met, but that the systems may still be found to be empirically quite stable for a long period of time. In order to draw a firm conclusion on the stability of a particular system, a more thorough investigation is needed, as well as data with a longer time base, which is beyond the scope of this paper.\n\nFor all stars with periodic radial velocity variations, we checked the Hipparcos (Perryman & ESA 1997) photometry. We checked periodograms for significant frequencies close to the obtained radial velocity period, and plotted the photometric values phased with the radial velocity period. None of the stars show photometric variations related to the observed radial velocity variations.\n\nThe mass distribution of our K giant sample is not known very well. The stellar masses are typically between 1 and 4 M ⊙ . Hence most of their main sequence progenitors should have been of A or F spectral class. The distribution of orbital parameters of sub-stellar companions orbiting A and F main sequence stars is still unknown. The core accretion model predicts more giant planets around more massive stars, so that the distribution of orbital parameters of sub-stellar companions orbiting F, G and K main-sequence stars probably cannot serve as a proxy. However, it should be instructive to compare the two distributions.\n\nSince the data presented here span ∼ 2500 days, radial velocity variations with longer periods are uncertain, and, therefore, not taken into consideration. Companions with periods exceeding the observation time span are also excluded from the F, G, and K main sequence star statistics. (Marcy et al. 2005), normalised to the number of stars in our sample. Fig. 6. Zoom in on the low end of the companion mass distribution of inferred sub-stellar companions orbiting K giant stars in our survey, shown as percentage of the total number of stars in the sample (gray histogram). The hatched histogram shows the distribution of companion masses orbiting main sequence stars as shown in Fig. 1 of Marcy et al. (2005), as a percentage of the total number of stars in their sample. The cross hatched area are main sequence stars with companions at a semi-major axis smaller than 0.3 AU (see Fig. 7). dwarf statistics around F, G and K main sequence stars for which only very few companions are found with m sin i > 15 M Jup around more than thousand stars. This is known as the brown dwarf desert and is possibly caused by migration and merging of brown dwarfs in a viscous disk with a mass at least comparable to the brown dwarf mass (Armitage & Bonnell 2002).\n\n\nMass distribution\n\nThe dashed line in Fig. 5 indicates the rise of sub-stellar companion masses M −1.05 from 10 M Jup down to Saturn masses for main sequence stars (Marcy et al. 2005), normalised Fig. 7. Semi-major axis distribution of inferred K giant companions in our sample (gray histogram), shown as a percentage of the total number of stars in the sample. The hatched histogram shows the distribution for F, G and K main sequence stars as shown in Fig. 2 of Marcy et al. (2005), as a percentage of the total number of stars in their sample. The main sequence stars with a companion orbiting at a semi-major axis smaller than 0.3 AU are cross-hatched. These are also indicated in Figs 6 and 8.\n\nto the number of stars in our sample. We use a two-sided Kolmogorov-Smirnov test (Press et al. 1992) (hereafter KStest) to compare the M −1.05 fit and the mass distribution (m sin i < 80 M Jup ) of our sample and find a probability of 0.05% (0.002% for m sin i < 28 M Jup ) that these are identical. This implies that inferred sub-stellar companions around K giants in our sample have higher masses compared to companions around F, G and K main sequence stars. Fig. 6 shows the low-mass companion distribution of our survey. Most K giant companions would have inferred masses between 2 and 8 M Jup , while the fraction of companions orbiting F, G and K dwarfs strongly decreases with increasing m sin i.\n\n\nSemi-major axis distribution\n\nThe distribution of the companions' semi-major axis is shown in Fig. 7. No inferred companions with semi-major axis smaller than 0.3 AU are present in the K giant sample, possibly due to increased stellar radii of giants. The fraction of stars with an inferred companion with a semi-major axis between 1 and 3 AU is much higher among the K giants compared to the F, G and K dwarfs. A comparison between the two distributions with a KStest reveals a probability for the two distributions to be identical of 11%. This increases to 32% when omitting the main sequence stars with semi-major axis < 0.3 AU. The increasing incompleteness beyond 3 AU, due to the limited time span of surveys, is present in both samples as the surveys cover a comparable amount of time. This incompleteness cannot cause the significant difference in the peak between 1 and 3 AU. Fig. 8. Period distribution of the observed radial velocity variations shown as a fraction of all significant periods (gray histogram). The period distribution of main sequence stars ) is shown in the hatched histogram, as a fraction of the total number of observed companions. The cross hatched area are main sequence stars with a semi-major axis smaller than 0.3 AU (see Fig. 7).\n\n\nPeriod distribution\n\nIn Fig. 8 the period distribution of the observed radial velocity variations is shown and compared with the companion period distribution of dwarfs. The large fraction of F, G and K dwarf companions with orbital periods shorter than 100 days corresponds to the ones with semi-major axis smaller than approximately 0.3 AU. The close-in short-period companions are not present around K giants, while about 80% of these stars with observed radial velocities have periods ranging between 400 and 800 days. A KS-test reveals a probability of less than 0.0001% for the two distributions to be identical. The probability remains below this level, when the companions orbiting main sequence stars with a semi-major axis < 0.3 AU are omitted. Fig. 9 shows the distribution of companion eccentricities for K giants in our sample and for dwarfs. Companions of dwarfs with periods less than 20 days are excluded, as these might be tidally circularised. The fraction of companion eccentricities < 0.3 for the giants is 75% compared to 50% for companions orbiting F, G and K dwarfs. The KS-test shows that these distributions are nearly identical (97%).\n\n\nEccentricity distribution\n\n\nIron abundance\n\nCompanion occurrence correlates strongly with the abundance of heavy elements (see for instance Gonzalez (1997), Fischer & Valenti (2005) and Santos et al. (2005)), such that F, G, and K dwarf stars with supersolar abundance are more likely to harbour sub-stellar companions (about 50% of the stars with 0.3 < [Fe/H] < 0.5). The increase of the fraction The hatched histogram is the eccentricity distribution of companions around main sequence stars shown as a fraction of all companions around main sequence stars. Companions with periods shorter than 20 days are excluded from the latter sample as these might be tidally circularised.\n\nof F, G and K dwarfs harbouring companions with increasing metallicity is well fitted with a power law, yielding a probability for such a star to harbour a companion to be: P = 0.03·[(N Fe /N H )/(N Fe /N H ) ⊙ ] 2.0 (Fischer & Valenti 2005).\n\nIn Fig. 10 the iron abundance distribution of stars with periodic radial velocity variations is shown as a percentage of the total number of observed stars with iron abundance in the same interval. The iron abundance is determined spectroscopically by imposing excitation and ionisation equilibrium in iron lines and is described by Hekker & Meléndez (2007). The maximum iron abundance of a K giant star in our sample is 0.29 and, therefore, we do not probe the high metallicity region in which F, G and K dwarfs are most likely to harbour a companion.\n\nThe mean metallicity of the K giants in the entire sample is −0.12 dex, while the mean metallicity of the stars with periodic radial velocity variations, presented in Fig. 10, is −0.13 dex. No correlation between companion occurrence and abundance, similar to the one which is present for dwarf stars, is found in this sample of giant stars.\n\n\nDiscussion and Conclusion\n\nThe tight correlation we found between log g and half of the peak-to-peak radial velocity variations seems to indicate that a large fraction of the observed radial velocity variations in our sample of K giants is induced by mechanism(s) intrinsic to the stars. We also present evidence that both intrinsic and extrinsic mechanisms play a role. The stars with a significant periodic signal are almost all located above the radial velocity amplitude vs. log g relation, but when the periodic signal is removed, the residuals show the same trend as for the non-periodic stars. Furthermore, no correlation is present between the amplitude of the periodic signal and log g. Fig. 10. Iron abundance [Fe/H] distribution of K giant stars with periodic radial velocity variations shown as a percentage of the total number of observed stars with iron abundance in the same interval. The numbers above each bar on the histogram indicate the ratio of stars with a significant periodic radial velocity variation to the total number of stars in each bin. The error bars are calculated assuming Poisson statistics (i.e., the percentage of stars with periodic radial velocities divided by the square root of the number of stars with periodic radial velocities). The dashed line is the power law derived for the increasing trend in the fraction of stars with companions as a function of metallicity of F, G and K main sequence stars (Fischer & Valenti 2005).\n\nAlmost all of the lowest log g stars show periodic variations. It may be possible that stars with such low surface gravity cannot be constant and that in these dilute atmospheres instabilities can occur very easily, and therefore may be periodic, but not extrinsic.\n\nBased on the evidence that extrinsic mechanism(s) play a role for K giant stars with periodic radial velocity variations we investigated the hypothesis that this periodic signal is caused by the reflex motion of sub-stellar companions orbiting these stars. We presented the characteristics of the orbital parameters of these companions and compared them with the known orbital parameters of sub-stellar companions orbiting F, G and K dwarfs.\n\nAbout 25% of the stars in our sample have radial velocity variations with significant periodicity, and could possibly harbour a sub-stellar companion, while approximately only 8% of the 1330 F, G and K main sequence stars investigated by Marcy et al. (2005) have a sub-stellar companion. Recently Johnson et al. (2007) and Lovis & Mayor (2007) showed that the number of companion harbouring stars increases with mass. The giants in the present sample have typical masses between 1 and 4 M ⊙ and are in general more massive than the main sequence stars investigated for companions. So, the high percentage is qualitatively in agreement with the results from the literature. Furthermore, Lovis & Mayor (2007) suggest that more massive stars form more massive planetary systems than lower mass stars. Figs 5 and 6 show that we find, in general, more massive companions around the more massive K giants than are present around F, G and K dwarfs.\n\nThe high percentage of more massive companions around the more massive K giant stars would also be compatible with the core accretion model. This model predicts very few giant planets, but a relatively large number of planets with the mass of Neptune or smaller around M dwarfs (Laughlin et al. 2004: Ida & Lin 2005. This is mainly the result of a much reduced surface density of the disk and the resulting shorter disk evolution timescales compared to those for more massive stars, implying that planet properties vary with the mass of their host stars. In particular, one would expect more sub-stellar companions with higher masses in our giant sample, as is indeed the case if we assume that the companion hypothesis is correct.\n\nThe mean metallicity of companion-hosting K giants would be similar to the mean metallicity of the total sample. This is in contrast with the correlation between companion occurrence and metallicity present in F, G and K dwarfs (e.g. Fischer & Valenti 2005). So far, several groups have investigated the correlation between companion occurrence and metallicity for giants with different results. Sadakane et al. (2005) and Pasquini et al. (2007) agree that companion-ahosting giants are on average not metal-rich, while Hekker & Meléndez (2007) find that giants with announced companions have higher metallicities than their total sample of giants. A detailed discussion about these different results is presented by Hekker & Meléndez (2007). They conclude that the samples on which the results are based are slightly different and the more metal-rich stars used in their study are lacking in the study by Pasquini et al. (2007). Furthermore, there is a difference in zero-point correction for the metallicities of announced companion-hosting stars from different surveys. All in all, these inferences are based on smallnumber statistics and all results have to be taken with caution.\n\nThe larger semi-major axis and long periods of the inferred companions orbiting K giant stars compared to companions orbiting dwarf stars are most likely due to the extended atmospheres of K giants. For the eccentricity no significant difference is found in the distribution between companions orbiting dwarfs or giants. Nevertheless, the high number of inferred companions around giants with eccentricities < 0.3 is striking. One could suspect that companion orbits circularise over time and that the companions in circular orbits are older than the eccentric ones, but there is no evidence for this hypothesis.\n\nIn principle, nearly sinusoidal radial velocity variations could also be caused by pulsations or spots. Although the periods of the radial velocity variations could well be the rotational periods of the stars, the presence of prominent spots is not very likely. In that case one would also expect photometric variations with periods correlated with the radial velocity variations. From the Hipparcos photometry (Perryman & ESA 1997) such correlations were not found.\n\nIn order to distinguish with certainty between companions and pulsations as the cause of the observed radial velocity variations, one needs to perform a spectral line profile analysis. A technique for doing this with very high-resolution spectra (R ≥ 100 000) will be presented separately (Hekker et al., in preparation) because the amount of such data at hand today is insufficient to add significantly to the conclusions of this paper.\n\nFig. 1 .\n1Radial velocity variations as a function of phase for a star (HIP34693) with a highly significant period (top), with its periodogram (bottom). The dashed line in the periodogram indicates the significance threshold.\n\nFig. 2 .\n2Radial velocity variations as a function of phase for a star (HIP7607) with a period close to the significance threshold (top), with its periodogram (bottom). The dashed line in the periodogram indicates the significance threshold.\n\nFig. 4 .\n4Top: Half of the peak-to-peak variation of the radial velocity as a function of surface gravity (log g), as inFig. 3, but showing only those stars with periodic radial velocity variations (blue •) and stellar binaries (black ⋄). The blue + and black × symbols indicate the amplitude of the radial velocity variations for these stars after subtraction of the Keplerian fits. (Colours are only visible in the online version of the paper.) The solid line indicates the linear fit through the stars with non-periodic radial velocity variations (fromFig. 3). Bottom: Amplitudes of the subtracted Keplerian fits as a function of log g.\n\nFig. 5\n5shows the distribution of inferred companion masses of our K giant sample. First, notice that 30% of the inferred companions would have masses in the brown dwarf regime 15 M Jup < m sin i < 80 M Jup . This is in sharp contrast to the brown\n\nFig. 5 .\n5A histogram of m sin i of inferred companion masses orbiting K giants in our sample. The dashed line indicates the rise of planet masses M −1.05 from 10 M Jup down to Saturn masses for sub-stellar companions around main sequence stars\n\nFig. 9 .\n9Distribution of eccentricities for possible companions around K giants in our sample (gray histogram) shown as a fraction of all possible sub-stellar companions in the sample.\n\nTable 1 .\n1Single stars with significant frequencies.star \nfrequency period frequency period \nµHz \ndays \nµHz \ndays \nHIP3419 \n0.0638 \n181 \n0.1197 \n97 \nHIP7607 \n0.0216 \n536 \nHIP7884 \n0.0184 \n629 \nHIP13905 \n0.0223 \n519 \nHIP16335 \n0.0197 \n588 \nHIP19011 \n0.0252 \n459 \nHIP21421 \n0.0203 \n570 \n0.0448 \n237 \nHIP23015 \n0.0151 \n767 \n0.0073 \n1586 \nHIP23123 \n0.0135 \n857 \nHIP31592 \n0.0144 \n804 \nHIP33160 \n0.0226 \n512 \nHIP34693 \n0.0380 \n305 \nHIP36616 \n0.0033 \n3507 \n0.0389 \n298 \nHIP37826 \n0.0194 \n597 \nHIP38253 \n0.0171 \n677 \nHIP39177 \n0.0133 \n870 \nHIP40526 \n0.0172 \n673 \nHIP46390 \n0.0233 \n497 \nHIP47959 \n0.0187 \n619 \nHIP53229 \n0.0022 \n5261 \nHIP53261 \n0.0149 \n777 \nHIP57399 \n0.0196 \n591 \nHIP64823 \n0.0021 \n5512 \nHIP69673 \n0.0382 \n303 \n0.0187 \n619 \nHIP73133 \n0.0014 \n8267 \n0.0157 \n737 \nHIP73620 \n0.0226 \n512 \nHIP74732 \n0.0233 \n497 \nHIP75458 \n0.0226 \n512 \nHIP79540 \n0.0203 \n570 \nHIP80693 \n0.0022 \n5261 \n0.0198 \n585 \nHIP84671 \n0.0251 \n461 \nHIP85139 \n0.2895 \n40 \nHIP85355 \n0.0258 \n449 \n0.0056 \n2067 \nHIP85693 \n0.0175 \n661 \nHIP87808 \n0.0153 \n757 \nHIP88048 \n0.0218 \n531 \n0.0029 \n3991 \nHIP91117 \n0.0303 \n382 \nHIP109023 \n0.0199 \n582 \nHIP109492 \n0.0217 \n533 \nHIP109602 \n0.0130 \n890 \nHIP109754 \n0.0189 \n612 \nHIP113562 \n0.0401 \n289 \n0.0472 \n245 \nHIP114855 \n0.0639 \n181 \n\n\nFor updated information on sub-stellar companions, see http://exoplanet.eu and http://exoplanets.org.\nAcknowledgements. We thank Debra Fischer, Geoff Marcy and Paul Butler for useful discussions and the development of the instrumentation and software for the determination of the radial velocities at Lick Observatory. In addition, we thank the entire staff at Lick Observatory for their excellent support. Finally, we would like to thank the anonymous referee for valuable comments and suggestions.\n. P J Armitage, I A Bonnell, MNRAS. 33011Armitage, P. J. & Bonnell, I. A. 2002, MNRAS, 330, L11\n\n. T R Bedding, R P Butler, F Carrier, ApJ. 647558Bedding, T. R., Butler, R. P., Carrier, F., et al. 2006, ApJ, 647, 558\n\n. R P Butler, G W Marcy, E Williams, PASP. 108500Butler, R. P., Marcy, G. W., Williams, E., et al. 1996, PASP, 108, 500\n\n. R P Butler, J T Wright, G W Marcy, ApJ. 646505Butler, R. P., Wright, J. T., Marcy, G. W., et al. 2006, ApJ, 646, 505\n\n. F Castelli, R G Gratton, R L Kurucz, A&A. 324432Castelli, F., Gratton, R. G., & Kurucz, R. L. 1997, A&A, 324, 432\n\n. A Cumming, G W Marcy, R P Butler, ApJ. 526890Cumming, A., Marcy, G. W., & Butler, R. P. 1999, ApJ, 526, 890\n\n. J De Ridder, C Barban, F Carrier, A&A. 448689De Ridder, J., Barban, C., Carrier, F., et al. 2006, A&A, 448, 689\n\n. M P Döllinger, A P Hatzes, L Pasquini, A&A. 472649Döllinger, M. P., Hatzes, A. P., Pasquini, L., et al. 2007, A&A, 472, 649\n\n. P Eggenberger, F Carrier, F Bouchy, A Blecha, A&A. 422247Eggenberger, P., Carrier, F., Bouchy, F., & Blecha, A. 2004, A&A, 422, 247\n\n. D A Fischer, J Valenti, ApJ. 6221102Fischer, D. A. & Valenti, J. 2005, ApJ, 622, 1102\n\n. S Frink, D S Mitchell, A Quirrenbach, ApJ. 576478Frink, S., Mitchell, D. S., Quirrenbach, A., et al. 2002, ApJ, 576, 478\n\n. S Frink, A Quirrenbach, D Fischer, S Röser, E Schilbach, PASP. 113173Frink, S., Quirrenbach, A., Fischer, D., Röser, S., & Schilbach, E. 2001, PASP, 113, 173\n\n. F Galland, A.-M Lagrange, S Udry, A&A. 452709Galland, F., Lagrange, A.-M., Udry, S., et al. 2006, A&A, 452, 709\n\n. B Gladman, Icarus. 106247Gladman, B. 1993, Icarus, 106, 247\n\n. G Gonzalez, MNRAS. 285403Gonzalez, G. 1997, MNRAS, 285, 403\n\nA P Hatzes, W D Cochran, Astronomical Society of the Pacific Conference Series. R. A. Donahue & J. A. Bookbinder154311Cool Stars, Stellar Systems, and the SunHatzes, A. P. & Cochran, W. D. 1998, in Astronomical Society of the Pacific Conference Series, Vol. 154, Cool Stars, Stellar Systems, and the Sun, ed. R. A. Donahue & J. A. Bookbinder, 311\n\n. A P Hatzes, W D Cochran, M Endl, A&A. 457335Hatzes, A. P., Cochran, W. D., Endl, M., et al. 2006, A&A, 457, 335\n\n. A P Hatzes, E W Guenther, M Endl, A&A. 437743Hatzes, A. P., Guenther, E. W., Endl, M., et al. 2005, A&A, 437, 743\n\n. S Hekker, J Meléndez, A&A. 4751003Hekker, S. & Meléndez, J. 2007, A&A, 475, 1003\n\n. S Hekker, S Reffert, A Quirrenbach, A&A. 454943Hekker, S., Reffert, S., Quirrenbach, A., et al. 2006, A&A, 454, 943\n\n. S Ida, D N C Lin, ApJ. 6261045Ida, S. & Lin, D. N. C. 2005, ApJ, 626, 1045\n\n. J A Johnson, D A Fischer, G W Marcy, ApJ. 665785Johnson, J. A., Fischer, D. A., Marcy, G. W., et al. 2007, ApJ, 665, 785\n\n. J A Johnson, G W Marcy, D A Fischer, ApJ. 6521724Johnson, J. A., Marcy, G. W., Fischer, D. A., et al. 2006, ApJ, 652, 1724\n\n. R Kuschnig, W W Weiss, R Gruber, P Y Bely, H Jenkner, A&A. 328544Kuschnig, R., Weiss, W. W., Gruber, R., Bely, P. Y., & Jenkner, H. 1997, A&A, 328, 544\n\n. G Laughlin, P Bodenheimer, F C Adams, ApJ. 61273Laughlin, G., Bodenheimer, P., & Adams, F. C. 2004, ApJ, 612, L73\n\nS Hekker, Precise radial velocities of giant stars. IV. S. Hekker et al.: Precise radial velocities of giant stars. IV.\n\n. C Lovis, M Mayor, G Marcy, R P Butler, D Fischer, Progress of Theoretical Physics Supplement. 70624A&ALovis, C. & Mayor, M. 2007, A&A, 706, in press Marcy, G., Butler, R. P., Fischer, D., et al. 2005, Progress of Theoretical Physics Supplement, 158, 24\n\n. G W Marcy, R P Butler, PASP. 104270Marcy, G. W. & Butler, R. P. 1992, PASP, 104, 270\n\n. G W Marcy, R P Butler, PASP. 112137Marcy, G. W. & Butler, R. P. 2000, PASP, 112, 137\n\n. G W Marcy, R P Butler, S S Vogt, ApJ. 555418Marcy, G. W., Butler, R. P., Vogt, S. S., et al. 2001, ApJ, 555, 418\n\n. M Martić, J.-C Lebrun, T Appourchaux, S G Korzennik, A&A. 418295Martić, M., Lebrun, J.-C., Appourchaux, T., & Korzennik, S. G. 2004, A&A, 418, 295\n\n. A Niedzielski, M Konacki, A Wolszczan, ApJ. 6691354Niedzielski, A., Konacki, M., Wolszczan, A., et al. 2007, ApJ, 669, 1354\n\n. L Pasquini, M P Döllinger, A Weiss, A&A. 473979Pasquini, L., Döllinger, M. P., Weiss, A., et al. 2007, A&A, 473, 979\n\nF Pepe, F Bouchy, D Queloz, M Mayor, Scientific Frontiers in Research on Extrasolar Planets. 29439Pepe, F., Bouchy, F., Queloz, D., & Mayor, M. 2003, in ASP Conf. Ser. 294: Scientific Frontiers in Research on Extrasolar Planets, 39\n\nThe HIPPARCOS and TYCHO catalogues. Astrometric and photometric star catalogues derived from the ESA HIPPARCOS Space Astrometry Mission Press. W H Teukolsky, S A Vetterling, W T Flannery, B P , Numerical recipes in C. The art of scientific computing. Perryman, M. A. C. & ESACambridge: University Press12002nd ed.Perryman, M. A. C. & ESA, eds. 1997, ESA Special Publication, Vol. 1200, The HIPPARCOS and TYCHO cata- logues. Astrometric and photometric star catalogues derived from the ESA HIPPARCOS Space Astrometry Mission Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, Numerical recipes in C. The art of scientific com- puting (Cambridge: University Press, Ic1992, 2nd ed.)\n\n. D Queloz, M Mayor, S Udry, The Messenger. 1051Queloz, D., Mayor, M., Udry, S., et al. 2001, The Messenger, 105, 1\n\n. S Reffert, A Quirrenbach, D S Mitchell, ApJ. 652661Reffert, S., Quirrenbach, A., Mitchell, D. S., et al. 2006, ApJ, 652, 661\n\n. K Sadakane, T Ohnishi, M Ohkubo, Y Takeda, PASJ. 57127Sadakane, K., Ohnishi, T., Ohkubo, M., & Takeda, Y. 2005, PASJ, 57, 127\n\n. N C Santos, G Israelian, M Mayor, A&A. 4371127Santos, N. C., Israelian, G., Mayor, M., et al. 2005, A&A, 437, 1127\n\n. B Sato, H Ando, E Kambe, ApJ. 597157Sato, B., Ando, H., Kambe, E., et al. 2003, ApJ, 597, L157\n\n. J D Scargle, ApJ. 263835Scargle, J. D. 1982, ApJ, 263, 835\n\n. J Setiawan, A P Hatzes, O Von Der Lühe, A&A. 39819Setiawan, J., Hatzes, A. P., von der Lühe, O., et al. 2003, A&A, 398, L19\n\n. J Setiawan, J Rodmann, L Da Silva, A&A. 43731Setiawan, J., Rodmann, J., da Silva, L., et al. 2005, A&A, 437, L31\n\n. C A Sneden, AA(University of Texas at AustinPhD thesisSneden, C. A. 1973, PhD thesis, AA(University of Texas at Austin)\n\n. J A Valenti, R P Butler, G W Marcy, PASP. 107966Valenti, J. A., Butler, R. P., & Marcy, G. W. 1995, PASP, 107, 966\n"},"corpus_id":{"kind":"number","value":33442610,"string":"33,442,610"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"0801.0741\",\n \"mag\": 2103449991,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1051/0004-6361:20078321\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/0801.0741v1.pdf\"\n ],\n \"pdf_sha\": \"11c7588046924474bb2d92e471addbf94e00406e\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"https://www.aanda.org/articles/aa/pdf/2008/10/aa8321-07.pdf\",\n \"status\": \"BRONZE\"\n }\n}"},"title":{"kind":"string","value":"Precise radial velocities of giant stars. IV. ⋆ A correlation between surface gravity and radial velocity variation and a statistical investigation of companion properties"},"authors":{"kind":"list like","value":["S Hekker \nLeiden Observatory\nLeiden University\nP.O. Box 95132300 RALeidenThe Netherlands\n\nRoyal Observatory of Belgium\nRinglaan 31180BrusselsBelgium","I A G Snellen \nLeiden Observatory\nLeiden University\nP.O. Box 95132300 RALeidenThe Netherlands","C Aerts \nInstituut voor Sterrenkunde\nKatholieke Universiteit Leuven\nCelestijnenlaan 200 D3001LeuvenBelgium\n\nDepartment of Astrophysics\nUniversity of Nijmegen\nP.O. Box 90106500 GLNijmegenThe Netherlands","A Quirrenbach \nLeiden Observatory\nLeiden University\nP.O. Box 95132300 RALeidenThe Netherlands\n\nZAH\nLandessternwarte Heidelberg, Königstuhl 12D-69117HeidelbergGermany","S Reffert \nZAH\nLandessternwarte Heidelberg, Königstuhl 12D-69117HeidelbergGermany","D S Mitchell \nCalifornia Polytechnic State University\n93407San Luis ObispoCAUSA"],"string":"[\n \"S Hekker \\nLeiden Observatory\\nLeiden University\\nP.O. Box 95132300 RALeidenThe Netherlands\\n\\nRoyal Observatory of Belgium\\nRinglaan 31180BrusselsBelgium\",\n \"I A G Snellen \\nLeiden Observatory\\nLeiden University\\nP.O. Box 95132300 RALeidenThe Netherlands\",\n \"C Aerts \\nInstituut voor Sterrenkunde\\nKatholieke Universiteit Leuven\\nCelestijnenlaan 200 D3001LeuvenBelgium\\n\\nDepartment of Astrophysics\\nUniversity of Nijmegen\\nP.O. Box 90106500 GLNijmegenThe Netherlands\",\n \"A Quirrenbach \\nLeiden Observatory\\nLeiden University\\nP.O. Box 95132300 RALeidenThe Netherlands\\n\\nZAH\\nLandessternwarte Heidelberg, Königstuhl 12D-69117HeidelbergGermany\",\n \"S Reffert \\nZAH\\nLandessternwarte Heidelberg, Königstuhl 12D-69117HeidelbergGermany\",\n \"D S Mitchell \\nCalifornia Polytechnic State University\\n93407San Luis ObispoCAUSA\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Context. Since 1999, we have been conducting a radial velocity survey of 179 K giants using the Coudé Auxiliary Telescope at UCO/Lick observatory. At present ∼20−100 measurements have been collected per star with a precision of 5 to 8 m s −1 . Of the stars monitored, 145 (80%) show radial velocity (RV) variations at a level >20 m s −1 , of which 43 exhibit significant periodicities. Aims. Our aim is to investigate possible mechanism(s) that cause these observed RV variations. We intend to test whether these variations are intrinsic in nature, or possibly induced by companions, or both. In addition, we aim to characterise the parameters of these companions. Methods. A relation between log g and the amplitude of the RV variations is investigated for all stars in the sample. Furthermore, the hypothesis that all periodic RV variations are caused by companions is investigated by comparing their inferred orbital statistics with the statistics of companions around main sequence F, G, and K dwarfs. Results. A strong relation is found between the amplitude of the RV variations and log g in K giant stars, as suggested earlier byHatzes & Cochran (1998). However, most of the stars exhibiting periodic variations are located above this relation. These RV variations can be split in a periodic component which is not correlated with log g and a random residual part which does correlate with log g. Compared to main-sequence dwarf stars, K giants frequently exhibit periodic RV variations. Interpreting these RV variations as being caused by companions, the orbital parameters are different from the companions orbiting dwarfs. Conclusions. Intrinsic mechanisms play an important role in producing RV variations in K giants stars, as suggested by their dependence on log g. However, it appears that periodic RV variations are additional to these intrinsic variations, consistent with them being caused by companions. If indeed the majority of the periodic RV variations in K giants is interpreted as due to substellar companions, then massive planets are significantly more common around K giants than around F, G, K main-sequence stars."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"For more than a decade, radial velocity observations with accuracies of order m s −1 have been within reach (see for instance Marcy & Butler (2000) and Queloz et al. (2001)). Even accuracies of less than 1 m s −1 (Pepe et al. 2003) are possible now. With these observations, more than 200 sub-stellar companions have been discovered by measuring the reflex motions of their parent stars. Most of these sub-stellar companions have been detected around F, G and K main sequence Send offprint requests to: S. Hekker, email: saskia@oma.be ⋆ Based on data obtained at UCO/Lick Observatory, USA stars, but detections around an A star (Galland et al. 2006) and several subgiants (Johnson et al. (2006), Johnson et al. (2007)) have also been reported recently. Moreover, 10 giant stars were reported to have sub-stellar companions (ι Draconis (K2III) Frink et al. (2002), HD104985 (G9III) Sato et al. (2003), HD47526 (K1III) Setiawan et al. (2003), HD13189 (K2II-III) Hatzes et al. (2005), HD11977 (G5III) Setiawan et al. (2005), Pollux (K0III) Hatzes et al. (2006), Reffert et al. (2006), 4UMa (K1III) Döllinger et al. (2007), NGC2423 No3 and NGC4349 No127 Lovis & Mayor (2007), and recently HD17092 (K0III) Niedzielski et al. (2007)) 1 . In addition to searches for extra-solar companions, radial velocity observations prove to be very useful for detecting solarlike oscillations in stars with turbulent atmospheres, such as the dwarf α Cen A (e.g. Bedding et al. 2006), the subgiant Procyon (e.g. Eggenberger et al. 2004: Martić et al. 2004) and the giant ǫ Ophiuchi (e.g. De Ridder et al. 2006). With techniques for accurate radial velocity observations at hand, a survey was started in 1999 to verify whether K giants are stable enough to be used as astrometric reference stars for SIM/PlanetQuest (Space Interferometry Mission) (Frink et al. 2001). This survey contains 179 stars and uses the Coudé Auxiliary Telescope (CAT) at University of California Observatories / Lick Observatory, in conjunction with the Hamilton Echelle Spectrograph. The survey has recently been expanded to about 380 giants and is still ongoing. For the analysis described in the present paper only data from the initial 179 stars are used.\nFrom this survey, companions have been announced for ι Draconis (Frink et al. 2002) and Pollux . Stars with radial velocity variations of less than 20 m s −1 have been presented as stable stars by Hekker et al. (2006). In addition, some binaries discovered with this survey, as well as an extensive overview of the sample, will be presented in forthcoming papers.\nAs almost all of the stars show significant radial velocity variations, we investigate here which mechanism causes these variations. Non-periodic radial velocity variations, of the order of the investigated timescales, are most likely caused by some intrinsic mechanism, while the periodic variability can also be caused by companions. We also investigate the characteristics of these companions.\nIn Sect. 2, the radial velocity observations are described in detail. In Sect. 3, the relation between the observed radial velocity amplitude and surface gravity is investigated. In Sect. 4, we explore the hypothesis that all periodic radial velocity variations are caused by sub-stellar companions, and we compare the inferred orbital parameters with those obtained for substellar companions orbiting main sequence stars. Our conclusions are presented in Sect. 5."},{"title":"Radial velocity variations","content":"The initial 179 stars selected for the radial velocity survey are used in the present work. These stars have been selected from the Hipparcos catalogue (Perryman & ESA 1997), based on the criteria described by Frink et al. (2001). The selected stars are all brighter than 6 mag, are presumably single and have photometric variations < 0.06 mag in V. The survey started in 1999 at Lick Observatory using the Coudé Auxiliary Telescope (CAT) in conjunction with the Hamilton Echelle Spectrograph (R=60 000). The system with an iodine cell in the light path has been described by Marcy & Butler (1992) and Valenti et al. (1995). With integration times of up to thirty minutes for the faintest stars (m v = 6 mag) we reach a signal to noise ratio of about 80 − 100 at λ = 5500 Å, yielding a radial velocity precision of 5 − 8 m s −1 . As we are looking for radial velocity variations of order 10 to 100 m s −1 , this is adequate and hence no attempt has been made to reach the 3 m s −1 accuracy which is in principle possible with this setup (Butler et al. 1996). For the determination of the radial velocities the pipeline described by Butler et al. (1996) is used. In this pipeline, a template iodine spectrum and a template spectrum of the target star obtained without an iodine cell in the lightpath are used to model the stellar observations with a superposed iodine spectrum. The Doppler shift is a free parameter in this model. Note that with this method the absolute radial velocity is not measured, but the radial velocity relative to the stellar template is obtained.\nAll 179 stars are subjected to a period search. The periodicity of the radial velocity variations is determined first of all from a classical Lomb-Scargle (LS) periodogram (Scargle 1982). The significance threshold is set to 6σ, where the noise level is determined from the average power of the residual Scargle periodogram for frequencies between 0 and 0.03 cycles per day (c/d) (0.35 µHz) and a frequency step of 0.00001 c/d (0.12 · 10 −3 µHz). We adopted the conventional method of iterative sinewave fitting ('prewhitening') to search for subsequent frequencies (Kuschnig et al. 1997). In Figs 1 and 2, the radial velocity variation as a function of phase is shown for two stars. The period of the star in Fig. 1 is highly significant, while the one in Fig. 2 is close to the significance threshold. Periodograms are shown in the bottom panels of these figures. As properly emphasized by Cumming et al. (1999), such a classical period search may not be appropriate for unevenly spaced sparse data, even though we set the significance level at a conservatively high level. In order to check this, we have done an additional LS analysis after prewhitening the original data by a linear polynomial. This led almost always to the same frequencies. We only accepted a frequency when it was found to meet the significance criterion for both these analyses. The significant frequencies are listed in Table 1. Hatzes & Cochran (1998) already investigated the origin of the observed radial velocities in K giant stars. Although their sample contained only 9 stars, they suggested that the amplitude of the radial velocity increases with decreasing surface gravity (log g). In lower surface gravity it takes longer to decrease the velocity of a moving parcel which results in larger amplitudes and the relation suggested by Hatzes & Cochran (1998) would therefore be evidence for pulsations or rotational modulation as the mechanism for these long period radial velocity variations. For the present sample, log g values were determined spectroscopically by Hekker & Meléndez (2007), by imposing excitation and ionisation equilibrium of iron lines through stellar models. The equivalent width of about two dozen carefully selected iron lines were used for a spectroscopic LTE analysis based on the 2002 version of MOOG (Sneden 1973) and Kurucz model atmospheres which include overshooting (Castelli et al. 1997). These authors estimated the error on log g to be 0.22 dex from the scatter found in a comparison with literature values. A detailed description of the stellar parameters for individual stars and a comparison with literature values are available in Hekker & Meléndez (2007) and is therefore omitted here. In Fig. 3, we show half of the peak-to-peak value of the observed radial velocity variations as a function of log g for K giants in our sample. A clear trend is visible between increasing radial velocity variations in single stars, and decreasing log g, which provides a strong indication that, at least for a large fraction of stars in our sample, the observed radial velocity vari-"},{"title":"Radial velocity amplitude -surface gravity relation","content":""},{"title":"Fig. 3.","content":"Half of the peak-to-peak variation of the radial velocity as a function of surface gravity (log g). The blue • indicate the single stars with periodic radial velocity variations (see text for periodicity criteria), the red indicate single stars with random radial velocity variations, and stellar binaries (companion mass > 100 M Jup ) are indicated with black ⋄ symbols. (Colours are only visible in the online version of the paper.) The solid line is the best fit through the random single stars, the dotted line indicates the 1σ interval around the best fit and the dashed line indicates the 3.5σ interval. Single stars with a higher radial velocity amplitude than expected based on their log g value (more than 3.5σ above the best fit) are indicated by arrows. Six of the 8 stars with periodic radial velocity variations and log g < 1.6 are classified bright giants or supergiants (Perryman & ESA 1997).\nations are induced by a mechanism intrinsic to the star. This trend is present for stars with random as well as stars with periodic radial velocity variations. Also, nearly all stars with periodic radial velocity variations and log g ≥ 1.6 are located above the fit in Fig. 3. Seven single stars have a higher radial velocity variation than expected based on their log g value, i.e. they are situated more than 3.5σ above the best fit for the relation obtained for single stars. These stars are indicated with arrows in Fig. 3. The radial velocity variation observed for HIP53229 may be due to a stellar companion in a wide orbit, with a period much longer than the observation time span. Due to this long period the companion mass, and, therefore, the (sub-)stellar nature, is still very uncertain. HIP33152 is classified as a supergiant. The observed radial velocity variations for HIP80693, HIP36616, and HIP88048 can be fitted with two Keplerian orbits, while HIP75458 can be explained by an eccentric substellar companion (Frink et al. 2002) and an additional linear trend, indicating a companion in a wide orbit. HIP34693 can be fitted very accurately with a Keplerian orbit of a single nearly sinusoidal sub-stellar companion.\nIn order to investigate the simultaneous occurrence of substellar companions and oscillations in giants, the best Keplerian fits are subtracted from the observed radial velocity variations of both the binaries and the single stars with significant periodic radial velocity variations. The half peak-to-peak values of these residuals are plotted as a function of surface gravity in Fig. 4, also showing the linear fit through the non-periodic stars, and the amplitudes of the subtracted periodic signals. There are several interesting points to make about this graph. Firstly, almost all periodic stars show larger radial velocity variations than predicted by the relation found for non-periodic stars, but when the periodicities are removed, their residual radial velocity variations follow the same relation as found for the non-periodic stars. This could be interpreted as evidence for both intrinsic (non-periodic radial velocity variation) and extrinsic (periodic variations) mechanisms playing a role in these stars. Secondly, there is no correlation between the amplitude of the subtracted periodic signal and log g, which provides additional evidence for the presence of companions. Thirdly, almost all (8 out of 9) stars with log g ≤ 1.6 exhibit periodic variations. If these indeed have an extrinsic mechanism, it would mean that ∼ 90% of these stars have sub-stellar companions. However, from an astrophysical point of view, stars with such low surface gravities are already very high on the giant branch or even on the asymptotic giant branch. At these low gravities, stars cannot be constant anymore, as the outer layers are so diluted that instabilities occur easily, either periodic or random. Stated differently, these stars are very close to the semi-regulars, which are on their way to become Mira variables. Hence, these periodic variations could well be intrinsic."},{"title":"Companion Interpretation","content":"From the analysis of the correlation between radial velocity amplitude and surface gravity we have evidence for the presence of both intrinsic variability and companions in at least a fraction of the K giant stars with periodic radial velocity variations. In order to study the characteristics of these companions, we take the hypothesis that the periodic radial velocity variations detected in 43 of our K giant stars, excluding binaries, are caused by sub-stellar companions. Under this hypothesis, we investigate the statistical properties of the orbital parameters of the sample and compare these with the statistical properties of companions orbiting F, G and K dwarfs.\nAccording to our analysis, 55 stars in the sample would have a single companion, and 11 stars multiple companions.\nTwenty-three (22 single and 1 in a multiple system) of these companions have m sin i larger than 100 M Jup and should be interpreted as stellar binaries. By advancing the multiple substellar systems forward in time via a Runga-Kutta integration, we investigated the stability of the systems, taking into account the mutual interactions of the companions. With the orbital parameters that minimise χ 2 taken at face value, we found that most of the inferred sub-stellar multiple systems would be 'likely unstable', with a change in semi-major axis > 1% and < 10%, or 'unstable' with a change in semi-major axis > 10% on a time scale of 100 years due to companion-companion interaction.\nHowever, the inferred stability depends on the starting epoch of the computations, as well as on the orbital parameters, which might change with an increasing number of observations. Furthermore, there is no guarantee that the obtained χ 2 minimum is a global minimum. Therefore, stars with multiple inferred sub-stellar companions that seem to be unstable, might also have stable solutions. One could also use the equations for dynamical stability described by Gladman (1993) and Marcy et al. (2001). Gladman (1993) also notes that the Hill stability criteria for companions in initially eccentric orbits may not be met, but that the systems may still be found to be empirically quite stable for a long period of time. In order to draw a firm conclusion on the stability of a particular system, a more thorough investigation is needed, as well as data with a longer time base, which is beyond the scope of this paper.\nFor all stars with periodic radial velocity variations, we checked the Hipparcos (Perryman & ESA 1997) photometry. We checked periodograms for significant frequencies close to the obtained radial velocity period, and plotted the photometric values phased with the radial velocity period. None of the stars show photometric variations related to the observed radial velocity variations.\nThe mass distribution of our K giant sample is not known very well. The stellar masses are typically between 1 and 4 M ⊙ . Hence most of their main sequence progenitors should have been of A or F spectral class. The distribution of orbital parameters of sub-stellar companions orbiting A and F main sequence stars is still unknown. The core accretion model predicts more giant planets around more massive stars, so that the distribution of orbital parameters of sub-stellar companions orbiting F, G and K main-sequence stars probably cannot serve as a proxy. However, it should be instructive to compare the two distributions.\nSince the data presented here span ∼ 2500 days, radial velocity variations with longer periods are uncertain, and, therefore, not taken into consideration. Companions with periods exceeding the observation time span are also excluded from the F, G, and K main sequence star statistics. (Marcy et al. 2005), normalised to the number of stars in our sample. Fig. 6. Zoom in on the low end of the companion mass distribution of inferred sub-stellar companions orbiting K giant stars in our survey, shown as percentage of the total number of stars in the sample (gray histogram). The hatched histogram shows the distribution of companion masses orbiting main sequence stars as shown in Fig. 1 of Marcy et al. (2005), as a percentage of the total number of stars in their sample. The cross hatched area are main sequence stars with companions at a semi-major axis smaller than 0.3 AU (see Fig. 7). dwarf statistics around F, G and K main sequence stars for which only very few companions are found with m sin i > 15 M Jup around more than thousand stars. This is known as the brown dwarf desert and is possibly caused by migration and merging of brown dwarfs in a viscous disk with a mass at least comparable to the brown dwarf mass (Armitage & Bonnell 2002)."},{"title":"Mass distribution","content":"The dashed line in Fig. 5 indicates the rise of sub-stellar companion masses M −1.05 from 10 M Jup down to Saturn masses for main sequence stars (Marcy et al. 2005), normalised Fig. 7. Semi-major axis distribution of inferred K giant companions in our sample (gray histogram), shown as a percentage of the total number of stars in the sample. The hatched histogram shows the distribution for F, G and K main sequence stars as shown in Fig. 2 of Marcy et al. (2005), as a percentage of the total number of stars in their sample. The main sequence stars with a companion orbiting at a semi-major axis smaller than 0.3 AU are cross-hatched. These are also indicated in Figs 6 and 8.\nto the number of stars in our sample. We use a two-sided Kolmogorov-Smirnov test (Press et al. 1992) (hereafter KStest) to compare the M −1.05 fit and the mass distribution (m sin i < 80 M Jup ) of our sample and find a probability of 0.05% (0.002% for m sin i < 28 M Jup ) that these are identical. This implies that inferred sub-stellar companions around K giants in our sample have higher masses compared to companions around F, G and K main sequence stars. Fig. 6 shows the low-mass companion distribution of our survey. Most K giant companions would have inferred masses between 2 and 8 M Jup , while the fraction of companions orbiting F, G and K dwarfs strongly decreases with increasing m sin i."},{"title":"Semi-major axis distribution","content":"The distribution of the companions' semi-major axis is shown in Fig. 7. No inferred companions with semi-major axis smaller than 0.3 AU are present in the K giant sample, possibly due to increased stellar radii of giants. The fraction of stars with an inferred companion with a semi-major axis between 1 and 3 AU is much higher among the K giants compared to the F, G and K dwarfs. A comparison between the two distributions with a KStest reveals a probability for the two distributions to be identical of 11%. This increases to 32% when omitting the main sequence stars with semi-major axis < 0.3 AU. The increasing incompleteness beyond 3 AU, due to the limited time span of surveys, is present in both samples as the surveys cover a comparable amount of time. This incompleteness cannot cause the significant difference in the peak between 1 and 3 AU. Fig. 8. Period distribution of the observed radial velocity variations shown as a fraction of all significant periods (gray histogram). The period distribution of main sequence stars ) is shown in the hatched histogram, as a fraction of the total number of observed companions. The cross hatched area are main sequence stars with a semi-major axis smaller than 0.3 AU (see Fig. 7)."},{"title":"Period distribution","content":"In Fig. 8 the period distribution of the observed radial velocity variations is shown and compared with the companion period distribution of dwarfs. The large fraction of F, G and K dwarf companions with orbital periods shorter than 100 days corresponds to the ones with semi-major axis smaller than approximately 0.3 AU. The close-in short-period companions are not present around K giants, while about 80% of these stars with observed radial velocities have periods ranging between 400 and 800 days. A KS-test reveals a probability of less than 0.0001% for the two distributions to be identical. The probability remains below this level, when the companions orbiting main sequence stars with a semi-major axis < 0.3 AU are omitted. Fig. 9 shows the distribution of companion eccentricities for K giants in our sample and for dwarfs. Companions of dwarfs with periods less than 20 days are excluded, as these might be tidally circularised. The fraction of companion eccentricities < 0.3 for the giants is 75% compared to 50% for companions orbiting F, G and K dwarfs. The KS-test shows that these distributions are nearly identical (97%)."},{"title":"Eccentricity distribution","content":""},{"title":"Iron abundance","content":"Companion occurrence correlates strongly with the abundance of heavy elements (see for instance Gonzalez (1997), Fischer & Valenti (2005) and Santos et al. (2005)), such that F, G, and K dwarf stars with supersolar abundance are more likely to harbour sub-stellar companions (about 50% of the stars with 0.3 < [Fe/H] < 0.5). The increase of the fraction The hatched histogram is the eccentricity distribution of companions around main sequence stars shown as a fraction of all companions around main sequence stars. Companions with periods shorter than 20 days are excluded from the latter sample as these might be tidally circularised.\nof F, G and K dwarfs harbouring companions with increasing metallicity is well fitted with a power law, yielding a probability for such a star to harbour a companion to be: P = 0.03·[(N Fe /N H )/(N Fe /N H ) ⊙ ] 2.0 (Fischer & Valenti 2005).\nIn Fig. 10 the iron abundance distribution of stars with periodic radial velocity variations is shown as a percentage of the total number of observed stars with iron abundance in the same interval. The iron abundance is determined spectroscopically by imposing excitation and ionisation equilibrium in iron lines and is described by Hekker & Meléndez (2007). The maximum iron abundance of a K giant star in our sample is 0.29 and, therefore, we do not probe the high metallicity region in which F, G and K dwarfs are most likely to harbour a companion.\nThe mean metallicity of the K giants in the entire sample is −0.12 dex, while the mean metallicity of the stars with periodic radial velocity variations, presented in Fig. 10, is −0.13 dex. No correlation between companion occurrence and abundance, similar to the one which is present for dwarf stars, is found in this sample of giant stars."},{"title":"Discussion and Conclusion","content":"The tight correlation we found between log g and half of the peak-to-peak radial velocity variations seems to indicate that a large fraction of the observed radial velocity variations in our sample of K giants is induced by mechanism(s) intrinsic to the stars. We also present evidence that both intrinsic and extrinsic mechanisms play a role. The stars with a significant periodic signal are almost all located above the radial velocity amplitude vs. log g relation, but when the periodic signal is removed, the residuals show the same trend as for the non-periodic stars. Furthermore, no correlation is present between the amplitude of the periodic signal and log g. Fig. 10. Iron abundance [Fe/H] distribution of K giant stars with periodic radial velocity variations shown as a percentage of the total number of observed stars with iron abundance in the same interval. The numbers above each bar on the histogram indicate the ratio of stars with a significant periodic radial velocity variation to the total number of stars in each bin. The error bars are calculated assuming Poisson statistics (i.e., the percentage of stars with periodic radial velocities divided by the square root of the number of stars with periodic radial velocities). The dashed line is the power law derived for the increasing trend in the fraction of stars with companions as a function of metallicity of F, G and K main sequence stars (Fischer & Valenti 2005).\nAlmost all of the lowest log g stars show periodic variations. It may be possible that stars with such low surface gravity cannot be constant and that in these dilute atmospheres instabilities can occur very easily, and therefore may be periodic, but not extrinsic.\nBased on the evidence that extrinsic mechanism(s) play a role for K giant stars with periodic radial velocity variations we investigated the hypothesis that this periodic signal is caused by the reflex motion of sub-stellar companions orbiting these stars. We presented the characteristics of the orbital parameters of these companions and compared them with the known orbital parameters of sub-stellar companions orbiting F, G and K dwarfs.\nAbout 25% of the stars in our sample have radial velocity variations with significant periodicity, and could possibly harbour a sub-stellar companion, while approximately only 8% of the 1330 F, G and K main sequence stars investigated by Marcy et al. (2005) have a sub-stellar companion. Recently Johnson et al. (2007) and Lovis & Mayor (2007) showed that the number of companion harbouring stars increases with mass. The giants in the present sample have typical masses between 1 and 4 M ⊙ and are in general more massive than the main sequence stars investigated for companions. So, the high percentage is qualitatively in agreement with the results from the literature. Furthermore, Lovis & Mayor (2007) suggest that more massive stars form more massive planetary systems than lower mass stars. Figs 5 and 6 show that we find, in general, more massive companions around the more massive K giants than are present around F, G and K dwarfs.\nThe high percentage of more massive companions around the more massive K giant stars would also be compatible with the core accretion model. This model predicts very few giant planets, but a relatively large number of planets with the mass of Neptune or smaller around M dwarfs (Laughlin et al. 2004: Ida & Lin 2005. This is mainly the result of a much reduced surface density of the disk and the resulting shorter disk evolution timescales compared to those for more massive stars, implying that planet properties vary with the mass of their host stars. In particular, one would expect more sub-stellar companions with higher masses in our giant sample, as is indeed the case if we assume that the companion hypothesis is correct.\nThe mean metallicity of companion-hosting K giants would be similar to the mean metallicity of the total sample. This is in contrast with the correlation between companion occurrence and metallicity present in F, G and K dwarfs (e.g. Fischer & Valenti 2005). So far, several groups have investigated the correlation between companion occurrence and metallicity for giants with different results. Sadakane et al. (2005) and Pasquini et al. (2007) agree that companion-ahosting giants are on average not metal-rich, while Hekker & Meléndez (2007) find that giants with announced companions have higher metallicities than their total sample of giants. A detailed discussion about these different results is presented by Hekker & Meléndez (2007). They conclude that the samples on which the results are based are slightly different and the more metal-rich stars used in their study are lacking in the study by Pasquini et al. (2007). Furthermore, there is a difference in zero-point correction for the metallicities of announced companion-hosting stars from different surveys. All in all, these inferences are based on smallnumber statistics and all results have to be taken with caution.\nThe larger semi-major axis and long periods of the inferred companions orbiting K giant stars compared to companions orbiting dwarf stars are most likely due to the extended atmospheres of K giants. For the eccentricity no significant difference is found in the distribution between companions orbiting dwarfs or giants. Nevertheless, the high number of inferred companions around giants with eccentricities < 0.3 is striking. One could suspect that companion orbits circularise over time and that the companions in circular orbits are older than the eccentric ones, but there is no evidence for this hypothesis.\nIn principle, nearly sinusoidal radial velocity variations could also be caused by pulsations or spots. Although the periods of the radial velocity variations could well be the rotational periods of the stars, the presence of prominent spots is not very likely. In that case one would also expect photometric variations with periods correlated with the radial velocity variations. From the Hipparcos photometry (Perryman & ESA 1997) such correlations were not found.\nIn order to distinguish with certainty between companions and pulsations as the cause of the observed radial velocity variations, one needs to perform a spectral line profile analysis. A technique for doing this with very high-resolution spectra (R ≥ 100 000) will be presented separately (Hekker et al., in preparation) because the amount of such data at hand today is insufficient to add significantly to the conclusions of this paper."},{"title":"Fig. 1 .","content":""},{"title":"Fig. 2 .","content":""},{"title":"Fig. 4 .","content":""},{"title":"Fig. 5","content":""},{"title":"Fig. 5 .","content":""},{"title":"Fig. 9 .","content":""},{"title":"Table 1 .","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"For more than a decade, radial velocity observations with accuracies of order m s −1 have been within reach (see for instance Marcy & Butler (2000) and Queloz et al. (2001)). Even accuracies of less than 1 m s −1 (Pepe et al. 2003) are possible now. With these observations, more than 200 sub-stellar companions have been discovered by measuring the reflex motions of their parent stars. Most of these sub-stellar companions have been detected around F, G and K main sequence Send offprint requests to: S. Hekker, email: saskia@oma.be ⋆ Based on data obtained at UCO/Lick Observatory, USA stars, but detections around an A star (Galland et al. 2006) and several subgiants (Johnson et al. (2006), Johnson et al. (2007)) have also been reported recently. Moreover, 10 giant stars were reported to have sub-stellar companions (ι Draconis (K2III) Frink et al. (2002), HD104985 (G9III) Sato et al. (2003), HD47526 (K1III) Setiawan et al. (2003), HD13189 (K2II-III) Hatzes et al. (2005), HD11977 (G5III) Setiawan et al. (2005), Pollux (K0III) Hatzes et al. (2006), Reffert et al. (2006), 4UMa (K1III) Döllinger et al. (2007), NGC2423 No3 and NGC4349 No127 Lovis & Mayor (2007), and recently HD17092 (K0III) Niedzielski et al. (2007)) 1 . In addition to searches for extra-solar companions, radial velocity observations prove to be very useful for detecting solarlike oscillations in stars with turbulent atmospheres, such as the dwarf α Cen A (e.g. Bedding et al. 2006), the subgiant Procyon (e.g. Eggenberger et al. 2004: Martić et al. 2004) and the giant ǫ Ophiuchi (e.g. De Ridder et al. 2006). With techniques for accurate radial velocity observations at hand, a survey was started in 1999 to verify whether K giants are stable enough to be used as astrometric reference stars for SIM/PlanetQuest (Space Interferometry Mission) (Frink et al. 2001). This survey contains 179 stars and uses the Coudé Auxiliary Telescope (CAT) at University of California Observatories / Lick Observatory, in conjunction with the Hamilton Echelle Spectrograph. The survey has recently been expanded to about 380 giants and is still ongoing. For the analysis described in the present paper only data from the initial 179 stars are used.\\nFrom this survey, companions have been announced for ι Draconis (Frink et al. 2002) and Pollux . Stars with radial velocity variations of less than 20 m s −1 have been presented as stable stars by Hekker et al. (2006). In addition, some binaries discovered with this survey, as well as an extensive overview of the sample, will be presented in forthcoming papers.\\nAs almost all of the stars show significant radial velocity variations, we investigate here which mechanism causes these variations. Non-periodic radial velocity variations, of the order of the investigated timescales, are most likely caused by some intrinsic mechanism, while the periodic variability can also be caused by companions. We also investigate the characteristics of these companions.\\nIn Sect. 2, the radial velocity observations are described in detail. In Sect. 3, the relation between the observed radial velocity amplitude and surface gravity is investigated. In Sect. 4, we explore the hypothesis that all periodic radial velocity variations are caused by sub-stellar companions, and we compare the inferred orbital parameters with those obtained for substellar companions orbiting main sequence stars. Our conclusions are presented in Sect. 5.\"\n },\n {\n \"title\": \"Radial velocity variations\",\n \"content\": \"The initial 179 stars selected for the radial velocity survey are used in the present work. These stars have been selected from the Hipparcos catalogue (Perryman & ESA 1997), based on the criteria described by Frink et al. (2001). The selected stars are all brighter than 6 mag, are presumably single and have photometric variations < 0.06 mag in V. The survey started in 1999 at Lick Observatory using the Coudé Auxiliary Telescope (CAT) in conjunction with the Hamilton Echelle Spectrograph (R=60 000). The system with an iodine cell in the light path has been described by Marcy & Butler (1992) and Valenti et al. (1995). With integration times of up to thirty minutes for the faintest stars (m v = 6 mag) we reach a signal to noise ratio of about 80 − 100 at λ = 5500 Å, yielding a radial velocity precision of 5 − 8 m s −1 . As we are looking for radial velocity variations of order 10 to 100 m s −1 , this is adequate and hence no attempt has been made to reach the 3 m s −1 accuracy which is in principle possible with this setup (Butler et al. 1996). For the determination of the radial velocities the pipeline described by Butler et al. (1996) is used. In this pipeline, a template iodine spectrum and a template spectrum of the target star obtained without an iodine cell in the lightpath are used to model the stellar observations with a superposed iodine spectrum. The Doppler shift is a free parameter in this model. Note that with this method the absolute radial velocity is not measured, but the radial velocity relative to the stellar template is obtained.\\nAll 179 stars are subjected to a period search. The periodicity of the radial velocity variations is determined first of all from a classical Lomb-Scargle (LS) periodogram (Scargle 1982). The significance threshold is set to 6σ, where the noise level is determined from the average power of the residual Scargle periodogram for frequencies between 0 and 0.03 cycles per day (c/d) (0.35 µHz) and a frequency step of 0.00001 c/d (0.12 · 10 −3 µHz). We adopted the conventional method of iterative sinewave fitting ('prewhitening') to search for subsequent frequencies (Kuschnig et al. 1997). In Figs 1 and 2, the radial velocity variation as a function of phase is shown for two stars. The period of the star in Fig. 1 is highly significant, while the one in Fig. 2 is close to the significance threshold. Periodograms are shown in the bottom panels of these figures. As properly emphasized by Cumming et al. (1999), such a classical period search may not be appropriate for unevenly spaced sparse data, even though we set the significance level at a conservatively high level. In order to check this, we have done an additional LS analysis after prewhitening the original data by a linear polynomial. This led almost always to the same frequencies. We only accepted a frequency when it was found to meet the significance criterion for both these analyses. The significant frequencies are listed in Table 1. Hatzes & Cochran (1998) already investigated the origin of the observed radial velocities in K giant stars. Although their sample contained only 9 stars, they suggested that the amplitude of the radial velocity increases with decreasing surface gravity (log g). In lower surface gravity it takes longer to decrease the velocity of a moving parcel which results in larger amplitudes and the relation suggested by Hatzes & Cochran (1998) would therefore be evidence for pulsations or rotational modulation as the mechanism for these long period radial velocity variations. For the present sample, log g values were determined spectroscopically by Hekker & Meléndez (2007), by imposing excitation and ionisation equilibrium of iron lines through stellar models. The equivalent width of about two dozen carefully selected iron lines were used for a spectroscopic LTE analysis based on the 2002 version of MOOG (Sneden 1973) and Kurucz model atmospheres which include overshooting (Castelli et al. 1997). These authors estimated the error on log g to be 0.22 dex from the scatter found in a comparison with literature values. A detailed description of the stellar parameters for individual stars and a comparison with literature values are available in Hekker & Meléndez (2007) and is therefore omitted here. In Fig. 3, we show half of the peak-to-peak value of the observed radial velocity variations as a function of log g for K giants in our sample. A clear trend is visible between increasing radial velocity variations in single stars, and decreasing log g, which provides a strong indication that, at least for a large fraction of stars in our sample, the observed radial velocity vari-\"\n },\n {\n \"title\": \"Radial velocity amplitude -surface gravity relation\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 3.\",\n \"content\": \"Half of the peak-to-peak variation of the radial velocity as a function of surface gravity (log g). The blue • indicate the single stars with periodic radial velocity variations (see text for periodicity criteria), the red indicate single stars with random radial velocity variations, and stellar binaries (companion mass > 100 M Jup ) are indicated with black ⋄ symbols. (Colours are only visible in the online version of the paper.) The solid line is the best fit through the random single stars, the dotted line indicates the 1σ interval around the best fit and the dashed line indicates the 3.5σ interval. Single stars with a higher radial velocity amplitude than expected based on their log g value (more than 3.5σ above the best fit) are indicated by arrows. Six of the 8 stars with periodic radial velocity variations and log g < 1.6 are classified bright giants or supergiants (Perryman & ESA 1997).\\nations are induced by a mechanism intrinsic to the star. This trend is present for stars with random as well as stars with periodic radial velocity variations. Also, nearly all stars with periodic radial velocity variations and log g ≥ 1.6 are located above the fit in Fig. 3. Seven single stars have a higher radial velocity variation than expected based on their log g value, i.e. they are situated more than 3.5σ above the best fit for the relation obtained for single stars. These stars are indicated with arrows in Fig. 3. The radial velocity variation observed for HIP53229 may be due to a stellar companion in a wide orbit, with a period much longer than the observation time span. Due to this long period the companion mass, and, therefore, the (sub-)stellar nature, is still very uncertain. HIP33152 is classified as a supergiant. The observed radial velocity variations for HIP80693, HIP36616, and HIP88048 can be fitted with two Keplerian orbits, while HIP75458 can be explained by an eccentric substellar companion (Frink et al. 2002) and an additional linear trend, indicating a companion in a wide orbit. HIP34693 can be fitted very accurately with a Keplerian orbit of a single nearly sinusoidal sub-stellar companion.\\nIn order to investigate the simultaneous occurrence of substellar companions and oscillations in giants, the best Keplerian fits are subtracted from the observed radial velocity variations of both the binaries and the single stars with significant periodic radial velocity variations. The half peak-to-peak values of these residuals are plotted as a function of surface gravity in Fig. 4, also showing the linear fit through the non-periodic stars, and the amplitudes of the subtracted periodic signals. There are several interesting points to make about this graph. Firstly, almost all periodic stars show larger radial velocity variations than predicted by the relation found for non-periodic stars, but when the periodicities are removed, their residual radial velocity variations follow the same relation as found for the non-periodic stars. This could be interpreted as evidence for both intrinsic (non-periodic radial velocity variation) and extrinsic (periodic variations) mechanisms playing a role in these stars. Secondly, there is no correlation between the amplitude of the subtracted periodic signal and log g, which provides additional evidence for the presence of companions. Thirdly, almost all (8 out of 9) stars with log g ≤ 1.6 exhibit periodic variations. If these indeed have an extrinsic mechanism, it would mean that ∼ 90% of these stars have sub-stellar companions. However, from an astrophysical point of view, stars with such low surface gravities are already very high on the giant branch or even on the asymptotic giant branch. At these low gravities, stars cannot be constant anymore, as the outer layers are so diluted that instabilities occur easily, either periodic or random. Stated differently, these stars are very close to the semi-regulars, which are on their way to become Mira variables. Hence, these periodic variations could well be intrinsic.\"\n },\n {\n \"title\": \"Companion Interpretation\",\n \"content\": \"From the analysis of the correlation between radial velocity amplitude and surface gravity we have evidence for the presence of both intrinsic variability and companions in at least a fraction of the K giant stars with periodic radial velocity variations. In order to study the characteristics of these companions, we take the hypothesis that the periodic radial velocity variations detected in 43 of our K giant stars, excluding binaries, are caused by sub-stellar companions. Under this hypothesis, we investigate the statistical properties of the orbital parameters of the sample and compare these with the statistical properties of companions orbiting F, G and K dwarfs.\\nAccording to our analysis, 55 stars in the sample would have a single companion, and 11 stars multiple companions.\\nTwenty-three (22 single and 1 in a multiple system) of these companions have m sin i larger than 100 M Jup and should be interpreted as stellar binaries. By advancing the multiple substellar systems forward in time via a Runga-Kutta integration, we investigated the stability of the systems, taking into account the mutual interactions of the companions. With the orbital parameters that minimise χ 2 taken at face value, we found that most of the inferred sub-stellar multiple systems would be 'likely unstable', with a change in semi-major axis > 1% and < 10%, or 'unstable' with a change in semi-major axis > 10% on a time scale of 100 years due to companion-companion interaction.\\nHowever, the inferred stability depends on the starting epoch of the computations, as well as on the orbital parameters, which might change with an increasing number of observations. Furthermore, there is no guarantee that the obtained χ 2 minimum is a global minimum. Therefore, stars with multiple inferred sub-stellar companions that seem to be unstable, might also have stable solutions. One could also use the equations for dynamical stability described by Gladman (1993) and Marcy et al. (2001). Gladman (1993) also notes that the Hill stability criteria for companions in initially eccentric orbits may not be met, but that the systems may still be found to be empirically quite stable for a long period of time. In order to draw a firm conclusion on the stability of a particular system, a more thorough investigation is needed, as well as data with a longer time base, which is beyond the scope of this paper.\\nFor all stars with periodic radial velocity variations, we checked the Hipparcos (Perryman & ESA 1997) photometry. We checked periodograms for significant frequencies close to the obtained radial velocity period, and plotted the photometric values phased with the radial velocity period. None of the stars show photometric variations related to the observed radial velocity variations.\\nThe mass distribution of our K giant sample is not known very well. The stellar masses are typically between 1 and 4 M ⊙ . Hence most of their main sequence progenitors should have been of A or F spectral class. The distribution of orbital parameters of sub-stellar companions orbiting A and F main sequence stars is still unknown. The core accretion model predicts more giant planets around more massive stars, so that the distribution of orbital parameters of sub-stellar companions orbiting F, G and K main-sequence stars probably cannot serve as a proxy. However, it should be instructive to compare the two distributions.\\nSince the data presented here span ∼ 2500 days, radial velocity variations with longer periods are uncertain, and, therefore, not taken into consideration. Companions with periods exceeding the observation time span are also excluded from the F, G, and K main sequence star statistics. (Marcy et al. 2005), normalised to the number of stars in our sample. Fig. 6. Zoom in on the low end of the companion mass distribution of inferred sub-stellar companions orbiting K giant stars in our survey, shown as percentage of the total number of stars in the sample (gray histogram). The hatched histogram shows the distribution of companion masses orbiting main sequence stars as shown in Fig. 1 of Marcy et al. (2005), as a percentage of the total number of stars in their sample. The cross hatched area are main sequence stars with companions at a semi-major axis smaller than 0.3 AU (see Fig. 7). dwarf statistics around F, G and K main sequence stars for which only very few companions are found with m sin i > 15 M Jup around more than thousand stars. This is known as the brown dwarf desert and is possibly caused by migration and merging of brown dwarfs in a viscous disk with a mass at least comparable to the brown dwarf mass (Armitage & Bonnell 2002).\"\n },\n {\n \"title\": \"Mass distribution\",\n \"content\": \"The dashed line in Fig. 5 indicates the rise of sub-stellar companion masses M −1.05 from 10 M Jup down to Saturn masses for main sequence stars (Marcy et al. 2005), normalised Fig. 7. Semi-major axis distribution of inferred K giant companions in our sample (gray histogram), shown as a percentage of the total number of stars in the sample. The hatched histogram shows the distribution for F, G and K main sequence stars as shown in Fig. 2 of Marcy et al. (2005), as a percentage of the total number of stars in their sample. The main sequence stars with a companion orbiting at a semi-major axis smaller than 0.3 AU are cross-hatched. These are also indicated in Figs 6 and 8.\\nto the number of stars in our sample. We use a two-sided Kolmogorov-Smirnov test (Press et al. 1992) (hereafter KStest) to compare the M −1.05 fit and the mass distribution (m sin i < 80 M Jup ) of our sample and find a probability of 0.05% (0.002% for m sin i < 28 M Jup ) that these are identical. This implies that inferred sub-stellar companions around K giants in our sample have higher masses compared to companions around F, G and K main sequence stars. Fig. 6 shows the low-mass companion distribution of our survey. Most K giant companions would have inferred masses between 2 and 8 M Jup , while the fraction of companions orbiting F, G and K dwarfs strongly decreases with increasing m sin i.\"\n },\n {\n \"title\": \"Semi-major axis distribution\",\n \"content\": \"The distribution of the companions' semi-major axis is shown in Fig. 7. No inferred companions with semi-major axis smaller than 0.3 AU are present in the K giant sample, possibly due to increased stellar radii of giants. The fraction of stars with an inferred companion with a semi-major axis between 1 and 3 AU is much higher among the K giants compared to the F, G and K dwarfs. A comparison between the two distributions with a KStest reveals a probability for the two distributions to be identical of 11%. This increases to 32% when omitting the main sequence stars with semi-major axis < 0.3 AU. The increasing incompleteness beyond 3 AU, due to the limited time span of surveys, is present in both samples as the surveys cover a comparable amount of time. This incompleteness cannot cause the significant difference in the peak between 1 and 3 AU. Fig. 8. Period distribution of the observed radial velocity variations shown as a fraction of all significant periods (gray histogram). The period distribution of main sequence stars ) is shown in the hatched histogram, as a fraction of the total number of observed companions. The cross hatched area are main sequence stars with a semi-major axis smaller than 0.3 AU (see Fig. 7).\"\n },\n {\n \"title\": \"Period distribution\",\n \"content\": \"In Fig. 8 the period distribution of the observed radial velocity variations is shown and compared with the companion period distribution of dwarfs. The large fraction of F, G and K dwarf companions with orbital periods shorter than 100 days corresponds to the ones with semi-major axis smaller than approximately 0.3 AU. The close-in short-period companions are not present around K giants, while about 80% of these stars with observed radial velocities have periods ranging between 400 and 800 days. A KS-test reveals a probability of less than 0.0001% for the two distributions to be identical. The probability remains below this level, when the companions orbiting main sequence stars with a semi-major axis < 0.3 AU are omitted. Fig. 9 shows the distribution of companion eccentricities for K giants in our sample and for dwarfs. Companions of dwarfs with periods less than 20 days are excluded, as these might be tidally circularised. The fraction of companion eccentricities < 0.3 for the giants is 75% compared to 50% for companions orbiting F, G and K dwarfs. The KS-test shows that these distributions are nearly identical (97%).\"\n },\n {\n \"title\": \"Eccentricity distribution\",\n \"content\": \"\"\n },\n {\n \"title\": \"Iron abundance\",\n \"content\": \"Companion occurrence correlates strongly with the abundance of heavy elements (see for instance Gonzalez (1997), Fischer & Valenti (2005) and Santos et al. (2005)), such that F, G, and K dwarf stars with supersolar abundance are more likely to harbour sub-stellar companions (about 50% of the stars with 0.3 < [Fe/H] < 0.5). The increase of the fraction The hatched histogram is the eccentricity distribution of companions around main sequence stars shown as a fraction of all companions around main sequence stars. Companions with periods shorter than 20 days are excluded from the latter sample as these might be tidally circularised.\\nof F, G and K dwarfs harbouring companions with increasing metallicity is well fitted with a power law, yielding a probability for such a star to harbour a companion to be: P = 0.03·[(N Fe /N H )/(N Fe /N H ) ⊙ ] 2.0 (Fischer & Valenti 2005).\\nIn Fig. 10 the iron abundance distribution of stars with periodic radial velocity variations is shown as a percentage of the total number of observed stars with iron abundance in the same interval. The iron abundance is determined spectroscopically by imposing excitation and ionisation equilibrium in iron lines and is described by Hekker & Meléndez (2007). The maximum iron abundance of a K giant star in our sample is 0.29 and, therefore, we do not probe the high metallicity region in which F, G and K dwarfs are most likely to harbour a companion.\\nThe mean metallicity of the K giants in the entire sample is −0.12 dex, while the mean metallicity of the stars with periodic radial velocity variations, presented in Fig. 10, is −0.13 dex. No correlation between companion occurrence and abundance, similar to the one which is present for dwarf stars, is found in this sample of giant stars.\"\n },\n {\n \"title\": \"Discussion and Conclusion\",\n \"content\": \"The tight correlation we found between log g and half of the peak-to-peak radial velocity variations seems to indicate that a large fraction of the observed radial velocity variations in our sample of K giants is induced by mechanism(s) intrinsic to the stars. We also present evidence that both intrinsic and extrinsic mechanisms play a role. The stars with a significant periodic signal are almost all located above the radial velocity amplitude vs. log g relation, but when the periodic signal is removed, the residuals show the same trend as for the non-periodic stars. Furthermore, no correlation is present between the amplitude of the periodic signal and log g. Fig. 10. Iron abundance [Fe/H] distribution of K giant stars with periodic radial velocity variations shown as a percentage of the total number of observed stars with iron abundance in the same interval. The numbers above each bar on the histogram indicate the ratio of stars with a significant periodic radial velocity variation to the total number of stars in each bin. The error bars are calculated assuming Poisson statistics (i.e., the percentage of stars with periodic radial velocities divided by the square root of the number of stars with periodic radial velocities). The dashed line is the power law derived for the increasing trend in the fraction of stars with companions as a function of metallicity of F, G and K main sequence stars (Fischer & Valenti 2005).\\nAlmost all of the lowest log g stars show periodic variations. It may be possible that stars with such low surface gravity cannot be constant and that in these dilute atmospheres instabilities can occur very easily, and therefore may be periodic, but not extrinsic.\\nBased on the evidence that extrinsic mechanism(s) play a role for K giant stars with periodic radial velocity variations we investigated the hypothesis that this periodic signal is caused by the reflex motion of sub-stellar companions orbiting these stars. We presented the characteristics of the orbital parameters of these companions and compared them with the known orbital parameters of sub-stellar companions orbiting F, G and K dwarfs.\\nAbout 25% of the stars in our sample have radial velocity variations with significant periodicity, and could possibly harbour a sub-stellar companion, while approximately only 8% of the 1330 F, G and K main sequence stars investigated by Marcy et al. (2005) have a sub-stellar companion. Recently Johnson et al. (2007) and Lovis & Mayor (2007) showed that the number of companion harbouring stars increases with mass. The giants in the present sample have typical masses between 1 and 4 M ⊙ and are in general more massive than the main sequence stars investigated for companions. So, the high percentage is qualitatively in agreement with the results from the literature. Furthermore, Lovis & Mayor (2007) suggest that more massive stars form more massive planetary systems than lower mass stars. Figs 5 and 6 show that we find, in general, more massive companions around the more massive K giants than are present around F, G and K dwarfs.\\nThe high percentage of more massive companions around the more massive K giant stars would also be compatible with the core accretion model. This model predicts very few giant planets, but a relatively large number of planets with the mass of Neptune or smaller around M dwarfs (Laughlin et al. 2004: Ida & Lin 2005. This is mainly the result of a much reduced surface density of the disk and the resulting shorter disk evolution timescales compared to those for more massive stars, implying that planet properties vary with the mass of their host stars. In particular, one would expect more sub-stellar companions with higher masses in our giant sample, as is indeed the case if we assume that the companion hypothesis is correct.\\nThe mean metallicity of companion-hosting K giants would be similar to the mean metallicity of the total sample. This is in contrast with the correlation between companion occurrence and metallicity present in F, G and K dwarfs (e.g. Fischer & Valenti 2005). So far, several groups have investigated the correlation between companion occurrence and metallicity for giants with different results. Sadakane et al. (2005) and Pasquini et al. (2007) agree that companion-ahosting giants are on average not metal-rich, while Hekker & Meléndez (2007) find that giants with announced companions have higher metallicities than their total sample of giants. A detailed discussion about these different results is presented by Hekker & Meléndez (2007). They conclude that the samples on which the results are based are slightly different and the more metal-rich stars used in their study are lacking in the study by Pasquini et al. (2007). Furthermore, there is a difference in zero-point correction for the metallicities of announced companion-hosting stars from different surveys. All in all, these inferences are based on smallnumber statistics and all results have to be taken with caution.\\nThe larger semi-major axis and long periods of the inferred companions orbiting K giant stars compared to companions orbiting dwarf stars are most likely due to the extended atmospheres of K giants. For the eccentricity no significant difference is found in the distribution between companions orbiting dwarfs or giants. Nevertheless, the high number of inferred companions around giants with eccentricities < 0.3 is striking. One could suspect that companion orbits circularise over time and that the companions in circular orbits are older than the eccentric ones, but there is no evidence for this hypothesis.\\nIn principle, nearly sinusoidal radial velocity variations could also be caused by pulsations or spots. Although the periods of the radial velocity variations could well be the rotational periods of the stars, the presence of prominent spots is not very likely. In that case one would also expect photometric variations with periods correlated with the radial velocity variations. From the Hipparcos photometry (Perryman & ESA 1997) such correlations were not found.\\nIn order to distinguish with certainty between companions and pulsations as the cause of the observed radial velocity variations, one needs to perform a spectral line profile analysis. A technique for doing this with very high-resolution spectra (R ≥ 100 000) will be presented separately (Hekker et al., in preparation) because the amount of such data at hand today is insufficient to add significantly to the conclusions of this paper.\"\n },\n {\n \"title\": \"Fig. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 5\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 9 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 1 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Radial velocity variations as a function of phase for a star (HIP34693) with a highly significant period (top), with its periodogram (bottom). The dashed line in the periodogram indicates the significance threshold.","content":"Fig. 1 .\n1Radial velocity variations as a function of phase for a star (HIP34693) with a highly significant period (top), with its periodogram (bottom). The dashed line in the periodogram indicates the significance threshold."},{"id":"","caption":"Radial velocity variations as a function of phase for a star (HIP7607) with a period close to the significance threshold (top), with its periodogram (bottom). The dashed line in the periodogram indicates the significance threshold.","content":"Fig. 2 .\n2Radial velocity variations as a function of phase for a star (HIP7607) with a period close to the significance threshold (top), with its periodogram (bottom). The dashed line in the periodogram indicates the significance threshold."},{"id":"","caption":"Top: Half of the peak-to-peak variation of the radial velocity as a function of surface gravity (log g), as inFig. 3, but showing only those stars with periodic radial velocity variations (blue •) and stellar binaries (black ⋄). The blue + and black × symbols indicate the amplitude of the radial velocity variations for these stars after subtraction of the Keplerian fits. (Colours are only visible in the online version of the paper.) The solid line indicates the linear fit through the stars with non-periodic radial velocity variations (fromFig. 3). Bottom: Amplitudes of the subtracted Keplerian fits as a function of log g.","content":"Fig. 4 .\n4Top: Half of the peak-to-peak variation of the radial velocity as a function of surface gravity (log g), as inFig. 3, but showing only those stars with periodic radial velocity variations (blue •) and stellar binaries (black ⋄). The blue + and black × symbols indicate the amplitude of the radial velocity variations for these stars after subtraction of the Keplerian fits. (Colours are only visible in the online version of the paper.) The solid line indicates the linear fit through the stars with non-periodic radial velocity variations (fromFig. 3). Bottom: Amplitudes of the subtracted Keplerian fits as a function of log g."},{"id":"","caption":"shows the distribution of inferred companion masses of our K giant sample. First, notice that 30% of the inferred companions would have masses in the brown dwarf regime 15 M Jup < m sin i < 80 M Jup . This is in sharp contrast to the brown","content":"Fig. 5\n5shows the distribution of inferred companion masses of our K giant sample. First, notice that 30% of the inferred companions would have masses in the brown dwarf regime 15 M Jup < m sin i < 80 M Jup . This is in sharp contrast to the brown"},{"id":"","caption":"A histogram of m sin i of inferred companion masses orbiting K giants in our sample. The dashed line indicates the rise of planet masses M −1.05 from 10 M Jup down to Saturn masses for sub-stellar companions around main sequence stars","content":"Fig. 5 .\n5A histogram of m sin i of inferred companion masses orbiting K giants in our sample. The dashed line indicates the rise of planet masses M −1.05 from 10 M Jup down to Saturn masses for sub-stellar companions around main sequence stars"},{"id":"","caption":"Distribution of eccentricities for possible companions around K giants in our sample (gray histogram) shown as a fraction of all possible sub-stellar companions in the sample.","content":"Fig. 9 .\n9Distribution of eccentricities for possible companions around K giants in our sample (gray histogram) shown as a fraction of all possible sub-stellar companions in the sample."},{"id":"","caption":"Single stars with significant frequencies.","content":"Table 1 .\n1Single stars with significant frequencies.star \nfrequency period frequency period \nµHz \ndays \nµHz \ndays \nHIP3419 \n0.0638 \n181 \n0.1197 \n97 \nHIP7607 \n0.0216 \n536 \nHIP7884 \n0.0184 \n629 \nHIP13905 \n0.0223 \n519 \nHIP16335 \n0.0197 \n588 \nHIP19011 \n0.0252 \n459 \nHIP21421 \n0.0203 \n570 \n0.0448 \n237 \nHIP23015 \n0.0151 \n767 \n0.0073 \n1586 \nHIP23123 \n0.0135 \n857 \nHIP31592 \n0.0144 \n804 \nHIP33160 \n0.0226 \n512 \nHIP34693 \n0.0380 \n305 \nHIP36616 \n0.0033 \n3507 \n0.0389 \n298 \nHIP37826 \n0.0194 \n597 \nHIP38253 \n0.0171 \n677 \nHIP39177 \n0.0133 \n870 \nHIP40526 \n0.0172 \n673 \nHIP46390 \n0.0233 \n497 \nHIP47959 \n0.0187 \n619 \nHIP53229 \n0.0022 \n5261 \nHIP53261 \n0.0149 \n777 \nHIP57399 \n0.0196 \n591 \nHIP64823 \n0.0021 \n5512 \nHIP69673 \n0.0382 \n303 \n0.0187 \n619 \nHIP73133 \n0.0014 \n8267 \n0.0157 \n737 \nHIP73620 \n0.0226 \n512 \nHIP74732 \n0.0233 \n497 \nHIP75458 \n0.0226 \n512 \nHIP79540 \n0.0203 \n570 \nHIP80693 \n0.0022 \n5261 \n0.0198 \n585 \nHIP84671 \n0.0251 \n461 \nHIP85139 \n0.2895 \n40 \nHIP85355 \n0.0258 \n449 \n0.0056 \n2067 \nHIP85693 \n0.0175 \n661 \nHIP87808 \n0.0153 \n757 \nHIP88048 \n0.0218 \n531 \n0.0029 \n3991 \nHIP91117 \n0.0303 \n382 \nHIP109023 \n0.0199 \n582 \nHIP109492 \n0.0217 \n533 \nHIP109602 \n0.0130 \n890 \nHIP109754 \n0.0189 \n612 \nHIP113562 \n0.0401 \n289 \n0.0472 \n245 \nHIP114855 \n0.0639 \n181"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Radial velocity variations as a function of phase for a star (HIP34693) with a highly significant period (top), with its periodogram (bottom). The dashed line in the periodogram indicates the significance threshold.\",\n \"content\": \"Fig. 1 .\\n1Radial velocity variations as a function of phase for a star (HIP34693) with a highly significant period (top), with its periodogram (bottom). The dashed line in the periodogram indicates the significance threshold.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Radial velocity variations as a function of phase for a star (HIP7607) with a period close to the significance threshold (top), with its periodogram (bottom). The dashed line in the periodogram indicates the significance threshold.\",\n \"content\": \"Fig. 2 .\\n2Radial velocity variations as a function of phase for a star (HIP7607) with a period close to the significance threshold (top), with its periodogram (bottom). The dashed line in the periodogram indicates the significance threshold.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Top: Half of the peak-to-peak variation of the radial velocity as a function of surface gravity (log g), as inFig. 3, but showing only those stars with periodic radial velocity variations (blue •) and stellar binaries (black ⋄). The blue + and black × symbols indicate the amplitude of the radial velocity variations for these stars after subtraction of the Keplerian fits. (Colours are only visible in the online version of the paper.) The solid line indicates the linear fit through the stars with non-periodic radial velocity variations (fromFig. 3). Bottom: Amplitudes of the subtracted Keplerian fits as a function of log g.\",\n \"content\": \"Fig. 4 .\\n4Top: Half of the peak-to-peak variation of the radial velocity as a function of surface gravity (log g), as inFig. 3, but showing only those stars with periodic radial velocity variations (blue •) and stellar binaries (black ⋄). The blue + and black × symbols indicate the amplitude of the radial velocity variations for these stars after subtraction of the Keplerian fits. (Colours are only visible in the online version of the paper.) The solid line indicates the linear fit through the stars with non-periodic radial velocity variations (fromFig. 3). Bottom: Amplitudes of the subtracted Keplerian fits as a function of log g.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"shows the distribution of inferred companion masses of our K giant sample. First, notice that 30% of the inferred companions would have masses in the brown dwarf regime 15 M Jup < m sin i < 80 M Jup . This is in sharp contrast to the brown\",\n \"content\": \"Fig. 5\\n5shows the distribution of inferred companion masses of our K giant sample. First, notice that 30% of the inferred companions would have masses in the brown dwarf regime 15 M Jup < m sin i < 80 M Jup . This is in sharp contrast to the brown\"\n },\n {\n \"id\": \"\",\n \"caption\": \"A histogram of m sin i of inferred companion masses orbiting K giants in our sample. The dashed line indicates the rise of planet masses M −1.05 from 10 M Jup down to Saturn masses for sub-stellar companions around main sequence stars\",\n \"content\": \"Fig. 5 .\\n5A histogram of m sin i of inferred companion masses orbiting K giants in our sample. The dashed line indicates the rise of planet masses M −1.05 from 10 M Jup down to Saturn masses for sub-stellar companions around main sequence stars\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Distribution of eccentricities for possible companions around K giants in our sample (gray histogram) shown as a fraction of all possible sub-stellar companions in the sample.\",\n \"content\": \"Fig. 9 .\\n9Distribution of eccentricities for possible companions around K giants in our sample (gray histogram) shown as a fraction of all possible sub-stellar companions in the sample.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Single stars with significant frequencies.\",\n \"content\": \"Table 1 .\\n1Single stars with significant frequencies.star \\nfrequency period frequency period \\nµHz \\ndays \\nµHz \\ndays \\nHIP3419 \\n0.0638 \\n181 \\n0.1197 \\n97 \\nHIP7607 \\n0.0216 \\n536 \\nHIP7884 \\n0.0184 \\n629 \\nHIP13905 \\n0.0223 \\n519 \\nHIP16335 \\n0.0197 \\n588 \\nHIP19011 \\n0.0252 \\n459 \\nHIP21421 \\n0.0203 \\n570 \\n0.0448 \\n237 \\nHIP23015 \\n0.0151 \\n767 \\n0.0073 \\n1586 \\nHIP23123 \\n0.0135 \\n857 \\nHIP31592 \\n0.0144 \\n804 \\nHIP33160 \\n0.0226 \\n512 \\nHIP34693 \\n0.0380 \\n305 \\nHIP36616 \\n0.0033 \\n3507 \\n0.0389 \\n298 \\nHIP37826 \\n0.0194 \\n597 \\nHIP38253 \\n0.0171 \\n677 \\nHIP39177 \\n0.0133 \\n870 \\nHIP40526 \\n0.0172 \\n673 \\nHIP46390 \\n0.0233 \\n497 \\nHIP47959 \\n0.0187 \\n619 \\nHIP53229 \\n0.0022 \\n5261 \\nHIP53261 \\n0.0149 \\n777 \\nHIP57399 \\n0.0196 \\n591 \\nHIP64823 \\n0.0021 \\n5512 \\nHIP69673 \\n0.0382 \\n303 \\n0.0187 \\n619 \\nHIP73133 \\n0.0014 \\n8267 \\n0.0157 \\n737 \\nHIP73620 \\n0.0226 \\n512 \\nHIP74732 \\n0.0233 \\n497 \\nHIP75458 \\n0.0226 \\n512 \\nHIP79540 \\n0.0203 \\n570 \\nHIP80693 \\n0.0022 \\n5261 \\n0.0198 \\n585 \\nHIP84671 \\n0.0251 \\n461 \\nHIP85139 \\n0.2895 \\n40 \\nHIP85355 \\n0.0258 \\n449 \\n0.0056 \\n2067 \\nHIP85693 \\n0.0175 \\n661 \\nHIP87808 \\n0.0153 \\n757 \\nHIP88048 \\n0.0218 \\n531 \\n0.0029 \\n3991 \\nHIP91117 \\n0.0303 \\n382 \\nHIP109023 \\n0.0199 \\n582 \\nHIP109492 \\n0.0217 \\n533 \\nHIP109602 \\n0.0130 \\n890 \\nHIP109754 \\n0.0189 \\n612 \\nHIP113562 \\n0.0401 \\n289 \\n0.0472 \\n245 \\nHIP114855 \\n0.0639 \\n181\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". P J Armitage, I A Bonnell, MNRAS. 33011Armitage, P. J. & Bonnell, I. A. 2002, MNRAS, 330, L11","doi":""},{"text":". T R Bedding, R P Butler, F Carrier, ApJ. 647558Bedding, T. R., Butler, R. P., Carrier, F., et al. 2006, ApJ, 647, 558","doi":""},{"text":". R P Butler, G W Marcy, E Williams, PASP. 108500Butler, R. P., Marcy, G. W., Williams, E., et al. 1996, PASP, 108, 500","doi":""},{"text":". R P Butler, J T Wright, G W Marcy, ApJ. 646505Butler, R. P., Wright, J. T., Marcy, G. W., et al. 2006, ApJ, 646, 505","doi":""},{"text":". F Castelli, R G Gratton, R L Kurucz, A&A. 324432Castelli, F., Gratton, R. G., & Kurucz, R. L. 1997, A&A, 324, 432","doi":""},{"text":". A Cumming, G W Marcy, R P Butler, ApJ. 526890Cumming, A., Marcy, G. W., & Butler, R. P. 1999, ApJ, 526, 890","doi":""},{"text":". J De Ridder, C Barban, F Carrier, A&A. 448689De Ridder, J., Barban, C., Carrier, F., et al. 2006, A&A, 448, 689","doi":""},{"text":". M P Döllinger, A P Hatzes, L Pasquini, A&A. 472649Döllinger, M. P., Hatzes, A. P., Pasquini, L., et al. 2007, A&A, 472, 649","doi":""},{"text":". P Eggenberger, F Carrier, F Bouchy, A Blecha, A&A. 422247Eggenberger, P., Carrier, F., Bouchy, F., & Blecha, A. 2004, A&A, 422, 247","doi":""},{"text":". D A Fischer, J Valenti, ApJ. 6221102Fischer, D. A. & Valenti, J. 2005, ApJ, 622, 1102","doi":""},{"text":". S Frink, D S Mitchell, A Quirrenbach, ApJ. 576478Frink, S., Mitchell, D. S., Quirrenbach, A., et al. 2002, ApJ, 576, 478","doi":""},{"text":". S Frink, A Quirrenbach, D Fischer, S Röser, E Schilbach, PASP. 113173Frink, S., Quirrenbach, A., Fischer, D., Röser, S., & Schilbach, E. 2001, PASP, 113, 173","doi":""},{"text":". F Galland, A.-M Lagrange, S Udry, A&A. 452709Galland, F., Lagrange, A.-M., Udry, S., et al. 2006, A&A, 452, 709","doi":""},{"text":". B Gladman, Icarus. 106247Gladman, B. 1993, Icarus, 106, 247","doi":""},{"text":". G Gonzalez, MNRAS. 285403Gonzalez, G. 1997, MNRAS, 285, 403","doi":""},{"text":"A P Hatzes, W D Cochran, Astronomical Society of the Pacific Conference Series. R. A. Donahue & J. A. Bookbinder154311Cool Stars, Stellar Systems, and the SunHatzes, A. P. & Cochran, W. D. 1998, in Astronomical Society of the Pacific Conference Series, Vol. 154, Cool Stars, Stellar Systems, and the Sun, ed. R. A. Donahue & J. A. Bookbinder, 311","doi":""},{"text":". A P Hatzes, W D Cochran, M Endl, A&A. 457335Hatzes, A. P., Cochran, W. D., Endl, M., et al. 2006, A&A, 457, 335","doi":""},{"text":". A P Hatzes, E W Guenther, M Endl, A&A. 437743Hatzes, A. P., Guenther, E. W., Endl, M., et al. 2005, A&A, 437, 743","doi":""},{"text":". S Hekker, J Meléndez, A&A. 4751003Hekker, S. & Meléndez, J. 2007, A&A, 475, 1003","doi":""},{"text":". S Hekker, S Reffert, A Quirrenbach, A&A. 454943Hekker, S., Reffert, S., Quirrenbach, A., et al. 2006, A&A, 454, 943","doi":""},{"text":". S Ida, D N C Lin, ApJ. 6261045Ida, S. & Lin, D. N. C. 2005, ApJ, 626, 1045","doi":""},{"text":". J A Johnson, D A Fischer, G W Marcy, ApJ. 665785Johnson, J. A., Fischer, D. A., Marcy, G. W., et al. 2007, ApJ, 665, 785","doi":""},{"text":". J A Johnson, G W Marcy, D A Fischer, ApJ. 6521724Johnson, J. A., Marcy, G. W., Fischer, D. A., et al. 2006, ApJ, 652, 1724","doi":""},{"text":". R Kuschnig, W W Weiss, R Gruber, P Y Bely, H Jenkner, A&A. 328544Kuschnig, R., Weiss, W. W., Gruber, R., Bely, P. Y., & Jenkner, H. 1997, A&A, 328, 544","doi":""},{"text":". G Laughlin, P Bodenheimer, F C Adams, ApJ. 61273Laughlin, G., Bodenheimer, P., & Adams, F. C. 2004, ApJ, 612, L73","doi":""},{"text":"S Hekker, Precise radial velocities of giant stars. IV. S. Hekker et al.: Precise radial velocities of giant stars. IV.","doi":""},{"text":". C Lovis, M Mayor, G Marcy, R P Butler, D Fischer, Progress of Theoretical Physics Supplement. 70624A&ALovis, C. & Mayor, M. 2007, A&A, 706, in press Marcy, G., Butler, R. P., Fischer, D., et al. 2005, Progress of Theoretical Physics Supplement, 158, 24","doi":""},{"text":". G W Marcy, R P Butler, PASP. 104270Marcy, G. W. & Butler, R. P. 1992, PASP, 104, 270","doi":""},{"text":". G W Marcy, R P Butler, PASP. 112137Marcy, G. W. & Butler, R. P. 2000, PASP, 112, 137","doi":""},{"text":". G W Marcy, R P Butler, S S Vogt, ApJ. 555418Marcy, G. W., Butler, R. P., Vogt, S. S., et al. 2001, ApJ, 555, 418","doi":""},{"text":". M Martić, J.-C Lebrun, T Appourchaux, S G Korzennik, A&A. 418295Martić, M., Lebrun, J.-C., Appourchaux, T., & Korzennik, S. G. 2004, A&A, 418, 295","doi":""},{"text":". A Niedzielski, M Konacki, A Wolszczan, ApJ. 6691354Niedzielski, A., Konacki, M., Wolszczan, A., et al. 2007, ApJ, 669, 1354","doi":""},{"text":". L Pasquini, M P Döllinger, A Weiss, A&A. 473979Pasquini, L., Döllinger, M. P., Weiss, A., et al. 2007, A&A, 473, 979","doi":""},{"text":"F Pepe, F Bouchy, D Queloz, M Mayor, Scientific Frontiers in Research on Extrasolar Planets. 29439Pepe, F., Bouchy, F., Queloz, D., & Mayor, M. 2003, in ASP Conf. Ser. 294: Scientific Frontiers in Research on Extrasolar Planets, 39","doi":""},{"text":"The HIPPARCOS and TYCHO catalogues. Astrometric and photometric star catalogues derived from the ESA HIPPARCOS Space Astrometry Mission Press. W H Teukolsky, S A Vetterling, W T Flannery, B P , Numerical recipes in C. The art of scientific computing. Perryman, M. A. C. & ESACambridge: University Press12002nd ed.Perryman, M. A. C. & ESA, eds. 1997, ESA Special Publication, Vol. 1200, The HIPPARCOS and TYCHO cata- logues. Astrometric and photometric star catalogues derived from the ESA HIPPARCOS Space Astrometry Mission Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, Numerical recipes in C. The art of scientific com- puting (Cambridge: University Press, Ic1992, 2nd ed.)","doi":""},{"text":". D Queloz, M Mayor, S Udry, The Messenger. 1051Queloz, D., Mayor, M., Udry, S., et al. 2001, The Messenger, 105, 1","doi":""},{"text":". S Reffert, A Quirrenbach, D S Mitchell, ApJ. 652661Reffert, S., Quirrenbach, A., Mitchell, D. S., et al. 2006, ApJ, 652, 661","doi":""},{"text":". K Sadakane, T Ohnishi, M Ohkubo, Y Takeda, PASJ. 57127Sadakane, K., Ohnishi, T., Ohkubo, M., & Takeda, Y. 2005, PASJ, 57, 127","doi":""},{"text":". N C Santos, G Israelian, M Mayor, A&A. 4371127Santos, N. C., Israelian, G., Mayor, M., et al. 2005, A&A, 437, 1127","doi":""},{"text":". B Sato, H Ando, E Kambe, ApJ. 597157Sato, B., Ando, H., Kambe, E., et al. 2003, ApJ, 597, L157","doi":""},{"text":". J D Scargle, ApJ. 263835Scargle, J. D. 1982, ApJ, 263, 835","doi":""},{"text":". J Setiawan, A P Hatzes, O Von Der Lühe, A&A. 39819Setiawan, J., Hatzes, A. P., von der Lühe, O., et al. 2003, A&A, 398, L19","doi":""},{"text":". J Setiawan, J Rodmann, L Da Silva, A&A. 43731Setiawan, J., Rodmann, J., da Silva, L., et al. 2005, A&A, 437, L31","doi":""},{"text":". C A Sneden, AA(University of Texas at AustinPhD thesisSneden, C. A. 1973, PhD thesis, AA(University of Texas at Austin)","doi":""},{"text":". J A Valenti, R P Butler, G W Marcy, PASP. 107966Valenti, J. A., Butler, R. P., & Marcy, G. W. 1995, PASP, 107, 966","doi":""}],"string":"[\n {\n \"text\": \". P J Armitage, I A Bonnell, MNRAS. 33011Armitage, P. J. & Bonnell, I. A. 2002, MNRAS, 330, L11\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T R Bedding, R P Butler, F Carrier, ApJ. 647558Bedding, T. R., Butler, R. P., Carrier, F., et al. 2006, ApJ, 647, 558\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R P Butler, G W Marcy, E Williams, PASP. 108500Butler, R. P., Marcy, G. W., Williams, E., et al. 1996, PASP, 108, 500\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R P Butler, J T Wright, G W Marcy, ApJ. 646505Butler, R. P., Wright, J. T., Marcy, G. W., et al. 2006, ApJ, 646, 505\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Castelli, R G Gratton, R L Kurucz, A&A. 324432Castelli, F., Gratton, R. G., & Kurucz, R. L. 1997, A&A, 324, 432\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Cumming, G W Marcy, R P Butler, ApJ. 526890Cumming, A., Marcy, G. W., & Butler, R. P. 1999, ApJ, 526, 890\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J De Ridder, C Barban, F Carrier, A&A. 448689De Ridder, J., Barban, C., Carrier, F., et al. 2006, A&A, 448, 689\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M P Döllinger, A P Hatzes, L Pasquini, A&A. 472649Döllinger, M. P., Hatzes, A. P., Pasquini, L., et al. 2007, A&A, 472, 649\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Eggenberger, F Carrier, F Bouchy, A Blecha, A&A. 422247Eggenberger, P., Carrier, F., Bouchy, F., & Blecha, A. 2004, A&A, 422, 247\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D A Fischer, J Valenti, ApJ. 6221102Fischer, D. A. & Valenti, J. 2005, ApJ, 622, 1102\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Frink, D S Mitchell, A Quirrenbach, ApJ. 576478Frink, S., Mitchell, D. S., Quirrenbach, A., et al. 2002, ApJ, 576, 478\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Frink, A Quirrenbach, D Fischer, S Röser, E Schilbach, PASP. 113173Frink, S., Quirrenbach, A., Fischer, D., Röser, S., & Schilbach, E. 2001, PASP, 113, 173\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Galland, A.-M Lagrange, S Udry, A&A. 452709Galland, F., Lagrange, A.-M., Udry, S., et al. 2006, A&A, 452, 709\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Gladman, Icarus. 106247Gladman, B. 1993, Icarus, 106, 247\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Gonzalez, MNRAS. 285403Gonzalez, G. 1997, MNRAS, 285, 403\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A P Hatzes, W D Cochran, Astronomical Society of the Pacific Conference Series. R. A. Donahue & J. A. Bookbinder154311Cool Stars, Stellar Systems, and the SunHatzes, A. P. & Cochran, W. D. 1998, in Astronomical Society of the Pacific Conference Series, Vol. 154, Cool Stars, Stellar Systems, and the Sun, ed. R. A. Donahue & J. A. Bookbinder, 311\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A P Hatzes, W D Cochran, M Endl, A&A. 457335Hatzes, A. P., Cochran, W. D., Endl, M., et al. 2006, A&A, 457, 335\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A P Hatzes, E W Guenther, M Endl, A&A. 437743Hatzes, A. P., Guenther, E. W., Endl, M., et al. 2005, A&A, 437, 743\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Hekker, J Meléndez, A&A. 4751003Hekker, S. & Meléndez, J. 2007, A&A, 475, 1003\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Hekker, S Reffert, A Quirrenbach, A&A. 454943Hekker, S., Reffert, S., Quirrenbach, A., et al. 2006, A&A, 454, 943\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Ida, D N C Lin, ApJ. 6261045Ida, S. & Lin, D. N. C. 2005, ApJ, 626, 1045\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J A Johnson, D A Fischer, G W Marcy, ApJ. 665785Johnson, J. A., Fischer, D. A., Marcy, G. W., et al. 2007, ApJ, 665, 785\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J A Johnson, G W Marcy, D A Fischer, ApJ. 6521724Johnson, J. A., Marcy, G. W., Fischer, D. A., et al. 2006, ApJ, 652, 1724\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Kuschnig, W W Weiss, R Gruber, P Y Bely, H Jenkner, A&A. 328544Kuschnig, R., Weiss, W. W., Gruber, R., Bely, P. Y., & Jenkner, H. 1997, A&A, 328, 544\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Laughlin, P Bodenheimer, F C Adams, ApJ. 61273Laughlin, G., Bodenheimer, P., & Adams, F. C. 2004, ApJ, 612, L73\",\n \"doi\": \"\"\n },\n {\n \"text\": \"S Hekker, Precise radial velocities of giant stars. IV. S. Hekker et al.: Precise radial velocities of giant stars. IV.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Lovis, M Mayor, G Marcy, R P Butler, D Fischer, Progress of Theoretical Physics Supplement. 70624A&ALovis, C. & Mayor, M. 2007, A&A, 706, in press Marcy, G., Butler, R. P., Fischer, D., et al. 2005, Progress of Theoretical Physics Supplement, 158, 24\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G W Marcy, R P Butler, PASP. 104270Marcy, G. W. & Butler, R. P. 1992, PASP, 104, 270\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G W Marcy, R P Butler, PASP. 112137Marcy, G. W. & Butler, R. P. 2000, PASP, 112, 137\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G W Marcy, R P Butler, S S Vogt, ApJ. 555418Marcy, G. W., Butler, R. P., Vogt, S. S., et al. 2001, ApJ, 555, 418\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Martić, J.-C Lebrun, T Appourchaux, S G Korzennik, A&A. 418295Martić, M., Lebrun, J.-C., Appourchaux, T., & Korzennik, S. G. 2004, A&A, 418, 295\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Niedzielski, M Konacki, A Wolszczan, ApJ. 6691354Niedzielski, A., Konacki, M., Wolszczan, A., et al. 2007, ApJ, 669, 1354\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Pasquini, M P Döllinger, A Weiss, A&A. 473979Pasquini, L., Döllinger, M. P., Weiss, A., et al. 2007, A&A, 473, 979\",\n \"doi\": \"\"\n },\n {\n \"text\": \"F Pepe, F Bouchy, D Queloz, M Mayor, Scientific Frontiers in Research on Extrasolar Planets. 29439Pepe, F., Bouchy, F., Queloz, D., & Mayor, M. 2003, in ASP Conf. Ser. 294: Scientific Frontiers in Research on Extrasolar Planets, 39\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The HIPPARCOS and TYCHO catalogues. Astrometric and photometric star catalogues derived from the ESA HIPPARCOS Space Astrometry Mission Press. W H Teukolsky, S A Vetterling, W T Flannery, B P , Numerical recipes in C. The art of scientific computing. Perryman, M. A. C. & ESACambridge: University Press12002nd ed.Perryman, M. A. C. & ESA, eds. 1997, ESA Special Publication, Vol. 1200, The HIPPARCOS and TYCHO cata- logues. Astrometric and photometric star catalogues derived from the ESA HIPPARCOS Space Astrometry Mission Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, Numerical recipes in C. The art of scientific com- puting (Cambridge: University Press, Ic1992, 2nd ed.)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Queloz, M Mayor, S Udry, The Messenger. 1051Queloz, D., Mayor, M., Udry, S., et al. 2001, The Messenger, 105, 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Reffert, A Quirrenbach, D S Mitchell, ApJ. 652661Reffert, S., Quirrenbach, A., Mitchell, D. S., et al. 2006, ApJ, 652, 661\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Sadakane, T Ohnishi, M Ohkubo, Y Takeda, PASJ. 57127Sadakane, K., Ohnishi, T., Ohkubo, M., & Takeda, Y. 2005, PASJ, 57, 127\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N C Santos, G Israelian, M Mayor, A&A. 4371127Santos, N. C., Israelian, G., Mayor, M., et al. 2005, A&A, 437, 1127\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Sato, H Ando, E Kambe, ApJ. 597157Sato, B., Ando, H., Kambe, E., et al. 2003, ApJ, 597, L157\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J D Scargle, ApJ. 263835Scargle, J. D. 1982, ApJ, 263, 835\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Setiawan, A P Hatzes, O Von Der Lühe, A&A. 39819Setiawan, J., Hatzes, A. P., von der Lühe, O., et al. 2003, A&A, 398, L19\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Setiawan, J Rodmann, L Da Silva, A&A. 43731Setiawan, J., Rodmann, J., da Silva, L., et al. 2005, A&A, 437, L31\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C A Sneden, AA(University of Texas at AustinPhD thesisSneden, C. A. 1973, PhD thesis, AA(University of Texas at Austin)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J A Valenti, R P Butler, G W Marcy, PASP. 107966Valenti, J. A., Butler, R. P., & Marcy, G. W. 1995, PASP, 107, 966\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":493,"cells":{"text":{"kind":"string","value":"\nMulti-wavelength analysis of the dust emission in the Small Magellanic Cloud ⋆\n4 May 2004 November 7, 2018\n\nC Bot \nCentre de Données Astronomiques de Strasbourg (CDS)\nUMR 7550\nUniversité Louis Pasteur\nF-67000STRASBOURGFrance\n\nInstitut d'Astrophysique Spatiale\nUniversité Paris-Sud\nF-91405OrsayFrance\n\nF Boulanger \nInstitut d'Astrophysique Spatiale\nUniversité Paris-Sud\nF-91405OrsayFrance\n\nG Lagache \nInstitut d'Astrophysique Spatiale\nUniversité Paris-Sud\nF-91405OrsayFrance\n\nL Cambrésy \nCentre de Données Astronomiques de Strasbourg (CDS)\nUMR 7550\nUniversité Louis Pasteur\nF-67000STRASBOURGFrance\n\nD Egret \nUMS 2201\nObservatoire de Paris\nF-75014ParisFrance\n\nMulti-wavelength analysis of the dust emission in the Small Magellanic Cloud ⋆\n4 May 2004 November 7, 2018Astronomy & Astrophysics manuscript no. 1449 (DOI: will be inserted by hand later) Received ...; accepted ...dust -Magellanic Clouds -ISM: abundances -evolution -Galaxies: ISM\nWe present an analysis of dust grain emission in the diffuse interstellar medium of the Small Magellanic Cloud (SMC). This study is motivated by the availability of 170 µm ISOPHOT data covering a large part of the SMC, with a resolution enabling to disentangle the diffuse medium from the star forming regions. After data reduction and subtraction of Galactic foreground emission, we used the ISOPHOT data together with HiRes IRAS data and ATCA/Parkes combined H column density maps to determine dust properties for the diffuse medium. We found a far infrared emissivity per hydrogen atom 30 times lower than the Solar Neighborhood value. The modeling of the spectral energy distribution of the dust, taking into account the enhanced interstellar radiation field, gives a similar conclusion for the smallest grains (PAHs and very small grains) emitting at shorter wavelength. Assuming Galactic dust composition in the SMC, this result implies a difference in the gas-to-dust ratio (GDR) 3 times larger than the difference in metallicity. This low depletion of heavy elements in dust could be specific of the diffuse ISM and not apply for the whole SMC dust if it results from efficient destruction of dust by supernovae explosions.\n\nIntroduction\n\nThe Small Magellanic Cloud (SMC) is a dwarf irregular galaxy at a distance of approximately 60 kpc (Westerlund 1991). Its relative proximity enables to have a global view of the galaxy, as well as a sufficiently good resolution to disentangle the diffuse medium from the star forming regions and molecular clouds. The SMC is an ideal target to study the low metallicity diffuse medium present not only in dwarf galaxies, but also in the external parts of large spirals.\n\nThe SMC contains several H regions and molecular clouds (Rubio et al. 1993a). It has a high interstellar radiation field (ISRF), between 4 and 10 times that of the Solar neighborhood (Lequeux 1979;Vangioni-Flam et al. 1980;Azzopardi et al. 1988). It is extremely rich in neutral atomic gas (Staveley-Smith et al. 1998) and has a low heavy element abundance (Dufour 1984;Sauvage & Vigroux 1991). All of these characteristics are indicative of a \" young\" galaxy, actively forming stars, which might in some sense be consid-Send offprint requests to: C.Bot, e-mail: bot@astro.u-strasbg.fr ⋆ based on observations with ISO, an ESA project with instruments funded by ESA Member States (especially the PI countries: France, Germany, The Netherlands and the UK) and with the participation of ISAS and NASA. ered as a local model of distant galaxies at the beginning of their evolution.\n\nDust grains are composed mainly of heavy atoms (e.g. carbon, silicium, oxygen, . . . ). In a low metallicity environment, such elements are lacking. How does this deficiency act on the dust grain abundances? The gas-to-dust ratio (GDR) is often assumed to be proportional to the metallicity (i.e. there is a fixed fraction of heavy elements in dust). Based on a model of the evolution of elemental abundances in gas and dust, Dwek (1998) concludes that this proportionality applies across the disk of the Galaxy. However, observations in dwarf irregular and blue compact dwarf galaxies show a large scatter in the GDR versus metallicity relation. This indicates that a simple scaling might be a too simplistic description of the phenomenon (Issa et al. 1990;Lisenfeld & Ferrara 1997, 1998. The GDR is expected to vary among ISM components because destruction of dust by shock waves is more efficient in the diffuse gas while accretion of heavy elements on dust dominates in molecular clouds. The integrated GDR is thus expected to depend on the relative fraction of matter in diffuse/dense components and on the supernova rate and thereby on the star formation history (Hirashita 1999).\n\nThe goal of our study is to characterize the dust properties in the diffuse medium of the SMC, and in particular the GDR. Comparing the GDR and the dust spectral energy distribution for the SMC and the Galaxy, we quantify the effect of the difference in metallicity on dust abundance and size distribution (e.g. the relative amounts of PAHs, very small grains and big grains).\n\nSince the gas and the dust masses are closely mixed, where the radiation is homogeneous the dust emission is expected to correlate with the distribution of the gas. In the diffuse medium, gas is mainly composed of neutral hydrogen and the GDR can be derived from the comparison between the dust infrared emission with the H column density, provided that the dust temperature is known. This is not the case in starforming regions, where the infrared emission from dust in H regions and molecular clouds closely associated with luminous stars can become dominant. Several authors have computed GDRs, comparing the IRAS infrared emission with the H column density, taking the 100µm/60µm intensity ratio to derive the dust temperature (Schwering & Israel 1989;Sauvage et al. 1990;Stanimirovic et al. 2000), but this ignores the contribution from small grains at 60µm. An additional difficulty comes from the variations in the intensity of the radiation field inside a beam, which casts doubt on any GDR estimate from studies where the angular resolution is too coarse to separate star forming regions from the diffuse emission (Li & Draine 2002;Aguirre et al. 2003) The availability of 170 µm ISOPHOT observations enables us to have a new view on dust properties of the SMC. Independently of our work, Wilke et al. (2003) have also reduced and analyzed these data but with a distinct focus on point sources. In a second paper (Wilke et al. 2004), these authors adress the same question as us but they investigate the main body (bar) of the SMC without discarding the star forming regions.\n\nAfter data reduction and validation (Sect. 2 and 3), the 170 µm map is compared to additional observations (Sect. 4), in particular the IRAS High Resolution map (HiRes) at 100 µm. Having obtained the 170/100 µm emission ratio, in Sect. 5.1 we derive a reference temperature for the big grains in the diffuse medium of the SMC. An H column density map from ATCA/Parkes combined data (Stanimirovic et al. 1999) is used to study the dust to gas correlation, allowing us to estimate the emissivity of the dust per hydrogen atom for the diffuse medium (Sect.5.2) . Finally, the spectral energy distribution of the dust grains in the diffuse medium of the SMC is fitted with the Desert et al. (1990) model to quantify the abundances of the different grain components (Sect. 5.4). The results are discussed in Sect. 6; Sect. 7 presents our conclusions.\n\n\nThe Data\n\nThis study relies on archive data. Some were ready-made as the IRAS maps at 12, 25, 60, and 100 µm, or the ATCA/Parkes combined data of the H line at 21 cm. The ISOPHOT data was imported from the ISO Data Archive 1 . But the quality of the ISOPHOT map at 170 µm in the ISO archive was not sufficient for this study, so we retrieved, reduced and calibrated 1 http://iso.vilspa.esa.es/ida/index.html the data. We first present the ISOPHOT data and the reduction processes. Then we describe the ancillary data that were used.\n\n\nThe ISOPHOT data\n\nThe Small Magellanic Cloud was almost fully covered by 10 observations with the ISOPHOT instrument (Lemke et al. 1996) on board the ISO satellite (Kessler et al. 1996;Laureijs et al. 2003). The observations were made using the P22 AOT (Astronomical Observation Template), i.e. a raster map on a two-dimensional regular grid, with the C200 detector (2x2 pixels) at an effective wavelength of 170 µm. The pixel field of view is 1.5 ′ and the raster pointing spacings are 180 ′′ in both in-scan and cross-scan directions. The ISOPHOT point spread function can be approximated by a 2D gaussian with 90 ′′ full width half maximum (FWHM) in both directions and is supposed to be constant across the map.\n\nWe reduced the data with PIA 2 (ISOPHOT Interactive Analysis) V10.0 (Gabriel et al. 1997(Gabriel et al. , 1998, except for the flatfield correction. In the following, we describe the specific treatments applied. Some observations showed specific problems on raw data (e.g. a pixel went into erratic state or a severe glitch affected a pixel until the end of the observation). We choose to remove the data for these pixels in the corresponding observations. If this removal is not done, the reduction processes will interpret these erratic fluxes as flat-fielding. This distorts the photometry of the other pixels for the whole observation.\n\nTo deal with the discarded pixels, we choose to perform the flat-field correction without the PIA tool as in Lagache & Dole (2001). The flat-field correction is on average 1.01 ± 0.05, 0.93 ± 0.03, 1.17 ± 0.09 and 0.92 ± 0.03 for pixel 1, 2, 3 and 4 respectively. Reproducible flat-field correction values for the ISOPHOT C 200 array have been published by Lagache & Dole (2001) for the carefull reduction of 3 deep fields (FIRBACK). We do not observe the same average flat field corrections factor and we find a higher dispersion from one observation to another. This difference may be because the responsivity of ISOPHOT C 200 pixels changes with the observed intensity, so that the flat-field correction computed with low fluxes is no longer valid for the brightest pixels (Pagani et al. 2003). This effect might affect the 170 µm flux determination for bright sources. The reproducible behavior of pixels observed by Lagache & Dole (2001) is then due to the relative flatness of deep fields.\n\nThe different observations have been done at different times, therefore the \"absolute calibration\" performed using the FCS measurement can be slightly different from one raster to another. This difference is observed as discontinuities between maps in the mosaic. To correct for these small absolute calibration differences (for a border between two maps, the mean difference in brightness observed is about 13%), we applied a \" general flat-field\" correction as in Lagache & Dole (2001).\n\nAfter concatenating the timelines, the raster is projected on the sky using each individual pixel coordinates, with a pixel size for the map of 10 ′′ to match the IRAS HiRes gridding. Because the raster step size of each observation was a full detector size, and the spacecraft Y-axis was not parallel to right ascensions or declinations, we observe gaps between the individual array pointings. The final map is presented at Fig. 1 after a bilinear interpolation to reconstruct the signal in the gaps.\n\n\nAncillary data\n\n\nIRAS HiRes data\n\nAs a complement to the ISOPHOT data at 170 µm, we used IRAS high resolution (HiRes) data at 12, 25, 60 and 100 µm. Maps of 2 • × 2 • centered at 00 h 57 m 29.12 s −72 • 32 ′ 29.2 ′′ in J2000 were obtained. The pixel size is 10 ′′ . The HiRes point spread function (PSF) varies spatially in a map. A gaussian approximation of the PSF is given with the HiRes maps and is defined by two spatially dependent FWHMs and a rotation angle. For more information on HiRes, please refer to Aumann et al. (1990) and Bontekoe et al. (1994). Since we are searching for extended structure properties, the HiRes IRAS maps were brought to the DC-calibration (Wheelock et al. 1994). This DC calibration is only valid for spatial scales greater than 40 ′ . As the SMC exhibits structure on various spatial scales, the DC calibration introduces some limitations that will be discussed in Sect. 3.2.\n\n\nATCA/Parkes combined SMC H data\n\nTo study the correlation between gas and dust in the SMC, we needed a H column density map to compare with the ISOPHOT map, in order to compute the abundances of dust grains relative to the hydrogen. Stanimirovic et al. (1999) have combined Parkes telescope observations of neutral hydrogen in the SMC with the Australia Telescope Compact Array (ATCA) aperture synthesis mosaic to obtain a set of images sensitive to all angular spatial scales between 98 ′′ and 4 • from 90 to 215 km.s −1 . The spatial resolution obtained is 98 ′′ and the velocity resolution is 1.65 km.s −1 . We used a column density map of the SMC, obtained by integrating over the whole velocity range and assuming that the neutral hydrogen is optically thin 3 .\n\n\nParkes H data for the Galactic foreground\n\nA Galactic filament exists in front of the SMC. Even though it is very smooth, large (see contours of Fig. 1) and uniform (column densities between 2.5 · 10 20 and 4 · 10 20 atom.cm −2 ), it can bias studies. The far-infrared fluxes observed in either the ISOPHOT or the IRAS HiRes maps represent the emission from the SMC dust, but also from this Galactic foreground. 3 The column density map and the data cube are available at ftp://ftp2.naic.edu/pub/ast/sstanimi/. However, this Galactic component of the emission is not observed in the ATCA/Parkes data, due to a velocity cut and spatial filtering. To study the SMC dust properties, it was thus necessary to remove this foreground component (see Sect. 3.3). In this step, we used a Galactic H column density map constructed from data of the Parkes narrow-band H survey of the tidal arms of the Magellanic system (Brüns et al. 2000). The column density map of the H Galactic foreground was obtained by integrating the emission from -60 to +50 km/s, outside the SMC velocities. The resolution is 15 ′ . Due to sidelobes effects, the map obtained can slightly overestimate the column density by 10 to 20 %.\n\n\nPhotometric consistency\n\nDIRBE provided infrared absolute sky brightness maps in 10 bands in the wavelength range 1.25 to 240 µm. This absolute photometry is used to check ISOPHOT mosaic and IRAS fluxes.\n\n\nISOPHOT data\n\nWe checked if the ISOPHOT 170 µm flux is consistent with the spectral energy distributions obtained from the 100, 140 and 240 µm DIRBE fluxes in the SMC. To do this we convolved the ISOPHOT map with the DIRBE beam. This gives 18 ISOPHOT fluxes in the SMC as if they were observed by DIRBE and we compare ISOPHOT and DIRBE fluxes by plotting the spectral energy distribution and fitting it with a modified black body with a spectral index of 2 on the DIRBE values. Fig. 2 shows that the DIRBE-convolved ISOPHOT flux integrated over the 18 positions observed by DIRBE, is consistent within the errors bars with the best modified blackbody fit (obtained for a temperature of 20.6K, an emissivity of ǫ H (250µm) = 3 · 10 −28 cm 2 for 10 20 hydrogen atoms.cm −2 ). Please note that these values correspond to an average over the whole galaxy and were obtained only for calibration purposes.\n\nWe have also plotted the individual fit values at 170 µm versus the corresponding 18 ISOPHOT values. This plot shows a significant scatter but is consistent with a linear correlation with unity slope and zero offset.\n\nA comparison with the IRAS high resolution data was also performed to check possible photometric errors introduced by the interpolation process. The correlation between IRAS and ISOPHOT data does not change when we restrict the comparison to the pixels measured by ISOPHOT rather than the full interpolated map. The interpolation required to fill the holes in the measurements does not seem to bias the image comparisons.\n\n\nIRAS data\n\nThe IRAS HiRes maps have the same photometric calibration as the IRAS ISSA maps. The IRAS detector had a dwell-time dependent responsivity change: the gain changes as a function of source size. This effect is known as the \" AC/DC effect\". A first set of corrections were taken into account in the IRAS Fig. 1. The ISOPHOT map at 170 µm (in MJy/sr) after reduction processes and a bilinear interpolation to fill the holes left by the mapping pattern. The flat-field has been efficiently corrected for the diffuse part of the map, but one can still see low level defects in the brightest parts. Contours of the H Galactic foreground are overplotted at column densities of 2.6, 2.8, 3., 3.2, 3.4 and 3.6·10 20 H atom.cm-2 (from light grey to black). data products. These corrections were later found to be insufficient. On the basis of a comparison of the IRAS and DIRBE sky brightness for extended emission, Wheelock et al. (1994) recommended applying additional correction factors of 0.88, 1.01, 0.82 and 0.74 at 12, 25, 60 and 100 µm respectively.\n\nHowever, these correction factors are only valid for spatial scales greater than 40 ′ . This is not true in the SMC where structures on various spatial scales are present. Stanimirovic et al. (2000) have also compared DIRBE and IRAS data by integrating fluxes over 6.25 •2 in the SMC. They found that integrated flux densities measured by DIRBE for 60 and 100 µm are higher than IRAS fluxes by about 10 to 20%, which is an opposite correction to that observed by Wheelock et al. (1994).\n\nWe have compared the IRAS (convolved to the DIRBE resolution) and the DIRBE 100 µm maps on a 10 • × 10 • scale around the SMC to understand this discrepancy. We see that the Wheelock et al. (1994) correction factor of 0.74 is only truly valid outside the galaxy and increases in the SMC up to a maximum value in the north of the optical bar (N66 region). For the diffuse medium of the SMC, the Wheelock et al. (1994) correction factor is sufficient. After correction, we estimate that the 100 µm IRAS brightness for the diffuse medium is underestimated by less than 15%.\n\n\nGalactic foreground removal\n\nUnlike the ATCA/Parkes H map, the far-infrared emission maps (ISOPHOT and IRAS) trace both the dust emission in the SMC and the dust in our Galaxy. To disentangle these two components in the far-infrared emission, we used a column density map of the H Galactic foreground presented in Sect. 2.2.3.\n\nDust grains absorb energy from the UV/optical radiation field that they re-emit in the infrared wavelength as thermal emission. Large grains are at a fixed temperature and their emitted intensity can be expressed as:\nI ν = N H ǫ H (ν)B ν (T dust )(1)\nwhere N H is the column density of hydrogen and ǫ H (ν) is the emissivity of grains per hydrogen atom. Assuming that the emissivity of dust grains in the far-infrared follows a power law: . 2. Comparison of the DIRBE-convolved ISOPHOT fluxes with the DIRBE ones (in MJy/sr), after an integration on the whole SMC. The modified black body traced was fit on the DIRBE values only and correspond to a temperature of 20.6 K and an emissivity of ǫ H (250µm) = 3 · 10 −28 cm 2 for 10 20 hydrogen atoms.cm −2 . We see that the ISOPHOT value at 170 µm is consistent with the DIRBE photometry.\nǫ H (ν) = ǫ H (ν 0 )( ν ν 0 ) β (2) Fig\nand taking β = 2 (as found for Galactic emission (Lagache et al. 1998(Lagache et al. , 1999 for wavelengths between 100 µm and 500 µm).\n\nUsing a temperature of 17.5 K and an emissivity per hydrogen atom of 10 −25 cm 2 (Boulanger et al. 1996a) 4 for the Galactic dust, we computed the Galactic thermal emission and removed this foreground emission from the IRAS and ISOPHOT maps to obtain the emission of the SMC dust only. At 12, 25 and 60 µm, we used the I 12 /I 100 , I 25 /I 100 and I 60 /I 100 ratios computed from IRAS maps outside the SMC, together with the computed foreground emission map at 100 µm to remove the Galactic foreground emission at these wavelengths. However, this foreground emission remains in the data noise at 12, 25 and 60 µm.\n\n\nSpatial image comparison\n\n\nProjection and resolution\n\nThe data set presented in Sect. 2 is composed of maps with different fields of view and different resolutions. We projected all of them onto a common grid: that of IRAS HiRes maps (2 • × 2 • field of view centered at 00 h 57 m 29.12 s − 72 • 32 ′ 29.2 ′′ (J2000) with 10 ′′ pixel size). These maps were inter-compared by pairs.\n\nTo compare the ISOPHOT map with one of the HiRes maps, the projected and re-sampled ISOPHOT map was convolved with a spatial varying kernel in order to be at the HiRes resolution when the point spread function of ISOPHOT was smaller than the one of HiRes (Kerton & Martin 2001). The HiRes map was also convolved with a spatially varying kernel to match the ISOPHOT resolution when the HiRes beam was smaller than the ISOPHOT one.\n\nTo compare the HI column density map with the ISOPHOT map, no convolution was necessary as long as their resolutions were comparable. Fig. 3 shows two of the transformed maps: the ISOPHOT map degraded to the IRAS 100 µm resolution and the projected ATCA/Parkes map.\n\n\nColors from scatterplots\n\nTo characterize the diffuse medium (i.e. the extended region where the infrared emission-H column density correlation is homogenous) spectral energy distribution, we developed a comparison method that is not biased by point sources and is also not affected by offsets in the data calibration.\n\nWe work with two maps (obtained after the treatment presented in Sect. 4). The general correlation between the two maps is first quantified by fitting a straight line to the scatterplot of the pixel-to-pixel comparison. We deduce a first ratio value and an error σ given by the dispersion in the correlation, which enables us to filter out localized sources (i.e. star-forming regions) by removing pixels that are 3σ above the correlation (there is no points below −3σ).\n\nFrom the \" filtered\" maps, we compute a ratio map by linearly fitting the pixel correlation in the scatterplot for a sliding box of 50×50 pixels. This corresponds to a 5.5×5.5 beams box which is a minimum size to have independent data points. The size of the box could not be greater, or we would not trace the local variations. In some cases there are too few points inside a box (too many points above 3σ) to do a meaningful fit. We assign no value to these positions. We also skip positions where the correlation coefficient is too low. Adjusting a gaussian on the histogram of the ratio values gives us a reference ratio and the associated dispersion. This dispersion contains photometric and fitting errors but also a true scatter in the ratio. Both phenomena cannot be easily disentangled.\n\nThe spatial distribution and the histogram with the gaussian fit obtained for the I 100µm /I 170µm ratio are presented in Fig. 4 as an example. We see that the comparison method avoids large areas in the main body (bar) of the SMC. These regions correspond to highly structured star-forming sites. In these regions, either the filtering technique has left too few independent positions in one box to compute a significant correlation or no clear correlation is observed.\n\n\nResults\n\n\nInfrared colors\n\nThe method presented in Sect. 4.2 was applied to the IRAS HiRes and ISOPHOT maps to compute the I 100µm /I 170µm , I 60µm /I 170µm , I 25µm /I 170µm , I 12µm /I 170µm ratios. Adjusting a gaussian on the histograms gives us reference ratios and their associated dispersions: \nI 100µm I 170µm = 1.0 ± 0.2(3)I 60µm I 170µm = 0.5 ± 0.1 (4) I 25µm I 170µm = 0.02 ± 0.01 (5) I 12µm I 170µm = 0.011 ± 0.006(6)\nUsing equation 1 and 2 we can derive from the I 100 µm I 170 µm ratio map the big grain equilibrium temperature distribution in the diffuse medium in the Small Magellanic Cloud. This leads to a reference temperature of: T dust = 22 ± 2 K. Note that this temperature is different from the one that would be inferred from the ratios of the mean I 100 µm and I 170 µm fluxes. Wilke et al. (2004) found a mean temperature for the main body of the SMC of 20.9 ± 1.9K, which is lower than the temperature we find for the diffuse medium. Their mean value falls within our range of temperatures, which is expected because the regions we study overlap. The difference in mean values could be due to the presence of cold atomic and molecular clouds in the main body of the SMC. Our studies seem to be consistent with each other and are more complementary than comparable.\n\nThe large dust grain temperature is related to the radiation field intensity and will be discussed in Sect. 6.\n\n\nGas to dust correlation\n\nFrom the ISOPHOT and the ATCA/Parkes maps, using the method described Sect. 4.2, we found the distribution presented on Fig. 5. This distribution exhibits a main peak and a distribution tail. The peak is characterized by:\nI 170µm N HI = 0.\n17 · 10 −20 ± 0.06 · 10 −20 MJy.sr −1 .cm 2 (7) Since the regions used to define the ratios are not exactly the same, the determination of the characteristics of the diffuse medium could be biased. We checked that this is not the case. If we use the region where I 170µm /N HI is computed to determine the colors, we find that the derived quantities are not affected by more than 15%. We see that the characteristics of the diffuse medium are not biased by the definition of the limits of the studied region. Comparing the region defined by equation 7 with the H α distribution, we see in Fig. 6 that the regions corresponding to the tail in the I 170µm /N HI (located mainly in the main body) are regions where the diffuse H α emission is important. Comparing the map in Fig. 4 and Fig. 6, we also see that the regions where no ratios can be computed correspond to important star-forming sites. Assuming that all the dust is in the H , we use the I 170µm /N HI ratio map and the temperature map computed above to obtain an histogram of the emissivity per hydrogen atom of the dust grains in the diffuse medium. We find a reference emissivity at 170 µm of:\n\nǫ H (λ) = 7.3 · 10 −27 ± 3.2 · 10 −27 cm 2 (8)\n\nThe error bar takes into account uncertainties in the temperature and I 170 µm /N HI . If the emissivity follows a power law with a spectral index of 2, the emissivity per H atom in the SMC at 250 µm is ǫ H (λ) = 3.4 · 10 −27 ± 1.5 · 10 −27 (λ/250 µm) −2 cm 2 . This corresponds to κ ν = 0.21(λ/250 µm) −2 cm 2 g −1 . Assuming similar dust optical properties in the SMC and the Galaxy, the emissivity is related to the GDR and the ratio between hydrogen column density and E(B − V) color excess by:\nǫ H (S MC) ǫ H (Galaxy) = M dust M gas (S MC) M dust M gas (Galaxy) = N(H I )+N(H 2 ) E(B−V) (Galaxy) N(H I )+N(H 2 ) E(B−V) (S MC)(9)\nThe solar neighborhood value for the emissivity is 10 −25 cm 2 at 250 µm (Boulanger et al. 1996a). The inferred SMC GDR is thus 30 times greater than in the Galaxy. Stanimirovic et al. (2000) have presented a similar study using the same H data together with HIRAS data. They came to the same conclusion that the GDR is a factor 30 higher in the SMC than in our Galaxy. However, this agreement might be a coincidence since 1) we did not use the same calibration of the IRAS data, 2) their temperature estimation based on the 60/100µm color is higher than ours, 3) the GDR they present is averaged over the whole galaxy (including star forming regions) and 4) their GDR is compared to a Galactic GDR of 375 instead of the standard value of 100 5 . Using the standard Galactic GDR value, the average SMC GDR found by Stanimirovic et al. (2000) becomes 100 times higher than in the Galaxy.\n\nOur estimate of the SMC's GDR 30 times higher than in the Galaxy, is significantly larger than the difference in metallicity, which is less than 10 (Dufour 1984). It is twice the observed difference in the N(H I )+N(H 2 )\n\n\nE(B−V)\n\nvalues presented by Tumlinson et al. (2002). \n\n\nInfrared excesses\n\nA noticeable result of the ISOPHOT and H comparison is the presence of a tail in the 170 µm/H histogram (see Fig. 5). This tail corresponds to pixels in the SMC optical bar with excess 170 µm emission with respect to N HI (see Fig. 7). We checked that the mean temperature for this region of excess is the same as that for the diffuse medium. These excesses may be due either to a variation in the dust abundances between the diffuse medium and the bar, or due to the presence of cold H or H 2 gas. We believe that the excesses are real but it is difficult to infer an amount of cold H and molecular gas from it. Firstly, it is difficult to disentangle these two possible contributions. Secondly, where some of these excesses are due to dust in cold H or H 2 we do not expect them to be correlated with H emission and another method is needed to quantify them. It is thus beyond the scope of this paper to translate these excesses into an estimate of cold gas that could be compared to other studies where this is attempted (Stanimirovic et al. 2000). Finally, we feel that the photometric uncertainties in the ISOPHOT and IRAS data are an additional difficulty, especially near starforming regions. Further studies with the Spitzer satellite could address this question with better photometry and angular resolution.\n\n\nSMC dust properties in diffuse H : modeling the spectral energy distribution\n\nThe different intensity ratios obtained in Sect. 5.1 enable us to trace the spectral energy distribution of dust in the diffuse medium of the SMC and to compare it with that of the solar neighborhood. The difference in temperature and in abundances is seen Fig. 9 as a shift along the wavelength and brightness axis respectively. The error bars include true variations in the dust properties across the SMC diffuse interstellar medium which are correlated over wavelengths.\n\nWe used the dust model of Desert et al. (1990) to quantify the grain dust abundances. In order to model the spectral energy distribution (SED), we need to estimate the interstellar radiation field (ISRF) in the diffuse medium of the SMC. In Sect. 5.1 we found the big dust grain equilibrium temperature to be 22 K. Using the dust thermal equilibrium temperature in the solar neighborhood of 17.5 K (Boulanger et al. 1996a) as a reference, the dust temperature in the SMC leads to a radiation field χ S MC = T 4+β dust (S MC)/T 4+β dust (Galaxy) = 4. This value is consistent with the estimate of the ISRF made from massive star counts by Lequeux (1979). Vangioni-Flam et al. (1980) also found the same ratio from UV measurements. We used the Mathis et al. (1983) 10 kpc Galactic ISRF and scaled it by a factor χ = 4 for the modeling. We have adjusted only the abundances of PAHs, very small grains (VSG) and big grains. For the other parameters defining the size distribution, we have kept the Galactic values. The best fit is presented on Fig. 8, it corresponds to under-anbundant PAHs, very small grains (VSG) and big grains (BG) compared to the Galaxy (36, 23 and 30 times less respectively). Mass abundances relative to hydrogen, compared with Galactic values (Desert et al. 1990) are summarized in table 1.\n\nThe modeling confirms the GDR estimated in Sect 5.2 to be around 30 times greater than in the Galaxy. Within the uncertainties, PAHs and very small grain abundances are also consistent with this factor.\n\nThe model is clearly below the 60 µm point. This could be due to our crude estimate of the ISRF. We have used a single ISRF while our data analysis shows temperature variations across the SMC diffuse medium. To take the variations in the radiation field into account, we transformed the temperature Fig. 9. Infrared spectral energy distribution of dust in the diffuse medium for the SMC and the Solar Neighborhood. The brightness corresponds to an interstellar gas column density of 10 20 cm −2 . The Galactic spectrum (in grey) is composed in the far-IR by COBE/FIRAS data from Boulanger et al. (1996a) (with the corresponding spectrum from grains in thermal equilibrium indicated by the dotted line), and at shorter wavelengths by the COBE/DIRBE (Bernard et al. 1994) and Arome balloon data (triangle at 3.3 µm) (Giard et al. 1994). An ISOCAM spectrum (Boulanger et al. 1996b) scaled to the 12 µm brightness is also shown. The SMC spectrum (in black) is the best fit obtained with the Desert et al. model, same as in figure 8. 2.0 · 10 −5 4.7 · 10 −4 BG 2.2 · 10 −4 6.4 · 10 −3 distribution obtained in Sect. 5.1 into a χ S MC distribution. For each value of the distribution, we computed a spectral energy distribution (SED) of the dust emission with the Desert et al. (1990) model, using the Mathis 10 kpc ISRF multiplied by the corresponding χ as an input (χ is a constant for the whole spectrum). We then computed the resulting total SED by adding the individual SEDs with weights according to the χ distribution shape. The resulting fit is not sufficiently broadened to encom-pass the 60 µm point. In addition, the mass abundances of dust grains have to be reduced by 8, 5 and 7 for the PAH, VSG and BG components respectively. Due to the high non-linearity of the Planck curve, even a small broadening of the χ distribution tends to reduce the abundances of dust grains significantly. These values are thus indicatives only but they show that the mass abundances represented in table 1 are more likely to be upper limits and that the emissivity could be even lower.\n\nIt is possible to fit the 60 µm value with a single ISRF scaled by χ = 10 (see Fig. 10). In this case, the dust grain mass abundances must then be reduced by 3, 2 and 2 for the PAHs, very small grains and big grains respectively. Such a high ISRF corresponds to a dust temperature of 25.6 K, which is higher than the estimate obtained in Sect. 5.1 but still compatible with the 100 µm and the 170 µm fluxes. This higher temperature would lead to an emissivity of 2.4 · 10 −27 cm 2 at 250 µm, which is even lower than the estimation obtained in Sect. 5.2 and results in a GDR 40 times higher than in the Galaxy. A higher radiation field could thus account for the 60µm excess without contra- dicting the main conclusion of this study, namely the fact that the GDR in the diffuse SMC ISM is much lower than the solar neighbourhood value scaled by the difference in metallicity.\n\nThe observed 60 µm excess could also be due to a change in the grain size distribution (Galliano et al. 2003). We assumed the Galactic grain sizes distribution to apply in the SMC. However, by changing the relative mass contributions of the different grain components, we enlarge the discontinuities in the size distribution between the three components of the model. It is necessary to further increase these discontinuities to account for the 60 µm excess.\n\nFurther investigations are needed to characterize the SMC ISRF and especially the 60 µm excess. The intensity and spectral shape of the ISRF are likely to be different than the Galactic one.\n\n\nDiscussion\n\nStudying the gas to dust correlation in the diffuse medium of the SMC (see Sect. 5.2), we found a GDR in mass around 30 times greater than in the solar neighborhood, larger by a factor of 3 than the difference in metallicity. This result confirms the value found by Stanimirovic et al. (2000) in a similar study. This lower depletion of heavy elements in dust was independently reported by Welty et al. (2001), who found Mg and Si to be essentially undepleted in the SMC. It was also reported for a sample of dwarf irregulars by Lisenfeld & Ferrara (1998). Our result in the SMC fits with the relation that they observed between GDR and metallicity.\n\nAn interpretation of this relation has been proposed by Hirashita et al. (2002). With a chemical evolution model, they illustrated the dependence of the GDR on the star formation history. Stars generate dust but also contribute to their destruction by supernova shocks. They propose a scenario in which an intermittent star formation history leads to a time variable GDR. At some epochs of the galaxy evolution, the formation of dust grains is less efficient than the destruction process by supernova shocks, so that the GDR no longer scales with the metallicity. This model could be applied to the SMC with its specific star formation history (Harris & Zaritsky 2003). It is also necessary to account for a possible difference in the GDR between the diffuse and the denser medium.\n\nExtinction studies lead to significantly smaller value of the GDR, closer to that expected from the Galactic value for a linear dependence with metallicity. Bouchet et al. (1985) used photometric observations of O-B stars in the SMC to derive their extinction curves. They found a GDR value that is about 8 times the Galactic value, which is consistent with the difference in metallicity. More recently, Tumlinson et al. (2002) measured\nN(H I )+N(H 2 ) E(B−V)\nfor a sample of stars observed with FUSE and found a mean value 16 times lower than the solar Neighborhood value. This is closer but still smaller by a factor 2 than our estimate. The difference may be due to the fact that extinction studies are based on line-of-sights with high hydrogen column density where dense interstellar medium is likely to be present. Most of the stars are in the optical bar and/or in star formation regions associated with molecular clouds. The difference in the GDRs inferred in our work and extinction studies might thus point to a dependence on the environment: a decrease of the ratio from the diffuse ISM to star forming regions that might arise from the condensation of heavy elements on dust grains in dense gas and their destruction by supernova shocks in the diffuse ISM. Stanimirovic et al. (2000) have also noted this possibility for the SMC, but it still requires confirmation by further studies.\n\n\nSummary and conclusions\n\nThis study was made feasible by the availability of 10 ISOPHOT observations at 170 µm, covering most of the SMC, in the ISO Data Archive. After reduction processes, mapping and Galactic foreground removal, the ISOPHOT data were compared with IRAS maps and a ATCA/Parkes combined H column density map to assess the properties of the dust in the diffuse medium of this low metallicity galaxy.\n\nFor the dust in the diffuse medium of the SMC, we found a reference equilibrium temperature of 22 ± 2 K and a reference dust emissivity of 3.4 · 10 −27 ± 1.5 · 10 −27 λ 250 µm cm 2 per hydrogen atom. This leads to a gas-to dust ratio 30 times greater than in the Galaxy. This high value reflects the SMC low metallicity, but also requires a lower depletion of dust elements than in the Galaxy. This shows that a simple linear dependence between the GDR and the metallicity does not apply for the diffuse medium of the SMC.\n\nThe spectral energy distribution is modeled with the Desert et al. (1990) model. The best fit obtained shows a similar decrease for PAHs and very small grains as for big grains. However, a 60 µm excess remains that can not be accounted for with simple assumptions. Further constraints, in particular on the ISRF, are needed to understand it.\n\nOur GDR estimate for the diffuse medium of the SMC is a factor 2 to 3 higher than that derived from extinction studies towards bright UV stars with high foreground hydrogen col-umn densities. This difference supports the notion of variations in the GDR, from diffuse regions where grains are more frequently destroyed by supernovae, to denser and more quiescent regions where grains re-accrete heavy elements.\n\nFig. 5 .\n5Histogram of the 170 micron flux to H column density ratio (in MJy.cm 2 /sr). The gaussian fit and the residuals are presented as continuous and dotted lines respectively\n\nFig. 4 .\n4Histogram and spatial distribution of the 100-170 µm colors. For the histogram, the gaussian fit and the residuals are presented as continuous and dotted lines respectively. For the map of the 100-170 µm colors, contours correspond to the 0.8 (black) and 1.2 (white) levels (one sigma limits of the gaussian fit). It should be noticed that the noise in the color map is not homogeneous.\n\nFig. 6 .\n6Uncalibrated H α map. As an illustration, the contour line corresponds to I 170 µm /N HI = 0.23 MJy.cm 2 /sr (see equation 7) as in figure 7.\n\nFig. 7 .\n7Spatial distribution of the I 170 µm /N HI ratio (in MJy.cm 2 /sr). Regions where no ratio is computed (see Sect. 4.2) are left blank. The contour line corresponds to a I 170 µm /N HI = 0.23 MJy.cm 2 /sr level. We see that the IR excesses are concentrated near the optical bar.\n\nFig. 8 .\n8Best fit for the dust infrared emission spectrum, obtained with the Desert et al. model. The brightness corresponds to an interstellar gas column density of 10 20 cm −2 . The IRAS and ISOPHOT data are represented by squares. The fit obtained with the model (continuous line) corresponds to the emission of three components: PAHs(dashed line), very small grains (mixed line) and big grains (dot-dot-dot-dashed line).\n\nFig. 10 .\n10Dust infrared emission spectrum fit obtained with the Desert et al. model to adjust the 60µm excess. The IRAS and ISOPHOT data are represented by squares. This spectrum is obtained with a single ISRF scaled by χ = 10 and reduced dust grain mass abundances for PAHs (dashed line), very small grains (mixed line) and big grains (dot-dot-dot-dashed line).\n\nTable 1 .\n1Mass abundances relative to hydrogen in the SMC and the Galaxy. The grain size distribution is assumed to be the same in both cases.Component SMC \nGalaxy \n\nm \nm H \n\nm \nm H \n\nPAH \n1.2 · 10 −5 4.3 · 10 −4 \nVSG \n\nThe ISOPHOT data presented in this paper were reduced using PIA, which is a joint development by the ESA Astrophysics Division and the ISOPHOT Consortium with the collaboration of the Infrared Processing and Analysis Center (IPAC). Contributing ISOPHOT Consortium institutes are DIAS, RAL, AIP, MPIK, and MPIA.\nWe checked that the observed IRAS and DIRBE far-infrared ratios around the SMC are consistent with these values.\nThis value can be checked with the interstellar abundance standards taken from table 2 ofSofia & Meyer (2001b) andSofia & Meyer (2001a).\nAcknowledgements. We acknowledge S. Stanimirović for providing us with the ATCA/Parkes combined H data and her help in using it, and C. Brüns for providing us the Galactic foreground H map. This study has been supported by the French national program PCMI (Physique et Chimie du Milieu Interstellaire, CNRS)\n. J E Aguirre, J J Bezaire, E S Cheng, ApJ. 596273Aguirre, J. E., Bezaire, J. J., Cheng, E. S., et al. 2003, ApJ, 596, 273\n\n. H H Aumann, J W Fowler, M Melnyk, AJ. 991674Aumann, H. H., Fowler, J. W., & Melnyk, M. 1990, AJ, 99, 1674\n\n. M Azzopardi, J Lequeux, A Maeder, A&A. 18934Azzopardi, M., Lequeux, J., & Maeder, A. 1988, A&A, 189, 34\n\n. J P Bernard, F Boulanger, F X Desert, A&A. 2915Bernard, J. P., Boulanger, F., Desert, F. X., et al. 1994, A&A, 291, L5\n\n. T R Bontekoe, E Koper, D J M Kester, A&A. 2841037Bontekoe, T. R., Koper, E., & Kester, D. J. M. 1994, A&A, 284, 1037\n\n. P Bouchet, J Lequeux, E Maurice, L Prevot, M L Prevot-Burnichon, A&A. 149330Bouchet, P., Lequeux, J., Maurice, E., Prevot, L., & Prevot- Burnichon, M. L. 1985, A&A, 149, 330\n\n. F Boulanger, A Abergel, J.-P Bernard, A&A. 312256Boulanger, F., Abergel, A., Bernard, J.-P., et al. 1996a, A&A, 312, 256\n\n. F Boulanger, W T Reach, A Abergel, A&A. 315325Boulanger, F., Reach, W. T., Abergel, A., et al. 1996b, A&A, 315, L325\n\nC Brüns, J Kerp, L Staveley-Smith, ASP Conf. Ser. 218: Mapping the Hidden Universe: The Universe behind the Milky Way -The Universe in HI. 349Brüns, C., Kerp, J., & Staveley-Smith, L. 2000, in ASP Conf. Ser. 218: Mapping the Hidden Universe: The Universe be- hind the Milky Way -The Universe in HI, 349\n\n. F.-X Desert, F Boulanger, J L Puget, A&A. 237215Desert, F.-X., Boulanger, F., & Puget, J. L. 1990, A&A, 237, 215\n\nR J Dufour, IAU Symp. 108: Structure and Evolution of the Magellanic Clouds. Dufour, R. J. 1984, in IAU Symp. 108: Structure and Evolution of the Magellanic Clouds, 353-360\n\n. E Dwek, ApJ. 501643Dwek, E. 1998, ApJ, 501, 643\n\n. C Gabriel, J Acosta-Puido, I Heinrichsen, A.S.P. Conference Series. 145165Gabriel, C., Acosta-Puido, J., & Heinrichsen, I. 1998, A.S.P. Conference Series, 145, 165\n\n. C Gabriel, J Acosta-Puido, U Kinkel, U Klaas, B Schulz, A.S.P. Conference Series. 125108Gabriel, C., Acosta-Puido, J., Kinkel, U., Klaas, U., & Schulz, B. 1997, A.S.P. Conference Series, 125, 108\n\n. F Galliano, S C Madden, A P Jones, A&A. 407159Galliano, F., Madden, S. C., Jones, A. P., et al. 2003, A&A, 407, 159\n\n. M Giard, J M Lamarre, F Pajot, G Serra, A&A. 286203Giard, M., Lamarre, J. M., Pajot, F., & Serra, G. 1994, A&A, 286, 203\n\n. J Harris, D Zaritsky, arXiv:astro-ph, 0312100Harris, J. & Zaritsky, D. 2003, arXiv:astro-ph, 0312100\n\n. H Hirashita, ApJ. 522220Hirashita, H. 1999, ApJ, 522, 220\n\n. H Hirashita, Y Y Tajiri, H Kamaya, A&A. 388439Hirashita, H., Tajiri, Y. Y., & Kamaya, H. 2002, A&A, 388, 439\n\n. M R Issa, I Maclaren, A W Wolfendale, A&A. 236237Issa, M. R., MacLaren, I., & Wolfendale, A. W. 1990, A&A, 236, 237\n\n. C R Kerton, P G Martin, PASP. 113872Kerton, C. R. & Martin, P. G. 2001, PASP, 113, 872\n\n. M F Kessler, J A Steinz, M E Anderegg, A&A. 31527Kessler, M. F., Steinz, J. A., Anderegg, M. E., et al. 1996, A&A, 315, L27\n\n. G Lagache, A Abergel, F Boulanger, F X Désert, J.-L Puget, A&A. 344322Lagache, G., Abergel, A., Boulanger, F., Désert, F. X., & Puget, J.-L. 1999, A&A, 344, 322\n\n. G Lagache, A Abergel, F Boulanger, J.-L Puget, A&A. 333709Lagache, G., Abergel, A., Boulanger, F., & Puget, J.-L. 1998, A&A, 333, 709\n\n. G Lagache, H Dole, A&A. 372702Lagache, G. & Dole, H. 2001, A&A, 372, 702\n\nR J Laureijs, U Klaas, P J Richards, B Schulz, The ISO Handbook, Volume IV -PHT -The Imaging Photo-Polarimeter. & Abraham, P.Laureijs, R. J., Klaas, U., Richards, P. J., Schulz, B., & Abraham, P., eds. 2003, The ISO Handbook, Volume IV - PHT -The Imaging Photo-Polarimeter\n\n. D Lemke, U Klaas, J Abolins, A&A. 3151A&ALemke, D., Klaas, U., Abolins, J., et al. 1996, A&A, 315, L64 Lequeux, J. 1979, A&A, 71, 1\n\n. A Li, B T Draine, ApJ. 576762Li, A. & Draine, B. T. 2002, ApJ, 576, 762\n\nU Lisenfeld, A Ferrara, Dwarf Galaxies: Probes for Galaxy Formation and Evolution. Kyoto, Japan2145ApJLisenfeld, U. & Ferrara, A. 1997, Dwarf Galaxies: Probes for Galaxy Formation and Evolution, Joint Discussion 2, 21 August 1997, Kyoto, Japan., 2 -. 1998, ApJ, 496, 145\n\n. J S Mathis, P G Mezger, N Panagia, A&A. 128212Mathis, J. S., Mezger, P. G., & Panagia, N. 1983, A&A, 128, 212\n\n. L Pagani, G Lagache, A Bacmann, A&A. 40659Pagani, L., Lagache, G., Bacmann, A., et al. 2003, A&A, 406, L59\n\n. M Rubio, J Lequeux, F Boulanger, A&A. 2719Rubio, M., Lequeux, J., & Boulanger, F. 1993a, A&A, 271, 9\n\n. M Rubio, J Lequeux, F Boulanger, A&A. 2711Rubio, M., Lequeux, J., Boulanger, F., et al. 1993b, A&A, 271, 1\n\nM Sauvage, L Vigroux, IAU Symp. 148: The Magellanic Clouds. 407Sauvage, M. & Vigroux, L. 1991, in IAU Symp. 148: The Magellanic Clouds, 407\n\n. M Sauvage, L Vigroux, T X Thuan, A&A. 237296Sauvage, M., Vigroux, L., & Thuan, T. X. 1990, A&A, 237, 296\n\n. P B W Schwering, F P Israel, A&AS. 7979Schwering, P. B. W. & Israel, F. P. 1989, A&AS, 79, 79\n\n. U J Sofia, D M Meyer, ApJ. 558221ApJSofia, U. J. & Meyer, D. M. 2001a, ApJ, 558, L147 -. 2001b, ApJ, 554, L221\n\n. S Stanimirovic, L Staveley-Smith, J M Dickey, R J Sault, S L Snowden, MNRAS. 302417Stanimirovic, S., Staveley-Smith, L., Dickey, J. M., Sault, R. J., & Snowden, S. L. 1999, MNRAS, 302, 417\n\n. S Stanimirovic, L Staveley-Smith, J M Van Der Hulst, MNRAS. 315791Stanimirovic, S., Staveley-Smith, L., van der Hulst, J. M., et al. 2000, MNRAS, 315, 791\n\n. L Staveley-Smith, S Kim, M Putman, S Stanimirović, Reviews of Modern Astronomy. 11117Staveley-Smith, L., Kim, S., Putman, M., & Stanimirović, S. 1998, Reviews of Modern Astronomy, 11, 117\n\n. J Tumlinson, J M Shull, B L Rachford, ApJ. 566857Tumlinson, J., Shull, J. M., Rachford, B. L., et al. 2002, ApJ, 566, 857\n\n. E Vangioni-Flam, J Lequeux, M Maucherat-Joubert, B Rocca-Volmerange, A&A. 9073Vangioni-Flam, E., Lequeux, J., Maucherat-Joubert, M., & Rocca-Volmerange, B. 1980, A&A, 90, 73\n\n. D E Welty, J T Lauroesch, J C Blades, L M Hobbs, D G York, ApJ Lett. 55475Welty, D. E., Lauroesch, J. T., Blades, J. C., Hobbs, L. M., & York, D. G. 2001, ApJ Lett., 554, L75\n\nB E Westerlund, IAU Symp. 148: The Magellanic Clouds. 15Westerlund, B. E. 1991, in IAU Symp. 148: The Magellanic Clouds, 15\n\n. S L Wheelock, T N Gautier, J Chillemi, NASA STI/Recon Technical Report N. 9522539Wheelock, S. L., Gautier, T. N., Chillemi, J., et al. 1994, NASA STI/Recon Technical Report N, 95, 22539\n\n. K Wilke, U Klaas, D Lemke, A&A. 41469Wilke, K., Klaas, U., Lemke, D., et al. 2004, A&A, 414, 69\n\n. K Wilke, M Stickel, M Haas, A&A. 401873Wilke, K., Stickel, M., Haas, M., et al. 2003, A&A, 401, 873\n"},"corpus_id":{"kind":"number","value":1432248,"string":"1,432,248"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"astro-ph/0405045\",\n \"mag\": 2127714165,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1051/0004-6361:20035918\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/astro-ph/0405045v1.pdf\"\n ],\n \"pdf_sha\": \"ccd3d843f2b79c5920f682d7ba1ff316c4e0d221\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"https://www.aanda.org/articles/aa/pdf/2004/32/aa0918.pdf\",\n \"status\": \"BRONZE\"\n }\n}"},"title":{"kind":"string","value":"Multi-wavelength analysis of the dust emission in the Small Magellanic Cloud ⋆"},"authors":{"kind":"list like","value":["C Bot \nCentre de Données Astronomiques de Strasbourg (CDS)\nUMR 7550\nUniversité Louis Pasteur\nF-67000STRASBOURGFrance\n\nInstitut d'Astrophysique Spatiale\nUniversité Paris-Sud\nF-91405OrsayFrance","F Boulanger \nInstitut d'Astrophysique Spatiale\nUniversité Paris-Sud\nF-91405OrsayFrance","G Lagache \nInstitut d'Astrophysique Spatiale\nUniversité Paris-Sud\nF-91405OrsayFrance","L Cambrésy \nCentre de Données Astronomiques de Strasbourg (CDS)\nUMR 7550\nUniversité Louis Pasteur\nF-67000STRASBOURGFrance","D Egret \nUMS 2201\nObservatoire de Paris\nF-75014ParisFrance"],"string":"[\n \"C Bot \\nCentre de Données Astronomiques de Strasbourg (CDS)\\nUMR 7550\\nUniversité Louis Pasteur\\nF-67000STRASBOURGFrance\\n\\nInstitut d'Astrophysique Spatiale\\nUniversité Paris-Sud\\nF-91405OrsayFrance\",\n \"F Boulanger \\nInstitut d'Astrophysique Spatiale\\nUniversité Paris-Sud\\nF-91405OrsayFrance\",\n \"G Lagache \\nInstitut d'Astrophysique Spatiale\\nUniversité Paris-Sud\\nF-91405OrsayFrance\",\n \"L Cambrésy \\nCentre de Données Astronomiques de Strasbourg (CDS)\\nUMR 7550\\nUniversité Louis Pasteur\\nF-67000STRASBOURGFrance\",\n \"D Egret \\nUMS 2201\\nObservatoire de Paris\\nF-75014ParisFrance\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We present an analysis of dust grain emission in the diffuse interstellar medium of the Small Magellanic Cloud (SMC). This study is motivated by the availability of 170 µm ISOPHOT data covering a large part of the SMC, with a resolution enabling to disentangle the diffuse medium from the star forming regions. After data reduction and subtraction of Galactic foreground emission, we used the ISOPHOT data together with HiRes IRAS data and ATCA/Parkes combined H column density maps to determine dust properties for the diffuse medium. We found a far infrared emissivity per hydrogen atom 30 times lower than the Solar Neighborhood value. The modeling of the spectral energy distribution of the dust, taking into account the enhanced interstellar radiation field, gives a similar conclusion for the smallest grains (PAHs and very small grains) emitting at shorter wavelength. Assuming Galactic dust composition in the SMC, this result implies a difference in the gas-to-dust ratio (GDR) 3 times larger than the difference in metallicity. This low depletion of heavy elements in dust could be specific of the diffuse ISM and not apply for the whole SMC dust if it results from efficient destruction of dust by supernovae explosions."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"The Small Magellanic Cloud (SMC) is a dwarf irregular galaxy at a distance of approximately 60 kpc (Westerlund 1991). Its relative proximity enables to have a global view of the galaxy, as well as a sufficiently good resolution to disentangle the diffuse medium from the star forming regions and molecular clouds. The SMC is an ideal target to study the low metallicity diffuse medium present not only in dwarf galaxies, but also in the external parts of large spirals.\nThe SMC contains several H regions and molecular clouds (Rubio et al. 1993a). It has a high interstellar radiation field (ISRF), between 4 and 10 times that of the Solar neighborhood (Lequeux 1979;Vangioni-Flam et al. 1980;Azzopardi et al. 1988). It is extremely rich in neutral atomic gas (Staveley-Smith et al. 1998) and has a low heavy element abundance (Dufour 1984;Sauvage & Vigroux 1991). All of these characteristics are indicative of a \" young\" galaxy, actively forming stars, which might in some sense be consid-Send offprint requests to: C.Bot, e-mail: bot@astro.u-strasbg.fr ⋆ based on observations with ISO, an ESA project with instruments funded by ESA Member States (especially the PI countries: France, Germany, The Netherlands and the UK) and with the participation of ISAS and NASA. ered as a local model of distant galaxies at the beginning of their evolution.\nDust grains are composed mainly of heavy atoms (e.g. carbon, silicium, oxygen, . . . ). In a low metallicity environment, such elements are lacking. How does this deficiency act on the dust grain abundances? The gas-to-dust ratio (GDR) is often assumed to be proportional to the metallicity (i.e. there is a fixed fraction of heavy elements in dust). Based on a model of the evolution of elemental abundances in gas and dust, Dwek (1998) concludes that this proportionality applies across the disk of the Galaxy. However, observations in dwarf irregular and blue compact dwarf galaxies show a large scatter in the GDR versus metallicity relation. This indicates that a simple scaling might be a too simplistic description of the phenomenon (Issa et al. 1990;Lisenfeld & Ferrara 1997, 1998. The GDR is expected to vary among ISM components because destruction of dust by shock waves is more efficient in the diffuse gas while accretion of heavy elements on dust dominates in molecular clouds. The integrated GDR is thus expected to depend on the relative fraction of matter in diffuse/dense components and on the supernova rate and thereby on the star formation history (Hirashita 1999).\nThe goal of our study is to characterize the dust properties in the diffuse medium of the SMC, and in particular the GDR. Comparing the GDR and the dust spectral energy distribution for the SMC and the Galaxy, we quantify the effect of the difference in metallicity on dust abundance and size distribution (e.g. the relative amounts of PAHs, very small grains and big grains).\nSince the gas and the dust masses are closely mixed, where the radiation is homogeneous the dust emission is expected to correlate with the distribution of the gas. In the diffuse medium, gas is mainly composed of neutral hydrogen and the GDR can be derived from the comparison between the dust infrared emission with the H column density, provided that the dust temperature is known. This is not the case in starforming regions, where the infrared emission from dust in H regions and molecular clouds closely associated with luminous stars can become dominant. Several authors have computed GDRs, comparing the IRAS infrared emission with the H column density, taking the 100µm/60µm intensity ratio to derive the dust temperature (Schwering & Israel 1989;Sauvage et al. 1990;Stanimirovic et al. 2000), but this ignores the contribution from small grains at 60µm. An additional difficulty comes from the variations in the intensity of the radiation field inside a beam, which casts doubt on any GDR estimate from studies where the angular resolution is too coarse to separate star forming regions from the diffuse emission (Li & Draine 2002;Aguirre et al. 2003) The availability of 170 µm ISOPHOT observations enables us to have a new view on dust properties of the SMC. Independently of our work, Wilke et al. (2003) have also reduced and analyzed these data but with a distinct focus on point sources. In a second paper (Wilke et al. 2004), these authors adress the same question as us but they investigate the main body (bar) of the SMC without discarding the star forming regions.\nAfter data reduction and validation (Sect. 2 and 3), the 170 µm map is compared to additional observations (Sect. 4), in particular the IRAS High Resolution map (HiRes) at 100 µm. Having obtained the 170/100 µm emission ratio, in Sect. 5.1 we derive a reference temperature for the big grains in the diffuse medium of the SMC. An H column density map from ATCA/Parkes combined data (Stanimirovic et al. 1999) is used to study the dust to gas correlation, allowing us to estimate the emissivity of the dust per hydrogen atom for the diffuse medium (Sect.5.2) . Finally, the spectral energy distribution of the dust grains in the diffuse medium of the SMC is fitted with the Desert et al. (1990) model to quantify the abundances of the different grain components (Sect. 5.4). The results are discussed in Sect. 6; Sect. 7 presents our conclusions."},{"title":"The Data","content":"This study relies on archive data. Some were ready-made as the IRAS maps at 12, 25, 60, and 100 µm, or the ATCA/Parkes combined data of the H line at 21 cm. The ISOPHOT data was imported from the ISO Data Archive 1 . But the quality of the ISOPHOT map at 170 µm in the ISO archive was not sufficient for this study, so we retrieved, reduced and calibrated 1 http://iso.vilspa.esa.es/ida/index.html the data. We first present the ISOPHOT data and the reduction processes. Then we describe the ancillary data that were used."},{"title":"The ISOPHOT data","content":"The Small Magellanic Cloud was almost fully covered by 10 observations with the ISOPHOT instrument (Lemke et al. 1996) on board the ISO satellite (Kessler et al. 1996;Laureijs et al. 2003). The observations were made using the P22 AOT (Astronomical Observation Template), i.e. a raster map on a two-dimensional regular grid, with the C200 detector (2x2 pixels) at an effective wavelength of 170 µm. The pixel field of view is 1.5 ′ and the raster pointing spacings are 180 ′′ in both in-scan and cross-scan directions. The ISOPHOT point spread function can be approximated by a 2D gaussian with 90 ′′ full width half maximum (FWHM) in both directions and is supposed to be constant across the map.\nWe reduced the data with PIA 2 (ISOPHOT Interactive Analysis) V10.0 (Gabriel et al. 1997(Gabriel et al. , 1998, except for the flatfield correction. In the following, we describe the specific treatments applied. Some observations showed specific problems on raw data (e.g. a pixel went into erratic state or a severe glitch affected a pixel until the end of the observation). We choose to remove the data for these pixels in the corresponding observations. If this removal is not done, the reduction processes will interpret these erratic fluxes as flat-fielding. This distorts the photometry of the other pixels for the whole observation.\nTo deal with the discarded pixels, we choose to perform the flat-field correction without the PIA tool as in Lagache & Dole (2001). The flat-field correction is on average 1.01 ± 0.05, 0.93 ± 0.03, 1.17 ± 0.09 and 0.92 ± 0.03 for pixel 1, 2, 3 and 4 respectively. Reproducible flat-field correction values for the ISOPHOT C 200 array have been published by Lagache & Dole (2001) for the carefull reduction of 3 deep fields (FIRBACK). We do not observe the same average flat field corrections factor and we find a higher dispersion from one observation to another. This difference may be because the responsivity of ISOPHOT C 200 pixels changes with the observed intensity, so that the flat-field correction computed with low fluxes is no longer valid for the brightest pixels (Pagani et al. 2003). This effect might affect the 170 µm flux determination for bright sources. The reproducible behavior of pixels observed by Lagache & Dole (2001) is then due to the relative flatness of deep fields.\nThe different observations have been done at different times, therefore the \"absolute calibration\" performed using the FCS measurement can be slightly different from one raster to another. This difference is observed as discontinuities between maps in the mosaic. To correct for these small absolute calibration differences (for a border between two maps, the mean difference in brightness observed is about 13%), we applied a \" general flat-field\" correction as in Lagache & Dole (2001).\nAfter concatenating the timelines, the raster is projected on the sky using each individual pixel coordinates, with a pixel size for the map of 10 ′′ to match the IRAS HiRes gridding. Because the raster step size of each observation was a full detector size, and the spacecraft Y-axis was not parallel to right ascensions or declinations, we observe gaps between the individual array pointings. The final map is presented at Fig. 1 after a bilinear interpolation to reconstruct the signal in the gaps."},{"title":"Ancillary data","content":""},{"title":"IRAS HiRes data","content":"As a complement to the ISOPHOT data at 170 µm, we used IRAS high resolution (HiRes) data at 12, 25, 60 and 100 µm. Maps of 2 • × 2 • centered at 00 h 57 m 29.12 s −72 • 32 ′ 29.2 ′′ in J2000 were obtained. The pixel size is 10 ′′ . The HiRes point spread function (PSF) varies spatially in a map. A gaussian approximation of the PSF is given with the HiRes maps and is defined by two spatially dependent FWHMs and a rotation angle. For more information on HiRes, please refer to Aumann et al. (1990) and Bontekoe et al. (1994). Since we are searching for extended structure properties, the HiRes IRAS maps were brought to the DC-calibration (Wheelock et al. 1994). This DC calibration is only valid for spatial scales greater than 40 ′ . As the SMC exhibits structure on various spatial scales, the DC calibration introduces some limitations that will be discussed in Sect. 3.2."},{"title":"ATCA/Parkes combined SMC H data","content":"To study the correlation between gas and dust in the SMC, we needed a H column density map to compare with the ISOPHOT map, in order to compute the abundances of dust grains relative to the hydrogen. Stanimirovic et al. (1999) have combined Parkes telescope observations of neutral hydrogen in the SMC with the Australia Telescope Compact Array (ATCA) aperture synthesis mosaic to obtain a set of images sensitive to all angular spatial scales between 98 ′′ and 4 • from 90 to 215 km.s −1 . The spatial resolution obtained is 98 ′′ and the velocity resolution is 1.65 km.s −1 . We used a column density map of the SMC, obtained by integrating over the whole velocity range and assuming that the neutral hydrogen is optically thin 3 ."},{"title":"Parkes H data for the Galactic foreground","content":"A Galactic filament exists in front of the SMC. Even though it is very smooth, large (see contours of Fig. 1) and uniform (column densities between 2.5 · 10 20 and 4 · 10 20 atom.cm −2 ), it can bias studies. The far-infrared fluxes observed in either the ISOPHOT or the IRAS HiRes maps represent the emission from the SMC dust, but also from this Galactic foreground. 3 The column density map and the data cube are available at ftp://ftp2.naic.edu/pub/ast/sstanimi/. However, this Galactic component of the emission is not observed in the ATCA/Parkes data, due to a velocity cut and spatial filtering. To study the SMC dust properties, it was thus necessary to remove this foreground component (see Sect. 3.3). In this step, we used a Galactic H column density map constructed from data of the Parkes narrow-band H survey of the tidal arms of the Magellanic system (Brüns et al. 2000). The column density map of the H Galactic foreground was obtained by integrating the emission from -60 to +50 km/s, outside the SMC velocities. The resolution is 15 ′ . Due to sidelobes effects, the map obtained can slightly overestimate the column density by 10 to 20 %."},{"title":"Photometric consistency","content":"DIRBE provided infrared absolute sky brightness maps in 10 bands in the wavelength range 1.25 to 240 µm. This absolute photometry is used to check ISOPHOT mosaic and IRAS fluxes."},{"title":"ISOPHOT data","content":"We checked if the ISOPHOT 170 µm flux is consistent with the spectral energy distributions obtained from the 100, 140 and 240 µm DIRBE fluxes in the SMC. To do this we convolved the ISOPHOT map with the DIRBE beam. This gives 18 ISOPHOT fluxes in the SMC as if they were observed by DIRBE and we compare ISOPHOT and DIRBE fluxes by plotting the spectral energy distribution and fitting it with a modified black body with a spectral index of 2 on the DIRBE values. Fig. 2 shows that the DIRBE-convolved ISOPHOT flux integrated over the 18 positions observed by DIRBE, is consistent within the errors bars with the best modified blackbody fit (obtained for a temperature of 20.6K, an emissivity of ǫ H (250µm) = 3 · 10 −28 cm 2 for 10 20 hydrogen atoms.cm −2 ). Please note that these values correspond to an average over the whole galaxy and were obtained only for calibration purposes.\nWe have also plotted the individual fit values at 170 µm versus the corresponding 18 ISOPHOT values. This plot shows a significant scatter but is consistent with a linear correlation with unity slope and zero offset.\nA comparison with the IRAS high resolution data was also performed to check possible photometric errors introduced by the interpolation process. The correlation between IRAS and ISOPHOT data does not change when we restrict the comparison to the pixels measured by ISOPHOT rather than the full interpolated map. The interpolation required to fill the holes in the measurements does not seem to bias the image comparisons."},{"title":"IRAS data","content":"The IRAS HiRes maps have the same photometric calibration as the IRAS ISSA maps. The IRAS detector had a dwell-time dependent responsivity change: the gain changes as a function of source size. This effect is known as the \" AC/DC effect\". A first set of corrections were taken into account in the IRAS Fig. 1. The ISOPHOT map at 170 µm (in MJy/sr) after reduction processes and a bilinear interpolation to fill the holes left by the mapping pattern. The flat-field has been efficiently corrected for the diffuse part of the map, but one can still see low level defects in the brightest parts. Contours of the H Galactic foreground are overplotted at column densities of 2.6, 2.8, 3., 3.2, 3.4 and 3.6·10 20 H atom.cm-2 (from light grey to black). data products. These corrections were later found to be insufficient. On the basis of a comparison of the IRAS and DIRBE sky brightness for extended emission, Wheelock et al. (1994) recommended applying additional correction factors of 0.88, 1.01, 0.82 and 0.74 at 12, 25, 60 and 100 µm respectively.\nHowever, these correction factors are only valid for spatial scales greater than 40 ′ . This is not true in the SMC where structures on various spatial scales are present. Stanimirovic et al. (2000) have also compared DIRBE and IRAS data by integrating fluxes over 6.25 •2 in the SMC. They found that integrated flux densities measured by DIRBE for 60 and 100 µm are higher than IRAS fluxes by about 10 to 20%, which is an opposite correction to that observed by Wheelock et al. (1994).\nWe have compared the IRAS (convolved to the DIRBE resolution) and the DIRBE 100 µm maps on a 10 • × 10 • scale around the SMC to understand this discrepancy. We see that the Wheelock et al. (1994) correction factor of 0.74 is only truly valid outside the galaxy and increases in the SMC up to a maximum value in the north of the optical bar (N66 region). For the diffuse medium of the SMC, the Wheelock et al. (1994) correction factor is sufficient. After correction, we estimate that the 100 µm IRAS brightness for the diffuse medium is underestimated by less than 15%."},{"title":"Galactic foreground removal","content":"Unlike the ATCA/Parkes H map, the far-infrared emission maps (ISOPHOT and IRAS) trace both the dust emission in the SMC and the dust in our Galaxy. To disentangle these two components in the far-infrared emission, we used a column density map of the H Galactic foreground presented in Sect. 2.2.3.\nDust grains absorb energy from the UV/optical radiation field that they re-emit in the infrared wavelength as thermal emission. Large grains are at a fixed temperature and their emitted intensity can be expressed as:\nwhere N H is the column density of hydrogen and ǫ H (ν) is the emissivity of grains per hydrogen atom. Assuming that the emissivity of dust grains in the far-infrared follows a power law: . 2. Comparison of the DIRBE-convolved ISOPHOT fluxes with the DIRBE ones (in MJy/sr), after an integration on the whole SMC. The modified black body traced was fit on the DIRBE values only and correspond to a temperature of 20.6 K and an emissivity of ǫ H (250µm) = 3 · 10 −28 cm 2 for 10 20 hydrogen atoms.cm −2 . We see that the ISOPHOT value at 170 µm is consistent with the DIRBE photometry.\nand taking β = 2 (as found for Galactic emission (Lagache et al. 1998(Lagache et al. , 1999 for wavelengths between 100 µm and 500 µm).\nUsing a temperature of 17.5 K and an emissivity per hydrogen atom of 10 −25 cm 2 (Boulanger et al. 1996a) 4 for the Galactic dust, we computed the Galactic thermal emission and removed this foreground emission from the IRAS and ISOPHOT maps to obtain the emission of the SMC dust only. At 12, 25 and 60 µm, we used the I 12 /I 100 , I 25 /I 100 and I 60 /I 100 ratios computed from IRAS maps outside the SMC, together with the computed foreground emission map at 100 µm to remove the Galactic foreground emission at these wavelengths. However, this foreground emission remains in the data noise at 12, 25 and 60 µm."},{"title":"Spatial image comparison","content":""},{"title":"Projection and resolution","content":"The data set presented in Sect. 2 is composed of maps with different fields of view and different resolutions. We projected all of them onto a common grid: that of IRAS HiRes maps (2 • × 2 • field of view centered at 00 h 57 m 29.12 s − 72 • 32 ′ 29.2 ′′ (J2000) with 10 ′′ pixel size). These maps were inter-compared by pairs.\nTo compare the ISOPHOT map with one of the HiRes maps, the projected and re-sampled ISOPHOT map was convolved with a spatial varying kernel in order to be at the HiRes resolution when the point spread function of ISOPHOT was smaller than the one of HiRes (Kerton & Martin 2001). The HiRes map was also convolved with a spatially varying kernel to match the ISOPHOT resolution when the HiRes beam was smaller than the ISOPHOT one.\nTo compare the HI column density map with the ISOPHOT map, no convolution was necessary as long as their resolutions were comparable. Fig. 3 shows two of the transformed maps: the ISOPHOT map degraded to the IRAS 100 µm resolution and the projected ATCA/Parkes map."},{"title":"Colors from scatterplots","content":"To characterize the diffuse medium (i.e. the extended region where the infrared emission-H column density correlation is homogenous) spectral energy distribution, we developed a comparison method that is not biased by point sources and is also not affected by offsets in the data calibration.\nWe work with two maps (obtained after the treatment presented in Sect. 4). The general correlation between the two maps is first quantified by fitting a straight line to the scatterplot of the pixel-to-pixel comparison. We deduce a first ratio value and an error σ given by the dispersion in the correlation, which enables us to filter out localized sources (i.e. star-forming regions) by removing pixels that are 3σ above the correlation (there is no points below −3σ).\nFrom the \" filtered\" maps, we compute a ratio map by linearly fitting the pixel correlation in the scatterplot for a sliding box of 50×50 pixels. This corresponds to a 5.5×5.5 beams box which is a minimum size to have independent data points. The size of the box could not be greater, or we would not trace the local variations. In some cases there are too few points inside a box (too many points above 3σ) to do a meaningful fit. We assign no value to these positions. We also skip positions where the correlation coefficient is too low. Adjusting a gaussian on the histogram of the ratio values gives us a reference ratio and the associated dispersion. This dispersion contains photometric and fitting errors but also a true scatter in the ratio. Both phenomena cannot be easily disentangled.\nThe spatial distribution and the histogram with the gaussian fit obtained for the I 100µm /I 170µm ratio are presented in Fig. 4 as an example. We see that the comparison method avoids large areas in the main body (bar) of the SMC. These regions correspond to highly structured star-forming sites. In these regions, either the filtering technique has left too few independent positions in one box to compute a significant correlation or no clear correlation is observed."},{"title":"Results","content":""},{"title":"Infrared colors","content":"The method presented in Sect. 4.2 was applied to the IRAS HiRes and ISOPHOT maps to compute the I 100µm /I 170µm , I 60µm /I 170µm , I 25µm /I 170µm , I 12µm /I 170µm ratios. Adjusting a gaussian on the histograms gives us reference ratios and their associated dispersions:\nUsing equation 1 and 2 we can derive from the I 100 µm I 170 µm ratio map the big grain equilibrium temperature distribution in the diffuse medium in the Small Magellanic Cloud. This leads to a reference temperature of: T dust = 22 ± 2 K. Note that this temperature is different from the one that would be inferred from the ratios of the mean I 100 µm and I 170 µm fluxes. Wilke et al. (2004) found a mean temperature for the main body of the SMC of 20.9 ± 1.9K, which is lower than the temperature we find for the diffuse medium. Their mean value falls within our range of temperatures, which is expected because the regions we study overlap. The difference in mean values could be due to the presence of cold atomic and molecular clouds in the main body of the SMC. Our studies seem to be consistent with each other and are more complementary than comparable.\nThe large dust grain temperature is related to the radiation field intensity and will be discussed in Sect. 6."},{"title":"Gas to dust correlation","content":"From the ISOPHOT and the ATCA/Parkes maps, using the method described Sect. 4.2, we found the distribution presented on Fig. 5. This distribution exhibits a main peak and a distribution tail. The peak is characterized by:\n17 · 10 −20 ± 0.06 · 10 −20 MJy.sr −1 .cm 2 (7) Since the regions used to define the ratios are not exactly the same, the determination of the characteristics of the diffuse medium could be biased. We checked that this is not the case. If we use the region where I 170µm /N HI is computed to determine the colors, we find that the derived quantities are not affected by more than 15%. We see that the characteristics of the diffuse medium are not biased by the definition of the limits of the studied region. Comparing the region defined by equation 7 with the H α distribution, we see in Fig. 6 that the regions corresponding to the tail in the I 170µm /N HI (located mainly in the main body) are regions where the diffuse H α emission is important. Comparing the map in Fig. 4 and Fig. 6, we also see that the regions where no ratios can be computed correspond to important star-forming sites. Assuming that all the dust is in the H , we use the I 170µm /N HI ratio map and the temperature map computed above to obtain an histogram of the emissivity per hydrogen atom of the dust grains in the diffuse medium. We find a reference emissivity at 170 µm of:\nǫ H (λ) = 7.3 · 10 −27 ± 3.2 · 10 −27 cm 2 (8)\nThe error bar takes into account uncertainties in the temperature and I 170 µm /N HI . If the emissivity follows a power law with a spectral index of 2, the emissivity per H atom in the SMC at 250 µm is ǫ H (λ) = 3.4 · 10 −27 ± 1.5 · 10 −27 (λ/250 µm) −2 cm 2 . This corresponds to κ ν = 0.21(λ/250 µm) −2 cm 2 g −1 . Assuming similar dust optical properties in the SMC and the Galaxy, the emissivity is related to the GDR and the ratio between hydrogen column density and E(B − V) color excess by:\nThe solar neighborhood value for the emissivity is 10 −25 cm 2 at 250 µm (Boulanger et al. 1996a). The inferred SMC GDR is thus 30 times greater than in the Galaxy. Stanimirovic et al. (2000) have presented a similar study using the same H data together with HIRAS data. They came to the same conclusion that the GDR is a factor 30 higher in the SMC than in our Galaxy. However, this agreement might be a coincidence since 1) we did not use the same calibration of the IRAS data, 2) their temperature estimation based on the 60/100µm color is higher than ours, 3) the GDR they present is averaged over the whole galaxy (including star forming regions) and 4) their GDR is compared to a Galactic GDR of 375 instead of the standard value of 100 5 . Using the standard Galactic GDR value, the average SMC GDR found by Stanimirovic et al. (2000) becomes 100 times higher than in the Galaxy.\nOur estimate of the SMC's GDR 30 times higher than in the Galaxy, is significantly larger than the difference in metallicity, which is less than 10 (Dufour 1984). It is twice the observed difference in the N(H I )+N(H 2 )"},{"title":"E(B−V)","content":"values presented by Tumlinson et al. (2002)."},{"title":"Infrared excesses","content":"A noticeable result of the ISOPHOT and H comparison is the presence of a tail in the 170 µm/H histogram (see Fig. 5). This tail corresponds to pixels in the SMC optical bar with excess 170 µm emission with respect to N HI (see Fig. 7). We checked that the mean temperature for this region of excess is the same as that for the diffuse medium. These excesses may be due either to a variation in the dust abundances between the diffuse medium and the bar, or due to the presence of cold H or H 2 gas. We believe that the excesses are real but it is difficult to infer an amount of cold H and molecular gas from it. Firstly, it is difficult to disentangle these two possible contributions. Secondly, where some of these excesses are due to dust in cold H or H 2 we do not expect them to be correlated with H emission and another method is needed to quantify them. It is thus beyond the scope of this paper to translate these excesses into an estimate of cold gas that could be compared to other studies where this is attempted (Stanimirovic et al. 2000). Finally, we feel that the photometric uncertainties in the ISOPHOT and IRAS data are an additional difficulty, especially near starforming regions. Further studies with the Spitzer satellite could address this question with better photometry and angular resolution."},{"title":"SMC dust properties in diffuse H : modeling the spectral energy distribution","content":"The different intensity ratios obtained in Sect. 5.1 enable us to trace the spectral energy distribution of dust in the diffuse medium of the SMC and to compare it with that of the solar neighborhood. The difference in temperature and in abundances is seen Fig. 9 as a shift along the wavelength and brightness axis respectively. The error bars include true variations in the dust properties across the SMC diffuse interstellar medium which are correlated over wavelengths.\nWe used the dust model of Desert et al. (1990) to quantify the grain dust abundances. In order to model the spectral energy distribution (SED), we need to estimate the interstellar radiation field (ISRF) in the diffuse medium of the SMC. In Sect. 5.1 we found the big dust grain equilibrium temperature to be 22 K. Using the dust thermal equilibrium temperature in the solar neighborhood of 17.5 K (Boulanger et al. 1996a) as a reference, the dust temperature in the SMC leads to a radiation field χ S MC = T 4+β dust (S MC)/T 4+β dust (Galaxy) = 4. This value is consistent with the estimate of the ISRF made from massive star counts by Lequeux (1979). Vangioni-Flam et al. (1980) also found the same ratio from UV measurements. We used the Mathis et al. (1983) 10 kpc Galactic ISRF and scaled it by a factor χ = 4 for the modeling. We have adjusted only the abundances of PAHs, very small grains (VSG) and big grains. For the other parameters defining the size distribution, we have kept the Galactic values. The best fit is presented on Fig. 8, it corresponds to under-anbundant PAHs, very small grains (VSG) and big grains (BG) compared to the Galaxy (36, 23 and 30 times less respectively). Mass abundances relative to hydrogen, compared with Galactic values (Desert et al. 1990) are summarized in table 1.\nThe modeling confirms the GDR estimated in Sect 5.2 to be around 30 times greater than in the Galaxy. Within the uncertainties, PAHs and very small grain abundances are also consistent with this factor.\nThe model is clearly below the 60 µm point. This could be due to our crude estimate of the ISRF. We have used a single ISRF while our data analysis shows temperature variations across the SMC diffuse medium. To take the variations in the radiation field into account, we transformed the temperature Fig. 9. Infrared spectral energy distribution of dust in the diffuse medium for the SMC and the Solar Neighborhood. The brightness corresponds to an interstellar gas column density of 10 20 cm −2 . The Galactic spectrum (in grey) is composed in the far-IR by COBE/FIRAS data from Boulanger et al. (1996a) (with the corresponding spectrum from grains in thermal equilibrium indicated by the dotted line), and at shorter wavelengths by the COBE/DIRBE (Bernard et al. 1994) and Arome balloon data (triangle at 3.3 µm) (Giard et al. 1994). An ISOCAM spectrum (Boulanger et al. 1996b) scaled to the 12 µm brightness is also shown. The SMC spectrum (in black) is the best fit obtained with the Desert et al. model, same as in figure 8. 2.0 · 10 −5 4.7 · 10 −4 BG 2.2 · 10 −4 6.4 · 10 −3 distribution obtained in Sect. 5.1 into a χ S MC distribution. For each value of the distribution, we computed a spectral energy distribution (SED) of the dust emission with the Desert et al. (1990) model, using the Mathis 10 kpc ISRF multiplied by the corresponding χ as an input (χ is a constant for the whole spectrum). We then computed the resulting total SED by adding the individual SEDs with weights according to the χ distribution shape. The resulting fit is not sufficiently broadened to encom-pass the 60 µm point. In addition, the mass abundances of dust grains have to be reduced by 8, 5 and 7 for the PAH, VSG and BG components respectively. Due to the high non-linearity of the Planck curve, even a small broadening of the χ distribution tends to reduce the abundances of dust grains significantly. These values are thus indicatives only but they show that the mass abundances represented in table 1 are more likely to be upper limits and that the emissivity could be even lower.\nIt is possible to fit the 60 µm value with a single ISRF scaled by χ = 10 (see Fig. 10). In this case, the dust grain mass abundances must then be reduced by 3, 2 and 2 for the PAHs, very small grains and big grains respectively. Such a high ISRF corresponds to a dust temperature of 25.6 K, which is higher than the estimate obtained in Sect. 5.1 but still compatible with the 100 µm and the 170 µm fluxes. This higher temperature would lead to an emissivity of 2.4 · 10 −27 cm 2 at 250 µm, which is even lower than the estimation obtained in Sect. 5.2 and results in a GDR 40 times higher than in the Galaxy. A higher radiation field could thus account for the 60µm excess without contra- dicting the main conclusion of this study, namely the fact that the GDR in the diffuse SMC ISM is much lower than the solar neighbourhood value scaled by the difference in metallicity.\nThe observed 60 µm excess could also be due to a change in the grain size distribution (Galliano et al. 2003). We assumed the Galactic grain sizes distribution to apply in the SMC. However, by changing the relative mass contributions of the different grain components, we enlarge the discontinuities in the size distribution between the three components of the model. It is necessary to further increase these discontinuities to account for the 60 µm excess.\nFurther investigations are needed to characterize the SMC ISRF and especially the 60 µm excess. The intensity and spectral shape of the ISRF are likely to be different than the Galactic one."},{"title":"Discussion","content":"Studying the gas to dust correlation in the diffuse medium of the SMC (see Sect. 5.2), we found a GDR in mass around 30 times greater than in the solar neighborhood, larger by a factor of 3 than the difference in metallicity. This result confirms the value found by Stanimirovic et al. (2000) in a similar study. This lower depletion of heavy elements in dust was independently reported by Welty et al. (2001), who found Mg and Si to be essentially undepleted in the SMC. It was also reported for a sample of dwarf irregulars by Lisenfeld & Ferrara (1998). Our result in the SMC fits with the relation that they observed between GDR and metallicity.\nAn interpretation of this relation has been proposed by Hirashita et al. (2002). With a chemical evolution model, they illustrated the dependence of the GDR on the star formation history. Stars generate dust but also contribute to their destruction by supernova shocks. They propose a scenario in which an intermittent star formation history leads to a time variable GDR. At some epochs of the galaxy evolution, the formation of dust grains is less efficient than the destruction process by supernova shocks, so that the GDR no longer scales with the metallicity. This model could be applied to the SMC with its specific star formation history (Harris & Zaritsky 2003). It is also necessary to account for a possible difference in the GDR between the diffuse and the denser medium.\nExtinction studies lead to significantly smaller value of the GDR, closer to that expected from the Galactic value for a linear dependence with metallicity. Bouchet et al. (1985) used photometric observations of O-B stars in the SMC to derive their extinction curves. They found a GDR value that is about 8 times the Galactic value, which is consistent with the difference in metallicity. More recently, Tumlinson et al. (2002) measured\nfor a sample of stars observed with FUSE and found a mean value 16 times lower than the solar Neighborhood value. This is closer but still smaller by a factor 2 than our estimate. The difference may be due to the fact that extinction studies are based on line-of-sights with high hydrogen column density where dense interstellar medium is likely to be present. Most of the stars are in the optical bar and/or in star formation regions associated with molecular clouds. The difference in the GDRs inferred in our work and extinction studies might thus point to a dependence on the environment: a decrease of the ratio from the diffuse ISM to star forming regions that might arise from the condensation of heavy elements on dust grains in dense gas and their destruction by supernova shocks in the diffuse ISM. Stanimirovic et al. (2000) have also noted this possibility for the SMC, but it still requires confirmation by further studies."},{"title":"Summary and conclusions","content":"This study was made feasible by the availability of 10 ISOPHOT observations at 170 µm, covering most of the SMC, in the ISO Data Archive. After reduction processes, mapping and Galactic foreground removal, the ISOPHOT data were compared with IRAS maps and a ATCA/Parkes combined H column density map to assess the properties of the dust in the diffuse medium of this low metallicity galaxy.\nFor the dust in the diffuse medium of the SMC, we found a reference equilibrium temperature of 22 ± 2 K and a reference dust emissivity of 3.4 · 10 −27 ± 1.5 · 10 −27 λ 250 µm cm 2 per hydrogen atom. This leads to a gas-to dust ratio 30 times greater than in the Galaxy. This high value reflects the SMC low metallicity, but also requires a lower depletion of dust elements than in the Galaxy. This shows that a simple linear dependence between the GDR and the metallicity does not apply for the diffuse medium of the SMC.\nThe spectral energy distribution is modeled with the Desert et al. (1990) model. The best fit obtained shows a similar decrease for PAHs and very small grains as for big grains. However, a 60 µm excess remains that can not be accounted for with simple assumptions. Further constraints, in particular on the ISRF, are needed to understand it.\nOur GDR estimate for the diffuse medium of the SMC is a factor 2 to 3 higher than that derived from extinction studies towards bright UV stars with high foreground hydrogen col-umn densities. This difference supports the notion of variations in the GDR, from diffuse regions where grains are more frequently destroyed by supernovae, to denser and more quiescent regions where grains re-accrete heavy elements."},{"title":"Fig. 5 .","content":""},{"title":"Fig. 4 .","content":""},{"title":"Fig. 6 .","content":""},{"title":"Fig. 7 .","content":""},{"title":"Fig. 8 .","content":""},{"title":"Fig. 10 .","content":""},{"title":"Table 1 .","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"The Small Magellanic Cloud (SMC) is a dwarf irregular galaxy at a distance of approximately 60 kpc (Westerlund 1991). Its relative proximity enables to have a global view of the galaxy, as well as a sufficiently good resolution to disentangle the diffuse medium from the star forming regions and molecular clouds. The SMC is an ideal target to study the low metallicity diffuse medium present not only in dwarf galaxies, but also in the external parts of large spirals.\\nThe SMC contains several H regions and molecular clouds (Rubio et al. 1993a). It has a high interstellar radiation field (ISRF), between 4 and 10 times that of the Solar neighborhood (Lequeux 1979;Vangioni-Flam et al. 1980;Azzopardi et al. 1988). It is extremely rich in neutral atomic gas (Staveley-Smith et al. 1998) and has a low heavy element abundance (Dufour 1984;Sauvage & Vigroux 1991). All of these characteristics are indicative of a \\\" young\\\" galaxy, actively forming stars, which might in some sense be consid-Send offprint requests to: C.Bot, e-mail: bot@astro.u-strasbg.fr ⋆ based on observations with ISO, an ESA project with instruments funded by ESA Member States (especially the PI countries: France, Germany, The Netherlands and the UK) and with the participation of ISAS and NASA. ered as a local model of distant galaxies at the beginning of their evolution.\\nDust grains are composed mainly of heavy atoms (e.g. carbon, silicium, oxygen, . . . ). In a low metallicity environment, such elements are lacking. How does this deficiency act on the dust grain abundances? The gas-to-dust ratio (GDR) is often assumed to be proportional to the metallicity (i.e. there is a fixed fraction of heavy elements in dust). Based on a model of the evolution of elemental abundances in gas and dust, Dwek (1998) concludes that this proportionality applies across the disk of the Galaxy. However, observations in dwarf irregular and blue compact dwarf galaxies show a large scatter in the GDR versus metallicity relation. This indicates that a simple scaling might be a too simplistic description of the phenomenon (Issa et al. 1990;Lisenfeld & Ferrara 1997, 1998. The GDR is expected to vary among ISM components because destruction of dust by shock waves is more efficient in the diffuse gas while accretion of heavy elements on dust dominates in molecular clouds. The integrated GDR is thus expected to depend on the relative fraction of matter in diffuse/dense components and on the supernova rate and thereby on the star formation history (Hirashita 1999).\\nThe goal of our study is to characterize the dust properties in the diffuse medium of the SMC, and in particular the GDR. Comparing the GDR and the dust spectral energy distribution for the SMC and the Galaxy, we quantify the effect of the difference in metallicity on dust abundance and size distribution (e.g. the relative amounts of PAHs, very small grains and big grains).\\nSince the gas and the dust masses are closely mixed, where the radiation is homogeneous the dust emission is expected to correlate with the distribution of the gas. In the diffuse medium, gas is mainly composed of neutral hydrogen and the GDR can be derived from the comparison between the dust infrared emission with the H column density, provided that the dust temperature is known. This is not the case in starforming regions, where the infrared emission from dust in H regions and molecular clouds closely associated with luminous stars can become dominant. Several authors have computed GDRs, comparing the IRAS infrared emission with the H column density, taking the 100µm/60µm intensity ratio to derive the dust temperature (Schwering & Israel 1989;Sauvage et al. 1990;Stanimirovic et al. 2000), but this ignores the contribution from small grains at 60µm. An additional difficulty comes from the variations in the intensity of the radiation field inside a beam, which casts doubt on any GDR estimate from studies where the angular resolution is too coarse to separate star forming regions from the diffuse emission (Li & Draine 2002;Aguirre et al. 2003) The availability of 170 µm ISOPHOT observations enables us to have a new view on dust properties of the SMC. Independently of our work, Wilke et al. (2003) have also reduced and analyzed these data but with a distinct focus on point sources. In a second paper (Wilke et al. 2004), these authors adress the same question as us but they investigate the main body (bar) of the SMC without discarding the star forming regions.\\nAfter data reduction and validation (Sect. 2 and 3), the 170 µm map is compared to additional observations (Sect. 4), in particular the IRAS High Resolution map (HiRes) at 100 µm. Having obtained the 170/100 µm emission ratio, in Sect. 5.1 we derive a reference temperature for the big grains in the diffuse medium of the SMC. An H column density map from ATCA/Parkes combined data (Stanimirovic et al. 1999) is used to study the dust to gas correlation, allowing us to estimate the emissivity of the dust per hydrogen atom for the diffuse medium (Sect.5.2) . Finally, the spectral energy distribution of the dust grains in the diffuse medium of the SMC is fitted with the Desert et al. (1990) model to quantify the abundances of the different grain components (Sect. 5.4). The results are discussed in Sect. 6; Sect. 7 presents our conclusions.\"\n },\n {\n \"title\": \"The Data\",\n \"content\": \"This study relies on archive data. Some were ready-made as the IRAS maps at 12, 25, 60, and 100 µm, or the ATCA/Parkes combined data of the H line at 21 cm. The ISOPHOT data was imported from the ISO Data Archive 1 . But the quality of the ISOPHOT map at 170 µm in the ISO archive was not sufficient for this study, so we retrieved, reduced and calibrated 1 http://iso.vilspa.esa.es/ida/index.html the data. We first present the ISOPHOT data and the reduction processes. Then we describe the ancillary data that were used.\"\n },\n {\n \"title\": \"The ISOPHOT data\",\n \"content\": \"The Small Magellanic Cloud was almost fully covered by 10 observations with the ISOPHOT instrument (Lemke et al. 1996) on board the ISO satellite (Kessler et al. 1996;Laureijs et al. 2003). The observations were made using the P22 AOT (Astronomical Observation Template), i.e. a raster map on a two-dimensional regular grid, with the C200 detector (2x2 pixels) at an effective wavelength of 170 µm. The pixel field of view is 1.5 ′ and the raster pointing spacings are 180 ′′ in both in-scan and cross-scan directions. The ISOPHOT point spread function can be approximated by a 2D gaussian with 90 ′′ full width half maximum (FWHM) in both directions and is supposed to be constant across the map.\\nWe reduced the data with PIA 2 (ISOPHOT Interactive Analysis) V10.0 (Gabriel et al. 1997(Gabriel et al. , 1998, except for the flatfield correction. In the following, we describe the specific treatments applied. Some observations showed specific problems on raw data (e.g. a pixel went into erratic state or a severe glitch affected a pixel until the end of the observation). We choose to remove the data for these pixels in the corresponding observations. If this removal is not done, the reduction processes will interpret these erratic fluxes as flat-fielding. This distorts the photometry of the other pixels for the whole observation.\\nTo deal with the discarded pixels, we choose to perform the flat-field correction without the PIA tool as in Lagache & Dole (2001). The flat-field correction is on average 1.01 ± 0.05, 0.93 ± 0.03, 1.17 ± 0.09 and 0.92 ± 0.03 for pixel 1, 2, 3 and 4 respectively. Reproducible flat-field correction values for the ISOPHOT C 200 array have been published by Lagache & Dole (2001) for the carefull reduction of 3 deep fields (FIRBACK). We do not observe the same average flat field corrections factor and we find a higher dispersion from one observation to another. This difference may be because the responsivity of ISOPHOT C 200 pixels changes with the observed intensity, so that the flat-field correction computed with low fluxes is no longer valid for the brightest pixels (Pagani et al. 2003). This effect might affect the 170 µm flux determination for bright sources. The reproducible behavior of pixels observed by Lagache & Dole (2001) is then due to the relative flatness of deep fields.\\nThe different observations have been done at different times, therefore the \\\"absolute calibration\\\" performed using the FCS measurement can be slightly different from one raster to another. This difference is observed as discontinuities between maps in the mosaic. To correct for these small absolute calibration differences (for a border between two maps, the mean difference in brightness observed is about 13%), we applied a \\\" general flat-field\\\" correction as in Lagache & Dole (2001).\\nAfter concatenating the timelines, the raster is projected on the sky using each individual pixel coordinates, with a pixel size for the map of 10 ′′ to match the IRAS HiRes gridding. Because the raster step size of each observation was a full detector size, and the spacecraft Y-axis was not parallel to right ascensions or declinations, we observe gaps between the individual array pointings. The final map is presented at Fig. 1 after a bilinear interpolation to reconstruct the signal in the gaps.\"\n },\n {\n \"title\": \"Ancillary data\",\n \"content\": \"\"\n },\n {\n \"title\": \"IRAS HiRes data\",\n \"content\": \"As a complement to the ISOPHOT data at 170 µm, we used IRAS high resolution (HiRes) data at 12, 25, 60 and 100 µm. Maps of 2 • × 2 • centered at 00 h 57 m 29.12 s −72 • 32 ′ 29.2 ′′ in J2000 were obtained. The pixel size is 10 ′′ . The HiRes point spread function (PSF) varies spatially in a map. A gaussian approximation of the PSF is given with the HiRes maps and is defined by two spatially dependent FWHMs and a rotation angle. For more information on HiRes, please refer to Aumann et al. (1990) and Bontekoe et al. (1994). Since we are searching for extended structure properties, the HiRes IRAS maps were brought to the DC-calibration (Wheelock et al. 1994). This DC calibration is only valid for spatial scales greater than 40 ′ . As the SMC exhibits structure on various spatial scales, the DC calibration introduces some limitations that will be discussed in Sect. 3.2.\"\n },\n {\n \"title\": \"ATCA/Parkes combined SMC H data\",\n \"content\": \"To study the correlation between gas and dust in the SMC, we needed a H column density map to compare with the ISOPHOT map, in order to compute the abundances of dust grains relative to the hydrogen. Stanimirovic et al. (1999) have combined Parkes telescope observations of neutral hydrogen in the SMC with the Australia Telescope Compact Array (ATCA) aperture synthesis mosaic to obtain a set of images sensitive to all angular spatial scales between 98 ′′ and 4 • from 90 to 215 km.s −1 . The spatial resolution obtained is 98 ′′ and the velocity resolution is 1.65 km.s −1 . We used a column density map of the SMC, obtained by integrating over the whole velocity range and assuming that the neutral hydrogen is optically thin 3 .\"\n },\n {\n \"title\": \"Parkes H data for the Galactic foreground\",\n \"content\": \"A Galactic filament exists in front of the SMC. Even though it is very smooth, large (see contours of Fig. 1) and uniform (column densities between 2.5 · 10 20 and 4 · 10 20 atom.cm −2 ), it can bias studies. The far-infrared fluxes observed in either the ISOPHOT or the IRAS HiRes maps represent the emission from the SMC dust, but also from this Galactic foreground. 3 The column density map and the data cube are available at ftp://ftp2.naic.edu/pub/ast/sstanimi/. However, this Galactic component of the emission is not observed in the ATCA/Parkes data, due to a velocity cut and spatial filtering. To study the SMC dust properties, it was thus necessary to remove this foreground component (see Sect. 3.3). In this step, we used a Galactic H column density map constructed from data of the Parkes narrow-band H survey of the tidal arms of the Magellanic system (Brüns et al. 2000). The column density map of the H Galactic foreground was obtained by integrating the emission from -60 to +50 km/s, outside the SMC velocities. The resolution is 15 ′ . Due to sidelobes effects, the map obtained can slightly overestimate the column density by 10 to 20 %.\"\n },\n {\n \"title\": \"Photometric consistency\",\n \"content\": \"DIRBE provided infrared absolute sky brightness maps in 10 bands in the wavelength range 1.25 to 240 µm. This absolute photometry is used to check ISOPHOT mosaic and IRAS fluxes.\"\n },\n {\n \"title\": \"ISOPHOT data\",\n \"content\": \"We checked if the ISOPHOT 170 µm flux is consistent with the spectral energy distributions obtained from the 100, 140 and 240 µm DIRBE fluxes in the SMC. To do this we convolved the ISOPHOT map with the DIRBE beam. This gives 18 ISOPHOT fluxes in the SMC as if they were observed by DIRBE and we compare ISOPHOT and DIRBE fluxes by plotting the spectral energy distribution and fitting it with a modified black body with a spectral index of 2 on the DIRBE values. Fig. 2 shows that the DIRBE-convolved ISOPHOT flux integrated over the 18 positions observed by DIRBE, is consistent within the errors bars with the best modified blackbody fit (obtained for a temperature of 20.6K, an emissivity of ǫ H (250µm) = 3 · 10 −28 cm 2 for 10 20 hydrogen atoms.cm −2 ). Please note that these values correspond to an average over the whole galaxy and were obtained only for calibration purposes.\\nWe have also plotted the individual fit values at 170 µm versus the corresponding 18 ISOPHOT values. This plot shows a significant scatter but is consistent with a linear correlation with unity slope and zero offset.\\nA comparison with the IRAS high resolution data was also performed to check possible photometric errors introduced by the interpolation process. The correlation between IRAS and ISOPHOT data does not change when we restrict the comparison to the pixels measured by ISOPHOT rather than the full interpolated map. The interpolation required to fill the holes in the measurements does not seem to bias the image comparisons.\"\n },\n {\n \"title\": \"IRAS data\",\n \"content\": \"The IRAS HiRes maps have the same photometric calibration as the IRAS ISSA maps. The IRAS detector had a dwell-time dependent responsivity change: the gain changes as a function of source size. This effect is known as the \\\" AC/DC effect\\\". A first set of corrections were taken into account in the IRAS Fig. 1. The ISOPHOT map at 170 µm (in MJy/sr) after reduction processes and a bilinear interpolation to fill the holes left by the mapping pattern. The flat-field has been efficiently corrected for the diffuse part of the map, but one can still see low level defects in the brightest parts. Contours of the H Galactic foreground are overplotted at column densities of 2.6, 2.8, 3., 3.2, 3.4 and 3.6·10 20 H atom.cm-2 (from light grey to black). data products. These corrections were later found to be insufficient. On the basis of a comparison of the IRAS and DIRBE sky brightness for extended emission, Wheelock et al. (1994) recommended applying additional correction factors of 0.88, 1.01, 0.82 and 0.74 at 12, 25, 60 and 100 µm respectively.\\nHowever, these correction factors are only valid for spatial scales greater than 40 ′ . This is not true in the SMC where structures on various spatial scales are present. Stanimirovic et al. (2000) have also compared DIRBE and IRAS data by integrating fluxes over 6.25 •2 in the SMC. They found that integrated flux densities measured by DIRBE for 60 and 100 µm are higher than IRAS fluxes by about 10 to 20%, which is an opposite correction to that observed by Wheelock et al. (1994).\\nWe have compared the IRAS (convolved to the DIRBE resolution) and the DIRBE 100 µm maps on a 10 • × 10 • scale around the SMC to understand this discrepancy. We see that the Wheelock et al. (1994) correction factor of 0.74 is only truly valid outside the galaxy and increases in the SMC up to a maximum value in the north of the optical bar (N66 region). For the diffuse medium of the SMC, the Wheelock et al. (1994) correction factor is sufficient. After correction, we estimate that the 100 µm IRAS brightness for the diffuse medium is underestimated by less than 15%.\"\n },\n {\n \"title\": \"Galactic foreground removal\",\n \"content\": \"Unlike the ATCA/Parkes H map, the far-infrared emission maps (ISOPHOT and IRAS) trace both the dust emission in the SMC and the dust in our Galaxy. To disentangle these two components in the far-infrared emission, we used a column density map of the H Galactic foreground presented in Sect. 2.2.3.\\nDust grains absorb energy from the UV/optical radiation field that they re-emit in the infrared wavelength as thermal emission. Large grains are at a fixed temperature and their emitted intensity can be expressed as:\\nwhere N H is the column density of hydrogen and ǫ H (ν) is the emissivity of grains per hydrogen atom. Assuming that the emissivity of dust grains in the far-infrared follows a power law: . 2. Comparison of the DIRBE-convolved ISOPHOT fluxes with the DIRBE ones (in MJy/sr), after an integration on the whole SMC. The modified black body traced was fit on the DIRBE values only and correspond to a temperature of 20.6 K and an emissivity of ǫ H (250µm) = 3 · 10 −28 cm 2 for 10 20 hydrogen atoms.cm −2 . We see that the ISOPHOT value at 170 µm is consistent with the DIRBE photometry.\\nand taking β = 2 (as found for Galactic emission (Lagache et al. 1998(Lagache et al. , 1999 for wavelengths between 100 µm and 500 µm).\\nUsing a temperature of 17.5 K and an emissivity per hydrogen atom of 10 −25 cm 2 (Boulanger et al. 1996a) 4 for the Galactic dust, we computed the Galactic thermal emission and removed this foreground emission from the IRAS and ISOPHOT maps to obtain the emission of the SMC dust only. At 12, 25 and 60 µm, we used the I 12 /I 100 , I 25 /I 100 and I 60 /I 100 ratios computed from IRAS maps outside the SMC, together with the computed foreground emission map at 100 µm to remove the Galactic foreground emission at these wavelengths. However, this foreground emission remains in the data noise at 12, 25 and 60 µm.\"\n },\n {\n \"title\": \"Spatial image comparison\",\n \"content\": \"\"\n },\n {\n \"title\": \"Projection and resolution\",\n \"content\": \"The data set presented in Sect. 2 is composed of maps with different fields of view and different resolutions. We projected all of them onto a common grid: that of IRAS HiRes maps (2 • × 2 • field of view centered at 00 h 57 m 29.12 s − 72 • 32 ′ 29.2 ′′ (J2000) with 10 ′′ pixel size). These maps were inter-compared by pairs.\\nTo compare the ISOPHOT map with one of the HiRes maps, the projected and re-sampled ISOPHOT map was convolved with a spatial varying kernel in order to be at the HiRes resolution when the point spread function of ISOPHOT was smaller than the one of HiRes (Kerton & Martin 2001). The HiRes map was also convolved with a spatially varying kernel to match the ISOPHOT resolution when the HiRes beam was smaller than the ISOPHOT one.\\nTo compare the HI column density map with the ISOPHOT map, no convolution was necessary as long as their resolutions were comparable. Fig. 3 shows two of the transformed maps: the ISOPHOT map degraded to the IRAS 100 µm resolution and the projected ATCA/Parkes map.\"\n },\n {\n \"title\": \"Colors from scatterplots\",\n \"content\": \"To characterize the diffuse medium (i.e. the extended region where the infrared emission-H column density correlation is homogenous) spectral energy distribution, we developed a comparison method that is not biased by point sources and is also not affected by offsets in the data calibration.\\nWe work with two maps (obtained after the treatment presented in Sect. 4). The general correlation between the two maps is first quantified by fitting a straight line to the scatterplot of the pixel-to-pixel comparison. We deduce a first ratio value and an error σ given by the dispersion in the correlation, which enables us to filter out localized sources (i.e. star-forming regions) by removing pixels that are 3σ above the correlation (there is no points below −3σ).\\nFrom the \\\" filtered\\\" maps, we compute a ratio map by linearly fitting the pixel correlation in the scatterplot for a sliding box of 50×50 pixels. This corresponds to a 5.5×5.5 beams box which is a minimum size to have independent data points. The size of the box could not be greater, or we would not trace the local variations. In some cases there are too few points inside a box (too many points above 3σ) to do a meaningful fit. We assign no value to these positions. We also skip positions where the correlation coefficient is too low. Adjusting a gaussian on the histogram of the ratio values gives us a reference ratio and the associated dispersion. This dispersion contains photometric and fitting errors but also a true scatter in the ratio. Both phenomena cannot be easily disentangled.\\nThe spatial distribution and the histogram with the gaussian fit obtained for the I 100µm /I 170µm ratio are presented in Fig. 4 as an example. We see that the comparison method avoids large areas in the main body (bar) of the SMC. These regions correspond to highly structured star-forming sites. In these regions, either the filtering technique has left too few independent positions in one box to compute a significant correlation or no clear correlation is observed.\"\n },\n {\n \"title\": \"Results\",\n \"content\": \"\"\n },\n {\n \"title\": \"Infrared colors\",\n \"content\": \"The method presented in Sect. 4.2 was applied to the IRAS HiRes and ISOPHOT maps to compute the I 100µm /I 170µm , I 60µm /I 170µm , I 25µm /I 170µm , I 12µm /I 170µm ratios. Adjusting a gaussian on the histograms gives us reference ratios and their associated dispersions:\\nUsing equation 1 and 2 we can derive from the I 100 µm I 170 µm ratio map the big grain equilibrium temperature distribution in the diffuse medium in the Small Magellanic Cloud. This leads to a reference temperature of: T dust = 22 ± 2 K. Note that this temperature is different from the one that would be inferred from the ratios of the mean I 100 µm and I 170 µm fluxes. Wilke et al. (2004) found a mean temperature for the main body of the SMC of 20.9 ± 1.9K, which is lower than the temperature we find for the diffuse medium. Their mean value falls within our range of temperatures, which is expected because the regions we study overlap. The difference in mean values could be due to the presence of cold atomic and molecular clouds in the main body of the SMC. Our studies seem to be consistent with each other and are more complementary than comparable.\\nThe large dust grain temperature is related to the radiation field intensity and will be discussed in Sect. 6.\"\n },\n {\n \"title\": \"Gas to dust correlation\",\n \"content\": \"From the ISOPHOT and the ATCA/Parkes maps, using the method described Sect. 4.2, we found the distribution presented on Fig. 5. This distribution exhibits a main peak and a distribution tail. The peak is characterized by:\\n17 · 10 −20 ± 0.06 · 10 −20 MJy.sr −1 .cm 2 (7) Since the regions used to define the ratios are not exactly the same, the determination of the characteristics of the diffuse medium could be biased. We checked that this is not the case. If we use the region where I 170µm /N HI is computed to determine the colors, we find that the derived quantities are not affected by more than 15%. We see that the characteristics of the diffuse medium are not biased by the definition of the limits of the studied region. Comparing the region defined by equation 7 with the H α distribution, we see in Fig. 6 that the regions corresponding to the tail in the I 170µm /N HI (located mainly in the main body) are regions where the diffuse H α emission is important. Comparing the map in Fig. 4 and Fig. 6, we also see that the regions where no ratios can be computed correspond to important star-forming sites. Assuming that all the dust is in the H , we use the I 170µm /N HI ratio map and the temperature map computed above to obtain an histogram of the emissivity per hydrogen atom of the dust grains in the diffuse medium. We find a reference emissivity at 170 µm of:\\nǫ H (λ) = 7.3 · 10 −27 ± 3.2 · 10 −27 cm 2 (8)\\nThe error bar takes into account uncertainties in the temperature and I 170 µm /N HI . If the emissivity follows a power law with a spectral index of 2, the emissivity per H atom in the SMC at 250 µm is ǫ H (λ) = 3.4 · 10 −27 ± 1.5 · 10 −27 (λ/250 µm) −2 cm 2 . This corresponds to κ ν = 0.21(λ/250 µm) −2 cm 2 g −1 . Assuming similar dust optical properties in the SMC and the Galaxy, the emissivity is related to the GDR and the ratio between hydrogen column density and E(B − V) color excess by:\\nThe solar neighborhood value for the emissivity is 10 −25 cm 2 at 250 µm (Boulanger et al. 1996a). The inferred SMC GDR is thus 30 times greater than in the Galaxy. Stanimirovic et al. (2000) have presented a similar study using the same H data together with HIRAS data. They came to the same conclusion that the GDR is a factor 30 higher in the SMC than in our Galaxy. However, this agreement might be a coincidence since 1) we did not use the same calibration of the IRAS data, 2) their temperature estimation based on the 60/100µm color is higher than ours, 3) the GDR they present is averaged over the whole galaxy (including star forming regions) and 4) their GDR is compared to a Galactic GDR of 375 instead of the standard value of 100 5 . Using the standard Galactic GDR value, the average SMC GDR found by Stanimirovic et al. (2000) becomes 100 times higher than in the Galaxy.\\nOur estimate of the SMC's GDR 30 times higher than in the Galaxy, is significantly larger than the difference in metallicity, which is less than 10 (Dufour 1984). It is twice the observed difference in the N(H I )+N(H 2 )\"\n },\n {\n \"title\": \"E(B−V)\",\n \"content\": \"values presented by Tumlinson et al. (2002).\"\n },\n {\n \"title\": \"Infrared excesses\",\n \"content\": \"A noticeable result of the ISOPHOT and H comparison is the presence of a tail in the 170 µm/H histogram (see Fig. 5). This tail corresponds to pixels in the SMC optical bar with excess 170 µm emission with respect to N HI (see Fig. 7). We checked that the mean temperature for this region of excess is the same as that for the diffuse medium. These excesses may be due either to a variation in the dust abundances between the diffuse medium and the bar, or due to the presence of cold H or H 2 gas. We believe that the excesses are real but it is difficult to infer an amount of cold H and molecular gas from it. Firstly, it is difficult to disentangle these two possible contributions. Secondly, where some of these excesses are due to dust in cold H or H 2 we do not expect them to be correlated with H emission and another method is needed to quantify them. It is thus beyond the scope of this paper to translate these excesses into an estimate of cold gas that could be compared to other studies where this is attempted (Stanimirovic et al. 2000). Finally, we feel that the photometric uncertainties in the ISOPHOT and IRAS data are an additional difficulty, especially near starforming regions. Further studies with the Spitzer satellite could address this question with better photometry and angular resolution.\"\n },\n {\n \"title\": \"SMC dust properties in diffuse H : modeling the spectral energy distribution\",\n \"content\": \"The different intensity ratios obtained in Sect. 5.1 enable us to trace the spectral energy distribution of dust in the diffuse medium of the SMC and to compare it with that of the solar neighborhood. The difference in temperature and in abundances is seen Fig. 9 as a shift along the wavelength and brightness axis respectively. The error bars include true variations in the dust properties across the SMC diffuse interstellar medium which are correlated over wavelengths.\\nWe used the dust model of Desert et al. (1990) to quantify the grain dust abundances. In order to model the spectral energy distribution (SED), we need to estimate the interstellar radiation field (ISRF) in the diffuse medium of the SMC. In Sect. 5.1 we found the big dust grain equilibrium temperature to be 22 K. Using the dust thermal equilibrium temperature in the solar neighborhood of 17.5 K (Boulanger et al. 1996a) as a reference, the dust temperature in the SMC leads to a radiation field χ S MC = T 4+β dust (S MC)/T 4+β dust (Galaxy) = 4. This value is consistent with the estimate of the ISRF made from massive star counts by Lequeux (1979). Vangioni-Flam et al. (1980) also found the same ratio from UV measurements. We used the Mathis et al. (1983) 10 kpc Galactic ISRF and scaled it by a factor χ = 4 for the modeling. We have adjusted only the abundances of PAHs, very small grains (VSG) and big grains. For the other parameters defining the size distribution, we have kept the Galactic values. The best fit is presented on Fig. 8, it corresponds to under-anbundant PAHs, very small grains (VSG) and big grains (BG) compared to the Galaxy (36, 23 and 30 times less respectively). Mass abundances relative to hydrogen, compared with Galactic values (Desert et al. 1990) are summarized in table 1.\\nThe modeling confirms the GDR estimated in Sect 5.2 to be around 30 times greater than in the Galaxy. Within the uncertainties, PAHs and very small grain abundances are also consistent with this factor.\\nThe model is clearly below the 60 µm point. This could be due to our crude estimate of the ISRF. We have used a single ISRF while our data analysis shows temperature variations across the SMC diffuse medium. To take the variations in the radiation field into account, we transformed the temperature Fig. 9. Infrared spectral energy distribution of dust in the diffuse medium for the SMC and the Solar Neighborhood. The brightness corresponds to an interstellar gas column density of 10 20 cm −2 . The Galactic spectrum (in grey) is composed in the far-IR by COBE/FIRAS data from Boulanger et al. (1996a) (with the corresponding spectrum from grains in thermal equilibrium indicated by the dotted line), and at shorter wavelengths by the COBE/DIRBE (Bernard et al. 1994) and Arome balloon data (triangle at 3.3 µm) (Giard et al. 1994). An ISOCAM spectrum (Boulanger et al. 1996b) scaled to the 12 µm brightness is also shown. The SMC spectrum (in black) is the best fit obtained with the Desert et al. model, same as in figure 8. 2.0 · 10 −5 4.7 · 10 −4 BG 2.2 · 10 −4 6.4 · 10 −3 distribution obtained in Sect. 5.1 into a χ S MC distribution. For each value of the distribution, we computed a spectral energy distribution (SED) of the dust emission with the Desert et al. (1990) model, using the Mathis 10 kpc ISRF multiplied by the corresponding χ as an input (χ is a constant for the whole spectrum). We then computed the resulting total SED by adding the individual SEDs with weights according to the χ distribution shape. The resulting fit is not sufficiently broadened to encom-pass the 60 µm point. In addition, the mass abundances of dust grains have to be reduced by 8, 5 and 7 for the PAH, VSG and BG components respectively. Due to the high non-linearity of the Planck curve, even a small broadening of the χ distribution tends to reduce the abundances of dust grains significantly. These values are thus indicatives only but they show that the mass abundances represented in table 1 are more likely to be upper limits and that the emissivity could be even lower.\\nIt is possible to fit the 60 µm value with a single ISRF scaled by χ = 10 (see Fig. 10). In this case, the dust grain mass abundances must then be reduced by 3, 2 and 2 for the PAHs, very small grains and big grains respectively. Such a high ISRF corresponds to a dust temperature of 25.6 K, which is higher than the estimate obtained in Sect. 5.1 but still compatible with the 100 µm and the 170 µm fluxes. This higher temperature would lead to an emissivity of 2.4 · 10 −27 cm 2 at 250 µm, which is even lower than the estimation obtained in Sect. 5.2 and results in a GDR 40 times higher than in the Galaxy. A higher radiation field could thus account for the 60µm excess without contra- dicting the main conclusion of this study, namely the fact that the GDR in the diffuse SMC ISM is much lower than the solar neighbourhood value scaled by the difference in metallicity.\\nThe observed 60 µm excess could also be due to a change in the grain size distribution (Galliano et al. 2003). We assumed the Galactic grain sizes distribution to apply in the SMC. However, by changing the relative mass contributions of the different grain components, we enlarge the discontinuities in the size distribution between the three components of the model. It is necessary to further increase these discontinuities to account for the 60 µm excess.\\nFurther investigations are needed to characterize the SMC ISRF and especially the 60 µm excess. The intensity and spectral shape of the ISRF are likely to be different than the Galactic one.\"\n },\n {\n \"title\": \"Discussion\",\n \"content\": \"Studying the gas to dust correlation in the diffuse medium of the SMC (see Sect. 5.2), we found a GDR in mass around 30 times greater than in the solar neighborhood, larger by a factor of 3 than the difference in metallicity. This result confirms the value found by Stanimirovic et al. (2000) in a similar study. This lower depletion of heavy elements in dust was independently reported by Welty et al. (2001), who found Mg and Si to be essentially undepleted in the SMC. It was also reported for a sample of dwarf irregulars by Lisenfeld & Ferrara (1998). Our result in the SMC fits with the relation that they observed between GDR and metallicity.\\nAn interpretation of this relation has been proposed by Hirashita et al. (2002). With a chemical evolution model, they illustrated the dependence of the GDR on the star formation history. Stars generate dust but also contribute to their destruction by supernova shocks. They propose a scenario in which an intermittent star formation history leads to a time variable GDR. At some epochs of the galaxy evolution, the formation of dust grains is less efficient than the destruction process by supernova shocks, so that the GDR no longer scales with the metallicity. This model could be applied to the SMC with its specific star formation history (Harris & Zaritsky 2003). It is also necessary to account for a possible difference in the GDR between the diffuse and the denser medium.\\nExtinction studies lead to significantly smaller value of the GDR, closer to that expected from the Galactic value for a linear dependence with metallicity. Bouchet et al. (1985) used photometric observations of O-B stars in the SMC to derive their extinction curves. They found a GDR value that is about 8 times the Galactic value, which is consistent with the difference in metallicity. More recently, Tumlinson et al. (2002) measured\\nfor a sample of stars observed with FUSE and found a mean value 16 times lower than the solar Neighborhood value. This is closer but still smaller by a factor 2 than our estimate. The difference may be due to the fact that extinction studies are based on line-of-sights with high hydrogen column density where dense interstellar medium is likely to be present. Most of the stars are in the optical bar and/or in star formation regions associated with molecular clouds. The difference in the GDRs inferred in our work and extinction studies might thus point to a dependence on the environment: a decrease of the ratio from the diffuse ISM to star forming regions that might arise from the condensation of heavy elements on dust grains in dense gas and their destruction by supernova shocks in the diffuse ISM. Stanimirovic et al. (2000) have also noted this possibility for the SMC, but it still requires confirmation by further studies.\"\n },\n {\n \"title\": \"Summary and conclusions\",\n \"content\": \"This study was made feasible by the availability of 10 ISOPHOT observations at 170 µm, covering most of the SMC, in the ISO Data Archive. After reduction processes, mapping and Galactic foreground removal, the ISOPHOT data were compared with IRAS maps and a ATCA/Parkes combined H column density map to assess the properties of the dust in the diffuse medium of this low metallicity galaxy.\\nFor the dust in the diffuse medium of the SMC, we found a reference equilibrium temperature of 22 ± 2 K and a reference dust emissivity of 3.4 · 10 −27 ± 1.5 · 10 −27 λ 250 µm cm 2 per hydrogen atom. This leads to a gas-to dust ratio 30 times greater than in the Galaxy. This high value reflects the SMC low metallicity, but also requires a lower depletion of dust elements than in the Galaxy. This shows that a simple linear dependence between the GDR and the metallicity does not apply for the diffuse medium of the SMC.\\nThe spectral energy distribution is modeled with the Desert et al. (1990) model. The best fit obtained shows a similar decrease for PAHs and very small grains as for big grains. However, a 60 µm excess remains that can not be accounted for with simple assumptions. Further constraints, in particular on the ISRF, are needed to understand it.\\nOur GDR estimate for the diffuse medium of the SMC is a factor 2 to 3 higher than that derived from extinction studies towards bright UV stars with high foreground hydrogen col-umn densities. This difference supports the notion of variations in the GDR, from diffuse regions where grains are more frequently destroyed by supernovae, to denser and more quiescent regions where grains re-accrete heavy elements.\"\n },\n {\n \"title\": \"Fig. 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 6 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 7 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 8 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 10 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 1 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Histogram of the 170 micron flux to H column density ratio (in MJy.cm 2 /sr). The gaussian fit and the residuals are presented as continuous and dotted lines respectively","content":"Fig. 5 .\n5Histogram of the 170 micron flux to H column density ratio (in MJy.cm 2 /sr). The gaussian fit and the residuals are presented as continuous and dotted lines respectively"},{"id":"","caption":"Histogram and spatial distribution of the 100-170 µm colors. For the histogram, the gaussian fit and the residuals are presented as continuous and dotted lines respectively. For the map of the 100-170 µm colors, contours correspond to the 0.8 (black) and 1.2 (white) levels (one sigma limits of the gaussian fit). It should be noticed that the noise in the color map is not homogeneous.","content":"Fig. 4 .\n4Histogram and spatial distribution of the 100-170 µm colors. For the histogram, the gaussian fit and the residuals are presented as continuous and dotted lines respectively. For the map of the 100-170 µm colors, contours correspond to the 0.8 (black) and 1.2 (white) levels (one sigma limits of the gaussian fit). It should be noticed that the noise in the color map is not homogeneous."},{"id":"","caption":"Uncalibrated H α map. As an illustration, the contour line corresponds to I 170 µm /N HI = 0.23 MJy.cm 2 /sr (see equation 7) as in figure 7.","content":"Fig. 6 .\n6Uncalibrated H α map. As an illustration, the contour line corresponds to I 170 µm /N HI = 0.23 MJy.cm 2 /sr (see equation 7) as in figure 7."},{"id":"","caption":"Spatial distribution of the I 170 µm /N HI ratio (in MJy.cm 2 /sr). Regions where no ratio is computed (see Sect. 4.2) are left blank. The contour line corresponds to a I 170 µm /N HI = 0.23 MJy.cm 2 /sr level. We see that the IR excesses are concentrated near the optical bar.","content":"Fig. 7 .\n7Spatial distribution of the I 170 µm /N HI ratio (in MJy.cm 2 /sr). Regions where no ratio is computed (see Sect. 4.2) are left blank. The contour line corresponds to a I 170 µm /N HI = 0.23 MJy.cm 2 /sr level. We see that the IR excesses are concentrated near the optical bar."},{"id":"","caption":"Best fit for the dust infrared emission spectrum, obtained with the Desert et al. model. The brightness corresponds to an interstellar gas column density of 10 20 cm −2 . The IRAS and ISOPHOT data are represented by squares. The fit obtained with the model (continuous line) corresponds to the emission of three components: PAHs(dashed line), very small grains (mixed line) and big grains (dot-dot-dot-dashed line).","content":"Fig. 8 .\n8Best fit for the dust infrared emission spectrum, obtained with the Desert et al. model. The brightness corresponds to an interstellar gas column density of 10 20 cm −2 . The IRAS and ISOPHOT data are represented by squares. The fit obtained with the model (continuous line) corresponds to the emission of three components: PAHs(dashed line), very small grains (mixed line) and big grains (dot-dot-dot-dashed line)."},{"id":"","caption":"Dust infrared emission spectrum fit obtained with the Desert et al. model to adjust the 60µm excess. The IRAS and ISOPHOT data are represented by squares. This spectrum is obtained with a single ISRF scaled by χ = 10 and reduced dust grain mass abundances for PAHs (dashed line), very small grains (mixed line) and big grains (dot-dot-dot-dashed line).","content":"Fig. 10 .\n10Dust infrared emission spectrum fit obtained with the Desert et al. model to adjust the 60µm excess. The IRAS and ISOPHOT data are represented by squares. This spectrum is obtained with a single ISRF scaled by χ = 10 and reduced dust grain mass abundances for PAHs (dashed line), very small grains (mixed line) and big grains (dot-dot-dot-dashed line)."},{"id":"","caption":"Mass abundances relative to hydrogen in the SMC and the Galaxy. The grain size distribution is assumed to be the same in both cases.","content":"Table 1 .\n1Mass abundances relative to hydrogen in the SMC and the Galaxy. The grain size distribution is assumed to be the same in both cases.Component SMC \nGalaxy \n\nm \nm H \n\nm \nm H \n\nPAH \n1.2 · 10 −5 4.3 · 10 −4 \nVSG"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Histogram of the 170 micron flux to H column density ratio (in MJy.cm 2 /sr). The gaussian fit and the residuals are presented as continuous and dotted lines respectively\",\n \"content\": \"Fig. 5 .\\n5Histogram of the 170 micron flux to H column density ratio (in MJy.cm 2 /sr). The gaussian fit and the residuals are presented as continuous and dotted lines respectively\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Histogram and spatial distribution of the 100-170 µm colors. For the histogram, the gaussian fit and the residuals are presented as continuous and dotted lines respectively. For the map of the 100-170 µm colors, contours correspond to the 0.8 (black) and 1.2 (white) levels (one sigma limits of the gaussian fit). It should be noticed that the noise in the color map is not homogeneous.\",\n \"content\": \"Fig. 4 .\\n4Histogram and spatial distribution of the 100-170 µm colors. For the histogram, the gaussian fit and the residuals are presented as continuous and dotted lines respectively. For the map of the 100-170 µm colors, contours correspond to the 0.8 (black) and 1.2 (white) levels (one sigma limits of the gaussian fit). It should be noticed that the noise in the color map is not homogeneous.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Uncalibrated H α map. As an illustration, the contour line corresponds to I 170 µm /N HI = 0.23 MJy.cm 2 /sr (see equation 7) as in figure 7.\",\n \"content\": \"Fig. 6 .\\n6Uncalibrated H α map. As an illustration, the contour line corresponds to I 170 µm /N HI = 0.23 MJy.cm 2 /sr (see equation 7) as in figure 7.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Spatial distribution of the I 170 µm /N HI ratio (in MJy.cm 2 /sr). Regions where no ratio is computed (see Sect. 4.2) are left blank. The contour line corresponds to a I 170 µm /N HI = 0.23 MJy.cm 2 /sr level. We see that the IR excesses are concentrated near the optical bar.\",\n \"content\": \"Fig. 7 .\\n7Spatial distribution of the I 170 µm /N HI ratio (in MJy.cm 2 /sr). Regions where no ratio is computed (see Sect. 4.2) are left blank. The contour line corresponds to a I 170 µm /N HI = 0.23 MJy.cm 2 /sr level. We see that the IR excesses are concentrated near the optical bar.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Best fit for the dust infrared emission spectrum, obtained with the Desert et al. model. The brightness corresponds to an interstellar gas column density of 10 20 cm −2 . The IRAS and ISOPHOT data are represented by squares. The fit obtained with the model (continuous line) corresponds to the emission of three components: PAHs(dashed line), very small grains (mixed line) and big grains (dot-dot-dot-dashed line).\",\n \"content\": \"Fig. 8 .\\n8Best fit for the dust infrared emission spectrum, obtained with the Desert et al. model. The brightness corresponds to an interstellar gas column density of 10 20 cm −2 . The IRAS and ISOPHOT data are represented by squares. The fit obtained with the model (continuous line) corresponds to the emission of three components: PAHs(dashed line), very small grains (mixed line) and big grains (dot-dot-dot-dashed line).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Dust infrared emission spectrum fit obtained with the Desert et al. model to adjust the 60µm excess. The IRAS and ISOPHOT data are represented by squares. This spectrum is obtained with a single ISRF scaled by χ = 10 and reduced dust grain mass abundances for PAHs (dashed line), very small grains (mixed line) and big grains (dot-dot-dot-dashed line).\",\n \"content\": \"Fig. 10 .\\n10Dust infrared emission spectrum fit obtained with the Desert et al. model to adjust the 60µm excess. The IRAS and ISOPHOT data are represented by squares. This spectrum is obtained with a single ISRF scaled by χ = 10 and reduced dust grain mass abundances for PAHs (dashed line), very small grains (mixed line) and big grains (dot-dot-dot-dashed line).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Mass abundances relative to hydrogen in the SMC and the Galaxy. The grain size distribution is assumed to be the same in both cases.\",\n \"content\": \"Table 1 .\\n1Mass abundances relative to hydrogen in the SMC and the Galaxy. The grain size distribution is assumed to be the same in both cases.Component SMC \\nGalaxy \\n\\nm \\nm H \\n\\nm \\nm H \\n\\nPAH \\n1.2 · 10 −5 4.3 · 10 −4 \\nVSG\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". J E Aguirre, J J Bezaire, E S Cheng, ApJ. 596273Aguirre, J. E., Bezaire, J. J., Cheng, E. S., et al. 2003, ApJ, 596, 273","doi":""},{"text":". H H Aumann, J W Fowler, M Melnyk, AJ. 991674Aumann, H. H., Fowler, J. W., & Melnyk, M. 1990, AJ, 99, 1674","doi":""},{"text":". M Azzopardi, J Lequeux, A Maeder, A&A. 18934Azzopardi, M., Lequeux, J., & Maeder, A. 1988, A&A, 189, 34","doi":""},{"text":". J P Bernard, F Boulanger, F X Desert, A&A. 2915Bernard, J. P., Boulanger, F., Desert, F. X., et al. 1994, A&A, 291, L5","doi":""},{"text":". T R Bontekoe, E Koper, D J M Kester, A&A. 2841037Bontekoe, T. R., Koper, E., & Kester, D. J. M. 1994, A&A, 284, 1037","doi":""},{"text":". P Bouchet, J Lequeux, E Maurice, L Prevot, M L Prevot-Burnichon, A&A. 149330Bouchet, P., Lequeux, J., Maurice, E., Prevot, L., & Prevot- Burnichon, M. L. 1985, A&A, 149, 330","doi":""},{"text":". F Boulanger, A Abergel, J.-P Bernard, A&A. 312256Boulanger, F., Abergel, A., Bernard, J.-P., et al. 1996a, A&A, 312, 256","doi":""},{"text":". F Boulanger, W T Reach, A Abergel, A&A. 315325Boulanger, F., Reach, W. T., Abergel, A., et al. 1996b, A&A, 315, L325","doi":""},{"text":"C Brüns, J Kerp, L Staveley-Smith, ASP Conf. Ser. 218: Mapping the Hidden Universe: The Universe behind the Milky Way -The Universe in HI. 349Brüns, C., Kerp, J., & Staveley-Smith, L. 2000, in ASP Conf. Ser. 218: Mapping the Hidden Universe: The Universe be- hind the Milky Way -The Universe in HI, 349","doi":""},{"text":". F.-X Desert, F Boulanger, J L Puget, A&A. 237215Desert, F.-X., Boulanger, F., & Puget, J. L. 1990, A&A, 237, 215","doi":""},{"text":"R J Dufour, IAU Symp. 108: Structure and Evolution of the Magellanic Clouds. Dufour, R. J. 1984, in IAU Symp. 108: Structure and Evolution of the Magellanic Clouds, 353-360","doi":""},{"text":". E Dwek, ApJ. 501643Dwek, E. 1998, ApJ, 501, 643","doi":""},{"text":". C Gabriel, J Acosta-Puido, I Heinrichsen, A.S.P. Conference Series. 145165Gabriel, C., Acosta-Puido, J., & Heinrichsen, I. 1998, A.S.P. Conference Series, 145, 165","doi":""},{"text":". C Gabriel, J Acosta-Puido, U Kinkel, U Klaas, B Schulz, A.S.P. Conference Series. 125108Gabriel, C., Acosta-Puido, J., Kinkel, U., Klaas, U., & Schulz, B. 1997, A.S.P. Conference Series, 125, 108","doi":""},{"text":". F Galliano, S C Madden, A P Jones, A&A. 407159Galliano, F., Madden, S. C., Jones, A. P., et al. 2003, A&A, 407, 159","doi":""},{"text":". M Giard, J M Lamarre, F Pajot, G Serra, A&A. 286203Giard, M., Lamarre, J. M., Pajot, F., & Serra, G. 1994, A&A, 286, 203","doi":""},{"text":". J Harris, D Zaritsky, arXiv:astro-ph, 0312100Harris, J. & Zaritsky, D. 2003, arXiv:astro-ph, 0312100","doi":""},{"text":". H Hirashita, ApJ. 522220Hirashita, H. 1999, ApJ, 522, 220","doi":""},{"text":". H Hirashita, Y Y Tajiri, H Kamaya, A&A. 388439Hirashita, H., Tajiri, Y. Y., & Kamaya, H. 2002, A&A, 388, 439","doi":""},{"text":". M R Issa, I Maclaren, A W Wolfendale, A&A. 236237Issa, M. R., MacLaren, I., & Wolfendale, A. W. 1990, A&A, 236, 237","doi":""},{"text":". C R Kerton, P G Martin, PASP. 113872Kerton, C. R. & Martin, P. G. 2001, PASP, 113, 872","doi":""},{"text":". M F Kessler, J A Steinz, M E Anderegg, A&A. 31527Kessler, M. F., Steinz, J. A., Anderegg, M. E., et al. 1996, A&A, 315, L27","doi":""},{"text":". G Lagache, A Abergel, F Boulanger, F X Désert, J.-L Puget, A&A. 344322Lagache, G., Abergel, A., Boulanger, F., Désert, F. X., & Puget, J.-L. 1999, A&A, 344, 322","doi":""},{"text":". G Lagache, A Abergel, F Boulanger, J.-L Puget, A&A. 333709Lagache, G., Abergel, A., Boulanger, F., & Puget, J.-L. 1998, A&A, 333, 709","doi":""},{"text":". G Lagache, H Dole, A&A. 372702Lagache, G. & Dole, H. 2001, A&A, 372, 702","doi":""},{"text":"R J Laureijs, U Klaas, P J Richards, B Schulz, The ISO Handbook, Volume IV -PHT -The Imaging Photo-Polarimeter. & Abraham, P.Laureijs, R. J., Klaas, U., Richards, P. J., Schulz, B., & Abraham, P., eds. 2003, The ISO Handbook, Volume IV - PHT -The Imaging Photo-Polarimeter","doi":""},{"text":". D Lemke, U Klaas, J Abolins, A&A. 3151A&ALemke, D., Klaas, U., Abolins, J., et al. 1996, A&A, 315, L64 Lequeux, J. 1979, A&A, 71, 1","doi":""},{"text":". A Li, B T Draine, ApJ. 576762Li, A. & Draine, B. T. 2002, ApJ, 576, 762","doi":""},{"text":"U Lisenfeld, A Ferrara, Dwarf Galaxies: Probes for Galaxy Formation and Evolution. Kyoto, Japan2145ApJLisenfeld, U. & Ferrara, A. 1997, Dwarf Galaxies: Probes for Galaxy Formation and Evolution, Joint Discussion 2, 21 August 1997, Kyoto, Japan., 2 -. 1998, ApJ, 496, 145","doi":""},{"text":". J S Mathis, P G Mezger, N Panagia, A&A. 128212Mathis, J. S., Mezger, P. G., & Panagia, N. 1983, A&A, 128, 212","doi":""},{"text":". L Pagani, G Lagache, A Bacmann, A&A. 40659Pagani, L., Lagache, G., Bacmann, A., et al. 2003, A&A, 406, L59","doi":""},{"text":". M Rubio, J Lequeux, F Boulanger, A&A. 2719Rubio, M., Lequeux, J., & Boulanger, F. 1993a, A&A, 271, 9","doi":""},{"text":". M Rubio, J Lequeux, F Boulanger, A&A. 2711Rubio, M., Lequeux, J., Boulanger, F., et al. 1993b, A&A, 271, 1","doi":""},{"text":"M Sauvage, L Vigroux, IAU Symp. 148: The Magellanic Clouds. 407Sauvage, M. & Vigroux, L. 1991, in IAU Symp. 148: The Magellanic Clouds, 407","doi":""},{"text":". M Sauvage, L Vigroux, T X Thuan, A&A. 237296Sauvage, M., Vigroux, L., & Thuan, T. X. 1990, A&A, 237, 296","doi":""},{"text":". P B W Schwering, F P Israel, A&AS. 7979Schwering, P. B. W. & Israel, F. P. 1989, A&AS, 79, 79","doi":""},{"text":". U J Sofia, D M Meyer, ApJ. 558221ApJSofia, U. J. & Meyer, D. M. 2001a, ApJ, 558, L147 -. 2001b, ApJ, 554, L221","doi":""},{"text":". S Stanimirovic, L Staveley-Smith, J M Dickey, R J Sault, S L Snowden, MNRAS. 302417Stanimirovic, S., Staveley-Smith, L., Dickey, J. M., Sault, R. J., & Snowden, S. L. 1999, MNRAS, 302, 417","doi":""},{"text":". S Stanimirovic, L Staveley-Smith, J M Van Der Hulst, MNRAS. 315791Stanimirovic, S., Staveley-Smith, L., van der Hulst, J. M., et al. 2000, MNRAS, 315, 791","doi":""},{"text":". L Staveley-Smith, S Kim, M Putman, S Stanimirović, Reviews of Modern Astronomy. 11117Staveley-Smith, L., Kim, S., Putman, M., & Stanimirović, S. 1998, Reviews of Modern Astronomy, 11, 117","doi":""},{"text":". J Tumlinson, J M Shull, B L Rachford, ApJ. 566857Tumlinson, J., Shull, J. M., Rachford, B. L., et al. 2002, ApJ, 566, 857","doi":""},{"text":". E Vangioni-Flam, J Lequeux, M Maucherat-Joubert, B Rocca-Volmerange, A&A. 9073Vangioni-Flam, E., Lequeux, J., Maucherat-Joubert, M., & Rocca-Volmerange, B. 1980, A&A, 90, 73","doi":""},{"text":". D E Welty, J T Lauroesch, J C Blades, L M Hobbs, D G York, ApJ Lett. 55475Welty, D. E., Lauroesch, J. T., Blades, J. C., Hobbs, L. M., & York, D. G. 2001, ApJ Lett., 554, L75","doi":""},{"text":"B E Westerlund, IAU Symp. 148: The Magellanic Clouds. 15Westerlund, B. E. 1991, in IAU Symp. 148: The Magellanic Clouds, 15","doi":""},{"text":". S L Wheelock, T N Gautier, J Chillemi, NASA STI/Recon Technical Report N. 9522539Wheelock, S. L., Gautier, T. N., Chillemi, J., et al. 1994, NASA STI/Recon Technical Report N, 95, 22539","doi":""},{"text":". K Wilke, U Klaas, D Lemke, A&A. 41469Wilke, K., Klaas, U., Lemke, D., et al. 2004, A&A, 414, 69","doi":""},{"text":". K Wilke, M Stickel, M Haas, A&A. 401873Wilke, K., Stickel, M., Haas, M., et al. 2003, A&A, 401, 873","doi":""}],"string":"[\n {\n \"text\": \". J E Aguirre, J J Bezaire, E S Cheng, ApJ. 596273Aguirre, J. E., Bezaire, J. J., Cheng, E. S., et al. 2003, ApJ, 596, 273\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H H Aumann, J W Fowler, M Melnyk, AJ. 991674Aumann, H. H., Fowler, J. W., & Melnyk, M. 1990, AJ, 99, 1674\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Azzopardi, J Lequeux, A Maeder, A&A. 18934Azzopardi, M., Lequeux, J., & Maeder, A. 1988, A&A, 189, 34\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J P Bernard, F Boulanger, F X Desert, A&A. 2915Bernard, J. P., Boulanger, F., Desert, F. X., et al. 1994, A&A, 291, L5\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T R Bontekoe, E Koper, D J M Kester, A&A. 2841037Bontekoe, T. R., Koper, E., & Kester, D. J. M. 1994, A&A, 284, 1037\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Bouchet, J Lequeux, E Maurice, L Prevot, M L Prevot-Burnichon, A&A. 149330Bouchet, P., Lequeux, J., Maurice, E., Prevot, L., & Prevot- Burnichon, M. L. 1985, A&A, 149, 330\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Boulanger, A Abergel, J.-P Bernard, A&A. 312256Boulanger, F., Abergel, A., Bernard, J.-P., et al. 1996a, A&A, 312, 256\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Boulanger, W T Reach, A Abergel, A&A. 315325Boulanger, F., Reach, W. T., Abergel, A., et al. 1996b, A&A, 315, L325\",\n \"doi\": \"\"\n },\n {\n \"text\": \"C Brüns, J Kerp, L Staveley-Smith, ASP Conf. Ser. 218: Mapping the Hidden Universe: The Universe behind the Milky Way -The Universe in HI. 349Brüns, C., Kerp, J., & Staveley-Smith, L. 2000, in ASP Conf. Ser. 218: Mapping the Hidden Universe: The Universe be- hind the Milky Way -The Universe in HI, 349\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F.-X Desert, F Boulanger, J L Puget, A&A. 237215Desert, F.-X., Boulanger, F., & Puget, J. L. 1990, A&A, 237, 215\",\n \"doi\": \"\"\n },\n {\n \"text\": \"R J Dufour, IAU Symp. 108: Structure and Evolution of the Magellanic Clouds. Dufour, R. J. 1984, in IAU Symp. 108: Structure and Evolution of the Magellanic Clouds, 353-360\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Dwek, ApJ. 501643Dwek, E. 1998, ApJ, 501, 643\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Gabriel, J Acosta-Puido, I Heinrichsen, A.S.P. Conference Series. 145165Gabriel, C., Acosta-Puido, J., & Heinrichsen, I. 1998, A.S.P. Conference Series, 145, 165\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Gabriel, J Acosta-Puido, U Kinkel, U Klaas, B Schulz, A.S.P. Conference Series. 125108Gabriel, C., Acosta-Puido, J., Kinkel, U., Klaas, U., & Schulz, B. 1997, A.S.P. Conference Series, 125, 108\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Galliano, S C Madden, A P Jones, A&A. 407159Galliano, F., Madden, S. C., Jones, A. P., et al. 2003, A&A, 407, 159\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Giard, J M Lamarre, F Pajot, G Serra, A&A. 286203Giard, M., Lamarre, J. M., Pajot, F., & Serra, G. 1994, A&A, 286, 203\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Harris, D Zaritsky, arXiv:astro-ph, 0312100Harris, J. & Zaritsky, D. 2003, arXiv:astro-ph, 0312100\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Hirashita, ApJ. 522220Hirashita, H. 1999, ApJ, 522, 220\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Hirashita, Y Y Tajiri, H Kamaya, A&A. 388439Hirashita, H., Tajiri, Y. Y., & Kamaya, H. 2002, A&A, 388, 439\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M R Issa, I Maclaren, A W Wolfendale, A&A. 236237Issa, M. R., MacLaren, I., & Wolfendale, A. W. 1990, A&A, 236, 237\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C R Kerton, P G Martin, PASP. 113872Kerton, C. R. & Martin, P. G. 2001, PASP, 113, 872\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M F Kessler, J A Steinz, M E Anderegg, A&A. 31527Kessler, M. F., Steinz, J. A., Anderegg, M. E., et al. 1996, A&A, 315, L27\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Lagache, A Abergel, F Boulanger, F X Désert, J.-L Puget, A&A. 344322Lagache, G., Abergel, A., Boulanger, F., Désert, F. X., & Puget, J.-L. 1999, A&A, 344, 322\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Lagache, A Abergel, F Boulanger, J.-L Puget, A&A. 333709Lagache, G., Abergel, A., Boulanger, F., & Puget, J.-L. 1998, A&A, 333, 709\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Lagache, H Dole, A&A. 372702Lagache, G. & Dole, H. 2001, A&A, 372, 702\",\n \"doi\": \"\"\n },\n {\n \"text\": \"R J Laureijs, U Klaas, P J Richards, B Schulz, The ISO Handbook, Volume IV -PHT -The Imaging Photo-Polarimeter. & Abraham, P.Laureijs, R. J., Klaas, U., Richards, P. J., Schulz, B., & Abraham, P., eds. 2003, The ISO Handbook, Volume IV - PHT -The Imaging Photo-Polarimeter\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Lemke, U Klaas, J Abolins, A&A. 3151A&ALemke, D., Klaas, U., Abolins, J., et al. 1996, A&A, 315, L64 Lequeux, J. 1979, A&A, 71, 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Li, B T Draine, ApJ. 576762Li, A. & Draine, B. T. 2002, ApJ, 576, 762\",\n \"doi\": \"\"\n },\n {\n \"text\": \"U Lisenfeld, A Ferrara, Dwarf Galaxies: Probes for Galaxy Formation and Evolution. Kyoto, Japan2145ApJLisenfeld, U. & Ferrara, A. 1997, Dwarf Galaxies: Probes for Galaxy Formation and Evolution, Joint Discussion 2, 21 August 1997, Kyoto, Japan., 2 -. 1998, ApJ, 496, 145\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J S Mathis, P G Mezger, N Panagia, A&A. 128212Mathis, J. S., Mezger, P. G., & Panagia, N. 1983, A&A, 128, 212\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Pagani, G Lagache, A Bacmann, A&A. 40659Pagani, L., Lagache, G., Bacmann, A., et al. 2003, A&A, 406, L59\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Rubio, J Lequeux, F Boulanger, A&A. 2719Rubio, M., Lequeux, J., & Boulanger, F. 1993a, A&A, 271, 9\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Rubio, J Lequeux, F Boulanger, A&A. 2711Rubio, M., Lequeux, J., Boulanger, F., et al. 1993b, A&A, 271, 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \"M Sauvage, L Vigroux, IAU Symp. 148: The Magellanic Clouds. 407Sauvage, M. & Vigroux, L. 1991, in IAU Symp. 148: The Magellanic Clouds, 407\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Sauvage, L Vigroux, T X Thuan, A&A. 237296Sauvage, M., Vigroux, L., & Thuan, T. X. 1990, A&A, 237, 296\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P B W Schwering, F P Israel, A&AS. 7979Schwering, P. B. W. & Israel, F. P. 1989, A&AS, 79, 79\",\n \"doi\": \"\"\n },\n {\n \"text\": \". U J Sofia, D M Meyer, ApJ. 558221ApJSofia, U. J. & Meyer, D. M. 2001a, ApJ, 558, L147 -. 2001b, ApJ, 554, L221\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Stanimirovic, L Staveley-Smith, J M Dickey, R J Sault, S L Snowden, MNRAS. 302417Stanimirovic, S., Staveley-Smith, L., Dickey, J. M., Sault, R. J., & Snowden, S. L. 1999, MNRAS, 302, 417\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Stanimirovic, L Staveley-Smith, J M Van Der Hulst, MNRAS. 315791Stanimirovic, S., Staveley-Smith, L., van der Hulst, J. M., et al. 2000, MNRAS, 315, 791\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Staveley-Smith, S Kim, M Putman, S Stanimirović, Reviews of Modern Astronomy. 11117Staveley-Smith, L., Kim, S., Putman, M., & Stanimirović, S. 1998, Reviews of Modern Astronomy, 11, 117\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Tumlinson, J M Shull, B L Rachford, ApJ. 566857Tumlinson, J., Shull, J. M., Rachford, B. L., et al. 2002, ApJ, 566, 857\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Vangioni-Flam, J Lequeux, M Maucherat-Joubert, B Rocca-Volmerange, A&A. 9073Vangioni-Flam, E., Lequeux, J., Maucherat-Joubert, M., & Rocca-Volmerange, B. 1980, A&A, 90, 73\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D E Welty, J T Lauroesch, J C Blades, L M Hobbs, D G York, ApJ Lett. 55475Welty, D. E., Lauroesch, J. T., Blades, J. C., Hobbs, L. M., & York, D. G. 2001, ApJ Lett., 554, L75\",\n \"doi\": \"\"\n },\n {\n \"text\": \"B E Westerlund, IAU Symp. 148: The Magellanic Clouds. 15Westerlund, B. E. 1991, in IAU Symp. 148: The Magellanic Clouds, 15\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S L Wheelock, T N Gautier, J Chillemi, NASA STI/Recon Technical Report N. 9522539Wheelock, S. L., Gautier, T. N., Chillemi, J., et al. 1994, NASA STI/Recon Technical Report N, 95, 22539\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Wilke, U Klaas, D Lemke, A&A. 41469Wilke, K., Klaas, U., Lemke, D., et al. 2004, A&A, 414, 69\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Wilke, M Stickel, M Haas, A&A. 401873Wilke, K., Stickel, M., Haas, M., et al. 2003, A&A, 401, 873\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":494,"cells":{"text":{"kind":"string","value":"\nLearning Features that Predict Cue Usage\n22 Oct 1997\n\nBarbara Di \nJohanna D Moore jmoore@cs.pitt.edu \nComputer Science Department\nand Learning Re-search & Development Center ‡ Intelligent Systems Program\n\n\nMassimo Paolucci paolucci@cs.pitt.edu \n\nUniversity of Pittsburgh Pittsburgh\n15260PAUSA\n\nLearning Features that Predict Cue Usage\n22 Oct 199777E3A39E2281831E25D00A205C2BD741arXiv:cmp-lg/9710006v1\nOur goal is to identify the features that predict the occurrence and placement of discourse cues in tutorial explanations in order to aid in the automatic generation of explanations.Previous attempts to devise rules for text generation were based on intuition or small numbers of constructed examples.We apply a machine learning program, C4.5, to induce decision trees for cue occurrence and placement from a corpus of data coded for a variety of features previously thought to affect cue usage.Our experiments enable us to identify the features with most predictive power, and show that machine learning can be used to induce decision trees useful for text generation.\n\nIntroduction\n\nDiscourse cues are words or phrases, such as because, first, and although, that mark structural and semantic relationships between discourse entities.They play a crucial role in many discourse processing tasks, including plan recognition (Litman and Allen, 1987), text comprehension (Cohen, 1984;Hobbs, 1985;Mann and Thompson, 1986;Reichman-Adar, 1984), and anaphora resolution (Grosz and Sidner, 1986).Moreover, research in reading comprehension indicates that felicitous use of cues improves comprehension and recall (Goldman, 1988), but that their indiscriminate use may have detrimental effects on recall (Millis, Graesser, and Haberlandt, 1993).\n\nOur goal is to identify general strategies for cue usage that can be implemented for automatic text generation.From the generation perspective, cue usage consists of three distinct, but interrelated problems: (1) occurrence: whether or not to include a cue in the generated text, (2) placement: where the cue should be placed in the text, and (3) selection: what lexical item(s) should be used.\n\nPrior work in text generation has focused on cue selection (McKeown and Elhadad, 1991;Elhadad and McKeown, 1990), or on the relation between cue occurrence and placement and specific rhetorical structures (Rösner and Stede, 1992; Scott and de Souza, 1990; Vander Linden and Martin, 1995).Other hypotheses about cue usage derive from work on discourse coherence and structure.Previous research (Hobbs, 1985;Grosz and Sidner, 1986;Schiffrin, 1987;Mann and Thompson, 1988;Elhadad and McKeown, 1990), which has been largely descriptive, suggests factors such as structural features of the discourse (e.g., level of embedding and segment complexity), intentional and informational relations in that structure, ordering of relata, and syntactic form of discourse constituents.Moser and Moore (1995;1997) coded a corpus of naturally occurring tutorial explanations for the range of features identified in prior work.Because they were also interested in the contrast between occurrence and non-occurrence of cues, they exhaustively coded for all of the factors thought to contribute to cue usage in all of the text.From their study, Moser and Moore identified several interesting correlations between particular features and specific aspects of cue usage, and were able to test specific hypotheses from the literature that were based on constructed examples.\n\nIn this paper, we focus on cue occurrence and placement, and present an empirical study of the hypotheses provided by previous research, which have never been systematically evaluated with naturally occurring data.We use a machine learning program, C4.5 (Quinlan, 1993), on the tagged corpus of Moser and Moore to induce decision trees.The number of coded features and their interactions makes the manual construction of rules that predict cue occurrence and placement an intractable task.\n\nOur results largely confirm the suggestions from the literature, and clarify them by highlighting the most influential features for a particular task.Discourse structure, in terms of both segment structure and levels of embedding, affects cue occurrence the most; intentional relations also play an important role.For cue placement, the most important factors are syntactic structure and segment complexity.\n\nThe paper is organized as follows.In Section 2 we discuss previous research in more detail.Section 3 provides an overview of Moser and Moore's coding scheme.In Section 4 we present our learning experiments, and in Section 5 we discuss our results and conclude.\n\n\nRelated Work\n\nMcKeown and Elhadad (1991; 1990) studied several connectives (e.g., but, since, because), and include many insightful hypotheses about cue selection; their observation that the distinction between but and although depends on the point of the move is related to the notion of core discussed below.However, they do not address the problem of cue occurrence.\n\nOther researchers (Rösner and Stede, 1992; Scott and de Souza, 1990) are concerned with generating text from \"RST trees\", hierarchical structures where leaf nodes contain content and internal nodes indicate the rhetorical relations, as defined in Rhetorical Structure Theory (RST) (Mann and Thompson, 1988), that exist between subtrees.They proposed heuristics for including and choosing cues based on the rhetorical relation between spans of text, the order of the relata, and the complexity of the related text spans.However, (Scott and de Souza, 1990) was based on a small number of constructed examples, and (Rösner and Stede, 1992) focused on a small number of RST relations.(Litman, 1996) and (Siegel and McKeown, 1994) have applied machine learning to disambiguate between the discourse and sentential usages of cues; however, they do not consider the issues of occurrence and placement, and approach the problem from the point of view of interpretation.We closely follow the approach in (Litman, 1996) in two ways.First, we use C4.5.Second, we experiment first with each feature individually, and then with \"interesting\" subsets of features.\n\n\nRelational Discourse Analysis\n\nThis section briefly describes Relational Discourse Analysis (RDA) (Moser, Moore, and Glendening, 1996), the coding scheme used to tag the data for our machine learning experiments. 1DA is a scheme devised for analyzing tutorial explanations in the domain of electronics troubleshooting.It synthesizes ideas from (Grosz and Sidner, 1986) and from RST (Mann and Thompson, 1988).\n\nCoders use RDA to exhaustively analyze each explanation in the corpus, i.e., every word in each explanation belongs to exactly one element in the analysis.An explanation may consist of multiple segments.Each segment originates with an intention of the speaker.Segments are internally structured and consist of a core, i.e., that element that most directly expresses the segment purpose, and any number of contributors, i.e. the remaining constituents.\n\nFor each contributor, one analyzes its relation to the core from an intentional perspective, i.e., how it is intended to support the core, and from an informational perspective, i.e., how its content relates to that of the core.The set of intentional relations in RDA is a modification of the presentational relations of RST, while informational relations are similar to the subject matter relations in RST.Each segment constituent, both core and contributors, may itself be a segment with a core:contributor structure.In some cases the core is not explicit.This is often the case with the whole tutor's explanation, since its purpose is to answer the student's explicit question.\n\nAs an example of the application of RDA, consider the partial tutor explanation in (1)2 .The purpose of this segment is to inform the student that she made the strategy error of testing inside part3 too soon.The constituent that makes the purpose obvious, in this case (1-B), is the core of the segment.The other constituents help to serve the segment purpose by contributing to it.(1-C) is an example of subsegment with its own core:contributor structure; its purpose is to give a reason for testing part2 first.\n\nThe RDA analysis of (1) is shown schematically in Figure 1.The core is depicted as the mother of all the relations it participates in.Each relation node is labeled with both its intentional and informational relation, with the order of relata in the label indicating the linear order in the discourse.Each relation node has up to two daughters: the cue, if any, and\n\n(1) Coders analyze each explanation in the corpus and enter their analyses into a database.The corpus consists of 854 clauses comprising 668 segments, for a total of 780 relations.Table 1 summarizes the distribution of different relations, and the number of cued relations in each category.Joints are segments comprising more than one core, but no contributor; clusters are multiunit structures with no recognizable core:contributor relation.(1-B) is a cluster composed of two units (the two clauses), related only at the informational level by a temporal relation.Both clauses describe actions, with the first action description embedded in a matrix (\"You should\").Cues are much more likely to occur in clusters, where only informational relations occur, than in core:contributor structures, where intentional and informational relations cooccur (χ 2 = 33.367,p <.001, df = 1).In the following, we will not discuss joints and clusters any further.\nAlthough A.\nAn important result pointed out by (Moser and Moore, 1995) is that cue placement depends on core position.When the core is first and a cue is associated with the relation, the cue never occurs with the core.In contrast, when the core is second, if a cue occurs, it can occur either on the core or on the contributor.\n\n\nLearning from the corpus\n\n\nThe algorithm\n\nWe chose the C4.5 learning algorithm (Quinlan, 1993) because it is well suited to a domain such as ours with discrete valued attributes.Moreover, C4.5 produces decision trees and rule sets, both often used in text generation to implement mappings from function features to forms. 3Finally, C4.5 is both readily available, and is a benchmark learning algorithm that has been extensively used in NLP applications, e.g.(Litman, 1996;Mooney, 1996;Vander Linden and Di Eugenio, 1996).\n\nAs our dataset is small, the results we report are based on cross-validation, which (Weiss and Kulikowski, 1991) recommends as the best method to evaluate decision trees on datasets whose cardinality is in the hundreds.Data for learning should be divided into training and test sets; however, for small datasets this has the disadvantage that a sizable portion of the data is not available for learning.Cross- N th used as the test set.The error rate of a tree obtained by using the whole dataset for training is then assumed to be the average error rate on the test set over the N runs.Further, as C4.5 prunes the initial tree it obtains to avoid overfitting, it computes both actual and estimated error rates for the pruned tree; see (Quinlan, 1993, Ch. 4) for details.Thus, below we will report the average estimated error rate on the test set, as computed by 10-fold cross-validation experiments.\n\n\nThe features\n\nEach data point in our dataset corresponds to a core:contributor relation, and is characterized by the following features, summarized in Table 2.\n\nSegment Structure.Three features capture the global structure of the segment in which the current core:contributor relation appears.\n\n• (Con)Trib(utor)-pos(ition) captures the position of a particular contributor within the larger segment in which it occurs, and encodes the structure of the segment in terms of how many contributors precede and follow the core.For example, contributor (1-D) in Figure 1 is labeled as B1A3-2after, as it is the second contributor following the core in a segment with 1 contributor before and 3 after the core.\n\n• Inten(tional)-structure indicates which contributors in the segment bear the same intentional relations to the core.\n\n• Infor(mational)-structure. Similar to intentional structure, but applied to informational relations.\n\nCore:contributor relation.These features more specifically characterize the current core:contributor relation.\n\n• Inten(tional)-rel(ation).One of concede, convince, enable.\n\n• Infor(mational)-rel(ation).About 30 informational relations have been coded for.However, as preliminary experiments showed that using them individually results in overfitting the data, we classify them according to the four classes proposed in (Moser, Moore, and Glendening, 1996): causality, similarity, elaboration, temporal .Temporal relations only appear in clusters, thus not in the data we discuss in this paper.\n\n• Syn(tactic)-rel(ation). Captures whether the core and contributor are independent units (segments or sentences); whether they are coordinated clauses; or which of the two is subordinate to the other.\n\n• Adjacency.Whether core and contributor are adjacent in linear order.\n\nEmbedding.These features capture segment embedding, Core-type and Trib-type qualitatively, and Above/Below quantitatively.\n\n• Core-type/(Con)Trib(utor)-type.Whether the core/the contributor is a segment, or a minimal unit (further subdivided into action, state, matrix).\n\n• Above/Below encode the number of relations hierarchically above and below the current relation.\n\n\nThe experiments\n\nInitially, we performed learning on all 406 instances of core:contributor relations.We quickly determined that this approach would not lead to useful decision trees.First, the trees we obtained were extremely complex (at least 50 nodes).Second, some of the subtrees corresponded to clearly identifiable subclasses of the data, such as relations with an implicit core, which suggested that we should apply learning to these independently identifiable subclasses.Thus, we subdivided the data into three subsets:\n\n• Core1 : core:contributor relations with the core in first position • Impl(icit)-core: core:contributor relations with an implicit core While this has the disadvantage of smaller training sets, the trees we obtain are more manageable and more meaningful.Table 3 summarizes the cardinality of these sets, and the frequencies of cue occurrence.\n\nWe ran four sets of experiments.In three of them we predict cue occurrence and in one cue placement.4\n\n\nCue Occurrence\n\nTable 4 summarizes our main results concerning cue occurrence, and includes the error rates associated with different feature sets.We adopt Litman's approach (1996) to determine whether two error rates E 1 and E 2 are significantly different.We compute 95% confidence intervals for the two error rates using a t -test.E 1 is significantly better than E 2 if the upper bound of the 95% confidence interval for E 1 is lower than the lower bound of the 95% confidence interval for E 2 .For each set of experiments, we report the following:\n\n1.A baseline measure obtained by choosing the majority class.E.g., for Core1 58.9% of the relations are not cued; thus, by deciding to never include a cue, one would be wrong 41.1% of the times.\n\n2. The best individual features whose predictive power is better than the baseline: as Table 4 makes apparent, individual features do not have much predictive power.For neither Core1 nor Impl-core does any individual feature perform better than the baseline, and for Core2 only one feature is sufficiently predictive.\n\n3. (One of) the best induced tree(s).For each tree, we list the number of nodes, and up to six of the features that appear highest in the tree, with their levels of embedding. 5Figure 2 shows the tree for Core2 (space constraints prevent us from including figures for each tree).In the figure, the numbers in parentheses indicate the number of cases correctly covered by the leaf, and the number of expected errors at that leaf.\n\nLearning turns out to be most useful for Core1, where the error reduction (as percentage) from baseline to the upper bound of the best result is 32%; error reduction is 19% for Core2 and only 3% for Impl-core.\n\nThe best tree was obtained partly by informed choice, partly by trial and error.Automatically trying out all the 2 11 = 2048 subsets of features would be possible, but it would require manual examination of about 2,000 sets of results, a daunting task.Thus, for each dataset we considered only the following subsets of features.\n\n1.All features.This always results in C4.5 selecting a few features (from 3 to 7) for the final tree.\n\n\nSubsets\n\n\nC4\n\n.5 on all features.\n\n3. In Table 2, three features -Trib-pos, Intenstruct, Infor-struct -concern segment structure, eight do not.We constructed three subsets by always including the eight features that do not concern segment structure, and adding one of those that does.The trees obtained by including Trib-pos, Inten-struct, Infor-struct at the same time are in general more complex, and not significantly better than other trees obtained by including only one of these three features.We attribute this to the fact that these features encode partly overlapping information.\n\nFinally, the best tree was obtained as follows.We build the set of trees that are statistically equivalent to the tree with the best error rate (i.e., with the lowest error rate upper bound).Among these trees, we choose the one that we deem the most perspicuous in terms of features and of complexity.Namely, we pick the simplest tree with Trib-Pos as the root if one exists, otherwise the simplest tree.Trees that have Trib-Pos as the root are the most useful for text generation, because, given a complex segment, Trib-Pos is the only attribute that unambiguously identifies a specific contributor.\n\nOur results make apparent that the structure of segments plays a fundamental role in determining cue occurrence.One of the three features concerning segment structure (Trib-Pos, Inten-Structure, Infor-Structure) appears as the root or just below the root in all trees in Table 4; more importantly, this same configuration occurs in all trees equivalent to the best tree (even if the specific feature encoding segment structure may change).The level of embedding in a segment, as encoded by Core-type, Trib-type, Above and Below also figures prominently.\n\nInten-rel appears in all trees, confirming the intuition that the speaker's purpose affects cue occurrence.More specifically, in Figure 2, Inten-rel distinguishes two different speaker purposes, convince and enable.The same split occurs in some of the best trees induced on Core1, with the same outcome: i.e., convince directly correlates with the occurrence of a cue, whereas for enable other features must be taken into account.6Informational relations do not appear as often as intentional relations; their discriminatory power seems more relevant for clusters.Preliminary experiments show that cue occurrence in clusters depends only on informational and syntactic relations.Finally, Adjacency does not seem to play any substantial role.\n\n\nCue Placement\n\nWhile cue occurrence and placement are interrelated problems, we performed learning on them separately.First, the issue of placement arises only in the case of Core2; for Core1, cues only occur on the contributor.Second, we attempted experiments on Core2 that discriminated between occurrence and placement at the same time, and the derived trees were complex and not perspicuous.Thus, we ran an experiment on the 100 cued relations from Core2 to investigate which factors affect placing the cue on the contributor in first position or on the core in second; see Table 5.\n\nWe ran the same trials discussed above on this dataset.In this case, the best tree -see Figure 3 -results from combining the two best individual features, and reduces the error rate by 50%.The most discriminant feature turns out to be the syntactic relation between the contributor and the core.However, segment structure still plays an important role, via Trib-pos.While the importance of Syn-rel for placement seems clear, its role concerning occurrence requires further exploration.It is interesting to note that the tree induced on Core1 -the only case in which Synrel is relevant for occurrence -includes the same distinction as in Figure 3: namely, if the contributor depends on the core, the contributor must be marked, otherwise other features have to be taken into account.Scott and de Souza (1990) point out that \"there is a strong correlation between the syntactic specification of a complex sentence and its perceived rhetorical structure.\"It seems that certain\n\n\nDiscussion and Conclusions\n\nWe have presented the results of machine learning experiments concerning cue occurrence and placement.As (Litman, 1996) observes, this sort of empirical work supports the utility of machine learning techniques applied to coded corpora.As our study shows, individual features have no predictive power for cue occurrence.Moreover, it is hard to see how the best combination of individual features could be found by manual inspection.\n\nOur results also provide guidance for those building text generation systems.This study clearly indicates that segment structure, most notably the ordering of core and contributor, is crucial for determining cue occurrence.Recall that it was only by considering Core1 and Core2 relations in distinct datasets that we were able to obtain perspicuous decision trees that significantly reduce the error rate.\n\nThis indicates that the representations produced by discourse planners should distinguish those elements that constitute the core of each discourse segment, in addition to representing the hierarchical structure of segments.Note that the notion of core is related to the notions of nucleus in RST, intended effect in (Young and Moore, 1994), and of point of a move in (Elhadad and McKeown, 1990), and that text generators representing these notions exist.\n\nMoreover, in order to use the decision trees derived here, decisions about whether or not to make the core explicit and how to order the core and con-tributor(s) must be made before deciding cue occurrence, e.g., by exploiting other factors such as focus (McKeown, 1985) and a discourse history.\n\nOnce decisions about core:contributor ordering and cue occurrence have been made, a generator must still determine where to place cues and select appropriate lexical items.A major focus of our future research is to explore the relationship between the selection and placement decisions.Elsewhere, we have found that particular lexical items tend to have a preferred location, defined in terms of functional (i.e., core or contributor) and linear (i.e., first or second relatum) criteria (Moser and Moore, 1997).Thus, if a generator uses decision trees such as the one shown in Figure 3 to determine where a cue should be placed, it can then select an appropriate cue from those that can mark the given intentional / informational relations, and are usually placed in that functional-linear location.To evaluate this strategy, we must do further work to understand whether there are important distinctions among cues (e.g., so, because) apart from their different preferred locations.The work of Elhadad (1990) and Knott (1996) will help in answering this question.\n\nFuture work comprises further probing into machine learning techniques, in particular investigating whether other learning algorithms are more appropriate for our problem (Mooney, 1996), especially algorithms that take into account some a priori knowledge about features and their dependencies.\n\nFigure 1 :\n1\nFigure 1: The RDA analysis of (1)\n\n\nFigure 2 :\n2\nFigure 2: Decision tree for Core2 -occurrence\n\n\nFigure 3 :\n3\nFigure 3: Decision tree for Core2 -placement\n\n\n\n\nyou know that part1 is good, B. you should eliminate part2 before troubleshooting inside part3.\nThis isbecauseC.1. part2 is moved frequently and thus 2. is more susceptible to damage than part3.Also,D. it is more work to open up part3 for testingandE. the process of opening drawers and extending cards in part3may induce problems which did not already exist.B. you should eliminate part2before troubleshooting inside part3d d d d dconcedeconvincecriterion:act ¨¨r r Although Aact:reason r ¨¨r r (This is) C.2becauseconvincecause:effect\n\nTable 3 :\n3\nDistributions of relations and cue occurrences\nDataset# of relations# of cued relationsCore112752Core2155100(on Trib: 43) (on Core: 57)Impl-core12429Total406181Core1Core2Impl-coreBaseline41.135.423.5Best features ∅Info-rel: 33.4±0.94 ∅Best tree25.6±1.24 (15) 27.4±1.28 (18)22.1±0.57 (10)0. Trib-pos0. Trib-Pos0. Core-type1. Trib-type1. Inten-rel1. Infor-struct2. Syn-rel2. Info-rel2. Inten-rel3. Core-type3. Above4. Above4. Core-type5. Inten-rel5. Below\nbuilt out of the 2 to 4 attributes appearing highest in the tree obtained by running\n\n\nTable 4 :\n4\nSummary of learning results\n\n\nTable 5 :\n5\nCue placement on Core2\n\nFor more detail about the RDA coding scheme see(Moser and Moore, 1995;Moser and Moore, 1997).\nTo make the example more intelligible, we replaced references to parts of the circuit with the labels part1, part2 and part3.\nAll our experiments are run with grouping turned on, so that C4.5 groups values together rather than creating a branch per value. The latter choice always results in trees overfitted to the data in our domain. Using classes of informational relations, rather than individual informational relations, constitutes a sort of a priori grouping.\nThe trees that C4.5 generates are right-branching, so this description is fairly adequate.\nWe can't draw any conclusions concerning concede, as there are only 24 occurrences of concede out of 406 core:contributor relations.\nAcknowledgementsThis research is supported by the Office of Naval Research, Cognitive and Neural Sciences Division (Grants N00014-91-J-1694 and N00014-93-I-0812).Thanks to Megan Moser for her prior work on this project and for comments on this paper; to Erin Glendening and Liina Pylkkänen for their coding efforts; to Haiqin Wang for running many experiments; to Giuseppe Carenini and Steffi Brüninghaus for discussions about machine learning.\nA computational theory of the function of clue words in argument understanding. Robin ; Cohen, Michael Elhadad, Kathleen Mckeown, Proceedings of COLING84. COLING84Stanford, CA; Helsinki, Finland1984. 1990Proceedings of COLING90\n\nThe role of sequence markers in reading and recall: Comparison of native and nonnative english speakers. Susan R Goldman, Barbara J Grosz, Candace L Sidner, Computational Linguistics. 1231988. 1986University of CaliforniaTechnical reportAttention, intention, and the structure of discourse\n\nOn the coherence and structure of discourse. Jerry R Hobbs, CSLI-85-371985Center for the Study of Language and Information, Stanford UniversityTechnical Report\n\nA Data-Driven methodology for motivating a set of coherence relations. Alistair Knott, 1996University of EdinburghPh.D. thesis\n\nA plan recognition model for subdialogues in conversations. Diane J Litman, Diane J ; Litman, F James, William C Allen ; Mann, Sandra A Thompson, Relational propositions in discourse. Discourse Processes. 1996. 1987. 19865Litman and Allen1987\n\nA contrastive evaluation of functional unification grammar for surface language generation: A case study in the choice of connectives. Thompson1988 ] Mann, William C Mann, Sandra A Thompson ; Mckeown, Kathleen R ; Mckeown, Kathleen R , Michael Elhadad, Natural Language Generation in Artificial Intelligence and Computational Linguistics. C L Paris, W R Swartout, W C Mann, Cambridge, England; BostonKluwer Academic Publishers1988. McKeown1985. 1985. 19918Text Generation: Using Discourse Strategies and Focus Constraints to Generate Natural Language Text\n\nThe impact of connectives on the memory for expository text. Graesser Millis, Millis, Arthur Keith, Karl Graesser, Haberlandt, Applied Cognitive Psychology. 71993\n\nComparative experiments on disambiguating word senses: An illustration of the role of bias in machine learning. Raymond J ; Mooney, Megan Moser, Johanna D Moore ; Moser, Megan , Johanna D Moore, Conference on Empirical Methods in Natural Language Processing. Boston, MA1996. 1995. 1997A corpus analysis of discourse cues and relational discourse structure. Submitted for publication\n\nInstructions for Coding Explanations: Identifying Segments, Relations and Minimal Units. Moore Moser, Moser, Johanna D Megan, Erin Moore, Glendening, 96-171996University of Pittsburgh, Department of Computer ScienceTechnical Report\n\nCustomizing RST for the automatic production of technical manuals. J Quinlan, Dietmar Ross ; Rösner, Manfred Stede, 6th International Workshop on Natural Language Generation. R Dale, E Hovy, D Rösner, O Stock, BerlinSpringer-Verlag1993. 1984. 199222C4.5: Programs for Machine Learning\n\nEmergent linguistic rules from inducing decision trees: Disambiguating discourse clue words. Deborah Schiffrin, Scott, ] De Souza1990, Donia Scott, Clarisse Sieckenius De Souza, ; Siegel, Eric V , Kathleen R Mckeown, Current Research in Natural Language Generation. R Dale, C Mellish, M Zock, New York; New YorkAcademic Press1987. 1990. 1994Proceedings of AAAI94\n\nExpressing rhetorical relations in instructional text: A case study of the purpose relation. Vander Linden, Di Eugenio1996] Vander Linden, Keith , Barbara Di, Eugenio ; , James H Martin, 8th International Workshop on Natural Language Generation. Sussex, UK. [Vander Linden and Martin1995] Vander Linden, Keith1996. 199521Learning micro-planning rules for preventative expressions\n\nComputer Systems that learn: classification and prediction methods from statistics, neural nets, machine learning, and expert systems. Kulikowski1991 ] Weiss, Sholom M Weiss, Casimir Kulikowski ; Young, R Michael, Johanna D Moore, 7th International Workshop on Natural Language Generation. Kennebunkport, MaineMorgan Kaufmann1991. 1994DPOCL: A Principled Approach to Discourse Planning\n"},"corpus_id":{"kind":"number","value":979,"string":"979"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"cmp-lg/9710006\",\n \"mag\": 2951950125,\n \"acl\": \"P97-1011\",\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"conf/acl/EugenioMP97\",\n \"doi\": \"10.3115/976909.979628\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/cmp-lg/9710006v1.pdf\"\n ],\n \"pdf_sha\": \"c5b2e949e7ad88c7d2922b8e5c5f016e178db493\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://dl.acm.org/ft_gateway.cfm?id=979628&type=pdf\",\n \"status\": \"BRONZE\"\n }\n}"},"title":{"kind":"string","value":"Learning Features that Predict Cue Usage"},"authors":{"kind":"list like","value":["Barbara Di","Johanna D Moore jmoore@cs.pitt.edu \nComputer Science Department\nand Learning Re-search & Development Center ‡ Intelligent Systems Program","Massimo Paolucci paolucci@cs.pitt.edu","University of Pittsburgh Pittsburgh\n15260PAUSA"],"string":"[\n \"Barbara Di\",\n \"Johanna D Moore jmoore@cs.pitt.edu \\nComputer Science Department\\nand Learning Re-search & Development Center ‡ Intelligent Systems Program\",\n \"Massimo Paolucci paolucci@cs.pitt.edu\",\n \"University of Pittsburgh Pittsburgh\\n15260PAUSA\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Our goal is to identify the features that predict the occurrence and placement of discourse cues in tutorial explanations in order to aid in the automatic generation of explanations.Previous attempts to devise rules for text generation were based on intuition or small numbers of constructed examples.We apply a machine learning program, C4.5, to induce decision trees for cue occurrence and placement from a corpus of data coded for a variety of features previously thought to affect cue usage.Our experiments enable us to identify the features with most predictive power, and show that machine learning can be used to induce decision trees useful for text generation."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Discourse cues are words or phrases, such as because, first, and although, that mark structural and semantic relationships between discourse entities.They play a crucial role in many discourse processing tasks, including plan recognition (Litman and Allen, 1987), text comprehension (Cohen, 1984;Hobbs, 1985;Mann and Thompson, 1986;Reichman-Adar, 1984), and anaphora resolution (Grosz and Sidner, 1986).Moreover, research in reading comprehension indicates that felicitous use of cues improves comprehension and recall (Goldman, 1988), but that their indiscriminate use may have detrimental effects on recall (Millis, Graesser, and Haberlandt, 1993).\nOur goal is to identify general strategies for cue usage that can be implemented for automatic text generation.From the generation perspective, cue usage consists of three distinct, but interrelated problems: (1) occurrence: whether or not to include a cue in the generated text, (2) placement: where the cue should be placed in the text, and (3) selection: what lexical item(s) should be used.\nPrior work in text generation has focused on cue selection (McKeown and Elhadad, 1991;Elhadad and McKeown, 1990), or on the relation between cue occurrence and placement and specific rhetorical structures (Rösner and Stede, 1992; Scott and de Souza, 1990; Vander Linden and Martin, 1995).Other hypotheses about cue usage derive from work on discourse coherence and structure.Previous research (Hobbs, 1985;Grosz and Sidner, 1986;Schiffrin, 1987;Mann and Thompson, 1988;Elhadad and McKeown, 1990), which has been largely descriptive, suggests factors such as structural features of the discourse (e.g., level of embedding and segment complexity), intentional and informational relations in that structure, ordering of relata, and syntactic form of discourse constituents.Moser and Moore (1995;1997) coded a corpus of naturally occurring tutorial explanations for the range of features identified in prior work.Because they were also interested in the contrast between occurrence and non-occurrence of cues, they exhaustively coded for all of the factors thought to contribute to cue usage in all of the text.From their study, Moser and Moore identified several interesting correlations between particular features and specific aspects of cue usage, and were able to test specific hypotheses from the literature that were based on constructed examples.\nIn this paper, we focus on cue occurrence and placement, and present an empirical study of the hypotheses provided by previous research, which have never been systematically evaluated with naturally occurring data.We use a machine learning program, C4.5 (Quinlan, 1993), on the tagged corpus of Moser and Moore to induce decision trees.The number of coded features and their interactions makes the manual construction of rules that predict cue occurrence and placement an intractable task.\nOur results largely confirm the suggestions from the literature, and clarify them by highlighting the most influential features for a particular task.Discourse structure, in terms of both segment structure and levels of embedding, affects cue occurrence the most; intentional relations also play an important role.For cue placement, the most important factors are syntactic structure and segment complexity.\nThe paper is organized as follows.In Section 2 we discuss previous research in more detail.Section 3 provides an overview of Moser and Moore's coding scheme.In Section 4 we present our learning experiments, and in Section 5 we discuss our results and conclude."},{"title":"Related Work","content":"McKeown and Elhadad (1991; 1990) studied several connectives (e.g., but, since, because), and include many insightful hypotheses about cue selection; their observation that the distinction between but and although depends on the point of the move is related to the notion of core discussed below.However, they do not address the problem of cue occurrence.\nOther researchers (Rösner and Stede, 1992; Scott and de Souza, 1990) are concerned with generating text from \"RST trees\", hierarchical structures where leaf nodes contain content and internal nodes indicate the rhetorical relations, as defined in Rhetorical Structure Theory (RST) (Mann and Thompson, 1988), that exist between subtrees.They proposed heuristics for including and choosing cues based on the rhetorical relation between spans of text, the order of the relata, and the complexity of the related text spans.However, (Scott and de Souza, 1990) was based on a small number of constructed examples, and (Rösner and Stede, 1992) focused on a small number of RST relations.(Litman, 1996) and (Siegel and McKeown, 1994) have applied machine learning to disambiguate between the discourse and sentential usages of cues; however, they do not consider the issues of occurrence and placement, and approach the problem from the point of view of interpretation.We closely follow the approach in (Litman, 1996) in two ways.First, we use C4.5.Second, we experiment first with each feature individually, and then with \"interesting\" subsets of features."},{"title":"Relational Discourse Analysis","content":"This section briefly describes Relational Discourse Analysis (RDA) (Moser, Moore, and Glendening, 1996), the coding scheme used to tag the data for our machine learning experiments. 1DA is a scheme devised for analyzing tutorial explanations in the domain of electronics troubleshooting.It synthesizes ideas from (Grosz and Sidner, 1986) and from RST (Mann and Thompson, 1988).\nCoders use RDA to exhaustively analyze each explanation in the corpus, i.e., every word in each explanation belongs to exactly one element in the analysis.An explanation may consist of multiple segments.Each segment originates with an intention of the speaker.Segments are internally structured and consist of a core, i.e., that element that most directly expresses the segment purpose, and any number of contributors, i.e. the remaining constituents.\nFor each contributor, one analyzes its relation to the core from an intentional perspective, i.e., how it is intended to support the core, and from an informational perspective, i.e., how its content relates to that of the core.The set of intentional relations in RDA is a modification of the presentational relations of RST, while informational relations are similar to the subject matter relations in RST.Each segment constituent, both core and contributors, may itself be a segment with a core:contributor structure.In some cases the core is not explicit.This is often the case with the whole tutor's explanation, since its purpose is to answer the student's explicit question.\nAs an example of the application of RDA, consider the partial tutor explanation in (1)2 .The purpose of this segment is to inform the student that she made the strategy error of testing inside part3 too soon.The constituent that makes the purpose obvious, in this case (1-B), is the core of the segment.The other constituents help to serve the segment purpose by contributing to it.(1-C) is an example of subsegment with its own core:contributor structure; its purpose is to give a reason for testing part2 first.\nThe RDA analysis of (1) is shown schematically in Figure 1.The core is depicted as the mother of all the relations it participates in.Each relation node is labeled with both its intentional and informational relation, with the order of relata in the label indicating the linear order in the discourse.Each relation node has up to two daughters: the cue, if any, and\n(1) Coders analyze each explanation in the corpus and enter their analyses into a database.The corpus consists of 854 clauses comprising 668 segments, for a total of 780 relations.Table 1 summarizes the distribution of different relations, and the number of cued relations in each category.Joints are segments comprising more than one core, but no contributor; clusters are multiunit structures with no recognizable core:contributor relation.(1-B) is a cluster composed of two units (the two clauses), related only at the informational level by a temporal relation.Both clauses describe actions, with the first action description embedded in a matrix (\"You should\").Cues are much more likely to occur in clusters, where only informational relations occur, than in core:contributor structures, where intentional and informational relations cooccur (χ 2 = 33.367,p <.001, df = 1).In the following, we will not discuss joints and clusters any further.\nAn important result pointed out by (Moser and Moore, 1995) is that cue placement depends on core position.When the core is first and a cue is associated with the relation, the cue never occurs with the core.In contrast, when the core is second, if a cue occurs, it can occur either on the core or on the contributor."},{"title":"Learning from the corpus","content":""},{"title":"The algorithm","content":"We chose the C4.5 learning algorithm (Quinlan, 1993) because it is well suited to a domain such as ours with discrete valued attributes.Moreover, C4.5 produces decision trees and rule sets, both often used in text generation to implement mappings from function features to forms. 3Finally, C4.5 is both readily available, and is a benchmark learning algorithm that has been extensively used in NLP applications, e.g.(Litman, 1996;Mooney, 1996;Vander Linden and Di Eugenio, 1996).\nAs our dataset is small, the results we report are based on cross-validation, which (Weiss and Kulikowski, 1991) recommends as the best method to evaluate decision trees on datasets whose cardinality is in the hundreds.Data for learning should be divided into training and test sets; however, for small datasets this has the disadvantage that a sizable portion of the data is not available for learning.Cross- N th used as the test set.The error rate of a tree obtained by using the whole dataset for training is then assumed to be the average error rate on the test set over the N runs.Further, as C4.5 prunes the initial tree it obtains to avoid overfitting, it computes both actual and estimated error rates for the pruned tree; see (Quinlan, 1993, Ch. 4) for details.Thus, below we will report the average estimated error rate on the test set, as computed by 10-fold cross-validation experiments."},{"title":"The features","content":"Each data point in our dataset corresponds to a core:contributor relation, and is characterized by the following features, summarized in Table 2.\nSegment Structure.Three features capture the global structure of the segment in which the current core:contributor relation appears.\n• (Con)Trib(utor)-pos(ition) captures the position of a particular contributor within the larger segment in which it occurs, and encodes the structure of the segment in terms of how many contributors precede and follow the core.For example, contributor (1-D) in Figure 1 is labeled as B1A3-2after, as it is the second contributor following the core in a segment with 1 contributor before and 3 after the core.\n• Inten(tional)-structure indicates which contributors in the segment bear the same intentional relations to the core.\n• Infor(mational)-structure. Similar to intentional structure, but applied to informational relations.\nCore:contributor relation.These features more specifically characterize the current core:contributor relation.\n• Inten(tional)-rel(ation).One of concede, convince, enable.\n• Infor(mational)-rel(ation).About 30 informational relations have been coded for.However, as preliminary experiments showed that using them individually results in overfitting the data, we classify them according to the four classes proposed in (Moser, Moore, and Glendening, 1996): causality, similarity, elaboration, temporal .Temporal relations only appear in clusters, thus not in the data we discuss in this paper.\n• Syn(tactic)-rel(ation). Captures whether the core and contributor are independent units (segments or sentences); whether they are coordinated clauses; or which of the two is subordinate to the other.\n• Adjacency.Whether core and contributor are adjacent in linear order.\nEmbedding.These features capture segment embedding, Core-type and Trib-type qualitatively, and Above/Below quantitatively.\n• Core-type/(Con)Trib(utor)-type.Whether the core/the contributor is a segment, or a minimal unit (further subdivided into action, state, matrix).\n• Above/Below encode the number of relations hierarchically above and below the current relation."},{"title":"The experiments","content":"Initially, we performed learning on all 406 instances of core:contributor relations.We quickly determined that this approach would not lead to useful decision trees.First, the trees we obtained were extremely complex (at least 50 nodes).Second, some of the subtrees corresponded to clearly identifiable subclasses of the data, such as relations with an implicit core, which suggested that we should apply learning to these independently identifiable subclasses.Thus, we subdivided the data into three subsets:\n• Core1 : core:contributor relations with the core in first position • Impl(icit)-core: core:contributor relations with an implicit core While this has the disadvantage of smaller training sets, the trees we obtain are more manageable and more meaningful.Table 3 summarizes the cardinality of these sets, and the frequencies of cue occurrence.\nWe ran four sets of experiments.In three of them we predict cue occurrence and in one cue placement.4"},{"title":"Cue Occurrence","content":"Table 4 summarizes our main results concerning cue occurrence, and includes the error rates associated with different feature sets.We adopt Litman's approach (1996) to determine whether two error rates E 1 and E 2 are significantly different.We compute 95% confidence intervals for the two error rates using a t -test.E 1 is significantly better than E 2 if the upper bound of the 95% confidence interval for E 1 is lower than the lower bound of the 95% confidence interval for E 2 .For each set of experiments, we report the following:\n1.A baseline measure obtained by choosing the majority class.E.g., for Core1 58.9% of the relations are not cued; thus, by deciding to never include a cue, one would be wrong 41.1% of the times.\n2. The best individual features whose predictive power is better than the baseline: as Table 4 makes apparent, individual features do not have much predictive power.For neither Core1 nor Impl-core does any individual feature perform better than the baseline, and for Core2 only one feature is sufficiently predictive.\n3. (One of) the best induced tree(s).For each tree, we list the number of nodes, and up to six of the features that appear highest in the tree, with their levels of embedding. 5Figure 2 shows the tree for Core2 (space constraints prevent us from including figures for each tree).In the figure, the numbers in parentheses indicate the number of cases correctly covered by the leaf, and the number of expected errors at that leaf.\nLearning turns out to be most useful for Core1, where the error reduction (as percentage) from baseline to the upper bound of the best result is 32%; error reduction is 19% for Core2 and only 3% for Impl-core.\nThe best tree was obtained partly by informed choice, partly by trial and error.Automatically trying out all the 2 11 = 2048 subsets of features would be possible, but it would require manual examination of about 2,000 sets of results, a daunting task.Thus, for each dataset we considered only the following subsets of features.\n1.All features.This always results in C4.5 selecting a few features (from 3 to 7) for the final tree."},{"title":"Subsets","content":""},{"title":"C4","content":".5 on all features.\n3. In Table 2, three features -Trib-pos, Intenstruct, Infor-struct -concern segment structure, eight do not.We constructed three subsets by always including the eight features that do not concern segment structure, and adding one of those that does.The trees obtained by including Trib-pos, Inten-struct, Infor-struct at the same time are in general more complex, and not significantly better than other trees obtained by including only one of these three features.We attribute this to the fact that these features encode partly overlapping information.\nFinally, the best tree was obtained as follows.We build the set of trees that are statistically equivalent to the tree with the best error rate (i.e., with the lowest error rate upper bound).Among these trees, we choose the one that we deem the most perspicuous in terms of features and of complexity.Namely, we pick the simplest tree with Trib-Pos as the root if one exists, otherwise the simplest tree.Trees that have Trib-Pos as the root are the most useful for text generation, because, given a complex segment, Trib-Pos is the only attribute that unambiguously identifies a specific contributor.\nOur results make apparent that the structure of segments plays a fundamental role in determining cue occurrence.One of the three features concerning segment structure (Trib-Pos, Inten-Structure, Infor-Structure) appears as the root or just below the root in all trees in Table 4; more importantly, this same configuration occurs in all trees equivalent to the best tree (even if the specific feature encoding segment structure may change).The level of embedding in a segment, as encoded by Core-type, Trib-type, Above and Below also figures prominently.\nInten-rel appears in all trees, confirming the intuition that the speaker's purpose affects cue occurrence.More specifically, in Figure 2, Inten-rel distinguishes two different speaker purposes, convince and enable.The same split occurs in some of the best trees induced on Core1, with the same outcome: i.e., convince directly correlates with the occurrence of a cue, whereas for enable other features must be taken into account.6Informational relations do not appear as often as intentional relations; their discriminatory power seems more relevant for clusters.Preliminary experiments show that cue occurrence in clusters depends only on informational and syntactic relations.Finally, Adjacency does not seem to play any substantial role."},{"title":"Cue Placement","content":"While cue occurrence and placement are interrelated problems, we performed learning on them separately.First, the issue of placement arises only in the case of Core2; for Core1, cues only occur on the contributor.Second, we attempted experiments on Core2 that discriminated between occurrence and placement at the same time, and the derived trees were complex and not perspicuous.Thus, we ran an experiment on the 100 cued relations from Core2 to investigate which factors affect placing the cue on the contributor in first position or on the core in second; see Table 5.\nWe ran the same trials discussed above on this dataset.In this case, the best tree -see Figure 3 -results from combining the two best individual features, and reduces the error rate by 50%.The most discriminant feature turns out to be the syntactic relation between the contributor and the core.However, segment structure still plays an important role, via Trib-pos.While the importance of Syn-rel for placement seems clear, its role concerning occurrence requires further exploration.It is interesting to note that the tree induced on Core1 -the only case in which Synrel is relevant for occurrence -includes the same distinction as in Figure 3: namely, if the contributor depends on the core, the contributor must be marked, otherwise other features have to be taken into account.Scott and de Souza (1990) point out that \"there is a strong correlation between the syntactic specification of a complex sentence and its perceived rhetorical structure.\"It seems that certain"},{"title":"Discussion and Conclusions","content":"We have presented the results of machine learning experiments concerning cue occurrence and placement.As (Litman, 1996) observes, this sort of empirical work supports the utility of machine learning techniques applied to coded corpora.As our study shows, individual features have no predictive power for cue occurrence.Moreover, it is hard to see how the best combination of individual features could be found by manual inspection.\nOur results also provide guidance for those building text generation systems.This study clearly indicates that segment structure, most notably the ordering of core and contributor, is crucial for determining cue occurrence.Recall that it was only by considering Core1 and Core2 relations in distinct datasets that we were able to obtain perspicuous decision trees that significantly reduce the error rate.\nThis indicates that the representations produced by discourse planners should distinguish those elements that constitute the core of each discourse segment, in addition to representing the hierarchical structure of segments.Note that the notion of core is related to the notions of nucleus in RST, intended effect in (Young and Moore, 1994), and of point of a move in (Elhadad and McKeown, 1990), and that text generators representing these notions exist.\nMoreover, in order to use the decision trees derived here, decisions about whether or not to make the core explicit and how to order the core and con-tributor(s) must be made before deciding cue occurrence, e.g., by exploiting other factors such as focus (McKeown, 1985) and a discourse history.\nOnce decisions about core:contributor ordering and cue occurrence have been made, a generator must still determine where to place cues and select appropriate lexical items.A major focus of our future research is to explore the relationship between the selection and placement decisions.Elsewhere, we have found that particular lexical items tend to have a preferred location, defined in terms of functional (i.e., core or contributor) and linear (i.e., first or second relatum) criteria (Moser and Moore, 1997).Thus, if a generator uses decision trees such as the one shown in Figure 3 to determine where a cue should be placed, it can then select an appropriate cue from those that can mark the given intentional / informational relations, and are usually placed in that functional-linear location.To evaluate this strategy, we must do further work to understand whether there are important distinctions among cues (e.g., so, because) apart from their different preferred locations.The work of Elhadad (1990) and Knott (1996) will help in answering this question.\nFuture work comprises further probing into machine learning techniques, in particular investigating whether other learning algorithms are more appropriate for our problem (Mooney, 1996), especially algorithms that take into account some a priori knowledge about features and their dependencies."},{"title":"Figure 1 :","content":"Figure 1: The RDA analysis of (1)"},{"title":"Figure 2 :","content":"Figure 2: Decision tree for Core2 -occurrence"},{"title":"Figure 3 :","content":"Figure 3: Decision tree for Core2 -placement\nyou know that part1 is good, B. you should eliminate part2 before troubleshooting inside part3."},{"title":"Table 3 :","content":"Distributions of relations and cue occurrences\nbuilt out of the 2 to 4 attributes appearing highest in the tree obtained by running"},{"title":"Table 4 :","content":"Summary of learning results"},{"title":"Table 5 :","content":"Cue placement on Core2"}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Discourse cues are words or phrases, such as because, first, and although, that mark structural and semantic relationships between discourse entities.They play a crucial role in many discourse processing tasks, including plan recognition (Litman and Allen, 1987), text comprehension (Cohen, 1984;Hobbs, 1985;Mann and Thompson, 1986;Reichman-Adar, 1984), and anaphora resolution (Grosz and Sidner, 1986).Moreover, research in reading comprehension indicates that felicitous use of cues improves comprehension and recall (Goldman, 1988), but that their indiscriminate use may have detrimental effects on recall (Millis, Graesser, and Haberlandt, 1993).\\nOur goal is to identify general strategies for cue usage that can be implemented for automatic text generation.From the generation perspective, cue usage consists of three distinct, but interrelated problems: (1) occurrence: whether or not to include a cue in the generated text, (2) placement: where the cue should be placed in the text, and (3) selection: what lexical item(s) should be used.\\nPrior work in text generation has focused on cue selection (McKeown and Elhadad, 1991;Elhadad and McKeown, 1990), or on the relation between cue occurrence and placement and specific rhetorical structures (Rösner and Stede, 1992; Scott and de Souza, 1990; Vander Linden and Martin, 1995).Other hypotheses about cue usage derive from work on discourse coherence and structure.Previous research (Hobbs, 1985;Grosz and Sidner, 1986;Schiffrin, 1987;Mann and Thompson, 1988;Elhadad and McKeown, 1990), which has been largely descriptive, suggests factors such as structural features of the discourse (e.g., level of embedding and segment complexity), intentional and informational relations in that structure, ordering of relata, and syntactic form of discourse constituents.Moser and Moore (1995;1997) coded a corpus of naturally occurring tutorial explanations for the range of features identified in prior work.Because they were also interested in the contrast between occurrence and non-occurrence of cues, they exhaustively coded for all of the factors thought to contribute to cue usage in all of the text.From their study, Moser and Moore identified several interesting correlations between particular features and specific aspects of cue usage, and were able to test specific hypotheses from the literature that were based on constructed examples.\\nIn this paper, we focus on cue occurrence and placement, and present an empirical study of the hypotheses provided by previous research, which have never been systematically evaluated with naturally occurring data.We use a machine learning program, C4.5 (Quinlan, 1993), on the tagged corpus of Moser and Moore to induce decision trees.The number of coded features and their interactions makes the manual construction of rules that predict cue occurrence and placement an intractable task.\\nOur results largely confirm the suggestions from the literature, and clarify them by highlighting the most influential features for a particular task.Discourse structure, in terms of both segment structure and levels of embedding, affects cue occurrence the most; intentional relations also play an important role.For cue placement, the most important factors are syntactic structure and segment complexity.\\nThe paper is organized as follows.In Section 2 we discuss previous research in more detail.Section 3 provides an overview of Moser and Moore's coding scheme.In Section 4 we present our learning experiments, and in Section 5 we discuss our results and conclude.\"\n },\n {\n \"title\": \"Related Work\",\n \"content\": \"McKeown and Elhadad (1991; 1990) studied several connectives (e.g., but, since, because), and include many insightful hypotheses about cue selection; their observation that the distinction between but and although depends on the point of the move is related to the notion of core discussed below.However, they do not address the problem of cue occurrence.\\nOther researchers (Rösner and Stede, 1992; Scott and de Souza, 1990) are concerned with generating text from \\\"RST trees\\\", hierarchical structures where leaf nodes contain content and internal nodes indicate the rhetorical relations, as defined in Rhetorical Structure Theory (RST) (Mann and Thompson, 1988), that exist between subtrees.They proposed heuristics for including and choosing cues based on the rhetorical relation between spans of text, the order of the relata, and the complexity of the related text spans.However, (Scott and de Souza, 1990) was based on a small number of constructed examples, and (Rösner and Stede, 1992) focused on a small number of RST relations.(Litman, 1996) and (Siegel and McKeown, 1994) have applied machine learning to disambiguate between the discourse and sentential usages of cues; however, they do not consider the issues of occurrence and placement, and approach the problem from the point of view of interpretation.We closely follow the approach in (Litman, 1996) in two ways.First, we use C4.5.Second, we experiment first with each feature individually, and then with \\\"interesting\\\" subsets of features.\"\n },\n {\n \"title\": \"Relational Discourse Analysis\",\n \"content\": \"This section briefly describes Relational Discourse Analysis (RDA) (Moser, Moore, and Glendening, 1996), the coding scheme used to tag the data for our machine learning experiments. 1DA is a scheme devised for analyzing tutorial explanations in the domain of electronics troubleshooting.It synthesizes ideas from (Grosz and Sidner, 1986) and from RST (Mann and Thompson, 1988).\\nCoders use RDA to exhaustively analyze each explanation in the corpus, i.e., every word in each explanation belongs to exactly one element in the analysis.An explanation may consist of multiple segments.Each segment originates with an intention of the speaker.Segments are internally structured and consist of a core, i.e., that element that most directly expresses the segment purpose, and any number of contributors, i.e. the remaining constituents.\\nFor each contributor, one analyzes its relation to the core from an intentional perspective, i.e., how it is intended to support the core, and from an informational perspective, i.e., how its content relates to that of the core.The set of intentional relations in RDA is a modification of the presentational relations of RST, while informational relations are similar to the subject matter relations in RST.Each segment constituent, both core and contributors, may itself be a segment with a core:contributor structure.In some cases the core is not explicit.This is often the case with the whole tutor's explanation, since its purpose is to answer the student's explicit question.\\nAs an example of the application of RDA, consider the partial tutor explanation in (1)2 .The purpose of this segment is to inform the student that she made the strategy error of testing inside part3 too soon.The constituent that makes the purpose obvious, in this case (1-B), is the core of the segment.The other constituents help to serve the segment purpose by contributing to it.(1-C) is an example of subsegment with its own core:contributor structure; its purpose is to give a reason for testing part2 first.\\nThe RDA analysis of (1) is shown schematically in Figure 1.The core is depicted as the mother of all the relations it participates in.Each relation node is labeled with both its intentional and informational relation, with the order of relata in the label indicating the linear order in the discourse.Each relation node has up to two daughters: the cue, if any, and\\n(1) Coders analyze each explanation in the corpus and enter their analyses into a database.The corpus consists of 854 clauses comprising 668 segments, for a total of 780 relations.Table 1 summarizes the distribution of different relations, and the number of cued relations in each category.Joints are segments comprising more than one core, but no contributor; clusters are multiunit structures with no recognizable core:contributor relation.(1-B) is a cluster composed of two units (the two clauses), related only at the informational level by a temporal relation.Both clauses describe actions, with the first action description embedded in a matrix (\\\"You should\\\").Cues are much more likely to occur in clusters, where only informational relations occur, than in core:contributor structures, where intentional and informational relations cooccur (χ 2 = 33.367,p <.001, df = 1).In the following, we will not discuss joints and clusters any further.\\nAn important result pointed out by (Moser and Moore, 1995) is that cue placement depends on core position.When the core is first and a cue is associated with the relation, the cue never occurs with the core.In contrast, when the core is second, if a cue occurs, it can occur either on the core or on the contributor.\"\n },\n {\n \"title\": \"Learning from the corpus\",\n \"content\": \"\"\n },\n {\n \"title\": \"The algorithm\",\n \"content\": \"We chose the C4.5 learning algorithm (Quinlan, 1993) because it is well suited to a domain such as ours with discrete valued attributes.Moreover, C4.5 produces decision trees and rule sets, both often used in text generation to implement mappings from function features to forms. 3Finally, C4.5 is both readily available, and is a benchmark learning algorithm that has been extensively used in NLP applications, e.g.(Litman, 1996;Mooney, 1996;Vander Linden and Di Eugenio, 1996).\\nAs our dataset is small, the results we report are based on cross-validation, which (Weiss and Kulikowski, 1991) recommends as the best method to evaluate decision trees on datasets whose cardinality is in the hundreds.Data for learning should be divided into training and test sets; however, for small datasets this has the disadvantage that a sizable portion of the data is not available for learning.Cross- N th used as the test set.The error rate of a tree obtained by using the whole dataset for training is then assumed to be the average error rate on the test set over the N runs.Further, as C4.5 prunes the initial tree it obtains to avoid overfitting, it computes both actual and estimated error rates for the pruned tree; see (Quinlan, 1993, Ch. 4) for details.Thus, below we will report the average estimated error rate on the test set, as computed by 10-fold cross-validation experiments.\"\n },\n {\n \"title\": \"The features\",\n \"content\": \"Each data point in our dataset corresponds to a core:contributor relation, and is characterized by the following features, summarized in Table 2.\\nSegment Structure.Three features capture the global structure of the segment in which the current core:contributor relation appears.\\n• (Con)Trib(utor)-pos(ition) captures the position of a particular contributor within the larger segment in which it occurs, and encodes the structure of the segment in terms of how many contributors precede and follow the core.For example, contributor (1-D) in Figure 1 is labeled as B1A3-2after, as it is the second contributor following the core in a segment with 1 contributor before and 3 after the core.\\n• Inten(tional)-structure indicates which contributors in the segment bear the same intentional relations to the core.\\n• Infor(mational)-structure. Similar to intentional structure, but applied to informational relations.\\nCore:contributor relation.These features more specifically characterize the current core:contributor relation.\\n• Inten(tional)-rel(ation).One of concede, convince, enable.\\n• Infor(mational)-rel(ation).About 30 informational relations have been coded for.However, as preliminary experiments showed that using them individually results in overfitting the data, we classify them according to the four classes proposed in (Moser, Moore, and Glendening, 1996): causality, similarity, elaboration, temporal .Temporal relations only appear in clusters, thus not in the data we discuss in this paper.\\n• Syn(tactic)-rel(ation). Captures whether the core and contributor are independent units (segments or sentences); whether they are coordinated clauses; or which of the two is subordinate to the other.\\n• Adjacency.Whether core and contributor are adjacent in linear order.\\nEmbedding.These features capture segment embedding, Core-type and Trib-type qualitatively, and Above/Below quantitatively.\\n• Core-type/(Con)Trib(utor)-type.Whether the core/the contributor is a segment, or a minimal unit (further subdivided into action, state, matrix).\\n• Above/Below encode the number of relations hierarchically above and below the current relation.\"\n },\n {\n \"title\": \"The experiments\",\n \"content\": \"Initially, we performed learning on all 406 instances of core:contributor relations.We quickly determined that this approach would not lead to useful decision trees.First, the trees we obtained were extremely complex (at least 50 nodes).Second, some of the subtrees corresponded to clearly identifiable subclasses of the data, such as relations with an implicit core, which suggested that we should apply learning to these independently identifiable subclasses.Thus, we subdivided the data into three subsets:\\n• Core1 : core:contributor relations with the core in first position • Impl(icit)-core: core:contributor relations with an implicit core While this has the disadvantage of smaller training sets, the trees we obtain are more manageable and more meaningful.Table 3 summarizes the cardinality of these sets, and the frequencies of cue occurrence.\\nWe ran four sets of experiments.In three of them we predict cue occurrence and in one cue placement.4\"\n },\n {\n \"title\": \"Cue Occurrence\",\n \"content\": \"Table 4 summarizes our main results concerning cue occurrence, and includes the error rates associated with different feature sets.We adopt Litman's approach (1996) to determine whether two error rates E 1 and E 2 are significantly different.We compute 95% confidence intervals for the two error rates using a t -test.E 1 is significantly better than E 2 if the upper bound of the 95% confidence interval for E 1 is lower than the lower bound of the 95% confidence interval for E 2 .For each set of experiments, we report the following:\\n1.A baseline measure obtained by choosing the majority class.E.g., for Core1 58.9% of the relations are not cued; thus, by deciding to never include a cue, one would be wrong 41.1% of the times.\\n2. The best individual features whose predictive power is better than the baseline: as Table 4 makes apparent, individual features do not have much predictive power.For neither Core1 nor Impl-core does any individual feature perform better than the baseline, and for Core2 only one feature is sufficiently predictive.\\n3. (One of) the best induced tree(s).For each tree, we list the number of nodes, and up to six of the features that appear highest in the tree, with their levels of embedding. 5Figure 2 shows the tree for Core2 (space constraints prevent us from including figures for each tree).In the figure, the numbers in parentheses indicate the number of cases correctly covered by the leaf, and the number of expected errors at that leaf.\\nLearning turns out to be most useful for Core1, where the error reduction (as percentage) from baseline to the upper bound of the best result is 32%; error reduction is 19% for Core2 and only 3% for Impl-core.\\nThe best tree was obtained partly by informed choice, partly by trial and error.Automatically trying out all the 2 11 = 2048 subsets of features would be possible, but it would require manual examination of about 2,000 sets of results, a daunting task.Thus, for each dataset we considered only the following subsets of features.\\n1.All features.This always results in C4.5 selecting a few features (from 3 to 7) for the final tree.\"\n },\n {\n \"title\": \"Subsets\",\n \"content\": \"\"\n },\n {\n \"title\": \"C4\",\n \"content\": \".5 on all features.\\n3. In Table 2, three features -Trib-pos, Intenstruct, Infor-struct -concern segment structure, eight do not.We constructed three subsets by always including the eight features that do not concern segment structure, and adding one of those that does.The trees obtained by including Trib-pos, Inten-struct, Infor-struct at the same time are in general more complex, and not significantly better than other trees obtained by including only one of these three features.We attribute this to the fact that these features encode partly overlapping information.\\nFinally, the best tree was obtained as follows.We build the set of trees that are statistically equivalent to the tree with the best error rate (i.e., with the lowest error rate upper bound).Among these trees, we choose the one that we deem the most perspicuous in terms of features and of complexity.Namely, we pick the simplest tree with Trib-Pos as the root if one exists, otherwise the simplest tree.Trees that have Trib-Pos as the root are the most useful for text generation, because, given a complex segment, Trib-Pos is the only attribute that unambiguously identifies a specific contributor.\\nOur results make apparent that the structure of segments plays a fundamental role in determining cue occurrence.One of the three features concerning segment structure (Trib-Pos, Inten-Structure, Infor-Structure) appears as the root or just below the root in all trees in Table 4; more importantly, this same configuration occurs in all trees equivalent to the best tree (even if the specific feature encoding segment structure may change).The level of embedding in a segment, as encoded by Core-type, Trib-type, Above and Below also figures prominently.\\nInten-rel appears in all trees, confirming the intuition that the speaker's purpose affects cue occurrence.More specifically, in Figure 2, Inten-rel distinguishes two different speaker purposes, convince and enable.The same split occurs in some of the best trees induced on Core1, with the same outcome: i.e., convince directly correlates with the occurrence of a cue, whereas for enable other features must be taken into account.6Informational relations do not appear as often as intentional relations; their discriminatory power seems more relevant for clusters.Preliminary experiments show that cue occurrence in clusters depends only on informational and syntactic relations.Finally, Adjacency does not seem to play any substantial role.\"\n },\n {\n \"title\": \"Cue Placement\",\n \"content\": \"While cue occurrence and placement are interrelated problems, we performed learning on them separately.First, the issue of placement arises only in the case of Core2; for Core1, cues only occur on the contributor.Second, we attempted experiments on Core2 that discriminated between occurrence and placement at the same time, and the derived trees were complex and not perspicuous.Thus, we ran an experiment on the 100 cued relations from Core2 to investigate which factors affect placing the cue on the contributor in first position or on the core in second; see Table 5.\\nWe ran the same trials discussed above on this dataset.In this case, the best tree -see Figure 3 -results from combining the two best individual features, and reduces the error rate by 50%.The most discriminant feature turns out to be the syntactic relation between the contributor and the core.However, segment structure still plays an important role, via Trib-pos.While the importance of Syn-rel for placement seems clear, its role concerning occurrence requires further exploration.It is interesting to note that the tree induced on Core1 -the only case in which Synrel is relevant for occurrence -includes the same distinction as in Figure 3: namely, if the contributor depends on the core, the contributor must be marked, otherwise other features have to be taken into account.Scott and de Souza (1990) point out that \\\"there is a strong correlation between the syntactic specification of a complex sentence and its perceived rhetorical structure.\\\"It seems that certain\"\n },\n {\n \"title\": \"Discussion and Conclusions\",\n \"content\": \"We have presented the results of machine learning experiments concerning cue occurrence and placement.As (Litman, 1996) observes, this sort of empirical work supports the utility of machine learning techniques applied to coded corpora.As our study shows, individual features have no predictive power for cue occurrence.Moreover, it is hard to see how the best combination of individual features could be found by manual inspection.\\nOur results also provide guidance for those building text generation systems.This study clearly indicates that segment structure, most notably the ordering of core and contributor, is crucial for determining cue occurrence.Recall that it was only by considering Core1 and Core2 relations in distinct datasets that we were able to obtain perspicuous decision trees that significantly reduce the error rate.\\nThis indicates that the representations produced by discourse planners should distinguish those elements that constitute the core of each discourse segment, in addition to representing the hierarchical structure of segments.Note that the notion of core is related to the notions of nucleus in RST, intended effect in (Young and Moore, 1994), and of point of a move in (Elhadad and McKeown, 1990), and that text generators representing these notions exist.\\nMoreover, in order to use the decision trees derived here, decisions about whether or not to make the core explicit and how to order the core and con-tributor(s) must be made before deciding cue occurrence, e.g., by exploiting other factors such as focus (McKeown, 1985) and a discourse history.\\nOnce decisions about core:contributor ordering and cue occurrence have been made, a generator must still determine where to place cues and select appropriate lexical items.A major focus of our future research is to explore the relationship between the selection and placement decisions.Elsewhere, we have found that particular lexical items tend to have a preferred location, defined in terms of functional (i.e., core or contributor) and linear (i.e., first or second relatum) criteria (Moser and Moore, 1997).Thus, if a generator uses decision trees such as the one shown in Figure 3 to determine where a cue should be placed, it can then select an appropriate cue from those that can mark the given intentional / informational relations, and are usually placed in that functional-linear location.To evaluate this strategy, we must do further work to understand whether there are important distinctions among cues (e.g., so, because) apart from their different preferred locations.The work of Elhadad (1990) and Knott (1996) will help in answering this question.\\nFuture work comprises further probing into machine learning techniques, in particular investigating whether other learning algorithms are more appropriate for our problem (Mooney, 1996), especially algorithms that take into account some a priori knowledge about features and their dependencies.\"\n },\n {\n \"title\": \"Figure 1 :\",\n \"content\": \"Figure 1: The RDA analysis of (1)\"\n },\n {\n \"title\": \"Figure 2 :\",\n \"content\": \"Figure 2: Decision tree for Core2 -occurrence\"\n },\n {\n \"title\": \"Figure 3 :\",\n \"content\": \"Figure 3: Decision tree for Core2 -placement\\nyou know that part1 is good, B. you should eliminate part2 before troubleshooting inside part3.\"\n },\n {\n \"title\": \"Table 3 :\",\n \"content\": \"Distributions of relations and cue occurrences\\nbuilt out of the 2 to 4 attributes appearing highest in the tree obtained by running\"\n },\n {\n \"title\": \"Table 4 :\",\n \"content\": \"Summary of learning results\"\n },\n {\n \"title\": \"Table 5 :\",\n \"content\": \"Cue placement on Core2\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Figure 1: The RDA analysis of (1)","content":"Figure 1 :\n1\nFigure 1: The RDA analysis of (1)"},{"id":"","caption":"Figure 2: Decision tree for Core2 -occurrence","content":"Figure 2 :\n2\nFigure 2: Decision tree for Core2 -occurrence"},{"id":"","caption":"Figure 3: Decision tree for Core2 -placement","content":"Figure 3 :\n3\nFigure 3: Decision tree for Core2 -placement"},{"id":"","caption":"you know that part1 is good, B. you should eliminate part2 before troubleshooting inside part3.","content":"you know that part1 is good, B. you should eliminate part2 before troubleshooting inside part3.\nThis isbecauseC.1. part2 is moved frequently and thus 2. is more susceptible to damage than part3.Also,D. it is more work to open up part3 for testingandE. the process of opening drawers and extending cards in part3may induce problems which did not already exist.B. you should eliminate part2before troubleshooting inside part3d d d d dconcedeconvincecriterion:act ¨¨r r Although Aact:reason r ¨¨r r (This is) C.2becauseconvincecause:effect"},{"id":"","caption":"Distributions of relations and cue occurrences","content":"Table 3 :\n3\nDistributions of relations and cue occurrences\nDataset# of relations# of cued relationsCore112752Core2155100(on Trib: 43) (on Core: 57)Impl-core12429Total406181Core1Core2Impl-coreBaseline41.135.423.5Best features ∅Info-rel: 33.4±0.94 ∅Best tree25.6±1.24 (15) 27.4±1.28 (18)22.1±0.57 (10)0. Trib-pos0. Trib-Pos0. Core-type1. Trib-type1. Inten-rel1. Infor-struct2. Syn-rel2. Info-rel2. Inten-rel3. Core-type3. Above4. Above4. Core-type5. Inten-rel5. Below\nbuilt out of the 2 to 4 attributes appearing highest in the tree obtained by running"},{"id":"","caption":"Summary of learning results","content":"Table 4 :\n4\nSummary of learning results"},{"id":"","caption":"Cue placement on Core2","content":"Table 5 :\n5\nCue placement on Core2"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Figure 1: The RDA analysis of (1)\",\n \"content\": \"Figure 1 :\\n1\\nFigure 1: The RDA analysis of (1)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 2: Decision tree for Core2 -occurrence\",\n \"content\": \"Figure 2 :\\n2\\nFigure 2: Decision tree for Core2 -occurrence\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 3: Decision tree for Core2 -placement\",\n \"content\": \"Figure 3 :\\n3\\nFigure 3: Decision tree for Core2 -placement\"\n },\n {\n \"id\": \"\",\n \"caption\": \"you know that part1 is good, B. you should eliminate part2 before troubleshooting inside part3.\",\n \"content\": \"you know that part1 is good, B. you should eliminate part2 before troubleshooting inside part3.\\nThis isbecauseC.1. part2 is moved frequently and thus 2. is more susceptible to damage than part3.Also,D. it is more work to open up part3 for testingandE. the process of opening drawers and extending cards in part3may induce problems which did not already exist.B. you should eliminate part2before troubleshooting inside part3d d d d dconcedeconvincecriterion:act ¨¨r r Although Aact:reason r ¨¨r r (This is) C.2becauseconvincecause:effect\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Distributions of relations and cue occurrences\",\n \"content\": \"Table 3 :\\n3\\nDistributions of relations and cue occurrences\\nDataset# of relations# of cued relationsCore112752Core2155100(on Trib: 43) (on Core: 57)Impl-core12429Total406181Core1Core2Impl-coreBaseline41.135.423.5Best features ∅Info-rel: 33.4±0.94 ∅Best tree25.6±1.24 (15) 27.4±1.28 (18)22.1±0.57 (10)0. Trib-pos0. Trib-Pos0. Core-type1. Trib-type1. Inten-rel1. Infor-struct2. Syn-rel2. Info-rel2. Inten-rel3. Core-type3. Above4. Above4. Core-type5. Inten-rel5. Below\\nbuilt out of the 2 to 4 attributes appearing highest in the tree obtained by running\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Summary of learning results\",\n \"content\": \"Table 4 :\\n4\\nSummary of learning results\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Cue placement on Core2\",\n \"content\": \"Table 5 :\\n5\\nCue placement on Core2\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"A computational theory of the function of clue words in argument understanding. Robin ; Cohen, Michael Elhadad, Kathleen Mckeown, Proceedings of COLING84. COLING84Stanford, CA; Helsinki, Finland1984. 1990Proceedings of COLING90","doi":""},{"text":"The role of sequence markers in reading and recall: Comparison of native and nonnative english speakers. Susan R Goldman, Barbara J Grosz, Candace L Sidner, Computational Linguistics. 1231988. 1986University of CaliforniaTechnical reportAttention, intention, and the structure of discourse","doi":""},{"text":"On the coherence and structure of discourse. Jerry R Hobbs, CSLI-85-371985Center for the Study of Language and Information, Stanford UniversityTechnical Report","doi":""},{"text":"A Data-Driven methodology for motivating a set of coherence relations. Alistair Knott, 1996University of EdinburghPh.D. thesis","doi":""},{"text":"A plan recognition model for subdialogues in conversations. Diane J Litman, Diane J ; Litman, F James, William C Allen ; Mann, Sandra A Thompson, Relational propositions in discourse. Discourse Processes. 1996. 1987. 19865Litman and Allen1987","doi":""},{"text":"A contrastive evaluation of functional unification grammar for surface language generation: A case study in the choice of connectives. Thompson1988 ] Mann, William C Mann, Sandra A Thompson ; Mckeown, Kathleen R ; Mckeown, Kathleen R , Michael Elhadad, Natural Language Generation in Artificial Intelligence and Computational Linguistics. C L Paris, W R Swartout, W C Mann, Cambridge, England; BostonKluwer Academic Publishers1988. McKeown1985. 1985. 19918Text Generation: Using Discourse Strategies and Focus Constraints to Generate Natural Language Text","doi":""},{"text":"The impact of connectives on the memory for expository text. Graesser Millis, Millis, Arthur Keith, Karl Graesser, Haberlandt, Applied Cognitive Psychology. 71993","doi":""},{"text":"Comparative experiments on disambiguating word senses: An illustration of the role of bias in machine learning. Raymond J ; Mooney, Megan Moser, Johanna D Moore ; Moser, Megan , Johanna D Moore, Conference on Empirical Methods in Natural Language Processing. Boston, MA1996. 1995. 1997A corpus analysis of discourse cues and relational discourse structure. Submitted for publication","doi":""},{"text":"Instructions for Coding Explanations: Identifying Segments, Relations and Minimal Units. Moore Moser, Moser, Johanna D Megan, Erin Moore, Glendening, 96-171996University of Pittsburgh, Department of Computer ScienceTechnical Report","doi":""},{"text":"Customizing RST for the automatic production of technical manuals. J Quinlan, Dietmar Ross ; Rösner, Manfred Stede, 6th International Workshop on Natural Language Generation. R Dale, E Hovy, D Rösner, O Stock, BerlinSpringer-Verlag1993. 1984. 199222C4.5: Programs for Machine Learning","doi":""},{"text":"Emergent linguistic rules from inducing decision trees: Disambiguating discourse clue words. Deborah Schiffrin, Scott, ] De Souza1990, Donia Scott, Clarisse Sieckenius De Souza, ; Siegel, Eric V , Kathleen R Mckeown, Current Research in Natural Language Generation. R Dale, C Mellish, M Zock, New York; New YorkAcademic Press1987. 1990. 1994Proceedings of AAAI94","doi":""},{"text":"Expressing rhetorical relations in instructional text: A case study of the purpose relation. Vander Linden, Di Eugenio1996] Vander Linden, Keith , Barbara Di, Eugenio ; , James H Martin, 8th International Workshop on Natural Language Generation. Sussex, UK. [Vander Linden and Martin1995] Vander Linden, Keith1996. 199521Learning micro-planning rules for preventative expressions","doi":""},{"text":"Computer Systems that learn: classification and prediction methods from statistics, neural nets, machine learning, and expert systems. Kulikowski1991 ] Weiss, Sholom M Weiss, Casimir Kulikowski ; Young, R Michael, Johanna D Moore, 7th International Workshop on Natural Language Generation. Kennebunkport, MaineMorgan Kaufmann1991. 1994DPOCL: A Principled Approach to Discourse Planning","doi":""}],"string":"[\n {\n \"text\": \"A computational theory of the function of clue words in argument understanding. Robin ; Cohen, Michael Elhadad, Kathleen Mckeown, Proceedings of COLING84. COLING84Stanford, CA; Helsinki, Finland1984. 1990Proceedings of COLING90\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The role of sequence markers in reading and recall: Comparison of native and nonnative english speakers. Susan R Goldman, Barbara J Grosz, Candace L Sidner, Computational Linguistics. 1231988. 1986University of CaliforniaTechnical reportAttention, intention, and the structure of discourse\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the coherence and structure of discourse. Jerry R Hobbs, CSLI-85-371985Center for the Study of Language and Information, Stanford UniversityTechnical Report\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Data-Driven methodology for motivating a set of coherence relations. Alistair Knott, 1996University of EdinburghPh.D. thesis\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A plan recognition model for subdialogues in conversations. Diane J Litman, Diane J ; Litman, F James, William C Allen ; Mann, Sandra A Thompson, Relational propositions in discourse. Discourse Processes. 1996. 1987. 19865Litman and Allen1987\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A contrastive evaluation of functional unification grammar for surface language generation: A case study in the choice of connectives. Thompson1988 ] Mann, William C Mann, Sandra A Thompson ; Mckeown, Kathleen R ; Mckeown, Kathleen R , Michael Elhadad, Natural Language Generation in Artificial Intelligence and Computational Linguistics. C L Paris, W R Swartout, W C Mann, Cambridge, England; BostonKluwer Academic Publishers1988. McKeown1985. 1985. 19918Text Generation: Using Discourse Strategies and Focus Constraints to Generate Natural Language Text\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The impact of connectives on the memory for expository text. Graesser Millis, Millis, Arthur Keith, Karl Graesser, Haberlandt, Applied Cognitive Psychology. 71993\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Comparative experiments on disambiguating word senses: An illustration of the role of bias in machine learning. Raymond J ; Mooney, Megan Moser, Johanna D Moore ; Moser, Megan , Johanna D Moore, Conference on Empirical Methods in Natural Language Processing. Boston, MA1996. 1995. 1997A corpus analysis of discourse cues and relational discourse structure. Submitted for publication\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Instructions for Coding Explanations: Identifying Segments, Relations and Minimal Units. Moore Moser, Moser, Johanna D Megan, Erin Moore, Glendening, 96-171996University of Pittsburgh, Department of Computer ScienceTechnical Report\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Customizing RST for the automatic production of technical manuals. J Quinlan, Dietmar Ross ; Rösner, Manfred Stede, 6th International Workshop on Natural Language Generation. R Dale, E Hovy, D Rösner, O Stock, BerlinSpringer-Verlag1993. 1984. 199222C4.5: Programs for Machine Learning\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Emergent linguistic rules from inducing decision trees: Disambiguating discourse clue words. Deborah Schiffrin, Scott, ] De Souza1990, Donia Scott, Clarisse Sieckenius De Souza, ; Siegel, Eric V , Kathleen R Mckeown, Current Research in Natural Language Generation. R Dale, C Mellish, M Zock, New York; New YorkAcademic Press1987. 1990. 1994Proceedings of AAAI94\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Expressing rhetorical relations in instructional text: A case study of the purpose relation. Vander Linden, Di Eugenio1996] Vander Linden, Keith , Barbara Di, Eugenio ; , James H Martin, 8th International Workshop on Natural Language Generation. Sussex, UK. [Vander Linden and Martin1995] Vander Linden, Keith1996. 199521Learning micro-planning rules for preventative expressions\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Computer Systems that learn: classification and prediction methods from statistics, neural nets, machine learning, and expert systems. Kulikowski1991 ] Weiss, Sholom M Weiss, Casimir Kulikowski ; Young, R Michael, Johanna D Moore, 7th International Workshop on Natural Language Generation. Kennebunkport, MaineMorgan Kaufmann1991. 1994DPOCL: A Principled Approach to Discourse Planning\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":495,"cells":{"text":{"kind":"string","value":"\nM-Theory and the Light Cone\nJun 1999\n\nJoseph Polchinski \nInstitute for Theoretical Physics\nUniversity of California\n93106-4030Santa BarbaraCA\n\nM-Theory and the Light Cone\nJun 1999arXiv:hep-th/9903165v2 14 1\nI discuss D0-brane quantum mechanics as a nonperturbative formulation of string theory, in particular the relation between the Banks-Fischler-Shenker-Susskind matrix model, the Maldacena conjecture for D0-branes, and type IIA/M-theory duality. Some features of the quantum mechanics of D0-branes are also discussed. §1. IntroductionThe main focus of this lecture is D0-brane quantum mechanics as a nonperturbative formulation of string theory. In particular I would like to discuss the relation between the Banks-Fischler-Shenker-Susskind matrix model, 1) the Maldacena conjecture for D0-branes 2), 3) , and type IIA/M-theory duality. 4) This has sometimes been a confusing subject, and it is one of which everyone has their own understanding. * * ) I hope that I will reduce rather than add to the confusion. I have certainly added to it in the past; the present version reflects my improved understanding.Figure 1is a schematic picture of the lessons of string duality, one that I have drawn perhaps a hundred times. It is intended to emphasize the limited range of conventional string theory, which only provides an asymptotic expansion at the five stringy cusps. This is true even when one includes D-branes -only asymptotically in the string coupling do these provide a quantitatively accurate description. To describe the complete physics at a finite distance from a cusp, and to reach the Mtheory point with 11-dimensional Lorentz invariance, requires a leap, a conjecture, to a new idea with a greater range of validity. This new idea can only be guessed, not deduced from information already in string theory.This has always been the intended message of figure 1, but as I will explain I now believe, rather surprising, that one can actually deduce a great deal from what is already known. The main outline of this talk is the description of four limits: the DKPS limit, the SSS limit, the BGLHKS limit, and the IMSY limit. Actually, these are all the same limit of type IIA string theory, but presented in different ways and with different emphases. I will then explain how these different descriptions illuminate the relations among the various ideas. At the end I discuss possible future directions in D0-brane quantum mechanics, and some properties of the D0-brane bound state. * ) joep@itp.ucsb.edu * * ) Other discussions of which I am aware, some of which overlap mine, are listed in ref. 5) .typeset using PTPT E X.sty <ver.0.8>\n\n§1. Introduction\n\nThe main focus of this lecture is D0-brane quantum mechanics as a nonperturbative formulation of string theory. In particular I would like to discuss the relation between the Banks-Fischler-Shenker-Susskind matrix model, 1) the Maldacena conjecture for D0-branes 2), 3) , and type IIA/M-theory duality. 4) This has sometimes been a confusing subject, and it is one of which everyone has their own understanding. * * ) I hope that I will reduce rather than add to the confusion. I have certainly added to it in the past; the present version reflects my improved understanding. Figure 1 is a schematic picture of the lessons of string duality, one that I have drawn perhaps a hundred times. It is intended to emphasize the limited range of conventional string theory, which only provides an asymptotic expansion at the five stringy cusps. This is true even when one includes D-branes -only asymptotically in the string coupling do these provide a quantitatively accurate description. To describe the complete physics at a finite distance from a cusp, and to reach the Mtheory point with 11-dimensional Lorentz invariance, requires a leap, a conjecture, to a new idea with a greater range of validity. This new idea can only be guessed, not deduced from information already in string theory.\n\nThis has always been the intended message of figure 1, but as I will explain I now believe, rather surprising, that one can actually deduce a great deal from what is already known. The main outline of this talk is the description of four limits: the DKPS limit, the SSS limit, the BGLHKS limit, and the IMSY limit. Actually, these are all the same limit of type IIA string theory, but presented in different ways and with different emphases. I will then explain how these different descriptions illuminate the relations among the various ideas. At the end I discuss possible future directions in D0-brane quantum mechanics, and some properties of the D0-brane bound state. Tr([X m , X n ] 2 ) + spin term + higher derivs. (2 . 1)\n\nHere g s is the dimensionless string coupling and l s = (α ′ ) 1/2 is the string length scale. The first term is the D0-brane rest energy. The second is the mass of excited open or closed strings. The third is from massless closed strings, with momenta of order q.\n\nThe important terms are in the second line. The first is a nonrelativistic kinetic term, P 2 /2m, except that it is a trace over N 2 terms rather than a sum over N terms. This is a reflection of the important non-Abelian geometry of D-branes. 7) The coordinates arise from the lightest open string state\nψ m −1/2 |ij . (2 . 2)\nThis state has a 9-dimensional spatial index m and so naturally corresponds to a collective coordinate. The ket |ij denotes a string that begins on the ith D0-brane and ends on the jth, so the coordinate X m ij is a matrix in the N -dimensional D0-brane index space. Then P m ij is the conjugate momentum. The potential term corrects this overcounting. It is nonnegative (the minus sign canceling an i 2 ) and vanishes precisely when all the X m commute and so can be diagonalized. Further, there is a U (N ) gauge invariance, so the state is independent of basis and the low energy physics reduces to the quantum mechanics of the 9N eigenvalues, as appropriate for N particles in 9 + 1 dimensions. The potential term has the same form as the 4-gluon Yang-Mills interaction Tr(F mn F mn ). This is no accident, the two being related by T -duality. The omitted spin terms are such that the system has 16 linearly realized supersymmetries, generating 256 states for each D0-brane. Now let us take the DKPS limit, 8) which is g s → 0 while holding fixed all of The IIA string is M-theory compactified on a circle of radius R 10 = g s l s , with eleven-dimensional Planck length M = g This scaling of the energy has a simple pictorial interpretation, as in figure 2. For any finite R 10 we can find a frame in which the spatial radius takes an arbitrary fixed value R ′ 10 ; the cost is that the periodicity is no longer at equal times but at nearly null separations. The Lorentz factor is γ = R ′ 10 /R 10 and so\nN , P , q , M = g −1/3 s l −1 s , H = 1 g s l s M H − N g s l s .(2 .(H − p 10 ) ′ = R ′ 10 R 10 (H − p 10 ) = M R ′ 10 H . (3 . 2)\nSince p ′ 10 = N/R ′ 10 is fixed, the DKPS limit is equivalent to holding fixed the energy in the boosted frame of figure 2b. Thus we have a new interpretation: 10), 11), 12) Summary: The matrix Hamiltonian (2 . 4) describes M-theory in a space compactified in a lightlike direction, in the sector with N units of compact momentum. This is defined as the limit of spacelike compactification, keeping all states that have finite energy in the limit. The eleven-dimensional Planck scale and the noncompact momenta are held fixed in the limit. §4. The Balasubramanian-Gopakumar-Larsen-Hyun-Kiem-Shin (BGLHKS) Limit\n\nAn important refinement of the preceding picture comes when we consider that the particles carrying p 10 must gravitate. With matter in the periodic space one would not expect that the compactified direction remain precisely null. A perturbation towards timelike compactification is implausible, so likely any gravitational effect is in the direction of making the periodicity spacelike. Indeed, as a function of distance r from the source, the radius of the compact dimension is given in terms of the black 0-brane dilaton solution 13) M R 10 = e 2Φ(r)/3 Here and in the metric below numerical constants have been ignored. In the last line we have taken the DKPS limit, g s → 0 at fixed M . The striking feature is that the radius R 10 (r) has a nonvanishing limit, and only when r → ∞ does the radius of the compact dimension go to zero. Thus the geometry is only asymptotically as depicted in figure 2b. 14), 15) Let us develop somewhat further the nature of the resulting space. The first obvious separation occurs at r ≈ N 1/7 /M , where e Φ(r) is of order one. At larger r, R 10 (r) is less than M −1 and so the effective description is in terms of weakly coupled IIA string theory. At smaller r, R 10 (r) is greater than M −1 and so the effective description is in terms of M-theory.\n\nTo go further examine the M-theory metric, obtained by lifting the black D0brane metric to M-theory in the coordinate system of figure 2b,\nds 2 = −dt ′2 + dx ′2 10 + N M 9 r 7 R ′2 10 (dt ′ − dx ′ 10 ) 2 + (dx ⊥ ) 2 . (4 . 2)\nThe periodicity is\n(t ′ , x ′ 10 ) ∼ = (t ′ − πR ′ 10 , x ′ 10 + πR ′ 10 ) . (4 . 3)\nThe metric appears flat at large r, but this deceptive because the radius of the periodic dimension and so the local string tension are going to zero. The latter is\nT (r) = M 3 R 10 (r) = N 1/2 M −3/2 r −7/2 . (4 . 4)\nThe corresponding length scale l s (r\n) = N −1/4 M 3/4 r 7/4 exceeds r when r > N 1/3 M −1 .\nAt this point geometry can no longer make sense: the strings are larger than the system (in terms of the string metric, which is curved, the curvature becomes large in string units).\n\nAnother key distance is r = N 1/9 M −1 , where R 10 (r) = r. At distances below this the metric (4 . 2) is no longer correct. Since generic sources will not be uniformly distributed in x 10 , the resulting metric will not be uniform either. This effect damps away when r > R 10 (r) (the space is then long in the r-direction and narrow in the x 10 -direction). Finally, when r < M −1 we would not expect any classical metric to be a good description. Collecting together these results gives 14), 15), 3)\nN 1/3 M −1 < r no geometry N 1/7 M −1 < r < N 1/3 M −1 IIA string in D0-brane metric N 1/9 M −1 < r < N 1/7 M −1 M-theory in lifted D0-brane metric M −1 < r < N 1/9 M −1 M-theory in inhomogeneous x 10 r < M −1 no geometry .\nWe have been treating the sources as pointlike. This is true, for example, for the ground state, which is a single graviton with N units of compact momentum. 4) For more distributed sources, the geometry below a given r would be smoothed out.\n\nSummary: The matrix Hamiltonian (2 . 4) describes M-theory in the sector with N units of compact momentum, where the compactified dimension is asymptotically lightlike. That is, it describes all states whose metric is asymptotically of the form (4 . 2). §5 \nP → M 1/2P , X → M −1/2X , (5 . 1)\nthen the kinetic term is canonically normalized and theX 4 interaction has a coefficient M 3 . The latter is the correct dimension for a Yang-Mills coupling-squared in 0 + 1 dimensions. The effective dimensionless coupling at an energy scale E is then are held fixed in the two limits. * ) Second, the IMSY limit is a statement about the physics as a function of energy scale. To relate this to the previous discussion, consider a standard observable, the expectation value of a product of local operators with some characteristic external energy E. If the reader is more comfortable in Euclidean space, he/she is free to analytically continue, since the quantum mechanics and the supergravity have a common global time (which in the supergravity is defined by the asymptotic translation invariance). A local operator in the QM is equivalent to some local disturbance in the supergravity. The only coordinate invariant local objects live at the boundary (coordinate transformations at the boundary are frozen by the specification of the asymptotic behavior), so by logic similar to the AdS/CFT case 17) we are led to the conclusion that these correspond to deformations of the boundary conditions. In this sense the QM 'lives' at the boundary at infinity, but it should be emphasized that it contains all states of the bulk theory. * ) Note that the IMSY g 2 YM differs from the value M 3 = g −1 s l 3 s given above because it is a dimensional quantity and the Hamiltonian has been rescaled.\ng 2 eff = N M 3 /E 3 ,(5 .\nThese boundary deformations do not change the D0-brane charge, and so correspond to boundary conditions that are invariant under translations in the periodic dimension: they depend on t ′ and x ′10 only in the combination t ′ + x ′10 , of the form exp iE ′ (t ′ + x ′10 ). Acting on these the wave operator reduces to\n− N M 9 r 7 R ′2 10 ∂ 2 t ′ + ∂ 2 ⊥ = − N E 2 M 7 r 7 + ∂ 2 ⊥ . (5 . 5)\nNote that the arbitrary R ′ 10 drops out after converting the SSS energy E ′ to the matrix QM energy E via eq. (3 . 2). From the relative scaling of the two terms in the wave operator it follows that the supergravity amplitude is dominated by the radial scale 18) r\n≈ N 1/5 E 2/5 M −7/5 ↔ E ≈ N −1/2 r 5/2 M 7/2 . (5 . 6)\nThis is the holographic energy-radius relation. 19) In particular, the energy E = N 1/3 M corresponds to r = N 1/3 M −1 . At higher energies/larger radii the geometric picture breaks down but matrix QM perturbation theory is valid. At lower energies/smaller radii, QM perturbation theory breaks down but the supergravity description is valid. At successively lower energies the physics is described by the successive supergravity pictures of the previous section. It is not clear how to understand E < N −1/2 M , where the supergravity breaks down at short distance, but we will argue in the next section that things become simple again at E ∼ N −1 M . Note that the only threshold visible in the 't Hooft (spherical) limit is E ≈ N 1/3 M corresponding to r ≈ N 1/3 M −1 . Summary: The matrix Hamiltonian (2 . 4) is equivalent to M-theory in the sector with N units of compact momentum, where the compactified dimension is asymptotically lightlike. In the preceding discussion we were using this to define M-theory, but we can also use it in the other direction: low energy supergravity is the effective theory for the strongly coupled QM at E < N 1/3 M . §6. Discussion\n\nLet us now consider the implications of the preceding results. One thing that I find particularly striking is the BGLHKS picture. We are conditioned to think of the g s → 0 limit as being strictly 10-dimensional. Now we see that in a sector of large but fixed D0-brane charge N , a finite bubble of 11-dimensional spacetime survives at N 1/9 M −1 < r < N 1/7 M −1 . This grows arbitrarily large at large N . Thus, the large-N matrix QM describes 11-dimensional bulk physics. The derivation appear to be quite reliable. On the QM side we have string theory with the coupling and the energy (in string units, above the BPS minimum) going to zero; we surely understand string theory in this regime. On the M-theory side, the geometry is quite smooth as g s → 0 in the indicated range of r. This is completely opposite to the mindset expressed in the introduction and figure 1, in that the nature of the 11-dimensional theory has been deduced from string perturbation theory. This is shown schematically in figure 3: the N -axis, rather than being perpendicular to the old parameter space, actually runs from the IIA vacuum toward the M-theory vacuum. Of course the key input is IIA/M-theory duality -this turns out to have a great deal more information than was initially evident. The IMSY result, or Maldacena duality for D0-branes, also appears to be a great deal of information from very little. It tells us about the behavior of a quantum mechanical system at strong coupling. Again, the key input is IIA/M-theory duality -with little additional information this implies the IMSY result. Now consider the BFSS matrix theory. 1) The statement above, that the large-N matrix QM describes 11-dimensional physics, would appear to be morally the same thing. However, the BFSS theory is more specific. * ) Given the above discussion, the natural way to define the 11-dimensional S-matrix would be to begin with N source D0-branes, to create the 11-dimensional bubble, and then to consider scattering of supergravity fields corresponding to local operators in the QM, aimed to intersect in the bubble. This is parallel to recent constructions in the AdS 5 case, with D3-branes replaced by D0-branes. 20), 21) The BFSS proposal is different, and more direct. Namely, scattering of matrix theory bound states is supposed to go directly over to the M-theory S-matrix in the large-N limit. The SSS picture suggests a derivation of this. 12) Consider gravitons (bound states) with D0-brane charges N 1 , N 2 , and so on. Scale the N i to infinity together, and take R ′ 10 to infinity at the same rate. The momenta p ′ 10,i = N i /R ′ 10 are then constant. The size of the box is going to infinity, in a frame in which the scattering event is held fixed. One might then expect that the limit would give the noncompact theory. For a limit of spacelike compactifications this would be expected, but lightlike * ) I would like to thank various members of the IAS seminar audience, particularly T. Banks, M. Douglas, H. Verlinde, and E. Witten, for making this point to me. compactification is surely more subtle, because the invariant radius of the compact dimension remains at zero. Indeed, the null compactification leads to specific effects that might complicate the limit. The first is zero tension strings, from membranes wrapping the periodic dimension. 22) The second is divergent fluctuations of modes with compact momentum zero. These arise because the effective 10-dimensional coupling in field theory would be g 2 11 /R 10 , which diverges in the lightlike limit. * ) In some field theory examples these effects dominate and the large-N limit does not give the noncompact theory. However, in the present case the BGLHKS effect seems to do away with potential problems. Note that the relevant distance scale p −1 and relevant energy E ∼ p 2 /N i are both quite small, where the graviton transverse momentum p is held fixed as N → ∞. (These do not satisfy the holographic relation (5 . 6) because they correspond to a different quantity in the QM theory.) At this distance the periodic dimension is very far from lightlike due to the gravitational effect of the D0-branes. Thus the wrapped membrane tension is large compared to the scale of the scattering, and so is the 10-dimensional Planck scale.\n\nOne might still have the following concern. At very large r (the 't Hooft scale) the spacetime interpretation does break down as we have argued earlier, due to a string tension that vanishes asymptotically and also to a 10-dimensional coupling that diverges asymptotically. The gravitons come in from infinite r, and so begin and end in this 'non-spacetime' region. How, then, can we get the physics of flat 11dimensional spacetime? The point is that at these asymptotic distances the gravitons are propagating freely, at energies small compared to the characteristic energies of the QM. All that matters then is the metric on moduli space, and this is simply flat as a consequence of supersymmetry -it does not matter that at the 't Hooft radius we stop interpreting it as the moduli space of particles in spacetime and begin interpreting as the moduli space of the quantum mechanics. * * )\n\nThus the potential objections to the Seiberg-Sen argument seem to be red herrings in this case. The BFSS matrix theory, like the IMSY duality, can indeed be deduced from perturbative string theory and IIA/M-theory duality. For the zero modes we can say that they cure their own problem, in that the BGLHKS effect is the expectation value of the gravitational zero mode.\n\nLet me conclude with some comments about the relation between perturbative matrix theory amplitudes and supergravity or M-theory amplitudes. From the literature one might get the impression that these are expected to agree in general: that if an amplitude has the same dependence on the parameters in the two theories (M , N , r, and v) then its form and coefficient must be the same. I can confess to sloppy thinking in this regard in the past. But in fact there is no general reason that there should be agreement. The ranges of validity are entirely different, E/M < N −1 vs E/M > N 1/3 , and as in any effective field theory there is no general reason to expect amplitudes at different scales to have any simple relation. For example, the exact amplitude could involve a function of E 3 /M 3 N which asymptotes to one value at * ) Zero mode divergences in light-cone field theory are a long story. For discussions in the context of the lightlike limit see refs. 23), 24) . * * ) Again I would like to thank the IAS seminar audience for straightening me out.\n\nhigh energy and to a different one at low energy. Of course some amplitudes are protected by supersymmetry nonrenormalization theorems and so must agree: if they did not it would falsify matrix theory and possibly IIA/M-theory duality as well. It is still interesting to compare calculations in the two limits, both as a way to infer new supersymmetric nonrenormalization theorems (since supersymmetry appears to be a strong restriction) and also because this is the route by which new dualities, such as the Maldacena duality, are sometimes discovered. §7. What Next?\n\nThe conclusion is that the large-N matrix QM gives the M-theory limit of string theory. It is one of many nonperturbative constructions of string theory, since we can take a similar large-N limit in any of the Maldacena dualities. Certainly these constructions are not yet in simplest form: string theory is constructed in states with specific, and unusual, boundary conditions. There should be some general framework describing all states of string theory -spacetimes without boundaries are especially puzzling from the present viewpoint. This will likely involve some new ideas, and look rather different from the current descriptions.\n\nStill, one way to try to proceed is to develop more fully the existing constructions. Much attention goes to the AdS 5 × S 5 and AdS 3 × S 3 cases because of their large symmetries, but the matrix QM/D0-brane case would seem to have advantages as well, due to its smaller number of degrees of freedom.\n\nAs an example, it would be nice to have a direct test of the Maldacena duality, comparing two calculations at the same point in parameter space without the aid of nonrenormalization results. The simplest place to look is the D0-brane QM, just below the 't Hooft energy. The duality predicts that the gauge theory entropy is given by the nonextremal D0-brane entropy as\nS = cN 2 T 9/5 (M 3 N ) −3/5 (7 . 1)\nwith c a known constant. 25) The exponent 9/5 indicates the nontriviality of the dynamics. As the temperature is reduced the effect of the commutator potential is to freeze out degrees of freedom in a gradual way, but the precise exponent appears to be difficult to get by analytic means. However, this entropy should be accessible by Monte Carlo (I should emphasize that this is very different from the index-like path integrals that have been calculated, in that the thermal boundary conditions break supersymmetry.). In field theory we would need to renormalize to maintain the supersymmetry in the cutoff theory, but QM is superrenormalizable. Also, at this relatively high temperature precise supersymmetric cancellations are not so relevant, and high temperature methods may also simplify the fermionic determinant. There is unlikely to be enough dynamic range at reasonable values of N to pick out the 9/5 accurately, but for the Monte Carlo calculation to come close to the value of c obtained in a black hole calculation would be very striking.\n\nOf course the IKKT matrix theory 26) should be even simpler than the BFSS model from this point of view, and it has been studied numerically. The question here is one of interpretation. The assumption in much of the literature appears to be that the IKKT model is a background-independent formulation of string theory, but an interpretation parallel to IMSY 3) seems more likely. The portion of the IKKT matrix integral where the eigenvalues are well-separated can be calculated perturbatively, but this breaks down when the eigenvalues become close. The natural extension of the IMSY analysis would be that there is an effective description of the latter part of the matrix integral in terms of supergravity in an asymptotically Dinstanton space. For a numerical test one would need to find a simple prediction parallel to the entropy (7 . 1).\n\nAnother advantage of being in QM is that renormalization is not needed, so we can add arbitrary perturbations to the Hamiltonian and so study more general spaces.\n\nA related advantage is that the wavefunction and operators have a direct meaning. Consider for example the size R of the D0-brane bound state. * ) In the BFSS paper 1) , two estimates are given,\nR ∼ N 1/3 M −1 (7 . 2)\nfrom 't Hooft scaling and\nR ∼ N 1/9 M −1 (7 . 3)\nfrom the quantum mechanics of the v 4 /r 7 potential. The relation between these was not resolved (though at least one of the authors [B] came to the conclusion that we will reach below). It was suggested that the 't Hooft result might be invalidated by IR divergences, but in fact one can show that it is rigorously correct. Start with the uncertainty principle, applied to one matrix element X 1 11 :\n\n0|(X 1 11 ) 2 |0 0|(P 1 11 ) 2 |0 ≥ 1 . Applying this to all 9N 2 elements gives * * ) 0|Tr(X 1 X 1 )|0 0|Tr(P n P n )|0 ≥ 9N 4 . (7 . 5)\n\nNow apply the virial theorem, relating the expectation value of the kinetic term to that of the potential. From 0|[X n P n , H]|0 = 0 one obtains for the expectation values −2K + 4P + F = 0, where K is the kinetic term, P is the potential, and F is the fermionic term (linear in X). On the other hand K + P + F = 0 by supersymmetry. Thus K : P : F = 1 : 1 : −2, and so\n0|Tr(P n P n )|0 = M 6 8π 2 m,n | 0|Tr([X m , X n ] 2 )|0 | . (7 . 6) and 0|Tr(X 1 X 1 )|0 m,n | 0|Tr([X m , X n ] 2 )|0 | ≥ 72N 4 π 2 M 6 . (7 . 7)\nFor Hermitean matrices the Schwarz inequality gives |Tr(ABAB)| < Tr(AABB) < 1 2 [Tr(A 4 ) + Tr(B 4 )], and so 0|Tr(X 1 X 1 )|0 0|Tr(X 1 X 1 X 1 X 1 )|0 ≥ N 4 π 2 2M 6 . (7 . 8)\n\nFor X 1 having an eigenvalue distribution of width R, the left-hand side is of order N 2 R 6 and so R ≥ O(N 1/3 M −1 ) . (7 . 9) This assumes that the expectation value of (X 1 ) 4 converges, as is indeed the case. This result, obtained from a Hamiltonian Born-Oppenheimer approximation in refs. 28) , can be nicely explained by an effective Hamiltonian analysis (this is adapted from an unpublished argument of L. Susskind, with comments from G. M. Graf). The dangerous direction is when the wavefunction separates into two blocks, along the flat directions. Here the relative coordinate is described by a free U (1) theory, so the Hamiltonian is just the Laplacian and the wavefunction is harmonic. For l = 0 the wavefunction falls as r −7 . The bound state has l = 2, because the relative spin plus motion gives an invariant, and so falls as r −9 . Then 0|Tr[(X 1 ) L ]|0 ∼ d 9 X |X| −18+L (7 . 10) converges for L < 9. Note that (7 . 9) is exactly the maximum radius in which the supergravity picture is valid. I therefore strongly expect that N 1/3 M −1 is the actual value, not just a lower bound.\n\nHow does this relate to the smaller estimate (7 . 3)? The bound (7 . 9) comes primarily from the quantum fluctuations of the off-diagonal modes of X n , while the smaller estimate refers to the diagonal motion on the moduli space. For the purpose of the BFSS theory, one is interested only in very small energies. The off-diagonal modes have much higher excitation energies and so are frozen into their ground states; the associated zero-point fluctuations are then not observable. Roughly speaking, Tr(X n X n ) = Tr(X n X n ) off−diag./high energy + Tr(X n X n ) diag./low energy . (7 . 11)\n\nThe off-diagonal part is much larger but is state-independent at low energy. In other words, Tr(X n X n ) in the low energy theory differs from the full Tr(X n X n ) by an additive renormalization. It seems hard to give a precise definition of the low energy part in terms of the underlying QM variables.\n\nIn conclusion, to find a description of the flat 11-dimensional M-theory (or the 10-dimensional IIB theory) we somehow need, in the large-N limit, to identify a subtheory contained in the intersection of all the field theory constructions. To find a description of the general background we need something which contains the union of all the field theory constructions.\n\nFig. 1 .\n1Space of string vacua. The cusps are limits in which a weakly coupled string description is possible (except for the M-theory limit). §2. The Douglas-Kabat-Pouliot-Shenker (DKPS) Limit Consider a state in the IIA string theory containing N D0-branes plus any collection of open and closed strings. * ) The energy of such a state is\n\n\n. The D0-brane charge is the number of units of compact momentum, N = p 10 R 10 . In terms of M-theory parameters, the DKPS limit takes R 10 → 0 while holding fixed all of p 10 R 10 , P , q , M , H = 1 M R10 (H − p 10 ) .(3 . 1)\n\nFig. 2 .\n2The same space in two frames. a) Identified point (indicated by arrows) are at equal times and small spatial separation. b) Identified points are at fixed spatial distance and almost null separation.\n\nNFig. 3 .\n3Space of string vacua. At gs = 0 but large N the IIA string has 11-dimensional physics.\n\n\nThus in this limit the IIA string theory becomes very simple, just the matrix quantum mechanics 9) (2 . 4). The string coupling is taken to zero, but because of the scaling of the energy the most relevant interaction does survive.Summary: The matrix Hamiltonian (2 . 4) describes the sector of IIA string theory with D0-brane-charge N , in the DKPS limit (2 . 3). §3. The Susskind-Seiberg-Sen (SSS) Limit3) \n\nIt follows from the energy (2 . 1) that all closed strings decouple, as do excited open \nstrings, leaving only massless open strings with \n\nH = \n1 \n2M \nTr(P 2 ) − \nM 5 \n16π 2 \n\nm,n \n\nTr([X m , X n ] 2 ) + spin term . \n(2 . 4) \n\n\n\n\n. The Itzhaki-Maldacena-Sonnenschein-Yankielowicz (IMSY) Limit Now let us think about the matrix Hamiltonian (2 . 4) directly. If we rescale to new canonical variables\n\n2 )\n2the N being from the usual 't Hooft counting.16) At E > N 1/3 M , the effective coupling is small and quantum mechanical perturbation theory is good. When E < N 1/3 M , perturbation theory breaks down -we have a strongly coupled problem. In the past this might have been a difficulty, but now that we are in the Age of Duality it presents an opportunity, to find a weakly coupled description. Indeed, this is what IMSY do. 3) By essentially the DKPS plus BGLHKS arguments, they show that the low energy physics of the matrix Hamiltonian is given by IIA/M-theory in the supergravity background(4 . 2).In making the connection, there are two points that should be clarified. First, the IMSY limit sounds different, in that it holds fixed3) \n* ) For reviews of D0-brane properties see refs.6) .\nAcknowledgementsI would like to thank the organizers and the city of Nishinomiya for a very stimulating meeting. I would also like to thank T. Banks This work was supported in part by NSF grants PHY94-07194 and PHY97-22022.\n. T Banks, W Fischler, S H Shenker, L Susskind, hep-th/9610043Phys. Rev. 55T. Banks, W. Fischler, S. H. Shenker, and L. Susskind, Phys. Rev. D55 (1961) 5112, hep-th/9610043.\n\n. J Maldacena, hep-th/9711200Adv. Theor. Math. Phys. 2J. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231, hep-th/9711200.\n\n. N Itzhaki, J M Maldacena, J Sonnenschein, S Yankielowicz, hep-th/9802042Phys. Rev. 58N. Itzhaki, J. M. Maldacena, J. Sonnenschein, and S. Yankielowicz, Phys. Rev. D58 (1998) 046004, hep-th/9802042.\n\n. P K Townsend, ; E Witten, hep-th/9503124Phys. Lett. 350Nucl. Phys.P. K. Townsend, Phys. Lett. B350 (1995) 184, hep-th/9501068. E. Witten, Nucl. Phys. B443 (1995) 85, hep-th/9503124.\n\nITP seminar. S Shenker, S. P. de Alwishep-th/9806188P. J. Silva. 5944029Phys. Rev.S. Shenker, ITP seminar, March 1998. S. P. de Alwis, Phys. Rev. D59 (1999) 044029, hep-th/9806178. P. J. Silva, hep-th/9806188.\n\n. V Sahakian, E Martinec, hep-th/9901033Phys. Rev. 59124005V. Sahakian and E. Martinec, Phys. Rev. D59 (1999) 124005, hep-th/9810224. I. Chepelev, hep-th/9901033.\n\n. J Polchinski, hep-th/9611050J. Polchinski, hep-th/9611050.\n\nJ Polchinski, String Theory, Vols. I and II. CambridgeCambridge University PressJ. Polchinski, String Theory, Vols. I and II (Cambridge University Press, Cambridge, 1998).\n\n. E Witten, hep-th/9510135Nucl. Phys. 460E. Witten, Nucl. Phys. B460 (1996) 335, hep-th/9510135.\n\n. M R Douglas, D Kabat, P Pouliot, S H Shenker, hep-th/9608024Nucl. Phys. 485M. R. Douglas, D. Kabat, P. Pouliot, and S. H. Shenker, Nucl. Phys. B485 (1997) 85, hep-th/9608024.\n\n. M Claudson, M B Halpern, ; R Flume, Annals Phys. 250189Nucl. Phys.M. Claudson and M. B. Halpern, Nucl. Phys. B250 689, 1985. R. Flume, Annals Phys. 164 189, 1985.\n\n. M Baake, M Reinicke, V Rittenberg, J. Math. Phys. 261070M. Baake, M. Reinicke, and V. Rittenberg, J. Math. Phys. 26 1070, 1985.\n\n. L Susskind, hep-th/9704080L. Susskind, hep-th/9704080.\n\n. K Becker, M Becker, J Polchinski, A Tseytlin, hep- th/9706072Phys. Rev. 56K. Becker, M. Becker, J. Polchinski, and A. Tseytlin, Phys. Rev. D56 (1997) 3174, hep- th/9706072.\n\n. N Seiberg, hep-th/9709220Adv. Theor. Math. Phys. 79Phys. Rev. Lett.N. Seiberg, Phys. Rev. Lett. 79 (1997) 3577, hep-th/9710009. A. Sen, Adv. Theor. Math. Phys. 2 (1998) 51, hep-th/9709220.\n\n. G T Horowitz, A Strominger, Nucl. Phys. 360197G. T. Horowitz and A. Strominger, Nucl. Phys. B360 (1991) 197.\n\n. V Balasubramanian, R Gopakumar, F Larsen, hep- th/9712077Nucl. Phys. 526V. Balasubramanian, R. Gopakumar, and F. Larsen, Nucl. Phys. B526 (1998) 415, hep- th/9712077.\n\n. S Hyun, Y Kiem, H Shin ; Hyun, Y Kiem, hep-th/9805136Phys. Lett. 5726003Phys. Rev.S. Hyun, Y. Kiem, and H. Shin, Phys. Rev. D57 (1998) 4856, hep-th/9712021. S. Hyun, Phys. Lett. B441 (1998) 116, hep-th/9802026. S. Hyun and Y. Kiem, Phys. Rev. D59 (1999) 026003, hep-th/9805136.\n\n. G Hooft, Nucl. Phys. 72461G. 't Hooft, Nucl. Phys. B72 (1974) 461.\n\n. S S Gubser, I R Klebanov, A M Polyakov, hep-th/9802150Adv. Theor. Math. Phys. 428Phys. Lett.S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Phys. Lett. B428 (1998) 105, hep- th/9802109. E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253, hep-th/9802150.\n\n. A Peet, J Polchinski, hep-th/9809022Phys. Rev. 59A. Peet and J. Polchinski, Phys. Rev. D59 (1999) 065006, hep-th/9809022.\n\n. L Susskind, E Witten, hep-th/9805114L. Susskind and E. Witten, hep-th/9805114.\n\nJ Polchinski, hep-th/9902182hep-th/9901076. L. Susskind, hep-th/9901079. J. Polchinski and L. Susskind. J. Polchinski, hep-th/9901076. L. Susskind, hep-th/9901079. J. Polchinski and L. Susskind, hep-th/9902182.\n\nV Balasubramanian, S B Giddings, A Lawrence, hep-th/9903048hep-th/9902052. S. B. Giddings. V. Balasubramanian, S. B. Giddings, and A. Lawrence, hep-th/9902052. S. B. Giddings, hep-th/9903048.\n\n. M R Douglas, H Ooguri, hep-th/9710178Phys. Lett. 425M. R. Douglas and H. Ooguri, Phys. Lett. B425 (1998) 71, hep-th/9710178.\n\n. H W L Naus, H J Pirner, T J Fields, J P Vary, hep-th/9704135Phys. Rev. 56H. W. L. Naus, H. J. Pirner, T. J. Fields, J. P. Vary, Phys. Rev. D56 (1997) 8062, hep-th/9704135.\n\n. S Hellerman, J Polchinski, hep-th/9711037S. Hellerman and J. Polchinski, hep-th/9711037.\n\n. I R Klebanov, A A Tseytlin, hep-th/9604089Nucl. Phys. 475I. R. Klebanov and A. A. Tseytlin, Nucl. Phys. B475 (1996) 164, hep-th/9604089.\n\n. N Ishibashi, H Kawai, Y Kitazawa, A Tsuchiya, hep-th/9612115Nucl. Phys. 498N. Ishibashi, H. Kawai, Y. Kitazawa, and A. Tsuchiya, Nucl. Phys. B498 (1997) 467, hep-th/9612115.\n\nM Li, hep-th/9902073hep-th/9901158. D. Kabat and G. Lifschytz. M. Li, hep-th/9901158. D. Kabat and G. Lifschytz, hep-th/9902073.\n\nM B Halpern, C Schwarz, ; J Frohlich, G M Graf, D Hasler, J Hoppe, S T Yau, hep-th/9712133hep-th/9904182 and references therein. 13M. B. Halpern and C. Schwarz, Int. J. Mod. Phys. A13 (1998) 4367, hep-th/9712133. J. Frohlich, G. M. Graf, D. Hasler, J. Hoppe, and S. T. Yau, hep-th/9904182 and references therein.\n"},"corpus_id":{"kind":"number","value":119417724,"string":"119,417,724"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"hep-th/9903165\",\n \"mag\": 2057905855,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1143/ptps.134.158\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/hep-th/9903165v2.pdf\"\n ],\n \"pdf_sha\": \"981ed619ad6cc1752e7dfadba808277c6bddf0cb\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"https://academic.oup.com/ptps/article-pdf/doi/10.1143/PTPS.134.158/5160623/134-158.pdf\",\n \"status\": \"BRONZE\"\n }\n}"},"title":{"kind":"string","value":"M-Theory and the Light Cone"},"authors":{"kind":"list like","value":["Joseph Polchinski \nInstitute for Theoretical Physics\nUniversity of California\n93106-4030Santa BarbaraCA"],"string":"[\n \"Joseph Polchinski \\nInstitute for Theoretical Physics\\nUniversity of California\\n93106-4030Santa BarbaraCA\"\n]","isConversational":false},"abstract":{"kind":"string","value":"I discuss D0-brane quantum mechanics as a nonperturbative formulation of string theory, in particular the relation between the Banks-Fischler-Shenker-Susskind matrix model, the Maldacena conjecture for D0-branes, and type IIA/M-theory duality. Some features of the quantum mechanics of D0-branes are also discussed. §1. IntroductionThe main focus of this lecture is D0-brane quantum mechanics as a nonperturbative formulation of string theory. In particular I would like to discuss the relation between the Banks-Fischler-Shenker-Susskind matrix model, 1) the Maldacena conjecture for D0-branes 2), 3) , and type IIA/M-theory duality. 4) This has sometimes been a confusing subject, and it is one of which everyone has their own understanding. * * ) I hope that I will reduce rather than add to the confusion. I have certainly added to it in the past; the present version reflects my improved understanding.Figure 1is a schematic picture of the lessons of string duality, one that I have drawn perhaps a hundred times. It is intended to emphasize the limited range of conventional string theory, which only provides an asymptotic expansion at the five stringy cusps. This is true even when one includes D-branes -only asymptotically in the string coupling do these provide a quantitatively accurate description. To describe the complete physics at a finite distance from a cusp, and to reach the Mtheory point with 11-dimensional Lorentz invariance, requires a leap, a conjecture, to a new idea with a greater range of validity. This new idea can only be guessed, not deduced from information already in string theory.This has always been the intended message of figure 1, but as I will explain I now believe, rather surprising, that one can actually deduce a great deal from what is already known. The main outline of this talk is the description of four limits: the DKPS limit, the SSS limit, the BGLHKS limit, and the IMSY limit. Actually, these are all the same limit of type IIA string theory, but presented in different ways and with different emphases. I will then explain how these different descriptions illuminate the relations among the various ideas. At the end I discuss possible future directions in D0-brane quantum mechanics, and some properties of the D0-brane bound state. * ) joep@itp.ucsb.edu * * ) Other discussions of which I am aware, some of which overlap mine, are listed in ref. 5) .typeset using PTPT E X.sty <ver.0.8>"},"sections":{"kind":"list like","value":[{"title":"§1. Introduction","content":"The main focus of this lecture is D0-brane quantum mechanics as a nonperturbative formulation of string theory. In particular I would like to discuss the relation between the Banks-Fischler-Shenker-Susskind matrix model, 1) the Maldacena conjecture for D0-branes 2), 3) , and type IIA/M-theory duality. 4) This has sometimes been a confusing subject, and it is one of which everyone has their own understanding. * * ) I hope that I will reduce rather than add to the confusion. I have certainly added to it in the past; the present version reflects my improved understanding. Figure 1 is a schematic picture of the lessons of string duality, one that I have drawn perhaps a hundred times. It is intended to emphasize the limited range of conventional string theory, which only provides an asymptotic expansion at the five stringy cusps. This is true even when one includes D-branes -only asymptotically in the string coupling do these provide a quantitatively accurate description. To describe the complete physics at a finite distance from a cusp, and to reach the Mtheory point with 11-dimensional Lorentz invariance, requires a leap, a conjecture, to a new idea with a greater range of validity. This new idea can only be guessed, not deduced from information already in string theory.\nThis has always been the intended message of figure 1, but as I will explain I now believe, rather surprising, that one can actually deduce a great deal from what is already known. The main outline of this talk is the description of four limits: the DKPS limit, the SSS limit, the BGLHKS limit, and the IMSY limit. Actually, these are all the same limit of type IIA string theory, but presented in different ways and with different emphases. I will then explain how these different descriptions illuminate the relations among the various ideas. At the end I discuss possible future directions in D0-brane quantum mechanics, and some properties of the D0-brane bound state. Tr([X m , X n ] 2 ) + spin term + higher derivs. (2 . 1)\nHere g s is the dimensionless string coupling and l s = (α ′ ) 1/2 is the string length scale. The first term is the D0-brane rest energy. The second is the mass of excited open or closed strings. The third is from massless closed strings, with momenta of order q.\nThe important terms are in the second line. The first is a nonrelativistic kinetic term, P 2 /2m, except that it is a trace over N 2 terms rather than a sum over N terms. This is a reflection of the important non-Abelian geometry of D-branes. 7) The coordinates arise from the lightest open string state\nThis state has a 9-dimensional spatial index m and so naturally corresponds to a collective coordinate. The ket |ij denotes a string that begins on the ith D0-brane and ends on the jth, so the coordinate X m ij is a matrix in the N -dimensional D0-brane index space. Then P m ij is the conjugate momentum. The potential term corrects this overcounting. It is nonnegative (the minus sign canceling an i 2 ) and vanishes precisely when all the X m commute and so can be diagonalized. Further, there is a U (N ) gauge invariance, so the state is independent of basis and the low energy physics reduces to the quantum mechanics of the 9N eigenvalues, as appropriate for N particles in 9 + 1 dimensions. The potential term has the same form as the 4-gluon Yang-Mills interaction Tr(F mn F mn ). This is no accident, the two being related by T -duality. The omitted spin terms are such that the system has 16 linearly realized supersymmetries, generating 256 states for each D0-brane. Now let us take the DKPS limit, 8) which is g s → 0 while holding fixed all of The IIA string is M-theory compactified on a circle of radius R 10 = g s l s , with eleven-dimensional Planck length M = g This scaling of the energy has a simple pictorial interpretation, as in figure 2. For any finite R 10 we can find a frame in which the spatial radius takes an arbitrary fixed value R ′ 10 ; the cost is that the periodicity is no longer at equal times but at nearly null separations. The Lorentz factor is γ = R ′ 10 /R 10 and so\nSince p ′ 10 = N/R ′ 10 is fixed, the DKPS limit is equivalent to holding fixed the energy in the boosted frame of figure 2b. Thus we have a new interpretation: 10), 11), 12) Summary: The matrix Hamiltonian (2 . 4) describes M-theory in a space compactified in a lightlike direction, in the sector with N units of compact momentum. This is defined as the limit of spacelike compactification, keeping all states that have finite energy in the limit. The eleven-dimensional Planck scale and the noncompact momenta are held fixed in the limit. §4. The Balasubramanian-Gopakumar-Larsen-Hyun-Kiem-Shin (BGLHKS) Limit\nAn important refinement of the preceding picture comes when we consider that the particles carrying p 10 must gravitate. With matter in the periodic space one would not expect that the compactified direction remain precisely null. A perturbation towards timelike compactification is implausible, so likely any gravitational effect is in the direction of making the periodicity spacelike. Indeed, as a function of distance r from the source, the radius of the compact dimension is given in terms of the black 0-brane dilaton solution 13) M R 10 = e 2Φ(r)/3 Here and in the metric below numerical constants have been ignored. In the last line we have taken the DKPS limit, g s → 0 at fixed M . The striking feature is that the radius R 10 (r) has a nonvanishing limit, and only when r → ∞ does the radius of the compact dimension go to zero. Thus the geometry is only asymptotically as depicted in figure 2b. 14), 15) Let us develop somewhat further the nature of the resulting space. The first obvious separation occurs at r ≈ N 1/7 /M , where e Φ(r) is of order one. At larger r, R 10 (r) is less than M −1 and so the effective description is in terms of weakly coupled IIA string theory. At smaller r, R 10 (r) is greater than M −1 and so the effective description is in terms of M-theory.\nTo go further examine the M-theory metric, obtained by lifting the black D0brane metric to M-theory in the coordinate system of figure 2b,\nThe periodicity is\nThe metric appears flat at large r, but this deceptive because the radius of the periodic dimension and so the local string tension are going to zero. The latter is\nThe corresponding length scale l s (r\nAt this point geometry can no longer make sense: the strings are larger than the system (in terms of the string metric, which is curved, the curvature becomes large in string units).\nAnother key distance is r = N 1/9 M −1 , where R 10 (r) = r. At distances below this the metric (4 . 2) is no longer correct. Since generic sources will not be uniformly distributed in x 10 , the resulting metric will not be uniform either. This effect damps away when r > R 10 (r) (the space is then long in the r-direction and narrow in the x 10 -direction). Finally, when r < M −1 we would not expect any classical metric to be a good description. Collecting together these results gives 14), 15), 3)\nWe have been treating the sources as pointlike. This is true, for example, for the ground state, which is a single graviton with N units of compact momentum. 4) For more distributed sources, the geometry below a given r would be smoothed out.\nSummary: The matrix Hamiltonian (2 . 4) describes M-theory in the sector with N units of compact momentum, where the compactified dimension is asymptotically lightlike. That is, it describes all states whose metric is asymptotically of the form (4 . 2). §5\nthen the kinetic term is canonically normalized and theX 4 interaction has a coefficient M 3 . The latter is the correct dimension for a Yang-Mills coupling-squared in 0 + 1 dimensions. The effective dimensionless coupling at an energy scale E is then are held fixed in the two limits. * ) Second, the IMSY limit is a statement about the physics as a function of energy scale. To relate this to the previous discussion, consider a standard observable, the expectation value of a product of local operators with some characteristic external energy E. If the reader is more comfortable in Euclidean space, he/she is free to analytically continue, since the quantum mechanics and the supergravity have a common global time (which in the supergravity is defined by the asymptotic translation invariance). A local operator in the QM is equivalent to some local disturbance in the supergravity. The only coordinate invariant local objects live at the boundary (coordinate transformations at the boundary are frozen by the specification of the asymptotic behavior), so by logic similar to the AdS/CFT case 17) we are led to the conclusion that these correspond to deformations of the boundary conditions. In this sense the QM 'lives' at the boundary at infinity, but it should be emphasized that it contains all states of the bulk theory. * ) Note that the IMSY g 2 YM differs from the value M 3 = g −1 s l 3 s given above because it is a dimensional quantity and the Hamiltonian has been rescaled.\nThese boundary deformations do not change the D0-brane charge, and so correspond to boundary conditions that are invariant under translations in the periodic dimension: they depend on t ′ and x ′10 only in the combination t ′ + x ′10 , of the form exp iE ′ (t ′ + x ′10 ). Acting on these the wave operator reduces to\nNote that the arbitrary R ′ 10 drops out after converting the SSS energy E ′ to the matrix QM energy E via eq. (3 . 2). From the relative scaling of the two terms in the wave operator it follows that the supergravity amplitude is dominated by the radial scale 18) r\nThis is the holographic energy-radius relation. 19) In particular, the energy E = N 1/3 M corresponds to r = N 1/3 M −1 . At higher energies/larger radii the geometric picture breaks down but matrix QM perturbation theory is valid. At lower energies/smaller radii, QM perturbation theory breaks down but the supergravity description is valid. At successively lower energies the physics is described by the successive supergravity pictures of the previous section. It is not clear how to understand E < N −1/2 M , where the supergravity breaks down at short distance, but we will argue in the next section that things become simple again at E ∼ N −1 M . Note that the only threshold visible in the 't Hooft (spherical) limit is E ≈ N 1/3 M corresponding to r ≈ N 1/3 M −1 . Summary: The matrix Hamiltonian (2 . 4) is equivalent to M-theory in the sector with N units of compact momentum, where the compactified dimension is asymptotically lightlike. In the preceding discussion we were using this to define M-theory, but we can also use it in the other direction: low energy supergravity is the effective theory for the strongly coupled QM at E < N 1/3 M . §6. Discussion\nLet us now consider the implications of the preceding results. One thing that I find particularly striking is the BGLHKS picture. We are conditioned to think of the g s → 0 limit as being strictly 10-dimensional. Now we see that in a sector of large but fixed D0-brane charge N , a finite bubble of 11-dimensional spacetime survives at N 1/9 M −1 < r < N 1/7 M −1 . This grows arbitrarily large at large N . Thus, the large-N matrix QM describes 11-dimensional bulk physics. The derivation appear to be quite reliable. On the QM side we have string theory with the coupling and the energy (in string units, above the BPS minimum) going to zero; we surely understand string theory in this regime. On the M-theory side, the geometry is quite smooth as g s → 0 in the indicated range of r. This is completely opposite to the mindset expressed in the introduction and figure 1, in that the nature of the 11-dimensional theory has been deduced from string perturbation theory. This is shown schematically in figure 3: the N -axis, rather than being perpendicular to the old parameter space, actually runs from the IIA vacuum toward the M-theory vacuum. Of course the key input is IIA/M-theory duality -this turns out to have a great deal more information than was initially evident. The IMSY result, or Maldacena duality for D0-branes, also appears to be a great deal of information from very little. It tells us about the behavior of a quantum mechanical system at strong coupling. Again, the key input is IIA/M-theory duality -with little additional information this implies the IMSY result. Now consider the BFSS matrix theory. 1) The statement above, that the large-N matrix QM describes 11-dimensional physics, would appear to be morally the same thing. However, the BFSS theory is more specific. * ) Given the above discussion, the natural way to define the 11-dimensional S-matrix would be to begin with N source D0-branes, to create the 11-dimensional bubble, and then to consider scattering of supergravity fields corresponding to local operators in the QM, aimed to intersect in the bubble. This is parallel to recent constructions in the AdS 5 case, with D3-branes replaced by D0-branes. 20), 21) The BFSS proposal is different, and more direct. Namely, scattering of matrix theory bound states is supposed to go directly over to the M-theory S-matrix in the large-N limit. The SSS picture suggests a derivation of this. 12) Consider gravitons (bound states) with D0-brane charges N 1 , N 2 , and so on. Scale the N i to infinity together, and take R ′ 10 to infinity at the same rate. The momenta p ′ 10,i = N i /R ′ 10 are then constant. The size of the box is going to infinity, in a frame in which the scattering event is held fixed. One might then expect that the limit would give the noncompact theory. For a limit of spacelike compactifications this would be expected, but lightlike * ) I would like to thank various members of the IAS seminar audience, particularly T. Banks, M. Douglas, H. Verlinde, and E. Witten, for making this point to me. compactification is surely more subtle, because the invariant radius of the compact dimension remains at zero. Indeed, the null compactification leads to specific effects that might complicate the limit. The first is zero tension strings, from membranes wrapping the periodic dimension. 22) The second is divergent fluctuations of modes with compact momentum zero. These arise because the effective 10-dimensional coupling in field theory would be g 2 11 /R 10 , which diverges in the lightlike limit. * ) In some field theory examples these effects dominate and the large-N limit does not give the noncompact theory. However, in the present case the BGLHKS effect seems to do away with potential problems. Note that the relevant distance scale p −1 and relevant energy E ∼ p 2 /N i are both quite small, where the graviton transverse momentum p is held fixed as N → ∞. (These do not satisfy the holographic relation (5 . 6) because they correspond to a different quantity in the QM theory.) At this distance the periodic dimension is very far from lightlike due to the gravitational effect of the D0-branes. Thus the wrapped membrane tension is large compared to the scale of the scattering, and so is the 10-dimensional Planck scale.\nOne might still have the following concern. At very large r (the 't Hooft scale) the spacetime interpretation does break down as we have argued earlier, due to a string tension that vanishes asymptotically and also to a 10-dimensional coupling that diverges asymptotically. The gravitons come in from infinite r, and so begin and end in this 'non-spacetime' region. How, then, can we get the physics of flat 11dimensional spacetime? The point is that at these asymptotic distances the gravitons are propagating freely, at energies small compared to the characteristic energies of the QM. All that matters then is the metric on moduli space, and this is simply flat as a consequence of supersymmetry -it does not matter that at the 't Hooft radius we stop interpreting it as the moduli space of particles in spacetime and begin interpreting as the moduli space of the quantum mechanics. * * )\nThus the potential objections to the Seiberg-Sen argument seem to be red herrings in this case. The BFSS matrix theory, like the IMSY duality, can indeed be deduced from perturbative string theory and IIA/M-theory duality. For the zero modes we can say that they cure their own problem, in that the BGLHKS effect is the expectation value of the gravitational zero mode.\nLet me conclude with some comments about the relation between perturbative matrix theory amplitudes and supergravity or M-theory amplitudes. From the literature one might get the impression that these are expected to agree in general: that if an amplitude has the same dependence on the parameters in the two theories (M , N , r, and v) then its form and coefficient must be the same. I can confess to sloppy thinking in this regard in the past. But in fact there is no general reason that there should be agreement. The ranges of validity are entirely different, E/M < N −1 vs E/M > N 1/3 , and as in any effective field theory there is no general reason to expect amplitudes at different scales to have any simple relation. For example, the exact amplitude could involve a function of E 3 /M 3 N which asymptotes to one value at * ) Zero mode divergences in light-cone field theory are a long story. For discussions in the context of the lightlike limit see refs. 23), 24) . * * ) Again I would like to thank the IAS seminar audience for straightening me out.\nhigh energy and to a different one at low energy. Of course some amplitudes are protected by supersymmetry nonrenormalization theorems and so must agree: if they did not it would falsify matrix theory and possibly IIA/M-theory duality as well. It is still interesting to compare calculations in the two limits, both as a way to infer new supersymmetric nonrenormalization theorems (since supersymmetry appears to be a strong restriction) and also because this is the route by which new dualities, such as the Maldacena duality, are sometimes discovered. §7. What Next?\nThe conclusion is that the large-N matrix QM gives the M-theory limit of string theory. It is one of many nonperturbative constructions of string theory, since we can take a similar large-N limit in any of the Maldacena dualities. Certainly these constructions are not yet in simplest form: string theory is constructed in states with specific, and unusual, boundary conditions. There should be some general framework describing all states of string theory -spacetimes without boundaries are especially puzzling from the present viewpoint. This will likely involve some new ideas, and look rather different from the current descriptions.\nStill, one way to try to proceed is to develop more fully the existing constructions. Much attention goes to the AdS 5 × S 5 and AdS 3 × S 3 cases because of their large symmetries, but the matrix QM/D0-brane case would seem to have advantages as well, due to its smaller number of degrees of freedom.\nAs an example, it would be nice to have a direct test of the Maldacena duality, comparing two calculations at the same point in parameter space without the aid of nonrenormalization results. The simplest place to look is the D0-brane QM, just below the 't Hooft energy. The duality predicts that the gauge theory entropy is given by the nonextremal D0-brane entropy as\nwith c a known constant. 25) The exponent 9/5 indicates the nontriviality of the dynamics. As the temperature is reduced the effect of the commutator potential is to freeze out degrees of freedom in a gradual way, but the precise exponent appears to be difficult to get by analytic means. However, this entropy should be accessible by Monte Carlo (I should emphasize that this is very different from the index-like path integrals that have been calculated, in that the thermal boundary conditions break supersymmetry.). In field theory we would need to renormalize to maintain the supersymmetry in the cutoff theory, but QM is superrenormalizable. Also, at this relatively high temperature precise supersymmetric cancellations are not so relevant, and high temperature methods may also simplify the fermionic determinant. There is unlikely to be enough dynamic range at reasonable values of N to pick out the 9/5 accurately, but for the Monte Carlo calculation to come close to the value of c obtained in a black hole calculation would be very striking.\nOf course the IKKT matrix theory 26) should be even simpler than the BFSS model from this point of view, and it has been studied numerically. The question here is one of interpretation. The assumption in much of the literature appears to be that the IKKT model is a background-independent formulation of string theory, but an interpretation parallel to IMSY 3) seems more likely. The portion of the IKKT matrix integral where the eigenvalues are well-separated can be calculated perturbatively, but this breaks down when the eigenvalues become close. The natural extension of the IMSY analysis would be that there is an effective description of the latter part of the matrix integral in terms of supergravity in an asymptotically Dinstanton space. For a numerical test one would need to find a simple prediction parallel to the entropy (7 . 1).\nAnother advantage of being in QM is that renormalization is not needed, so we can add arbitrary perturbations to the Hamiltonian and so study more general spaces.\nA related advantage is that the wavefunction and operators have a direct meaning. Consider for example the size R of the D0-brane bound state. * ) In the BFSS paper 1) , two estimates are given,\nfrom 't Hooft scaling and\nfrom the quantum mechanics of the v 4 /r 7 potential. The relation between these was not resolved (though at least one of the authors [B] came to the conclusion that we will reach below). It was suggested that the 't Hooft result might be invalidated by IR divergences, but in fact one can show that it is rigorously correct. Start with the uncertainty principle, applied to one matrix element X 1 11 :\n0|(X 1 11 ) 2 |0 0|(P 1 11 ) 2 |0 ≥ 1 . Applying this to all 9N 2 elements gives * * ) 0|Tr(X 1 X 1 )|0 0|Tr(P n P n )|0 ≥ 9N 4 . (7 . 5)\nNow apply the virial theorem, relating the expectation value of the kinetic term to that of the potential. From 0|[X n P n , H]|0 = 0 one obtains for the expectation values −2K + 4P + F = 0, where K is the kinetic term, P is the potential, and F is the fermionic term (linear in X). On the other hand K + P + F = 0 by supersymmetry. Thus K : P : F = 1 : 1 : −2, and so\nFor Hermitean matrices the Schwarz inequality gives |Tr(ABAB)| < Tr(AABB) < 1 2 [Tr(A 4 ) + Tr(B 4 )], and so 0|Tr(X 1 X 1 )|0 0|Tr(X 1 X 1 X 1 X 1 )|0 ≥ N 4 π 2 2M 6 . (7 . 8)\nFor X 1 having an eigenvalue distribution of width R, the left-hand side is of order N 2 R 6 and so R ≥ O(N 1/3 M −1 ) . (7 . 9) This assumes that the expectation value of (X 1 ) 4 converges, as is indeed the case. This result, obtained from a Hamiltonian Born-Oppenheimer approximation in refs. 28) , can be nicely explained by an effective Hamiltonian analysis (this is adapted from an unpublished argument of L. Susskind, with comments from G. M. Graf). The dangerous direction is when the wavefunction separates into two blocks, along the flat directions. Here the relative coordinate is described by a free U (1) theory, so the Hamiltonian is just the Laplacian and the wavefunction is harmonic. For l = 0 the wavefunction falls as r −7 . The bound state has l = 2, because the relative spin plus motion gives an invariant, and so falls as r −9 . Then 0|Tr[(X 1 ) L ]|0 ∼ d 9 X |X| −18+L (7 . 10) converges for L < 9. Note that (7 . 9) is exactly the maximum radius in which the supergravity picture is valid. I therefore strongly expect that N 1/3 M −1 is the actual value, not just a lower bound.\nHow does this relate to the smaller estimate (7 . 3)? The bound (7 . 9) comes primarily from the quantum fluctuations of the off-diagonal modes of X n , while the smaller estimate refers to the diagonal motion on the moduli space. For the purpose of the BFSS theory, one is interested only in very small energies. The off-diagonal modes have much higher excitation energies and so are frozen into their ground states; the associated zero-point fluctuations are then not observable. Roughly speaking, Tr(X n X n ) = Tr(X n X n ) off−diag./high energy + Tr(X n X n ) diag./low energy . (7 . 11)\nThe off-diagonal part is much larger but is state-independent at low energy. In other words, Tr(X n X n ) in the low energy theory differs from the full Tr(X n X n ) by an additive renormalization. It seems hard to give a precise definition of the low energy part in terms of the underlying QM variables.\nIn conclusion, to find a description of the flat 11-dimensional M-theory (or the 10-dimensional IIB theory) we somehow need, in the large-N limit, to identify a subtheory contained in the intersection of all the field theory constructions. To find a description of the general background we need something which contains the union of all the field theory constructions."},{"title":"Fig. 1 .","content":""},{"title":"Fig. 2 .","content":""},{"title":"NFig. 3 .","content":""},{"title":"2 )","content":""}],"string":"[\n {\n \"title\": \"§1. Introduction\",\n \"content\": \"The main focus of this lecture is D0-brane quantum mechanics as a nonperturbative formulation of string theory. In particular I would like to discuss the relation between the Banks-Fischler-Shenker-Susskind matrix model, 1) the Maldacena conjecture for D0-branes 2), 3) , and type IIA/M-theory duality. 4) This has sometimes been a confusing subject, and it is one of which everyone has their own understanding. * * ) I hope that I will reduce rather than add to the confusion. I have certainly added to it in the past; the present version reflects my improved understanding. Figure 1 is a schematic picture of the lessons of string duality, one that I have drawn perhaps a hundred times. It is intended to emphasize the limited range of conventional string theory, which only provides an asymptotic expansion at the five stringy cusps. This is true even when one includes D-branes -only asymptotically in the string coupling do these provide a quantitatively accurate description. To describe the complete physics at a finite distance from a cusp, and to reach the Mtheory point with 11-dimensional Lorentz invariance, requires a leap, a conjecture, to a new idea with a greater range of validity. This new idea can only be guessed, not deduced from information already in string theory.\\nThis has always been the intended message of figure 1, but as I will explain I now believe, rather surprising, that one can actually deduce a great deal from what is already known. The main outline of this talk is the description of four limits: the DKPS limit, the SSS limit, the BGLHKS limit, and the IMSY limit. Actually, these are all the same limit of type IIA string theory, but presented in different ways and with different emphases. I will then explain how these different descriptions illuminate the relations among the various ideas. At the end I discuss possible future directions in D0-brane quantum mechanics, and some properties of the D0-brane bound state. Tr([X m , X n ] 2 ) + spin term + higher derivs. (2 . 1)\\nHere g s is the dimensionless string coupling and l s = (α ′ ) 1/2 is the string length scale. The first term is the D0-brane rest energy. The second is the mass of excited open or closed strings. The third is from massless closed strings, with momenta of order q.\\nThe important terms are in the second line. The first is a nonrelativistic kinetic term, P 2 /2m, except that it is a trace over N 2 terms rather than a sum over N terms. This is a reflection of the important non-Abelian geometry of D-branes. 7) The coordinates arise from the lightest open string state\\nThis state has a 9-dimensional spatial index m and so naturally corresponds to a collective coordinate. The ket |ij denotes a string that begins on the ith D0-brane and ends on the jth, so the coordinate X m ij is a matrix in the N -dimensional D0-brane index space. Then P m ij is the conjugate momentum. The potential term corrects this overcounting. It is nonnegative (the minus sign canceling an i 2 ) and vanishes precisely when all the X m commute and so can be diagonalized. Further, there is a U (N ) gauge invariance, so the state is independent of basis and the low energy physics reduces to the quantum mechanics of the 9N eigenvalues, as appropriate for N particles in 9 + 1 dimensions. The potential term has the same form as the 4-gluon Yang-Mills interaction Tr(F mn F mn ). This is no accident, the two being related by T -duality. The omitted spin terms are such that the system has 16 linearly realized supersymmetries, generating 256 states for each D0-brane. Now let us take the DKPS limit, 8) which is g s → 0 while holding fixed all of The IIA string is M-theory compactified on a circle of radius R 10 = g s l s , with eleven-dimensional Planck length M = g This scaling of the energy has a simple pictorial interpretation, as in figure 2. For any finite R 10 we can find a frame in which the spatial radius takes an arbitrary fixed value R ′ 10 ; the cost is that the periodicity is no longer at equal times but at nearly null separations. The Lorentz factor is γ = R ′ 10 /R 10 and so\\nSince p ′ 10 = N/R ′ 10 is fixed, the DKPS limit is equivalent to holding fixed the energy in the boosted frame of figure 2b. Thus we have a new interpretation: 10), 11), 12) Summary: The matrix Hamiltonian (2 . 4) describes M-theory in a space compactified in a lightlike direction, in the sector with N units of compact momentum. This is defined as the limit of spacelike compactification, keeping all states that have finite energy in the limit. The eleven-dimensional Planck scale and the noncompact momenta are held fixed in the limit. §4. The Balasubramanian-Gopakumar-Larsen-Hyun-Kiem-Shin (BGLHKS) Limit\\nAn important refinement of the preceding picture comes when we consider that the particles carrying p 10 must gravitate. With matter in the periodic space one would not expect that the compactified direction remain precisely null. A perturbation towards timelike compactification is implausible, so likely any gravitational effect is in the direction of making the periodicity spacelike. Indeed, as a function of distance r from the source, the radius of the compact dimension is given in terms of the black 0-brane dilaton solution 13) M R 10 = e 2Φ(r)/3 Here and in the metric below numerical constants have been ignored. In the last line we have taken the DKPS limit, g s → 0 at fixed M . The striking feature is that the radius R 10 (r) has a nonvanishing limit, and only when r → ∞ does the radius of the compact dimension go to zero. Thus the geometry is only asymptotically as depicted in figure 2b. 14), 15) Let us develop somewhat further the nature of the resulting space. The first obvious separation occurs at r ≈ N 1/7 /M , where e Φ(r) is of order one. At larger r, R 10 (r) is less than M −1 and so the effective description is in terms of weakly coupled IIA string theory. At smaller r, R 10 (r) is greater than M −1 and so the effective description is in terms of M-theory.\\nTo go further examine the M-theory metric, obtained by lifting the black D0brane metric to M-theory in the coordinate system of figure 2b,\\nThe periodicity is\\nThe metric appears flat at large r, but this deceptive because the radius of the periodic dimension and so the local string tension are going to zero. The latter is\\nThe corresponding length scale l s (r\\nAt this point geometry can no longer make sense: the strings are larger than the system (in terms of the string metric, which is curved, the curvature becomes large in string units).\\nAnother key distance is r = N 1/9 M −1 , where R 10 (r) = r. At distances below this the metric (4 . 2) is no longer correct. Since generic sources will not be uniformly distributed in x 10 , the resulting metric will not be uniform either. This effect damps away when r > R 10 (r) (the space is then long in the r-direction and narrow in the x 10 -direction). Finally, when r < M −1 we would not expect any classical metric to be a good description. Collecting together these results gives 14), 15), 3)\\nWe have been treating the sources as pointlike. This is true, for example, for the ground state, which is a single graviton with N units of compact momentum. 4) For more distributed sources, the geometry below a given r would be smoothed out.\\nSummary: The matrix Hamiltonian (2 . 4) describes M-theory in the sector with N units of compact momentum, where the compactified dimension is asymptotically lightlike. That is, it describes all states whose metric is asymptotically of the form (4 . 2). §5\\nthen the kinetic term is canonically normalized and theX 4 interaction has a coefficient M 3 . The latter is the correct dimension for a Yang-Mills coupling-squared in 0 + 1 dimensions. The effective dimensionless coupling at an energy scale E is then are held fixed in the two limits. * ) Second, the IMSY limit is a statement about the physics as a function of energy scale. To relate this to the previous discussion, consider a standard observable, the expectation value of a product of local operators with some characteristic external energy E. If the reader is more comfortable in Euclidean space, he/she is free to analytically continue, since the quantum mechanics and the supergravity have a common global time (which in the supergravity is defined by the asymptotic translation invariance). A local operator in the QM is equivalent to some local disturbance in the supergravity. The only coordinate invariant local objects live at the boundary (coordinate transformations at the boundary are frozen by the specification of the asymptotic behavior), so by logic similar to the AdS/CFT case 17) we are led to the conclusion that these correspond to deformations of the boundary conditions. In this sense the QM 'lives' at the boundary at infinity, but it should be emphasized that it contains all states of the bulk theory. * ) Note that the IMSY g 2 YM differs from the value M 3 = g −1 s l 3 s given above because it is a dimensional quantity and the Hamiltonian has been rescaled.\\nThese boundary deformations do not change the D0-brane charge, and so correspond to boundary conditions that are invariant under translations in the periodic dimension: they depend on t ′ and x ′10 only in the combination t ′ + x ′10 , of the form exp iE ′ (t ′ + x ′10 ). Acting on these the wave operator reduces to\\nNote that the arbitrary R ′ 10 drops out after converting the SSS energy E ′ to the matrix QM energy E via eq. (3 . 2). From the relative scaling of the two terms in the wave operator it follows that the supergravity amplitude is dominated by the radial scale 18) r\\nThis is the holographic energy-radius relation. 19) In particular, the energy E = N 1/3 M corresponds to r = N 1/3 M −1 . At higher energies/larger radii the geometric picture breaks down but matrix QM perturbation theory is valid. At lower energies/smaller radii, QM perturbation theory breaks down but the supergravity description is valid. At successively lower energies the physics is described by the successive supergravity pictures of the previous section. It is not clear how to understand E < N −1/2 M , where the supergravity breaks down at short distance, but we will argue in the next section that things become simple again at E ∼ N −1 M . Note that the only threshold visible in the 't Hooft (spherical) limit is E ≈ N 1/3 M corresponding to r ≈ N 1/3 M −1 . Summary: The matrix Hamiltonian (2 . 4) is equivalent to M-theory in the sector with N units of compact momentum, where the compactified dimension is asymptotically lightlike. In the preceding discussion we were using this to define M-theory, but we can also use it in the other direction: low energy supergravity is the effective theory for the strongly coupled QM at E < N 1/3 M . §6. Discussion\\nLet us now consider the implications of the preceding results. One thing that I find particularly striking is the BGLHKS picture. We are conditioned to think of the g s → 0 limit as being strictly 10-dimensional. Now we see that in a sector of large but fixed D0-brane charge N , a finite bubble of 11-dimensional spacetime survives at N 1/9 M −1 < r < N 1/7 M −1 . This grows arbitrarily large at large N . Thus, the large-N matrix QM describes 11-dimensional bulk physics. The derivation appear to be quite reliable. On the QM side we have string theory with the coupling and the energy (in string units, above the BPS minimum) going to zero; we surely understand string theory in this regime. On the M-theory side, the geometry is quite smooth as g s → 0 in the indicated range of r. This is completely opposite to the mindset expressed in the introduction and figure 1, in that the nature of the 11-dimensional theory has been deduced from string perturbation theory. This is shown schematically in figure 3: the N -axis, rather than being perpendicular to the old parameter space, actually runs from the IIA vacuum toward the M-theory vacuum. Of course the key input is IIA/M-theory duality -this turns out to have a great deal more information than was initially evident. The IMSY result, or Maldacena duality for D0-branes, also appears to be a great deal of information from very little. It tells us about the behavior of a quantum mechanical system at strong coupling. Again, the key input is IIA/M-theory duality -with little additional information this implies the IMSY result. Now consider the BFSS matrix theory. 1) The statement above, that the large-N matrix QM describes 11-dimensional physics, would appear to be morally the same thing. However, the BFSS theory is more specific. * ) Given the above discussion, the natural way to define the 11-dimensional S-matrix would be to begin with N source D0-branes, to create the 11-dimensional bubble, and then to consider scattering of supergravity fields corresponding to local operators in the QM, aimed to intersect in the bubble. This is parallel to recent constructions in the AdS 5 case, with D3-branes replaced by D0-branes. 20), 21) The BFSS proposal is different, and more direct. Namely, scattering of matrix theory bound states is supposed to go directly over to the M-theory S-matrix in the large-N limit. The SSS picture suggests a derivation of this. 12) Consider gravitons (bound states) with D0-brane charges N 1 , N 2 , and so on. Scale the N i to infinity together, and take R ′ 10 to infinity at the same rate. The momenta p ′ 10,i = N i /R ′ 10 are then constant. The size of the box is going to infinity, in a frame in which the scattering event is held fixed. One might then expect that the limit would give the noncompact theory. For a limit of spacelike compactifications this would be expected, but lightlike * ) I would like to thank various members of the IAS seminar audience, particularly T. Banks, M. Douglas, H. Verlinde, and E. Witten, for making this point to me. compactification is surely more subtle, because the invariant radius of the compact dimension remains at zero. Indeed, the null compactification leads to specific effects that might complicate the limit. The first is zero tension strings, from membranes wrapping the periodic dimension. 22) The second is divergent fluctuations of modes with compact momentum zero. These arise because the effective 10-dimensional coupling in field theory would be g 2 11 /R 10 , which diverges in the lightlike limit. * ) In some field theory examples these effects dominate and the large-N limit does not give the noncompact theory. However, in the present case the BGLHKS effect seems to do away with potential problems. Note that the relevant distance scale p −1 and relevant energy E ∼ p 2 /N i are both quite small, where the graviton transverse momentum p is held fixed as N → ∞. (These do not satisfy the holographic relation (5 . 6) because they correspond to a different quantity in the QM theory.) At this distance the periodic dimension is very far from lightlike due to the gravitational effect of the D0-branes. Thus the wrapped membrane tension is large compared to the scale of the scattering, and so is the 10-dimensional Planck scale.\\nOne might still have the following concern. At very large r (the 't Hooft scale) the spacetime interpretation does break down as we have argued earlier, due to a string tension that vanishes asymptotically and also to a 10-dimensional coupling that diverges asymptotically. The gravitons come in from infinite r, and so begin and end in this 'non-spacetime' region. How, then, can we get the physics of flat 11dimensional spacetime? The point is that at these asymptotic distances the gravitons are propagating freely, at energies small compared to the characteristic energies of the QM. All that matters then is the metric on moduli space, and this is simply flat as a consequence of supersymmetry -it does not matter that at the 't Hooft radius we stop interpreting it as the moduli space of particles in spacetime and begin interpreting as the moduli space of the quantum mechanics. * * )\\nThus the potential objections to the Seiberg-Sen argument seem to be red herrings in this case. The BFSS matrix theory, like the IMSY duality, can indeed be deduced from perturbative string theory and IIA/M-theory duality. For the zero modes we can say that they cure their own problem, in that the BGLHKS effect is the expectation value of the gravitational zero mode.\\nLet me conclude with some comments about the relation between perturbative matrix theory amplitudes and supergravity or M-theory amplitudes. From the literature one might get the impression that these are expected to agree in general: that if an amplitude has the same dependence on the parameters in the two theories (M , N , r, and v) then its form and coefficient must be the same. I can confess to sloppy thinking in this regard in the past. But in fact there is no general reason that there should be agreement. The ranges of validity are entirely different, E/M < N −1 vs E/M > N 1/3 , and as in any effective field theory there is no general reason to expect amplitudes at different scales to have any simple relation. For example, the exact amplitude could involve a function of E 3 /M 3 N which asymptotes to one value at * ) Zero mode divergences in light-cone field theory are a long story. For discussions in the context of the lightlike limit see refs. 23), 24) . * * ) Again I would like to thank the IAS seminar audience for straightening me out.\\nhigh energy and to a different one at low energy. Of course some amplitudes are protected by supersymmetry nonrenormalization theorems and so must agree: if they did not it would falsify matrix theory and possibly IIA/M-theory duality as well. It is still interesting to compare calculations in the two limits, both as a way to infer new supersymmetric nonrenormalization theorems (since supersymmetry appears to be a strong restriction) and also because this is the route by which new dualities, such as the Maldacena duality, are sometimes discovered. §7. What Next?\\nThe conclusion is that the large-N matrix QM gives the M-theory limit of string theory. It is one of many nonperturbative constructions of string theory, since we can take a similar large-N limit in any of the Maldacena dualities. Certainly these constructions are not yet in simplest form: string theory is constructed in states with specific, and unusual, boundary conditions. There should be some general framework describing all states of string theory -spacetimes without boundaries are especially puzzling from the present viewpoint. This will likely involve some new ideas, and look rather different from the current descriptions.\\nStill, one way to try to proceed is to develop more fully the existing constructions. Much attention goes to the AdS 5 × S 5 and AdS 3 × S 3 cases because of their large symmetries, but the matrix QM/D0-brane case would seem to have advantages as well, due to its smaller number of degrees of freedom.\\nAs an example, it would be nice to have a direct test of the Maldacena duality, comparing two calculations at the same point in parameter space without the aid of nonrenormalization results. The simplest place to look is the D0-brane QM, just below the 't Hooft energy. The duality predicts that the gauge theory entropy is given by the nonextremal D0-brane entropy as\\nwith c a known constant. 25) The exponent 9/5 indicates the nontriviality of the dynamics. As the temperature is reduced the effect of the commutator potential is to freeze out degrees of freedom in a gradual way, but the precise exponent appears to be difficult to get by analytic means. However, this entropy should be accessible by Monte Carlo (I should emphasize that this is very different from the index-like path integrals that have been calculated, in that the thermal boundary conditions break supersymmetry.). In field theory we would need to renormalize to maintain the supersymmetry in the cutoff theory, but QM is superrenormalizable. Also, at this relatively high temperature precise supersymmetric cancellations are not so relevant, and high temperature methods may also simplify the fermionic determinant. There is unlikely to be enough dynamic range at reasonable values of N to pick out the 9/5 accurately, but for the Monte Carlo calculation to come close to the value of c obtained in a black hole calculation would be very striking.\\nOf course the IKKT matrix theory 26) should be even simpler than the BFSS model from this point of view, and it has been studied numerically. The question here is one of interpretation. The assumption in much of the literature appears to be that the IKKT model is a background-independent formulation of string theory, but an interpretation parallel to IMSY 3) seems more likely. The portion of the IKKT matrix integral where the eigenvalues are well-separated can be calculated perturbatively, but this breaks down when the eigenvalues become close. The natural extension of the IMSY analysis would be that there is an effective description of the latter part of the matrix integral in terms of supergravity in an asymptotically Dinstanton space. For a numerical test one would need to find a simple prediction parallel to the entropy (7 . 1).\\nAnother advantage of being in QM is that renormalization is not needed, so we can add arbitrary perturbations to the Hamiltonian and so study more general spaces.\\nA related advantage is that the wavefunction and operators have a direct meaning. Consider for example the size R of the D0-brane bound state. * ) In the BFSS paper 1) , two estimates are given,\\nfrom 't Hooft scaling and\\nfrom the quantum mechanics of the v 4 /r 7 potential. The relation between these was not resolved (though at least one of the authors [B] came to the conclusion that we will reach below). It was suggested that the 't Hooft result might be invalidated by IR divergences, but in fact one can show that it is rigorously correct. Start with the uncertainty principle, applied to one matrix element X 1 11 :\\n0|(X 1 11 ) 2 |0 0|(P 1 11 ) 2 |0 ≥ 1 . Applying this to all 9N 2 elements gives * * ) 0|Tr(X 1 X 1 )|0 0|Tr(P n P n )|0 ≥ 9N 4 . (7 . 5)\\nNow apply the virial theorem, relating the expectation value of the kinetic term to that of the potential. From 0|[X n P n , H]|0 = 0 one obtains for the expectation values −2K + 4P + F = 0, where K is the kinetic term, P is the potential, and F is the fermionic term (linear in X). On the other hand K + P + F = 0 by supersymmetry. Thus K : P : F = 1 : 1 : −2, and so\\nFor Hermitean matrices the Schwarz inequality gives |Tr(ABAB)| < Tr(AABB) < 1 2 [Tr(A 4 ) + Tr(B 4 )], and so 0|Tr(X 1 X 1 )|0 0|Tr(X 1 X 1 X 1 X 1 )|0 ≥ N 4 π 2 2M 6 . (7 . 8)\\nFor X 1 having an eigenvalue distribution of width R, the left-hand side is of order N 2 R 6 and so R ≥ O(N 1/3 M −1 ) . (7 . 9) This assumes that the expectation value of (X 1 ) 4 converges, as is indeed the case. This result, obtained from a Hamiltonian Born-Oppenheimer approximation in refs. 28) , can be nicely explained by an effective Hamiltonian analysis (this is adapted from an unpublished argument of L. Susskind, with comments from G. M. Graf). The dangerous direction is when the wavefunction separates into two blocks, along the flat directions. Here the relative coordinate is described by a free U (1) theory, so the Hamiltonian is just the Laplacian and the wavefunction is harmonic. For l = 0 the wavefunction falls as r −7 . The bound state has l = 2, because the relative spin plus motion gives an invariant, and so falls as r −9 . Then 0|Tr[(X 1 ) L ]|0 ∼ d 9 X |X| −18+L (7 . 10) converges for L < 9. Note that (7 . 9) is exactly the maximum radius in which the supergravity picture is valid. I therefore strongly expect that N 1/3 M −1 is the actual value, not just a lower bound.\\nHow does this relate to the smaller estimate (7 . 3)? The bound (7 . 9) comes primarily from the quantum fluctuations of the off-diagonal modes of X n , while the smaller estimate refers to the diagonal motion on the moduli space. For the purpose of the BFSS theory, one is interested only in very small energies. The off-diagonal modes have much higher excitation energies and so are frozen into their ground states; the associated zero-point fluctuations are then not observable. Roughly speaking, Tr(X n X n ) = Tr(X n X n ) off−diag./high energy + Tr(X n X n ) diag./low energy . (7 . 11)\\nThe off-diagonal part is much larger but is state-independent at low energy. In other words, Tr(X n X n ) in the low energy theory differs from the full Tr(X n X n ) by an additive renormalization. It seems hard to give a precise definition of the low energy part in terms of the underlying QM variables.\\nIn conclusion, to find a description of the flat 11-dimensional M-theory (or the 10-dimensional IIB theory) we somehow need, in the large-N limit, to identify a subtheory contained in the intersection of all the field theory constructions. To find a description of the general background we need something which contains the union of all the field theory constructions.\"\n },\n {\n \"title\": \"Fig. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"NFig. 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"2 )\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Space of string vacua. The cusps are limits in which a weakly coupled string description is possible (except for the M-theory limit). §2. The Douglas-Kabat-Pouliot-Shenker (DKPS) Limit Consider a state in the IIA string theory containing N D0-branes plus any collection of open and closed strings. * ) The energy of such a state is","content":"Fig. 1 .\n1Space of string vacua. The cusps are limits in which a weakly coupled string description is possible (except for the M-theory limit). §2. The Douglas-Kabat-Pouliot-Shenker (DKPS) Limit Consider a state in the IIA string theory containing N D0-branes plus any collection of open and closed strings. * ) The energy of such a state is"},{"id":"","caption":". The D0-brane charge is the number of units of compact momentum, N = p 10 R 10 . In terms of M-theory parameters, the DKPS limit takes R 10 → 0 while holding fixed all of p 10 R 10 , P , q , M , H = 1 M R10 (H − p 10 ) .(3 . 1)","content":". The D0-brane charge is the number of units of compact momentum, N = p 10 R 10 . In terms of M-theory parameters, the DKPS limit takes R 10 → 0 while holding fixed all of p 10 R 10 , P , q , M , H = 1 M R10 (H − p 10 ) .(3 . 1)"},{"id":"","caption":"The same space in two frames. a) Identified point (indicated by arrows) are at equal times and small spatial separation. b) Identified points are at fixed spatial distance and almost null separation.","content":"Fig. 2 .\n2The same space in two frames. a) Identified point (indicated by arrows) are at equal times and small spatial separation. b) Identified points are at fixed spatial distance and almost null separation."},{"id":"","caption":"Space of string vacua. At gs = 0 but large N the IIA string has 11-dimensional physics.","content":"NFig. 3 .\n3Space of string vacua. At gs = 0 but large N the IIA string has 11-dimensional physics."},{"id":"","caption":"Thus in this limit the IIA string theory becomes very simple, just the matrix quantum mechanics 9) (2 . 4). The string coupling is taken to zero, but because of the scaling of the energy the most relevant interaction does survive.Summary: The matrix Hamiltonian (2 . 4) describes the sector of IIA string theory with D0-brane-charge N , in the DKPS limit (2 . 3). §3. The Susskind-Seiberg-Sen (SSS) Limit","content":"Thus in this limit the IIA string theory becomes very simple, just the matrix quantum mechanics 9) (2 . 4). The string coupling is taken to zero, but because of the scaling of the energy the most relevant interaction does survive.Summary: The matrix Hamiltonian (2 . 4) describes the sector of IIA string theory with D0-brane-charge N , in the DKPS limit (2 . 3). §3. The Susskind-Seiberg-Sen (SSS) Limit3) \n\nIt follows from the energy (2 . 1) that all closed strings decouple, as do excited open \nstrings, leaving only massless open strings with \n\nH = \n1 \n2M \nTr(P 2 ) − \nM 5 \n16π 2 \n\nm,n \n\nTr([X m , X n ] 2 ) + spin term . \n(2 . 4)"},{"id":"","caption":". The Itzhaki-Maldacena-Sonnenschein-Yankielowicz (IMSY) Limit Now let us think about the matrix Hamiltonian (2 . 4) directly. If we rescale to new canonical variables","content":". The Itzhaki-Maldacena-Sonnenschein-Yankielowicz (IMSY) Limit Now let us think about the matrix Hamiltonian (2 . 4) directly. If we rescale to new canonical variables"},{"id":"","caption":"the N being from the usual 't Hooft counting.16) At E > N 1/3 M , the effective coupling is small and quantum mechanical perturbation theory is good. When E < N 1/3 M , perturbation theory breaks down -we have a strongly coupled problem. In the past this might have been a difficulty, but now that we are in the Age of Duality it presents an opportunity, to find a weakly coupled description. Indeed, this is what IMSY do. 3) By essentially the DKPS plus BGLHKS arguments, they show that the low energy physics of the matrix Hamiltonian is given by IIA/M-theory in the supergravity background(4 . 2).In making the connection, there are two points that should be clarified. First, the IMSY limit sounds different, in that it holds fixed3)","content":"2 )\n2the N being from the usual 't Hooft counting.16) At E > N 1/3 M , the effective coupling is small and quantum mechanical perturbation theory is good. When E < N 1/3 M , perturbation theory breaks down -we have a strongly coupled problem. In the past this might have been a difficulty, but now that we are in the Age of Duality it presents an opportunity, to find a weakly coupled description. Indeed, this is what IMSY do. 3) By essentially the DKPS plus BGLHKS arguments, they show that the low energy physics of the matrix Hamiltonian is given by IIA/M-theory in the supergravity background(4 . 2).In making the connection, there are two points that should be clarified. First, the IMSY limit sounds different, in that it holds fixed3)"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Space of string vacua. The cusps are limits in which a weakly coupled string description is possible (except for the M-theory limit). §2. The Douglas-Kabat-Pouliot-Shenker (DKPS) Limit Consider a state in the IIA string theory containing N D0-branes plus any collection of open and closed strings. * ) The energy of such a state is\",\n \"content\": \"Fig. 1 .\\n1Space of string vacua. The cusps are limits in which a weakly coupled string description is possible (except for the M-theory limit). §2. The Douglas-Kabat-Pouliot-Shenker (DKPS) Limit Consider a state in the IIA string theory containing N D0-branes plus any collection of open and closed strings. * ) The energy of such a state is\"\n },\n {\n \"id\": \"\",\n \"caption\": \". The D0-brane charge is the number of units of compact momentum, N = p 10 R 10 . In terms of M-theory parameters, the DKPS limit takes R 10 → 0 while holding fixed all of p 10 R 10 , P , q , M , H = 1 M R10 (H − p 10 ) .(3 . 1)\",\n \"content\": \". The D0-brane charge is the number of units of compact momentum, N = p 10 R 10 . In terms of M-theory parameters, the DKPS limit takes R 10 → 0 while holding fixed all of p 10 R 10 , P , q , M , H = 1 M R10 (H − p 10 ) .(3 . 1)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The same space in two frames. a) Identified point (indicated by arrows) are at equal times and small spatial separation. b) Identified points are at fixed spatial distance and almost null separation.\",\n \"content\": \"Fig. 2 .\\n2The same space in two frames. a) Identified point (indicated by arrows) are at equal times and small spatial separation. b) Identified points are at fixed spatial distance and almost null separation.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Space of string vacua. At gs = 0 but large N the IIA string has 11-dimensional physics.\",\n \"content\": \"NFig. 3 .\\n3Space of string vacua. At gs = 0 but large N the IIA string has 11-dimensional physics.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Thus in this limit the IIA string theory becomes very simple, just the matrix quantum mechanics 9) (2 . 4). The string coupling is taken to zero, but because of the scaling of the energy the most relevant interaction does survive.Summary: The matrix Hamiltonian (2 . 4) describes the sector of IIA string theory with D0-brane-charge N , in the DKPS limit (2 . 3). §3. The Susskind-Seiberg-Sen (SSS) Limit\",\n \"content\": \"Thus in this limit the IIA string theory becomes very simple, just the matrix quantum mechanics 9) (2 . 4). The string coupling is taken to zero, but because of the scaling of the energy the most relevant interaction does survive.Summary: The matrix Hamiltonian (2 . 4) describes the sector of IIA string theory with D0-brane-charge N , in the DKPS limit (2 . 3). §3. The Susskind-Seiberg-Sen (SSS) Limit3) \\n\\nIt follows from the energy (2 . 1) that all closed strings decouple, as do excited open \\nstrings, leaving only massless open strings with \\n\\nH = \\n1 \\n2M \\nTr(P 2 ) − \\nM 5 \\n16π 2 \\n\\nm,n \\n\\nTr([X m , X n ] 2 ) + spin term . \\n(2 . 4)\"\n },\n {\n \"id\": \"\",\n \"caption\": \". The Itzhaki-Maldacena-Sonnenschein-Yankielowicz (IMSY) Limit Now let us think about the matrix Hamiltonian (2 . 4) directly. If we rescale to new canonical variables\",\n \"content\": \". The Itzhaki-Maldacena-Sonnenschein-Yankielowicz (IMSY) Limit Now let us think about the matrix Hamiltonian (2 . 4) directly. If we rescale to new canonical variables\"\n },\n {\n \"id\": \"\",\n \"caption\": \"the N being from the usual 't Hooft counting.16) At E > N 1/3 M , the effective coupling is small and quantum mechanical perturbation theory is good. When E < N 1/3 M , perturbation theory breaks down -we have a strongly coupled problem. In the past this might have been a difficulty, but now that we are in the Age of Duality it presents an opportunity, to find a weakly coupled description. Indeed, this is what IMSY do. 3) By essentially the DKPS plus BGLHKS arguments, they show that the low energy physics of the matrix Hamiltonian is given by IIA/M-theory in the supergravity background(4 . 2).In making the connection, there are two points that should be clarified. First, the IMSY limit sounds different, in that it holds fixed3)\",\n \"content\": \"2 )\\n2the N being from the usual 't Hooft counting.16) At E > N 1/3 M , the effective coupling is small and quantum mechanical perturbation theory is good. When E < N 1/3 M , perturbation theory breaks down -we have a strongly coupled problem. In the past this might have been a difficulty, but now that we are in the Age of Duality it presents an opportunity, to find a weakly coupled description. Indeed, this is what IMSY do. 3) By essentially the DKPS plus BGLHKS arguments, they show that the low energy physics of the matrix Hamiltonian is given by IIA/M-theory in the supergravity background(4 . 2).In making the connection, there are two points that should be clarified. First, the IMSY limit sounds different, in that it holds fixed3)\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". T Banks, W Fischler, S H Shenker, L Susskind, hep-th/9610043Phys. Rev. 55T. Banks, W. Fischler, S. H. Shenker, and L. Susskind, Phys. Rev. D55 (1961) 5112, hep-th/9610043.","doi":"hep-th/9610043"},{"text":". J Maldacena, hep-th/9711200Adv. Theor. Math. Phys. 2J. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231, hep-th/9711200.","doi":"hep-th/9711200"},{"text":". N Itzhaki, J M Maldacena, J Sonnenschein, S Yankielowicz, hep-th/9802042Phys. Rev. 58N. Itzhaki, J. M. Maldacena, J. Sonnenschein, and S. Yankielowicz, Phys. Rev. D58 (1998) 046004, hep-th/9802042.","doi":"hep-th/9802042"},{"text":". P K Townsend, ; E Witten, hep-th/9503124Phys. Lett. 350Nucl. Phys.P. K. Townsend, Phys. Lett. B350 (1995) 184, hep-th/9501068. E. Witten, Nucl. Phys. B443 (1995) 85, hep-th/9503124.","doi":"hep-th/9503124"},{"text":"ITP seminar. S Shenker, S. P. de Alwishep-th/9806188P. J. Silva. 5944029Phys. Rev.S. Shenker, ITP seminar, March 1998. S. P. de Alwis, Phys. Rev. D59 (1999) 044029, hep-th/9806178. P. J. Silva, hep-th/9806188.","doi":"hep-th/9806188"},{"text":". V Sahakian, E Martinec, hep-th/9901033Phys. Rev. 59124005V. Sahakian and E. Martinec, Phys. Rev. D59 (1999) 124005, hep-th/9810224. I. Chepelev, hep-th/9901033.","doi":"hep-th/9901033"},{"text":". J Polchinski, hep-th/9611050J. Polchinski, hep-th/9611050.","doi":"hep-th/9611050"},{"text":"J Polchinski, String Theory, Vols. I and II. CambridgeCambridge University PressJ. Polchinski, String Theory, Vols. I and II (Cambridge University Press, Cambridge, 1998).","doi":""},{"text":". E Witten, hep-th/9510135Nucl. Phys. 460E. Witten, Nucl. Phys. B460 (1996) 335, hep-th/9510135.","doi":"hep-th/9510135"},{"text":". M R Douglas, D Kabat, P Pouliot, S H Shenker, hep-th/9608024Nucl. Phys. 485M. R. Douglas, D. Kabat, P. Pouliot, and S. H. Shenker, Nucl. Phys. B485 (1997) 85, hep-th/9608024.","doi":"hep-th/9608024"},{"text":". M Claudson, M B Halpern, ; R Flume, Annals Phys. 250189Nucl. Phys.M. Claudson and M. B. Halpern, Nucl. Phys. B250 689, 1985. R. Flume, Annals Phys. 164 189, 1985.","doi":""},{"text":". M Baake, M Reinicke, V Rittenberg, J. Math. Phys. 261070M. Baake, M. Reinicke, and V. Rittenberg, J. Math. Phys. 26 1070, 1985.","doi":""},{"text":". L Susskind, hep-th/9704080L. Susskind, hep-th/9704080.","doi":"hep-th/9704080"},{"text":". K Becker, M Becker, J Polchinski, A Tseytlin, hep- th/9706072Phys. Rev. 56K. Becker, M. Becker, J. Polchinski, and A. Tseytlin, Phys. Rev. D56 (1997) 3174, hep- th/9706072.","doi":"hep- th/9706072"},{"text":". N Seiberg, hep-th/9709220Adv. Theor. Math. Phys. 79Phys. Rev. Lett.N. Seiberg, Phys. Rev. Lett. 79 (1997) 3577, hep-th/9710009. A. Sen, Adv. Theor. Math. Phys. 2 (1998) 51, hep-th/9709220.","doi":"hep-th/9709220"},{"text":". G T Horowitz, A Strominger, Nucl. Phys. 360197G. T. Horowitz and A. Strominger, Nucl. Phys. B360 (1991) 197.","doi":""},{"text":". V Balasubramanian, R Gopakumar, F Larsen, hep- th/9712077Nucl. Phys. 526V. Balasubramanian, R. Gopakumar, and F. Larsen, Nucl. Phys. B526 (1998) 415, hep- th/9712077.","doi":"hep- th/9712077"},{"text":". S Hyun, Y Kiem, H Shin ; Hyun, Y Kiem, hep-th/9805136Phys. Lett. 5726003Phys. Rev.S. Hyun, Y. Kiem, and H. Shin, Phys. Rev. D57 (1998) 4856, hep-th/9712021. S. Hyun, Phys. Lett. B441 (1998) 116, hep-th/9802026. S. Hyun and Y. Kiem, Phys. Rev. D59 (1999) 026003, hep-th/9805136.","doi":"hep-th/9805136"},{"text":". G Hooft, Nucl. Phys. 72461G. 't Hooft, Nucl. Phys. B72 (1974) 461.","doi":""},{"text":". S S Gubser, I R Klebanov, A M Polyakov, hep-th/9802150Adv. Theor. Math. Phys. 428Phys. Lett.S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Phys. Lett. B428 (1998) 105, hep- th/9802109. E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253, hep-th/9802150.","doi":"hep-th/9802150"},{"text":". A Peet, J Polchinski, hep-th/9809022Phys. Rev. 59A. Peet and J. Polchinski, Phys. Rev. D59 (1999) 065006, hep-th/9809022.","doi":"hep-th/9809022"},{"text":". L Susskind, E Witten, hep-th/9805114L. Susskind and E. Witten, hep-th/9805114.","doi":"hep-th/9805114"},{"text":"J Polchinski, hep-th/9902182hep-th/9901076. L. Susskind, hep-th/9901079. J. Polchinski and L. Susskind. J. Polchinski, hep-th/9901076. L. Susskind, hep-th/9901079. J. Polchinski and L. Susskind, hep-th/9902182.","doi":"hep-th/9902182"},{"text":"V Balasubramanian, S B Giddings, A Lawrence, hep-th/9903048hep-th/9902052. S. B. Giddings. V. Balasubramanian, S. B. Giddings, and A. Lawrence, hep-th/9902052. S. B. Giddings, hep-th/9903048.","doi":"hep-th/9903048"},{"text":". M R Douglas, H Ooguri, hep-th/9710178Phys. Lett. 425M. R. Douglas and H. Ooguri, Phys. Lett. B425 (1998) 71, hep-th/9710178.","doi":"hep-th/9710178"},{"text":". H W L Naus, H J Pirner, T J Fields, J P Vary, hep-th/9704135Phys. Rev. 56H. W. L. Naus, H. J. Pirner, T. J. Fields, J. P. Vary, Phys. Rev. D56 (1997) 8062, hep-th/9704135.","doi":"hep-th/9704135"},{"text":". S Hellerman, J Polchinski, hep-th/9711037S. Hellerman and J. Polchinski, hep-th/9711037.","doi":"hep-th/9711037"},{"text":". I R Klebanov, A A Tseytlin, hep-th/9604089Nucl. Phys. 475I. R. Klebanov and A. A. Tseytlin, Nucl. Phys. B475 (1996) 164, hep-th/9604089.","doi":"hep-th/9604089"},{"text":". N Ishibashi, H Kawai, Y Kitazawa, A Tsuchiya, hep-th/9612115Nucl. Phys. 498N. Ishibashi, H. Kawai, Y. Kitazawa, and A. Tsuchiya, Nucl. Phys. B498 (1997) 467, hep-th/9612115.","doi":"hep-th/9612115"},{"text":"M Li, hep-th/9902073hep-th/9901158. D. Kabat and G. Lifschytz. M. Li, hep-th/9901158. D. Kabat and G. Lifschytz, hep-th/9902073.","doi":"hep-th/9902073"},{"text":"M B Halpern, C Schwarz, ; J Frohlich, G M Graf, D Hasler, J Hoppe, S T Yau, hep-th/9712133hep-th/9904182 and references therein. 13M. B. Halpern and C. Schwarz, Int. J. Mod. Phys. A13 (1998) 4367, hep-th/9712133. J. Frohlich, G. M. Graf, D. Hasler, J. Hoppe, and S. T. Yau, hep-th/9904182 and references therein.","doi":"hep-th/9712133"}],"string":"[\n {\n \"text\": \". T Banks, W Fischler, S H Shenker, L Susskind, hep-th/9610043Phys. Rev. 55T. Banks, W. Fischler, S. H. Shenker, and L. Susskind, Phys. Rev. D55 (1961) 5112, hep-th/9610043.\",\n \"doi\": \"hep-th/9610043\"\n },\n {\n \"text\": \". J Maldacena, hep-th/9711200Adv. Theor. Math. Phys. 2J. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231, hep-th/9711200.\",\n \"doi\": \"hep-th/9711200\"\n },\n {\n \"text\": \". N Itzhaki, J M Maldacena, J Sonnenschein, S Yankielowicz, hep-th/9802042Phys. Rev. 58N. Itzhaki, J. M. Maldacena, J. Sonnenschein, and S. Yankielowicz, Phys. Rev. D58 (1998) 046004, hep-th/9802042.\",\n \"doi\": \"hep-th/9802042\"\n },\n {\n \"text\": \". P K Townsend, ; E Witten, hep-th/9503124Phys. Lett. 350Nucl. Phys.P. K. Townsend, Phys. Lett. B350 (1995) 184, hep-th/9501068. E. Witten, Nucl. Phys. B443 (1995) 85, hep-th/9503124.\",\n \"doi\": \"hep-th/9503124\"\n },\n {\n \"text\": \"ITP seminar. S Shenker, S. P. de Alwishep-th/9806188P. J. Silva. 5944029Phys. Rev.S. Shenker, ITP seminar, March 1998. S. P. de Alwis, Phys. Rev. D59 (1999) 044029, hep-th/9806178. P. J. Silva, hep-th/9806188.\",\n \"doi\": \"hep-th/9806188\"\n },\n {\n \"text\": \". V Sahakian, E Martinec, hep-th/9901033Phys. Rev. 59124005V. Sahakian and E. Martinec, Phys. Rev. D59 (1999) 124005, hep-th/9810224. I. Chepelev, hep-th/9901033.\",\n \"doi\": \"hep-th/9901033\"\n },\n {\n \"text\": \". J Polchinski, hep-th/9611050J. Polchinski, hep-th/9611050.\",\n \"doi\": \"hep-th/9611050\"\n },\n {\n \"text\": \"J Polchinski, String Theory, Vols. I and II. CambridgeCambridge University PressJ. Polchinski, String Theory, Vols. I and II (Cambridge University Press, Cambridge, 1998).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Witten, hep-th/9510135Nucl. Phys. 460E. Witten, Nucl. Phys. B460 (1996) 335, hep-th/9510135.\",\n \"doi\": \"hep-th/9510135\"\n },\n {\n \"text\": \". M R Douglas, D Kabat, P Pouliot, S H Shenker, hep-th/9608024Nucl. Phys. 485M. R. Douglas, D. Kabat, P. Pouliot, and S. H. Shenker, Nucl. Phys. B485 (1997) 85, hep-th/9608024.\",\n \"doi\": \"hep-th/9608024\"\n },\n {\n \"text\": \". M Claudson, M B Halpern, ; R Flume, Annals Phys. 250189Nucl. Phys.M. Claudson and M. B. Halpern, Nucl. Phys. B250 689, 1985. R. Flume, Annals Phys. 164 189, 1985.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Baake, M Reinicke, V Rittenberg, J. Math. Phys. 261070M. Baake, M. Reinicke, and V. Rittenberg, J. Math. Phys. 26 1070, 1985.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Susskind, hep-th/9704080L. Susskind, hep-th/9704080.\",\n \"doi\": \"hep-th/9704080\"\n },\n {\n \"text\": \". K Becker, M Becker, J Polchinski, A Tseytlin, hep- th/9706072Phys. Rev. 56K. Becker, M. Becker, J. Polchinski, and A. Tseytlin, Phys. Rev. D56 (1997) 3174, hep- th/9706072.\",\n \"doi\": \"hep- th/9706072\"\n },\n {\n \"text\": \". N Seiberg, hep-th/9709220Adv. Theor. Math. Phys. 79Phys. Rev. Lett.N. Seiberg, Phys. Rev. Lett. 79 (1997) 3577, hep-th/9710009. A. Sen, Adv. Theor. Math. Phys. 2 (1998) 51, hep-th/9709220.\",\n \"doi\": \"hep-th/9709220\"\n },\n {\n \"text\": \". G T Horowitz, A Strominger, Nucl. Phys. 360197G. T. Horowitz and A. Strominger, Nucl. Phys. B360 (1991) 197.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Balasubramanian, R Gopakumar, F Larsen, hep- th/9712077Nucl. Phys. 526V. Balasubramanian, R. Gopakumar, and F. Larsen, Nucl. Phys. B526 (1998) 415, hep- th/9712077.\",\n \"doi\": \"hep- th/9712077\"\n },\n {\n \"text\": \". S Hyun, Y Kiem, H Shin ; Hyun, Y Kiem, hep-th/9805136Phys. Lett. 5726003Phys. Rev.S. Hyun, Y. Kiem, and H. Shin, Phys. Rev. D57 (1998) 4856, hep-th/9712021. S. Hyun, Phys. Lett. B441 (1998) 116, hep-th/9802026. S. Hyun and Y. Kiem, Phys. Rev. D59 (1999) 026003, hep-th/9805136.\",\n \"doi\": \"hep-th/9805136\"\n },\n {\n \"text\": \". G Hooft, Nucl. Phys. 72461G. 't Hooft, Nucl. Phys. B72 (1974) 461.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S S Gubser, I R Klebanov, A M Polyakov, hep-th/9802150Adv. Theor. Math. Phys. 428Phys. Lett.S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Phys. Lett. B428 (1998) 105, hep- th/9802109. E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253, hep-th/9802150.\",\n \"doi\": \"hep-th/9802150\"\n },\n {\n \"text\": \". A Peet, J Polchinski, hep-th/9809022Phys. Rev. 59A. Peet and J. Polchinski, Phys. Rev. D59 (1999) 065006, hep-th/9809022.\",\n \"doi\": \"hep-th/9809022\"\n },\n {\n \"text\": \". L Susskind, E Witten, hep-th/9805114L. Susskind and E. Witten, hep-th/9805114.\",\n \"doi\": \"hep-th/9805114\"\n },\n {\n \"text\": \"J Polchinski, hep-th/9902182hep-th/9901076. L. Susskind, hep-th/9901079. J. Polchinski and L. Susskind. J. Polchinski, hep-th/9901076. L. Susskind, hep-th/9901079. J. Polchinski and L. Susskind, hep-th/9902182.\",\n \"doi\": \"hep-th/9902182\"\n },\n {\n \"text\": \"V Balasubramanian, S B Giddings, A Lawrence, hep-th/9903048hep-th/9902052. S. B. Giddings. V. Balasubramanian, S. B. Giddings, and A. Lawrence, hep-th/9902052. S. B. Giddings, hep-th/9903048.\",\n \"doi\": \"hep-th/9903048\"\n },\n {\n \"text\": \". M R Douglas, H Ooguri, hep-th/9710178Phys. Lett. 425M. R. Douglas and H. Ooguri, Phys. Lett. B425 (1998) 71, hep-th/9710178.\",\n \"doi\": \"hep-th/9710178\"\n },\n {\n \"text\": \". H W L Naus, H J Pirner, T J Fields, J P Vary, hep-th/9704135Phys. Rev. 56H. W. L. Naus, H. J. Pirner, T. J. Fields, J. P. Vary, Phys. Rev. D56 (1997) 8062, hep-th/9704135.\",\n \"doi\": \"hep-th/9704135\"\n },\n {\n \"text\": \". S Hellerman, J Polchinski, hep-th/9711037S. Hellerman and J. Polchinski, hep-th/9711037.\",\n \"doi\": \"hep-th/9711037\"\n },\n {\n \"text\": \". I R Klebanov, A A Tseytlin, hep-th/9604089Nucl. Phys. 475I. R. Klebanov and A. A. Tseytlin, Nucl. Phys. B475 (1996) 164, hep-th/9604089.\",\n \"doi\": \"hep-th/9604089\"\n },\n {\n \"text\": \". N Ishibashi, H Kawai, Y Kitazawa, A Tsuchiya, hep-th/9612115Nucl. Phys. 498N. Ishibashi, H. Kawai, Y. Kitazawa, and A. Tsuchiya, Nucl. Phys. B498 (1997) 467, hep-th/9612115.\",\n \"doi\": \"hep-th/9612115\"\n },\n {\n \"text\": \"M Li, hep-th/9902073hep-th/9901158. D. Kabat and G. Lifschytz. M. Li, hep-th/9901158. D. Kabat and G. Lifschytz, hep-th/9902073.\",\n \"doi\": \"hep-th/9902073\"\n },\n {\n \"text\": \"M B Halpern, C Schwarz, ; J Frohlich, G M Graf, D Hasler, J Hoppe, S T Yau, hep-th/9712133hep-th/9904182 and references therein. 13M. B. Halpern and C. Schwarz, Int. J. Mod. Phys. A13 (1998) 4367, hep-th/9712133. J. Frohlich, G. M. Graf, D. Hasler, J. Hoppe, and S. T. Yau, hep-th/9904182 and references therein.\",\n \"doi\": \"hep-th/9712133\"\n }\n]","isConversational":false}}},{"rowIdx":496,"cells":{"text":{"kind":"string","value":"\nMade-to-measure N -body systems\n18 September 2018\n\nD Syer \nCanadian Institute for Theoretical Astrophysics\nMcLennan Labs\n60 St. George StreetM5S 1A7TorontoOntarioCanada\n\nMax-Planck-Institut für Astrophysik\nKarl-Schwarzschild-Straße 185748Garching-bei-MünchenGermany\n\nS Tremaine \nCanadian Institute for Theoretical Astrophysics\nMcLennan Labs\n60 St. George StreetM5S 1A7TorontoOntarioCanada\n\nMade-to-measure N -body systems\n\nMon. Not. R. Astron. Soc\n00018 September 2018arXiv:astro-ph/9605061v1 13 (MN plain T E X macros v1.6)methods: numerical -galaxies: kinematics and dynamics\nWe describe an algorithm for constructing N -body realisations of equilibrium stellar systems. The algorithm complements existing orbit-based modelling techniques using linear programming or other optimization algorithms. The equilibria are constructed by integrating an N -body system while slowly adjusting the masses of the particles until the time-averaged density field and other observables converge to a prescribed value. The procedure can be arranged to maximise a linear combination of the entropy of the system and the χ 2 statistic for the observables. The equilibria so produced may be useful as initial conditions for N -body simulations or for modelling observations of individual galaxies.\n\nINTRODUCTION\n\nOne of the central problems in stellar dynamics is to construct made-to-measure stellar systems. For example, (i) when modelling observations of an elliptical galaxy we wish to find a phase-space distribution function f (r, v) (hereafter DF) that solves Poisson's equation and the collisionless Boltzman equation and reproduces the observed surfacebrightness distribution, rotation curve, velocity-dispersion profile, etc. (in the sense of minimising χ 2 , the mean-square deviation between the observations and the model); (ii) when conducting simulations we wish to construct initial states that are N -body realisations of equilibrium stellar systems with given density profile, rotation curve, bulge/disc ratio, etc.\n\nExisting methods for constructing made-to-meaure stellar systems can be classified as follows:\n\n• DF-based methods, which solve directly for the DF f (r, v). These generally require that all the integrals of motion are known explicitly (e.g. if the potential is spherical or has Stäckel form) or that the DF is assumed to depend only on known analytic integrals. These include methods that fit observations to few-parameter models with analytic DFs such as spherical Michie or King models; a difficulty with such methods is that the dependence of the derived properties of the galaxy on the choice of the model can be large and uncertain (e.g. Merritt & Tremblay 1994). A more flexible DF-based method is described by Dejonghe (1989), who expands the DF for a spherical system in a truncated series of basis functions and minimises χ 2 subject to the constraint that the DF is positive, using quadratic programming. DF-based methods can be applied to axisymmetric galaxies, and are completely general so long as the DF does not depend on a third integral (e.g. Qian et al . 1995, Kuijken 1995. • Moment-based methods, which find solutions of the Jeans equations (or higher-order velocity moments of the collisionless Boltzmann equation) that minimise χ 2 . This method was used by Binney & Mamon (1982) to construct spherical models of M87 and has been applied to axisymmetric models by Binney et al. (1990), van der Marel et al. (1994), and Magorrian & Binney (1994). A drawback is that this procedure does not guarantee that there is a positive-definite DF with the required velocity moments. • Orbit-based methods (Schwarzschild 1979, Schwarzschild 1993 compute the density distribution of a large library of orbits in a fixed potential, and then determine the weight each orbit must have in order to reproduce the desired final state. In this method the DF or other integrals of motion are not explicitly required, although the DF can be regarded as a sum of deltafunctions on the phase-space surfaces covered by the orbits. Orbit-based methods are generally ill-conditioned. The ill-conditioning can be removed by iterating from a smooth initial guess for the orbit weights using the Richardson-Lucy method (Newton & Binney 1984), but the final weights will then depend in a complicated way on the initial guess. A better procedure is to minimise χ 2 minus a profit function that measures the smoothness of the distribution of orbit weights. The profit function may be an entropy (Richstone & Tremaine 1988) or any other function that is large when the DF is smooth (Merritt 1993). The most flexible of these are orbit-based methods, since they do not require that the integrals of motion are known and the approximation they provide to the DF is known to be positive if the orbit weights are positive.\n\nThe goal of this paper is to introduce a novel class of methods for constructing made-to-measure N -body realisations of stellar systems. In the classification above, these methods might be called 'particle-based'; they work by sculpting an initial N -body system until it matches the prescribed density field and other observables.\n\n\nTHE ALGORITHM\n\nIn most of our discussion we restrict ourselves to constructing stellar systems in a fixed potential Φ(r). We suppose that Φ admits a collisionless equilibrium configuration specified by a distribution function in phase space f (r, v). Thus f satisfies the time-independent collisionless Boltzmann equation:\nv · ∂f ∂r − ∇Φ · ∂f ∂v = 0.(1)\nAn 'observable' of the stellar system is a quantity of the form\nYj = Kj (z)f (z) d 6 z,(2)\nwhere z = (r, v) and Kj is a known kernel. Suitable observables include the surface or volume density at a given point, the surface density times the mean line-of-sight velocity, the surface density times any moment of the line-of-sight velocity, etc. Now consider a system of N particles having weights wi and phase-space positions zi(t) (i ≤ N ). The observables of this system are\nyj(t) = N i=1 wiKj [zi(t)].(3)\nOur goal is the following: given a set of distinct observables Yj, j = 1, . . . , J, construct a system of N particles orbiting in the potential Φ(r) whose time-averaged observables yj(t) are equal to Yj. We hope that this paper will stimulate interest in seeking better particle-based algorithms than the one we describe.\n\n\nThe force of change\n\nThe heart of the algorithm is a prescription for changing the weights {wi(t)} as the particles proceed along their fixed orbits in the potential Φ(r). The prescription is similar to that employed by Syer & Tremaine (1995) in a different context. It consists of applying gentle pressure on wi according to the value of ∆j ≡ yj(t)/Yj − 1: if ∆j < 0 then increase wi, and if ∆j > 0 then decrease wi. More precisely we let\ndwi(t) dt = −ǫwi(t) J j=1 Kj [zi(t)] Zj ∆j (t).(4)\nwhere ǫ is small and positive, and Z is so far arbitrary. The factor wi on the right side ensures that dwi/dt → 0 as wi → 0 so that wi cannot become negative. The factor Kj /Zj ensures that the difference ∆j changes the weight wi only if particle i is contributing to observable j. Equation (4) is closely related to Lucy's (1974) method for solving integral equations with noisy data. Some insight into the solutions of equation (4) is offered by the following argument. Since ǫ is small the weights wi change only over many orbits; thus we may orbit average:\ndwi(t) dt = −ǫwi(t) J j=1 Kji Zj ∆j .(5)\nHere · denotes the time-average over an interval that is much longer than a typical orbital period but much less than ǫ −1 periods, and Kji is shorthand for the time-independent quantity Kj[zi(t)] . We have also assumed that the fluctuations in Kj [zi(t)] and ∆j(t) are not correlated, which is plausible if many particles contribute to ∆j. We now define\nΘ k (t) ≡ Y k Z k 1/2 ∆ k ,(6)\nwhich obeys the differential equation\ndΘ k dt = −ǫ ij wi(t) K ki Kji (Y k Z k YjZj ) 1/2 Θj .(7)\nWhen we are close to convergence (|Θ| ≪ 1), the behaviour of the right side is dominated by changes in Θj rather than changes in wi, so we may replace wi by a constant, w 0 i . Then the vector Θ satisfies the matrix equation\ndΘ dt = −ǫA · Θ,(8)\nwhere the matrix A has components\nA kj = N i=1 w 0 i K ki Kji (Y k Z k YjZj ) 1/2 .(9)\nThe solutions to equation (8) have the form\nΘ k = J m=1 a km exp(−λmt),(10)\nwhere the eigenvalues {λm} are solutions of the equation\ndet(ǫA − λI) = 0.(11)\nSince A is positive-definite by construction (i.e. x t · A · x > 0 for all x), all of its eigenvalues are positive so λm > 0. This argument suggests that all observables converge to the desired values (| ∆ | → 0) on O(ǫ −1 ) orbital periods, if ǫ is sufficiently small and we start close to the correct final state. For comparison, orbit-based methods evaluate and store the entire matrix Kji (by following the orbits for a fixed time that is much longer than the orbital period; the matrix K is often called the \"orbit library\"), then solve the matrix equation\nYj = N i=1 Kji wi;(12)\nif N > J the matrix equation is ill-conditioned and must be solved subject to a constraint that maximizes some profit function such as entropy. The storage needed by particlebased methods is O(N J) whereas the storage needed by particle-based methods is only O(N ).\n\n\nThe kernel\n\nSuppose that the observable yr is the density at r. What is the appropriate kernel Kr(r ′ , v ′ )? One approach would be to use a smooth kernel (à la SPH) with\nKr(r ′ , v ′ ) = W (r − r ′ , h),(13)\nwhere W is some smooth function, and h is the resolution length (possibly a function of r); W should be normalised so that W (r, h) d 3 r = 1. If h is too large resolution is lost, while if h is too small the observables yj fluctuate strongly because too few particles contribute to a given observable. The smooth-kernel approach is expensive because nearest neighbours have to be found at each timestep and we have therefore adopted a different method. We first divide the coordinate space into bins. Then we set Kr(r ′ , v ′ ) to zero if r and r ′ are not in the same bin, and equal to the inverse of the volume of the bin otherwise. Obviously Kr 1 and Kr 2 are the same if r1 and r2 are in the same bin, so there can be at most one density observable per bin. We are still free to choose the parameter Zj in (4); a simple choice is to set Zj equal to the inverse of the volume of the bin. Thus Kj (zi)/Zj is unity if ri is in bin j, and zero otherwise.\n\nAn improvement to this simple scheme, which we employ here, is to borrow from the smooth-kernel approach and to smear each particle into neighbouring bins. Each particle is replaced by a gaussian distribution at the bin centre closest to the particle position, with dispersion equal to half the bin width. Neighbouring bins are assigned a weight corresponding to the integral of the gaussian over those bins (normalized so that the total contribution over all bins is wi). The bin closest to the particle position thus contains about two thirds of its weight. This procedure corresponds nicely to the phenomenon of seeing in the case of a projected observable.\n\n\nResolution and smoothing\n\nSuppose that we divide the d-dimensional coordinate space into M d bins, and each density observable corresponds to the mass per unit volume in a single bin. Then at any given time there are on average N/M d particles contributing to each observable, and the rms statistical fluctuations in ∆j will be of order δ ∼ (M d /N ) 1/2 . These fluctuations can be kept small if d = 1 (spherical or one-dimensional systems); for example if M = 30 and N = 1000 we have δ ∼ 0.2. However, for triaxial systems (d = 3) the fluctuations will be much larger for reasonable values of M and N .\n\nTo improve this situation we employ a form of temporal smoothing which effectively boosts N without any need for extra storage or computation per time step. This is implemented by replacing ∆j (t) in equation (4) with ∆j (t), where\n∆j(t) = α ∞ 0 ∆j (t − τ )e −ατ dτ,(14)\nand α is small and positive. This quantity is most easily calculated using the equivalent differential equation\nd ∆ dt = α ∆ − ∆ .(15)\nEach particle is smeared backwards along its trajectory and represents a set of virtual or ghost particles strung out along the orbit with ever decreasing weights. In effect, temporal smoothing increases the effective number of particles from N to\nN eff = N t 1 2 ∆t ,(16)\nwhere ∆t is the timestep and t 1 2 = (ln 2)/α is the half-life of the ghost particles.\n\nSome insight into the effect of this smoothing procedure is given by the following argument. As in Section 2.1, let · denote the time-average over an interval that is much longer than a typical orbital period but much less than α −1 . The time-averaged version of equation (15) is\nd ∆ dt = α ∆ − ∆ or d Θ dt = α Θ − Θ ,(17)\nwhere Θ is defined by replacing ∆ by ∆ in (6). The evolution of Θ is described by equation (8), with Θ replaced by Θ on the right side. The solution to equations (8) and (17) has the form (10), with eigenvalues λ that satisfy\ndet(ǫA − λI + λ 2 α −1 I) = 0.(18)\nIn the simple case of a single observable, we have λ = 1 2 α ± 1 2 (α 2 − 4αǫA) 1/2 , where A > 0. For α ≫ ǫA, we find λ ≃ ǫA, which is the same convergence rate that would obtain without temporal smoothing. However, for α < 4ǫA, λ is complex, so the observables execute damped oscillations rather than converging smoothly, and the convergence rate is 1 2 Re(α) which is slower than the convergence rate without smoothing whenever α < 2ǫA.\n\nWe conclude that excessive temporal smoothing is undesirable and that the maximum smoothing time α −1 should satisfy\nǫ < α,(19)\nassuming |A| = O(1).\n\n\nMaximum entropy\n\nIf the number of particles exceeds the number of observables, the differential equations (4) are ill-conditioned. In practice this means that the observables {yj (t)} will converge fairly rapidly to {Yj}, but that the individual particle weights {wi} will continue to drift long after the observables have converged. Such behaviour is undesirable as we would like to use the particle weights to predict other properties of the stationary stellar system. To remove the ill-conditioning, we can maximise some form of profit function, such as the entropy\nS = − i wi log(wi/mi),(20)\nwhere {mi} is a pre-determined set of weights (the 'prior'). Thus we maximise the function\nF = µS − 1 2 χ 2(21)\nwhere\nχ 2 = 1 J J j=1 ∆ 2 j .(22)\nEquation (4) is replaced by\ndwi(t) dt = ǫwi(t) µ ∂S ∂wi − J j=1 Kj [zi(t)] Yj ∆j (t) .(23)\nThe factor µ is a measure of the relative contribution of χ 2 and S to the final state: if µ is large we get a smooth solution (in the sense that the {wi} are close to the {mi}) but a large χ 2 , while if µ is small the solution is not smooth but χ 2 is likely to be smaller. The parameter µ can be specified at the start of the calculation, or adjusted as the calculation proceeds using a prescription such as\ndµ(t) dt = µη D 2 − χ 2 (t) ,(24)\nwith 0 < η ≪ 1; in this case χ 2 will converge to the specified value D 2 . In the simulations described here we have kept µ constant. The condition N > J (number of particles exceeds the number of bins) is neither a strict criterion for illconditioning nor a necessary condition for a sensible result, since different observables are not independent (both because of the spatial smoothing described in Section 2.2 and because a single orbit contributes to many observables).\n\n\nRESULTS\n\n\nOne-dimensional results\n\nWe first present the results of experiments in one dimension, to illustrate the effects of the various parameters in the algorithm. We use for the background potential\nΦ(x) = − 1 (b 2 + x 2 ) 1/2(25)\nwith b = 1 3 , and for the observables we use the density distribution\nν(x) = 1 (b 2 + x 2 ) 5/2 .(26)\nThe period of a low-energy orbit is 2πb 3/2 = 1.209. We use a fourth-order leapfrog with timestep ∆t = 0.1 to integrate the particle orbits, and adjust the weights {wi} after every timestep according to equation (23). To measure the density we use a uniform grid with 16 bins in the range x ∈ [−1, 1]. With b = 1 3 there is a density contrast of about 200 between the inner and outermost bins. We evaluate the success of Figure 2. The rms fractional density error χ (eq. 22) for simulation A (bottom curve), and simulation B (top curve). Simulation B has no orbit averaging. The initial state has small χ 2 , but is not in equilibrium, so χ 2 immediately grows as soon as the particles start to move. The initial state has small χ 2 , but iis not in equilibrium, so χ 2 immediately grows as soon as the particles start to move. Figure 1 shows the evolution of the weights {wi} for simulation A, in which α = 0.0524, ǫ = 0.025 and µ = 0.01. All the {wi} evolve smoothly, and converge in a time of order ǫ −1 . We note that the final equilibrium has a wide range of masses, which suggests that some gain in efficiency might be possible if the particles with smaller w could be replaced with a smaller number of particles with a larger w (for example, we could discard particles whose weights fall below a threshold and replace them with new particles on randomly chosen orbits, or combine low-weight particles with nearby orbits). Figure 2 shows the evolution of χ 2 in simulation A and compares with simulation B in which the orbit averaging is removed (α = ∞). Both the final value of χ 2 and its fractional fluctuation are smaller in A, illustrating the benefits of orbit-averaging. Figure 3 shows χ 2 in simulations A, C and D. Simulations C and A differ only in the parameter ǫ: C has larger ǫ and hence converges more rapidly, although the final states in A and C are very similar. Simulation D and C differ only in the parameter µ: D has µ = 0 (no entropy constraint). The final value of χ 2 is lower in simulation Dand the fluctuations are larger-since the simulation does not have to trade higher χ 2 for lower entropy. and do not converge smoothly to their final values. Comparing simulations C and D we see that this noise is worse for small w when the entropy constraint is removed. Note that the larger w are not much different in C and D. The smaller w do not converge in D, but they do not contribute much to χ 2 . In C the entropy constraint has tied down the smaller values of w. Figure 5 compares the effects of orbit averaging and particle number. Simulation E has 10 times the number of particles as A, but the smoothing time α −1 is 10 times smaller; as a consequence the noise in the two simulations is about the same. The shorter smoothing time in E allows a larger ǫ so that E converges faster than A; note that the time coordinate in Figure 5 has been shrunk by a factor of 10 for simulation A. Thus there is a tradeoff between number of particles and convergence time; simulations A and E each take about 2 minutes of CPU on a DEC Alpha 3000/300. Figure 6 shows the same 6 particle weights as in Figure 4.\n\n\nThree-dimensional results\n\nWe have performed a number of three-dimensional simulations. The parameters for the algorithm are in each case the same, as given in the last row of Table 1.\n\n\nMass models\n\nThe mass models were of three types: PS Plummer sphere, with density law given by equation (26), with x denoting the radius and 4πG = 3b 2 . We denote this model 'PS'. PT Triaxial Plummer model with density law given by equation (26) with x denoting the triaxial radius, s, defined by\ns 2 = x 2 A 2 + y 2 B 2 + z 2 C 2 ,(27)\nand (A, B, C) = (1.41, 1.12, 1.00). ST Schwarzschild's (1979) model with aixs ratios given by (A, B, C) = (2.00, 1.25, 1.00).\n\n\nForce calculation\n\nThe force calculation is carried out in one of two ways: A Analytically. In the case of PT the analytic potential is not that which would be self-consistently generated by the mass model, but rather the Plummer potential with s replacing x in equation (25). F Numerically using the Fourier Convolution Theorem and FFT on a 16 3 grid. Mass is assigned to the grid from the desired mass model, and the forces and potential are calculated once only at the beginning of the simulation. In the ST models the numerical and analytic potentials are significantly different (40% on average) because the model has inifinite mass and has been truncated at finite radius. In the PS models the forces agree to within a few percent.\n\n\nInitial condition\n\nWe also use three different types of initial condition: B Particles uniformly distributed in r < 1, with velocities uniform in the range allowed by the condition that their orbits remain in r < 1. In the Schwarzschild potential this initial condition produces a preponderence of box orbits, hence we refer to it as 'box-dominated'. T Particles uniformly distributed in r < 1. Additionally 20% have their velocities chosen as for B, and 80% are given velocities perpendicular to their radius vector with magnitude equal to that of a circular orbit in a spherical potential with the same total force. In the Schwarzschild potential this initial condition produces more tube orbits than B, hence we refer to it as 'tubedominated'. E Particles distributed in r < 1 according to the required mass model. Thus there are extra particles in the inner regions as compared with B or T. Velocities chosen as for T. To summarise the notation by example, simulation PSAB is a Plummer sphere with analytic forces, and box-dominated Table 2. The results of the three-dimensional simulations. Column 1 is the simulation type; column 2 the initial percentage by mass of tube orbits (as defined in the text); column 3 the final percentage by mass; column 4 the final percentage by number; and column 5 the final value of χ. initial condition; simulation STFE is a Schwarzschild triaxial model with Fourier numerical forces, and extra sampling of the inner regions.\n\nIn each case we measure the density using a uniform cubical grid with M = 16 elements on a side for a total of 2 12 observables. The core radius of alll the models b = 1 3 , and we restrict particles to orbits with r < 1. Simulation PSAB took about 200 minutes of CPU on the same machine as simulation A, a factor of ∼ 100 longer; most of this factor reflects the factor of 40 increase in the number of particles, while the rest reflects the increased complexity of the differential equations (which must follow 6 phase-space coordinates instead of 2) and increased access time to the much larger array which represents the observables.\n\nOrbit classification is carried out by recording the maximum and mininum values of the angular momentum L about the z-axis (the shortest principal axis of the triaxial models). Box orbits tend to reverse direction and 'retrace' their paths, so Lmax ≃ −Lmin. Figure 7 shows the distribution of Lmax and Lmin for simulation PTFT. The box orbits essentially lie in the upper left quadrant. Tube orbits are here defined as those with LmaxLmin > 0. Table 2 gives the percentage of tube orbits in each of the three-dimensional simulations.\n\nWith the exception of STFB all simulations converge well: none give final values of χ that are more than a factor of 2 different from the preferred value D = 0.05. Note that the initial condition B, although it works fine for the Plummer models, fails to produce enough tube orbits to support the Schwarzschild potential, and hence the final value of χ is higher than the preferred value D = 0.05. The algorithm converges, but it is forced to substitute smoothness for accuracy because it has too few tube orbits. Also note that the various realisations of the Schwarzschild model have different final weights of tube orbits. This is not surprising as it is expected that the degeneracy in a model which only matches the volume density might lead to a variety of box/tube mixtures.\n\nThe behaviour of the weights {wi(t)} and of χ as a function of time look very similar to the one-dimensional case and they are not reproduced here. In Figure 8 we show the density along the principal axes in two of the simulations.\n\n\nDISCUSSION\n\n\nChoosing the parameters\n\nHow should the parameters, (N, M, α, ǫ, µ) be chosen to optimise the investment in CPU? The arguments given in Section 2 lead to the following guidelines.\n\nFirst choose D, the value of χ which is as large as can be tolerated-in the simulations above we had χ of a few times 10 −2 . Then choose the resolution required, via the number of resolution elements M d . This informs our choice of N and α through (cf. equation 16)\nχ 2 ∼ M d N eff ∼ M d N α∆t.(28)\nThus, if N is limited by storage requirements, equation (28) determines the desired value of α.\n\nOnce we have α then we must choose ǫ smaller than α (equation 19), bearing in mind that the convergence time will be of order ǫ −1 orbital periods. In the simulations we found that ǫ ∼ 0.5α is about right.\n\nThe final step is to choose µ. This is largely a matter of taste since µ determines the balance between smoothness and accuracy (Merritt & Tremblay 1992 discuss this issue in another context). The natural choice for µ is the one for which χ 2 reflects the observational errors; and this is the value which will be obtained if µ is allowed to vary according to equation (24). We do not recommend equation (24) for general use however because convergence of µ can be rather slow. One method, which has the advantage of being at least semi-quantitative, is to do a run including equation (24), with D equal to the preferred value of χ. Far from having to converge, it only has to get µ into the right ballpark, and then we go back to constant µ for an extended run.\n\n\nThe initial condition and prior\n\nThe initial condition of the particles should be chosen to sample phase space as well as possible. Considerable effort is sometimes devoted to this choice in orbit-based calculations. In the simulations presented here we have chosen simply to sample phase-space uniformly (Section 3). With detailed knowledge of the potential and its orbit families, more informed choices could be made.\n\nWe can think of the prior {mi}, together with the particle positions and velocities, as a random realisation of some known DF f0(r, v). In the simulations presented here we use {mi} which are all equal. In those where the initial particle positions and velocities sample phase space uniformly, for example, we are effectively using f0 which is initially equal to a constant. If the initial condition is well mixed, as we may reasonably hope, then f0 is independent of time. The smoothness constraint has the effect of driving the DF of the system towards f0.\n\nWith a suitable choice of prior {mi}, the final equilibrium should not depend on the initial condition. It should merely reflect the choice of µ, which determines the balance between smoothness and accuracy. The resolution of the final equilibrium in phase space, however, may be affected by the choice of initial condition. The wide range of weights in Figure 1 is a reflection of this fact. However, only perfect knowledge of the target DF would allow one to set up an initial condition which led to an equilibrium with all the {wi} equal. This problem could be alleviated 'on the fly' by methods that kill particles with low weights and split particles with high weights into several daughter particles with similar orbits.\n\n\nComparison with orbit-based methods\n\nThe particle-based method we have described here has several advantages over orbit-based methods.\n\n• Particle-based methods use less storage: if there are N particles (or orbits) and J < N observables, orbit-based methods must store O(N J) variables (the contribution of each orbit to each observable), while a particle-based method stores only O(N ) variables (the particle weights at a given time). This advantage is particularly important in systems with a large number of observables (triaxial systems, or systems in which the entire line-ofsight velocity distribution is observed). • Although we have only discussed constructing stellar systems in a fixed potential, it should be possible to generalise particle-based methods so that the potential is determined self-consistently by the particles. Perhaps the best approach would be to expand the potential as a linear combination of a set of basis functions (which can be chosen to preserve any desired symmetries, e.g. spherical symmetry). The coefficient of each basis function, determined from the evolving weights of the particles, could be orbit averaged (cf. equation 14) to reduce the effects of relaxation. • Model construction with orbit-based methods is a multistep process: first compute the luminosity density from the surface brightness; then solve Poisson's equation assuming (say) constant mass-to-light ratio; then integrate orbits in this potential to construct an orbit library (the matrix containing the contribution of each orbit to each observable); then use some inversion/optimization method such as maximum entropy, Lucy's method, or linear or quadratic programming to determine the orbit weights. In a self-consistent particle-based methods all of these steps could be done at the same time.\n\nParticle-based methods also have disadvantages compared to orbit-based methods: they use more computing cycles per orbit because the orbits must be followed for a longer time; as in all Monte Carlo methods, accuracy only scales as N 1/2 ; poorly chosen values of parameters such as α and ǫ can cripple the method.\n\nThe simple experiments we have described show that particle-based methods might be able to compete with orbitbased methods. Possible improvements include: (i) nonuniform spatial grids so that the resolution is highest near the centre of the galaxy; (ii) methods that kill particles with low weights and split particles with high weights into several daughter particles with similar orbits; (iii) gridless methods based on smooth kernels (Section 2.2); (iv) determining the potential self-consistently from the gravitational field of the particles.\n\nFigure 1 .\n1The weights {w i } as a function of time for simulation A.\n\nFigure 3 .\n3The density error χ for simulations A (top curve), C (middle curve) and D (bottom curve).\n\nFigure 4 .Figure 5 .\n45A representative sample of 6 of the {w i } for each of the simulations A (top left), B (top right), C (bottom left) and D The density error χ for simulations E and A (with the time coordinate for A shrunk by a factor of ten to facilitate the comparison).\n\nFigure 4 Figure 6 .\n46shows the evolution of the same 6 particles from simulations A, B, C and D. Comparing A (ǫ = 0.48α) with C and D (ǫ = 95α), we see the effect of excessive temporal smoothing (equation 19): the trajectories wi are noisy As inFigure 4but for simulation E.\n\nFigure 7 .Figure 8\n78A scatter plot of Lmax versus L min for simulation PTFT (only 10% of the total number of particles are shown). The values of L plotted are relative to the average magnitude of the angular momentum of each particle. (a). The density along the principal axes in simulation PTFE showing the target mass model (solid) and the final state observed distribution (diamonds).\n\nFigure 8\n8(b). As inFigure 8(a) for simulation STFE.\n\nTable 1 .\n1The parameter values used in the simulations.N \n10 2 α \n10 2 ǫ \n10 2 µ \nD \nA \n100 \n5.24 \n2.5 \n1 \n0.05 \nB \n100 \n∞ \n2.5 \n1 \n0.05 \nC \n100 \n5.24 \n500 \n1 \n0.05 \nD \n100 \n5.24 \n500 \n0 \n-\nE \n1000 \n51.9 \n25 \n1 \n0.05 \n3D \n4000 \n5.24 \n2.5 \n0.1 \n0.05 \n\nthe algorithm by examining the time evolution of the {wi} \nand of χ 2 . The parameter values we use are summarised \nin Table 1. In the initial state the particles were uniformly \ndistributed in x ∈ [−1, 1] with equal weights. The veloci-\nties were uniformly distributed in the range allowed by the \ncondition that their orbits are restricted to x ∈ [−1, 1]. \n\nc 0000 RAS, MNRAS 000, 0-0\nThis paper has been produced using the Royal Astronomical Society/Blackwell Science T E X macros.\n. J Binney, G A Mamon, MNRAS. 361Binney, J., & Mamon, G. A. 1982,MNRAS, 200, 361\n\n. J J Binney, R L Davies, G D Illingworth, ApJ. 36178Binney, J. J., Davies, R. L., & Illingworth, G. D. (1990), ApJ, 361, 78\n\n. H Dejonghe, ApJ. 343113Dejonghe, H. (1989),ApJ, 343, 113\n\n. K Kuijken, ApJ. 446194Kuijken, K. (1995), ApJ, 446, 194\n\n. L B Lucy, AJ. 79745Lucy, L. B. 1974, AJ, 79, 745\n\n. J Magorrian, J Binney, MNRAS. 271949Magorrian, J., & Binney, J. (1994), MNRAS, 271, 949\n\n. E E Qian, P T De Zeeuw, R P Van Der Marel, C Hunter, MNRAS. 274602Qian, E. E., de Zeeuw, P. T., van der Marel, R. P., & Hunter, C. (1995), MNRAS, 274, 602\n\n. D Merritt, ApJ. 41379Merritt, D. (1993), ApJ, 413, 79\n\n. D Merritt, B Tremblay, AJ. 108514Merritt, D., & Tremblay, B. (1994), AJ, 108, 514\n\n. A J Newton, J Binney, MNRAS. 210711Newton, A. J., & Binney, J. (1984), MNRAS, 210, 711\n\n. D O Richstone, S Tremaine, ApJ. 32782Richstone, D. O., & Tremaine, S. (1988), ApJ, 327, 82\n\n. D Syer, S Tremaine, MNRAS. 232236ApJSyer, D., & Tremaine, S. (1995), MNRAS, 000, in press Schwarzschild, M. 1979, ApJ, 232, 236\n\n. M Schwarzschild, ApJ. 409563Schwarzschild, M. 1993, ApJ, 409, 563\n"},"corpus_id":{"kind":"number","value":3256508,"string":"3,256,508"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"astro-ph/9605061\",\n \"mag\": 2112706506,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1093/mnras/282.1.223\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/astro-ph/9605061v1.pdf\"\n ],\n \"pdf_sha\": \"612a335a9ddf0fb303994b5eabf821cc41d89fca\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"https://academic.oup.com/mnras/article-pdf/282/1/223/18199992/282-1-223.pdf\",\n \"status\": \"BRONZE\"\n }\n}"},"title":{"kind":"string","value":"Made-to-measure N -body systems"},"authors":{"kind":"list like","value":["D Syer \nCanadian Institute for Theoretical Astrophysics\nMcLennan Labs\n60 St. George StreetM5S 1A7TorontoOntarioCanada\n\nMax-Planck-Institut für Astrophysik\nKarl-Schwarzschild-Straße 185748Garching-bei-MünchenGermany","S Tremaine \nCanadian Institute for Theoretical Astrophysics\nMcLennan Labs\n60 St. George StreetM5S 1A7TorontoOntarioCanada"],"string":"[\n \"D Syer \\nCanadian Institute for Theoretical Astrophysics\\nMcLennan Labs\\n60 St. George StreetM5S 1A7TorontoOntarioCanada\\n\\nMax-Planck-Institut für Astrophysik\\nKarl-Schwarzschild-Straße 185748Garching-bei-MünchenGermany\",\n \"S Tremaine \\nCanadian Institute for Theoretical Astrophysics\\nMcLennan Labs\\n60 St. George StreetM5S 1A7TorontoOntarioCanada\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We describe an algorithm for constructing N -body realisations of equilibrium stellar systems. The algorithm complements existing orbit-based modelling techniques using linear programming or other optimization algorithms. The equilibria are constructed by integrating an N -body system while slowly adjusting the masses of the particles until the time-averaged density field and other observables converge to a prescribed value. The procedure can be arranged to maximise a linear combination of the entropy of the system and the χ 2 statistic for the observables. The equilibria so produced may be useful as initial conditions for N -body simulations or for modelling observations of individual galaxies."},"sections":{"kind":"list like","value":[{"title":"INTRODUCTION","content":"One of the central problems in stellar dynamics is to construct made-to-measure stellar systems. For example, (i) when modelling observations of an elliptical galaxy we wish to find a phase-space distribution function f (r, v) (hereafter DF) that solves Poisson's equation and the collisionless Boltzman equation and reproduces the observed surfacebrightness distribution, rotation curve, velocity-dispersion profile, etc. (in the sense of minimising χ 2 , the mean-square deviation between the observations and the model); (ii) when conducting simulations we wish to construct initial states that are N -body realisations of equilibrium stellar systems with given density profile, rotation curve, bulge/disc ratio, etc.\nExisting methods for constructing made-to-meaure stellar systems can be classified as follows:\n• DF-based methods, which solve directly for the DF f (r, v). These generally require that all the integrals of motion are known explicitly (e.g. if the potential is spherical or has Stäckel form) or that the DF is assumed to depend only on known analytic integrals. These include methods that fit observations to few-parameter models with analytic DFs such as spherical Michie or King models; a difficulty with such methods is that the dependence of the derived properties of the galaxy on the choice of the model can be large and uncertain (e.g. Merritt & Tremblay 1994). A more flexible DF-based method is described by Dejonghe (1989), who expands the DF for a spherical system in a truncated series of basis functions and minimises χ 2 subject to the constraint that the DF is positive, using quadratic programming. DF-based methods can be applied to axisymmetric galaxies, and are completely general so long as the DF does not depend on a third integral (e.g. Qian et al . 1995, Kuijken 1995. • Moment-based methods, which find solutions of the Jeans equations (or higher-order velocity moments of the collisionless Boltzmann equation) that minimise χ 2 . This method was used by Binney & Mamon (1982) to construct spherical models of M87 and has been applied to axisymmetric models by Binney et al. (1990), van der Marel et al. (1994), and Magorrian & Binney (1994). A drawback is that this procedure does not guarantee that there is a positive-definite DF with the required velocity moments. • Orbit-based methods (Schwarzschild 1979, Schwarzschild 1993 compute the density distribution of a large library of orbits in a fixed potential, and then determine the weight each orbit must have in order to reproduce the desired final state. In this method the DF or other integrals of motion are not explicitly required, although the DF can be regarded as a sum of deltafunctions on the phase-space surfaces covered by the orbits. Orbit-based methods are generally ill-conditioned. The ill-conditioning can be removed by iterating from a smooth initial guess for the orbit weights using the Richardson-Lucy method (Newton & Binney 1984), but the final weights will then depend in a complicated way on the initial guess. A better procedure is to minimise χ 2 minus a profit function that measures the smoothness of the distribution of orbit weights. The profit function may be an entropy (Richstone & Tremaine 1988) or any other function that is large when the DF is smooth (Merritt 1993). The most flexible of these are orbit-based methods, since they do not require that the integrals of motion are known and the approximation they provide to the DF is known to be positive if the orbit weights are positive.\nThe goal of this paper is to introduce a novel class of methods for constructing made-to-measure N -body realisations of stellar systems. In the classification above, these methods might be called 'particle-based'; they work by sculpting an initial N -body system until it matches the prescribed density field and other observables."},{"title":"THE ALGORITHM","content":"In most of our discussion we restrict ourselves to constructing stellar systems in a fixed potential Φ(r). We suppose that Φ admits a collisionless equilibrium configuration specified by a distribution function in phase space f (r, v). Thus f satisfies the time-independent collisionless Boltzmann equation:\nAn 'observable' of the stellar system is a quantity of the form\nwhere z = (r, v) and Kj is a known kernel. Suitable observables include the surface or volume density at a given point, the surface density times the mean line-of-sight velocity, the surface density times any moment of the line-of-sight velocity, etc. Now consider a system of N particles having weights wi and phase-space positions zi(t) (i ≤ N ). The observables of this system are\nOur goal is the following: given a set of distinct observables Yj, j = 1, . . . , J, construct a system of N particles orbiting in the potential Φ(r) whose time-averaged observables yj(t) are equal to Yj. We hope that this paper will stimulate interest in seeking better particle-based algorithms than the one we describe."},{"title":"The force of change","content":"The heart of the algorithm is a prescription for changing the weights {wi(t)} as the particles proceed along their fixed orbits in the potential Φ(r). The prescription is similar to that employed by Syer & Tremaine (1995) in a different context. It consists of applying gentle pressure on wi according to the value of ∆j ≡ yj(t)/Yj − 1: if ∆j < 0 then increase wi, and if ∆j > 0 then decrease wi. More precisely we let\nwhere ǫ is small and positive, and Z is so far arbitrary. The factor wi on the right side ensures that dwi/dt → 0 as wi → 0 so that wi cannot become negative. The factor Kj /Zj ensures that the difference ∆j changes the weight wi only if particle i is contributing to observable j. Equation (4) is closely related to Lucy's (1974) method for solving integral equations with noisy data. Some insight into the solutions of equation (4) is offered by the following argument. Since ǫ is small the weights wi change only over many orbits; thus we may orbit average:\nHere · denotes the time-average over an interval that is much longer than a typical orbital period but much less than ǫ −1 periods, and Kji is shorthand for the time-independent quantity Kj[zi(t)] . We have also assumed that the fluctuations in Kj [zi(t)] and ∆j(t) are not correlated, which is plausible if many particles contribute to ∆j. We now define\nwhich obeys the differential equation\nWhen we are close to convergence (|Θ| ≪ 1), the behaviour of the right side is dominated by changes in Θj rather than changes in wi, so we may replace wi by a constant, w 0 i . Then the vector Θ satisfies the matrix equation\nwhere the matrix A has components\nThe solutions to equation (8) have the form\nwhere the eigenvalues {λm} are solutions of the equation\nSince A is positive-definite by construction (i.e. x t · A · x > 0 for all x), all of its eigenvalues are positive so λm > 0. This argument suggests that all observables converge to the desired values (| ∆ | → 0) on O(ǫ −1 ) orbital periods, if ǫ is sufficiently small and we start close to the correct final state. For comparison, orbit-based methods evaluate and store the entire matrix Kji (by following the orbits for a fixed time that is much longer than the orbital period; the matrix K is often called the \"orbit library\"), then solve the matrix equation\nif N > J the matrix equation is ill-conditioned and must be solved subject to a constraint that maximizes some profit function such as entropy. The storage needed by particlebased methods is O(N J) whereas the storage needed by particle-based methods is only O(N )."},{"title":"The kernel","content":"Suppose that the observable yr is the density at r. What is the appropriate kernel Kr(r ′ , v ′ )? One approach would be to use a smooth kernel (à la SPH) with\nwhere W is some smooth function, and h is the resolution length (possibly a function of r); W should be normalised so that W (r, h) d 3 r = 1. If h is too large resolution is lost, while if h is too small the observables yj fluctuate strongly because too few particles contribute to a given observable. The smooth-kernel approach is expensive because nearest neighbours have to be found at each timestep and we have therefore adopted a different method. We first divide the coordinate space into bins. Then we set Kr(r ′ , v ′ ) to zero if r and r ′ are not in the same bin, and equal to the inverse of the volume of the bin otherwise. Obviously Kr 1 and Kr 2 are the same if r1 and r2 are in the same bin, so there can be at most one density observable per bin. We are still free to choose the parameter Zj in (4); a simple choice is to set Zj equal to the inverse of the volume of the bin. Thus Kj (zi)/Zj is unity if ri is in bin j, and zero otherwise.\nAn improvement to this simple scheme, which we employ here, is to borrow from the smooth-kernel approach and to smear each particle into neighbouring bins. Each particle is replaced by a gaussian distribution at the bin centre closest to the particle position, with dispersion equal to half the bin width. Neighbouring bins are assigned a weight corresponding to the integral of the gaussian over those bins (normalized so that the total contribution over all bins is wi). The bin closest to the particle position thus contains about two thirds of its weight. This procedure corresponds nicely to the phenomenon of seeing in the case of a projected observable."},{"title":"Resolution and smoothing","content":"Suppose that we divide the d-dimensional coordinate space into M d bins, and each density observable corresponds to the mass per unit volume in a single bin. Then at any given time there are on average N/M d particles contributing to each observable, and the rms statistical fluctuations in ∆j will be of order δ ∼ (M d /N ) 1/2 . These fluctuations can be kept small if d = 1 (spherical or one-dimensional systems); for example if M = 30 and N = 1000 we have δ ∼ 0.2. However, for triaxial systems (d = 3) the fluctuations will be much larger for reasonable values of M and N .\nTo improve this situation we employ a form of temporal smoothing which effectively boosts N without any need for extra storage or computation per time step. This is implemented by replacing ∆j (t) in equation (4) with ∆j (t), where\nand α is small and positive. This quantity is most easily calculated using the equivalent differential equation\nEach particle is smeared backwards along its trajectory and represents a set of virtual or ghost particles strung out along the orbit with ever decreasing weights. In effect, temporal smoothing increases the effective number of particles from N to\nwhere ∆t is the timestep and t 1 2 = (ln 2)/α is the half-life of the ghost particles.\nSome insight into the effect of this smoothing procedure is given by the following argument. As in Section 2.1, let · denote the time-average over an interval that is much longer than a typical orbital period but much less than α −1 . The time-averaged version of equation (15) is\nwhere Θ is defined by replacing ∆ by ∆ in (6). The evolution of Θ is described by equation (8), with Θ replaced by Θ on the right side. The solution to equations (8) and (17) has the form (10), with eigenvalues λ that satisfy\nIn the simple case of a single observable, we have λ = 1 2 α ± 1 2 (α 2 − 4αǫA) 1/2 , where A > 0. For α ≫ ǫA, we find λ ≃ ǫA, which is the same convergence rate that would obtain without temporal smoothing. However, for α < 4ǫA, λ is complex, so the observables execute damped oscillations rather than converging smoothly, and the convergence rate is 1 2 Re(α) which is slower than the convergence rate without smoothing whenever α < 2ǫA.\nWe conclude that excessive temporal smoothing is undesirable and that the maximum smoothing time α −1 should satisfy\nassuming |A| = O(1)."},{"title":"Maximum entropy","content":"If the number of particles exceeds the number of observables, the differential equations (4) are ill-conditioned. In practice this means that the observables {yj (t)} will converge fairly rapidly to {Yj}, but that the individual particle weights {wi} will continue to drift long after the observables have converged. Such behaviour is undesirable as we would like to use the particle weights to predict other properties of the stationary stellar system. To remove the ill-conditioning, we can maximise some form of profit function, such as the entropy\nwhere {mi} is a pre-determined set of weights (the 'prior'). Thus we maximise the function\nwhere\nEquation (4) is replaced by\nThe factor µ is a measure of the relative contribution of χ 2 and S to the final state: if µ is large we get a smooth solution (in the sense that the {wi} are close to the {mi}) but a large χ 2 , while if µ is small the solution is not smooth but χ 2 is likely to be smaller. The parameter µ can be specified at the start of the calculation, or adjusted as the calculation proceeds using a prescription such as\nwith 0 < η ≪ 1; in this case χ 2 will converge to the specified value D 2 . In the simulations described here we have kept µ constant. The condition N > J (number of particles exceeds the number of bins) is neither a strict criterion for illconditioning nor a necessary condition for a sensible result, since different observables are not independent (both because of the spatial smoothing described in Section 2.2 and because a single orbit contributes to many observables)."},{"title":"RESULTS","content":""},{"title":"One-dimensional results","content":"We first present the results of experiments in one dimension, to illustrate the effects of the various parameters in the algorithm. We use for the background potential\nwith b = 1 3 , and for the observables we use the density distribution\nThe period of a low-energy orbit is 2πb 3/2 = 1.209. We use a fourth-order leapfrog with timestep ∆t = 0.1 to integrate the particle orbits, and adjust the weights {wi} after every timestep according to equation (23). To measure the density we use a uniform grid with 16 bins in the range x ∈ [−1, 1]. With b = 1 3 there is a density contrast of about 200 between the inner and outermost bins. We evaluate the success of Figure 2. The rms fractional density error χ (eq. 22) for simulation A (bottom curve), and simulation B (top curve). Simulation B has no orbit averaging. The initial state has small χ 2 , but is not in equilibrium, so χ 2 immediately grows as soon as the particles start to move. The initial state has small χ 2 , but iis not in equilibrium, so χ 2 immediately grows as soon as the particles start to move. Figure 1 shows the evolution of the weights {wi} for simulation A, in which α = 0.0524, ǫ = 0.025 and µ = 0.01. All the {wi} evolve smoothly, and converge in a time of order ǫ −1 . We note that the final equilibrium has a wide range of masses, which suggests that some gain in efficiency might be possible if the particles with smaller w could be replaced with a smaller number of particles with a larger w (for example, we could discard particles whose weights fall below a threshold and replace them with new particles on randomly chosen orbits, or combine low-weight particles with nearby orbits). Figure 2 shows the evolution of χ 2 in simulation A and compares with simulation B in which the orbit averaging is removed (α = ∞). Both the final value of χ 2 and its fractional fluctuation are smaller in A, illustrating the benefits of orbit-averaging. Figure 3 shows χ 2 in simulations A, C and D. Simulations C and A differ only in the parameter ǫ: C has larger ǫ and hence converges more rapidly, although the final states in A and C are very similar. Simulation D and C differ only in the parameter µ: D has µ = 0 (no entropy constraint). The final value of χ 2 is lower in simulation Dand the fluctuations are larger-since the simulation does not have to trade higher χ 2 for lower entropy. and do not converge smoothly to their final values. Comparing simulations C and D we see that this noise is worse for small w when the entropy constraint is removed. Note that the larger w are not much different in C and D. The smaller w do not converge in D, but they do not contribute much to χ 2 . In C the entropy constraint has tied down the smaller values of w. Figure 5 compares the effects of orbit averaging and particle number. Simulation E has 10 times the number of particles as A, but the smoothing time α −1 is 10 times smaller; as a consequence the noise in the two simulations is about the same. The shorter smoothing time in E allows a larger ǫ so that E converges faster than A; note that the time coordinate in Figure 5 has been shrunk by a factor of 10 for simulation A. Thus there is a tradeoff between number of particles and convergence time; simulations A and E each take about 2 minutes of CPU on a DEC Alpha 3000/300. Figure 6 shows the same 6 particle weights as in Figure 4."},{"title":"Three-dimensional results","content":"We have performed a number of three-dimensional simulations. The parameters for the algorithm are in each case the same, as given in the last row of Table 1."},{"title":"Mass models","content":"The mass models were of three types: PS Plummer sphere, with density law given by equation (26), with x denoting the radius and 4πG = 3b 2 . We denote this model 'PS'. PT Triaxial Plummer model with density law given by equation (26) with x denoting the triaxial radius, s, defined by\nand (A, B, C) = (1.41, 1.12, 1.00). ST Schwarzschild's (1979) model with aixs ratios given by (A, B, C) = (2.00, 1.25, 1.00)."},{"title":"Force calculation","content":"The force calculation is carried out in one of two ways: A Analytically. In the case of PT the analytic potential is not that which would be self-consistently generated by the mass model, but rather the Plummer potential with s replacing x in equation (25). F Numerically using the Fourier Convolution Theorem and FFT on a 16 3 grid. Mass is assigned to the grid from the desired mass model, and the forces and potential are calculated once only at the beginning of the simulation. In the ST models the numerical and analytic potentials are significantly different (40% on average) because the model has inifinite mass and has been truncated at finite radius. In the PS models the forces agree to within a few percent."},{"title":"Initial condition","content":"We also use three different types of initial condition: B Particles uniformly distributed in r < 1, with velocities uniform in the range allowed by the condition that their orbits remain in r < 1. In the Schwarzschild potential this initial condition produces a preponderence of box orbits, hence we refer to it as 'box-dominated'. T Particles uniformly distributed in r < 1. Additionally 20% have their velocities chosen as for B, and 80% are given velocities perpendicular to their radius vector with magnitude equal to that of a circular orbit in a spherical potential with the same total force. In the Schwarzschild potential this initial condition produces more tube orbits than B, hence we refer to it as 'tubedominated'. E Particles distributed in r < 1 according to the required mass model. Thus there are extra particles in the inner regions as compared with B or T. Velocities chosen as for T. To summarise the notation by example, simulation PSAB is a Plummer sphere with analytic forces, and box-dominated Table 2. The results of the three-dimensional simulations. Column 1 is the simulation type; column 2 the initial percentage by mass of tube orbits (as defined in the text); column 3 the final percentage by mass; column 4 the final percentage by number; and column 5 the final value of χ. initial condition; simulation STFE is a Schwarzschild triaxial model with Fourier numerical forces, and extra sampling of the inner regions.\nIn each case we measure the density using a uniform cubical grid with M = 16 elements on a side for a total of 2 12 observables. The core radius of alll the models b = 1 3 , and we restrict particles to orbits with r < 1. Simulation PSAB took about 200 minutes of CPU on the same machine as simulation A, a factor of ∼ 100 longer; most of this factor reflects the factor of 40 increase in the number of particles, while the rest reflects the increased complexity of the differential equations (which must follow 6 phase-space coordinates instead of 2) and increased access time to the much larger array which represents the observables.\nOrbit classification is carried out by recording the maximum and mininum values of the angular momentum L about the z-axis (the shortest principal axis of the triaxial models). Box orbits tend to reverse direction and 'retrace' their paths, so Lmax ≃ −Lmin. Figure 7 shows the distribution of Lmax and Lmin for simulation PTFT. The box orbits essentially lie in the upper left quadrant. Tube orbits are here defined as those with LmaxLmin > 0. Table 2 gives the percentage of tube orbits in each of the three-dimensional simulations.\nWith the exception of STFB all simulations converge well: none give final values of χ that are more than a factor of 2 different from the preferred value D = 0.05. Note that the initial condition B, although it works fine for the Plummer models, fails to produce enough tube orbits to support the Schwarzschild potential, and hence the final value of χ is higher than the preferred value D = 0.05. The algorithm converges, but it is forced to substitute smoothness for accuracy because it has too few tube orbits. Also note that the various realisations of the Schwarzschild model have different final weights of tube orbits. This is not surprising as it is expected that the degeneracy in a model which only matches the volume density might lead to a variety of box/tube mixtures.\nThe behaviour of the weights {wi(t)} and of χ as a function of time look very similar to the one-dimensional case and they are not reproduced here. In Figure 8 we show the density along the principal axes in two of the simulations."},{"title":"DISCUSSION","content":""},{"title":"Choosing the parameters","content":"How should the parameters, (N, M, α, ǫ, µ) be chosen to optimise the investment in CPU? The arguments given in Section 2 lead to the following guidelines.\nFirst choose D, the value of χ which is as large as can be tolerated-in the simulations above we had χ of a few times 10 −2 . Then choose the resolution required, via the number of resolution elements M d . This informs our choice of N and α through (cf. equation 16)\nThus, if N is limited by storage requirements, equation (28) determines the desired value of α.\nOnce we have α then we must choose ǫ smaller than α (equation 19), bearing in mind that the convergence time will be of order ǫ −1 orbital periods. In the simulations we found that ǫ ∼ 0.5α is about right.\nThe final step is to choose µ. This is largely a matter of taste since µ determines the balance between smoothness and accuracy (Merritt & Tremblay 1992 discuss this issue in another context). The natural choice for µ is the one for which χ 2 reflects the observational errors; and this is the value which will be obtained if µ is allowed to vary according to equation (24). We do not recommend equation (24) for general use however because convergence of µ can be rather slow. One method, which has the advantage of being at least semi-quantitative, is to do a run including equation (24), with D equal to the preferred value of χ. Far from having to converge, it only has to get µ into the right ballpark, and then we go back to constant µ for an extended run."},{"title":"The initial condition and prior","content":"The initial condition of the particles should be chosen to sample phase space as well as possible. Considerable effort is sometimes devoted to this choice in orbit-based calculations. In the simulations presented here we have chosen simply to sample phase-space uniformly (Section 3). With detailed knowledge of the potential and its orbit families, more informed choices could be made.\nWe can think of the prior {mi}, together with the particle positions and velocities, as a random realisation of some known DF f0(r, v). In the simulations presented here we use {mi} which are all equal. In those where the initial particle positions and velocities sample phase space uniformly, for example, we are effectively using f0 which is initially equal to a constant. If the initial condition is well mixed, as we may reasonably hope, then f0 is independent of time. The smoothness constraint has the effect of driving the DF of the system towards f0.\nWith a suitable choice of prior {mi}, the final equilibrium should not depend on the initial condition. It should merely reflect the choice of µ, which determines the balance between smoothness and accuracy. The resolution of the final equilibrium in phase space, however, may be affected by the choice of initial condition. The wide range of weights in Figure 1 is a reflection of this fact. However, only perfect knowledge of the target DF would allow one to set up an initial condition which led to an equilibrium with all the {wi} equal. This problem could be alleviated 'on the fly' by methods that kill particles with low weights and split particles with high weights into several daughter particles with similar orbits."},{"title":"Comparison with orbit-based methods","content":"The particle-based method we have described here has several advantages over orbit-based methods.\n• Particle-based methods use less storage: if there are N particles (or orbits) and J < N observables, orbit-based methods must store O(N J) variables (the contribution of each orbit to each observable), while a particle-based method stores only O(N ) variables (the particle weights at a given time). This advantage is particularly important in systems with a large number of observables (triaxial systems, or systems in which the entire line-ofsight velocity distribution is observed). • Although we have only discussed constructing stellar systems in a fixed potential, it should be possible to generalise particle-based methods so that the potential is determined self-consistently by the particles. Perhaps the best approach would be to expand the potential as a linear combination of a set of basis functions (which can be chosen to preserve any desired symmetries, e.g. spherical symmetry). The coefficient of each basis function, determined from the evolving weights of the particles, could be orbit averaged (cf. equation 14) to reduce the effects of relaxation. • Model construction with orbit-based methods is a multistep process: first compute the luminosity density from the surface brightness; then solve Poisson's equation assuming (say) constant mass-to-light ratio; then integrate orbits in this potential to construct an orbit library (the matrix containing the contribution of each orbit to each observable); then use some inversion/optimization method such as maximum entropy, Lucy's method, or linear or quadratic programming to determine the orbit weights. In a self-consistent particle-based methods all of these steps could be done at the same time.\nParticle-based methods also have disadvantages compared to orbit-based methods: they use more computing cycles per orbit because the orbits must be followed for a longer time; as in all Monte Carlo methods, accuracy only scales as N 1/2 ; poorly chosen values of parameters such as α and ǫ can cripple the method.\nThe simple experiments we have described show that particle-based methods might be able to compete with orbitbased methods. Possible improvements include: (i) nonuniform spatial grids so that the resolution is highest near the centre of the galaxy; (ii) methods that kill particles with low weights and split particles with high weights into several daughter particles with similar orbits; (iii) gridless methods based on smooth kernels (Section 2.2); (iv) determining the potential self-consistently from the gravitational field of the particles."},{"title":"Figure 1 .","content":""},{"title":"Figure 3 .","content":""},{"title":"Figure 4 .Figure 5 .","content":""},{"title":"Figure 4 Figure 6 .","content":""},{"title":"Figure 7 .Figure 8","content":""},{"title":"Figure 8","content":""},{"title":"Table 1 .","content":""}],"string":"[\n {\n \"title\": \"INTRODUCTION\",\n \"content\": \"One of the central problems in stellar dynamics is to construct made-to-measure stellar systems. For example, (i) when modelling observations of an elliptical galaxy we wish to find a phase-space distribution function f (r, v) (hereafter DF) that solves Poisson's equation and the collisionless Boltzman equation and reproduces the observed surfacebrightness distribution, rotation curve, velocity-dispersion profile, etc. (in the sense of minimising χ 2 , the mean-square deviation between the observations and the model); (ii) when conducting simulations we wish to construct initial states that are N -body realisations of equilibrium stellar systems with given density profile, rotation curve, bulge/disc ratio, etc.\\nExisting methods for constructing made-to-meaure stellar systems can be classified as follows:\\n• DF-based methods, which solve directly for the DF f (r, v). These generally require that all the integrals of motion are known explicitly (e.g. if the potential is spherical or has Stäckel form) or that the DF is assumed to depend only on known analytic integrals. These include methods that fit observations to few-parameter models with analytic DFs such as spherical Michie or King models; a difficulty with such methods is that the dependence of the derived properties of the galaxy on the choice of the model can be large and uncertain (e.g. Merritt & Tremblay 1994). A more flexible DF-based method is described by Dejonghe (1989), who expands the DF for a spherical system in a truncated series of basis functions and minimises χ 2 subject to the constraint that the DF is positive, using quadratic programming. DF-based methods can be applied to axisymmetric galaxies, and are completely general so long as the DF does not depend on a third integral (e.g. Qian et al . 1995, Kuijken 1995. • Moment-based methods, which find solutions of the Jeans equations (or higher-order velocity moments of the collisionless Boltzmann equation) that minimise χ 2 . This method was used by Binney & Mamon (1982) to construct spherical models of M87 and has been applied to axisymmetric models by Binney et al. (1990), van der Marel et al. (1994), and Magorrian & Binney (1994). A drawback is that this procedure does not guarantee that there is a positive-definite DF with the required velocity moments. • Orbit-based methods (Schwarzschild 1979, Schwarzschild 1993 compute the density distribution of a large library of orbits in a fixed potential, and then determine the weight each orbit must have in order to reproduce the desired final state. In this method the DF or other integrals of motion are not explicitly required, although the DF can be regarded as a sum of deltafunctions on the phase-space surfaces covered by the orbits. Orbit-based methods are generally ill-conditioned. The ill-conditioning can be removed by iterating from a smooth initial guess for the orbit weights using the Richardson-Lucy method (Newton & Binney 1984), but the final weights will then depend in a complicated way on the initial guess. A better procedure is to minimise χ 2 minus a profit function that measures the smoothness of the distribution of orbit weights. The profit function may be an entropy (Richstone & Tremaine 1988) or any other function that is large when the DF is smooth (Merritt 1993). The most flexible of these are orbit-based methods, since they do not require that the integrals of motion are known and the approximation they provide to the DF is known to be positive if the orbit weights are positive.\\nThe goal of this paper is to introduce a novel class of methods for constructing made-to-measure N -body realisations of stellar systems. In the classification above, these methods might be called 'particle-based'; they work by sculpting an initial N -body system until it matches the prescribed density field and other observables.\"\n },\n {\n \"title\": \"THE ALGORITHM\",\n \"content\": \"In most of our discussion we restrict ourselves to constructing stellar systems in a fixed potential Φ(r). We suppose that Φ admits a collisionless equilibrium configuration specified by a distribution function in phase space f (r, v). Thus f satisfies the time-independent collisionless Boltzmann equation:\\nAn 'observable' of the stellar system is a quantity of the form\\nwhere z = (r, v) and Kj is a known kernel. Suitable observables include the surface or volume density at a given point, the surface density times the mean line-of-sight velocity, the surface density times any moment of the line-of-sight velocity, etc. Now consider a system of N particles having weights wi and phase-space positions zi(t) (i ≤ N ). The observables of this system are\\nOur goal is the following: given a set of distinct observables Yj, j = 1, . . . , J, construct a system of N particles orbiting in the potential Φ(r) whose time-averaged observables yj(t) are equal to Yj. We hope that this paper will stimulate interest in seeking better particle-based algorithms than the one we describe.\"\n },\n {\n \"title\": \"The force of change\",\n \"content\": \"The heart of the algorithm is a prescription for changing the weights {wi(t)} as the particles proceed along their fixed orbits in the potential Φ(r). The prescription is similar to that employed by Syer & Tremaine (1995) in a different context. It consists of applying gentle pressure on wi according to the value of ∆j ≡ yj(t)/Yj − 1: if ∆j < 0 then increase wi, and if ∆j > 0 then decrease wi. More precisely we let\\nwhere ǫ is small and positive, and Z is so far arbitrary. The factor wi on the right side ensures that dwi/dt → 0 as wi → 0 so that wi cannot become negative. The factor Kj /Zj ensures that the difference ∆j changes the weight wi only if particle i is contributing to observable j. Equation (4) is closely related to Lucy's (1974) method for solving integral equations with noisy data. Some insight into the solutions of equation (4) is offered by the following argument. Since ǫ is small the weights wi change only over many orbits; thus we may orbit average:\\nHere · denotes the time-average over an interval that is much longer than a typical orbital period but much less than ǫ −1 periods, and Kji is shorthand for the time-independent quantity Kj[zi(t)] . We have also assumed that the fluctuations in Kj [zi(t)] and ∆j(t) are not correlated, which is plausible if many particles contribute to ∆j. We now define\\nwhich obeys the differential equation\\nWhen we are close to convergence (|Θ| ≪ 1), the behaviour of the right side is dominated by changes in Θj rather than changes in wi, so we may replace wi by a constant, w 0 i . Then the vector Θ satisfies the matrix equation\\nwhere the matrix A has components\\nThe solutions to equation (8) have the form\\nwhere the eigenvalues {λm} are solutions of the equation\\nSince A is positive-definite by construction (i.e. x t · A · x > 0 for all x), all of its eigenvalues are positive so λm > 0. This argument suggests that all observables converge to the desired values (| ∆ | → 0) on O(ǫ −1 ) orbital periods, if ǫ is sufficiently small and we start close to the correct final state. For comparison, orbit-based methods evaluate and store the entire matrix Kji (by following the orbits for a fixed time that is much longer than the orbital period; the matrix K is often called the \\\"orbit library\\\"), then solve the matrix equation\\nif N > J the matrix equation is ill-conditioned and must be solved subject to a constraint that maximizes some profit function such as entropy. The storage needed by particlebased methods is O(N J) whereas the storage needed by particle-based methods is only O(N ).\"\n },\n {\n \"title\": \"The kernel\",\n \"content\": \"Suppose that the observable yr is the density at r. What is the appropriate kernel Kr(r ′ , v ′ )? One approach would be to use a smooth kernel (à la SPH) with\\nwhere W is some smooth function, and h is the resolution length (possibly a function of r); W should be normalised so that W (r, h) d 3 r = 1. If h is too large resolution is lost, while if h is too small the observables yj fluctuate strongly because too few particles contribute to a given observable. The smooth-kernel approach is expensive because nearest neighbours have to be found at each timestep and we have therefore adopted a different method. We first divide the coordinate space into bins. Then we set Kr(r ′ , v ′ ) to zero if r and r ′ are not in the same bin, and equal to the inverse of the volume of the bin otherwise. Obviously Kr 1 and Kr 2 are the same if r1 and r2 are in the same bin, so there can be at most one density observable per bin. We are still free to choose the parameter Zj in (4); a simple choice is to set Zj equal to the inverse of the volume of the bin. Thus Kj (zi)/Zj is unity if ri is in bin j, and zero otherwise.\\nAn improvement to this simple scheme, which we employ here, is to borrow from the smooth-kernel approach and to smear each particle into neighbouring bins. Each particle is replaced by a gaussian distribution at the bin centre closest to the particle position, with dispersion equal to half the bin width. Neighbouring bins are assigned a weight corresponding to the integral of the gaussian over those bins (normalized so that the total contribution over all bins is wi). The bin closest to the particle position thus contains about two thirds of its weight. This procedure corresponds nicely to the phenomenon of seeing in the case of a projected observable.\"\n },\n {\n \"title\": \"Resolution and smoothing\",\n \"content\": \"Suppose that we divide the d-dimensional coordinate space into M d bins, and each density observable corresponds to the mass per unit volume in a single bin. Then at any given time there are on average N/M d particles contributing to each observable, and the rms statistical fluctuations in ∆j will be of order δ ∼ (M d /N ) 1/2 . These fluctuations can be kept small if d = 1 (spherical or one-dimensional systems); for example if M = 30 and N = 1000 we have δ ∼ 0.2. However, for triaxial systems (d = 3) the fluctuations will be much larger for reasonable values of M and N .\\nTo improve this situation we employ a form of temporal smoothing which effectively boosts N without any need for extra storage or computation per time step. This is implemented by replacing ∆j (t) in equation (4) with ∆j (t), where\\nand α is small and positive. This quantity is most easily calculated using the equivalent differential equation\\nEach particle is smeared backwards along its trajectory and represents a set of virtual or ghost particles strung out along the orbit with ever decreasing weights. In effect, temporal smoothing increases the effective number of particles from N to\\nwhere ∆t is the timestep and t 1 2 = (ln 2)/α is the half-life of the ghost particles.\\nSome insight into the effect of this smoothing procedure is given by the following argument. As in Section 2.1, let · denote the time-average over an interval that is much longer than a typical orbital period but much less than α −1 . The time-averaged version of equation (15) is\\nwhere Θ is defined by replacing ∆ by ∆ in (6). The evolution of Θ is described by equation (8), with Θ replaced by Θ on the right side. The solution to equations (8) and (17) has the form (10), with eigenvalues λ that satisfy\\nIn the simple case of a single observable, we have λ = 1 2 α ± 1 2 (α 2 − 4αǫA) 1/2 , where A > 0. For α ≫ ǫA, we find λ ≃ ǫA, which is the same convergence rate that would obtain without temporal smoothing. However, for α < 4ǫA, λ is complex, so the observables execute damped oscillations rather than converging smoothly, and the convergence rate is 1 2 Re(α) which is slower than the convergence rate without smoothing whenever α < 2ǫA.\\nWe conclude that excessive temporal smoothing is undesirable and that the maximum smoothing time α −1 should satisfy\\nassuming |A| = O(1).\"\n },\n {\n \"title\": \"Maximum entropy\",\n \"content\": \"If the number of particles exceeds the number of observables, the differential equations (4) are ill-conditioned. In practice this means that the observables {yj (t)} will converge fairly rapidly to {Yj}, but that the individual particle weights {wi} will continue to drift long after the observables have converged. Such behaviour is undesirable as we would like to use the particle weights to predict other properties of the stationary stellar system. To remove the ill-conditioning, we can maximise some form of profit function, such as the entropy\\nwhere {mi} is a pre-determined set of weights (the 'prior'). Thus we maximise the function\\nwhere\\nEquation (4) is replaced by\\nThe factor µ is a measure of the relative contribution of χ 2 and S to the final state: if µ is large we get a smooth solution (in the sense that the {wi} are close to the {mi}) but a large χ 2 , while if µ is small the solution is not smooth but χ 2 is likely to be smaller. The parameter µ can be specified at the start of the calculation, or adjusted as the calculation proceeds using a prescription such as\\nwith 0 < η ≪ 1; in this case χ 2 will converge to the specified value D 2 . In the simulations described here we have kept µ constant. The condition N > J (number of particles exceeds the number of bins) is neither a strict criterion for illconditioning nor a necessary condition for a sensible result, since different observables are not independent (both because of the spatial smoothing described in Section 2.2 and because a single orbit contributes to many observables).\"\n },\n {\n \"title\": \"RESULTS\",\n \"content\": \"\"\n },\n {\n \"title\": \"One-dimensional results\",\n \"content\": \"We first present the results of experiments in one dimension, to illustrate the effects of the various parameters in the algorithm. We use for the background potential\\nwith b = 1 3 , and for the observables we use the density distribution\\nThe period of a low-energy orbit is 2πb 3/2 = 1.209. We use a fourth-order leapfrog with timestep ∆t = 0.1 to integrate the particle orbits, and adjust the weights {wi} after every timestep according to equation (23). To measure the density we use a uniform grid with 16 bins in the range x ∈ [−1, 1]. With b = 1 3 there is a density contrast of about 200 between the inner and outermost bins. We evaluate the success of Figure 2. The rms fractional density error χ (eq. 22) for simulation A (bottom curve), and simulation B (top curve). Simulation B has no orbit averaging. The initial state has small χ 2 , but is not in equilibrium, so χ 2 immediately grows as soon as the particles start to move. The initial state has small χ 2 , but iis not in equilibrium, so χ 2 immediately grows as soon as the particles start to move. Figure 1 shows the evolution of the weights {wi} for simulation A, in which α = 0.0524, ǫ = 0.025 and µ = 0.01. All the {wi} evolve smoothly, and converge in a time of order ǫ −1 . We note that the final equilibrium has a wide range of masses, which suggests that some gain in efficiency might be possible if the particles with smaller w could be replaced with a smaller number of particles with a larger w (for example, we could discard particles whose weights fall below a threshold and replace them with new particles on randomly chosen orbits, or combine low-weight particles with nearby orbits). Figure 2 shows the evolution of χ 2 in simulation A and compares with simulation B in which the orbit averaging is removed (α = ∞). Both the final value of χ 2 and its fractional fluctuation are smaller in A, illustrating the benefits of orbit-averaging. Figure 3 shows χ 2 in simulations A, C and D. Simulations C and A differ only in the parameter ǫ: C has larger ǫ and hence converges more rapidly, although the final states in A and C are very similar. Simulation D and C differ only in the parameter µ: D has µ = 0 (no entropy constraint). The final value of χ 2 is lower in simulation Dand the fluctuations are larger-since the simulation does not have to trade higher χ 2 for lower entropy. and do not converge smoothly to their final values. Comparing simulations C and D we see that this noise is worse for small w when the entropy constraint is removed. Note that the larger w are not much different in C and D. The smaller w do not converge in D, but they do not contribute much to χ 2 . In C the entropy constraint has tied down the smaller values of w. Figure 5 compares the effects of orbit averaging and particle number. Simulation E has 10 times the number of particles as A, but the smoothing time α −1 is 10 times smaller; as a consequence the noise in the two simulations is about the same. The shorter smoothing time in E allows a larger ǫ so that E converges faster than A; note that the time coordinate in Figure 5 has been shrunk by a factor of 10 for simulation A. Thus there is a tradeoff between number of particles and convergence time; simulations A and E each take about 2 minutes of CPU on a DEC Alpha 3000/300. Figure 6 shows the same 6 particle weights as in Figure 4.\"\n },\n {\n \"title\": \"Three-dimensional results\",\n \"content\": \"We have performed a number of three-dimensional simulations. The parameters for the algorithm are in each case the same, as given in the last row of Table 1.\"\n },\n {\n \"title\": \"Mass models\",\n \"content\": \"The mass models were of three types: PS Plummer sphere, with density law given by equation (26), with x denoting the radius and 4πG = 3b 2 . We denote this model 'PS'. PT Triaxial Plummer model with density law given by equation (26) with x denoting the triaxial radius, s, defined by\\nand (A, B, C) = (1.41, 1.12, 1.00). ST Schwarzschild's (1979) model with aixs ratios given by (A, B, C) = (2.00, 1.25, 1.00).\"\n },\n {\n \"title\": \"Force calculation\",\n \"content\": \"The force calculation is carried out in one of two ways: A Analytically. In the case of PT the analytic potential is not that which would be self-consistently generated by the mass model, but rather the Plummer potential with s replacing x in equation (25). F Numerically using the Fourier Convolution Theorem and FFT on a 16 3 grid. Mass is assigned to the grid from the desired mass model, and the forces and potential are calculated once only at the beginning of the simulation. In the ST models the numerical and analytic potentials are significantly different (40% on average) because the model has inifinite mass and has been truncated at finite radius. In the PS models the forces agree to within a few percent.\"\n },\n {\n \"title\": \"Initial condition\",\n \"content\": \"We also use three different types of initial condition: B Particles uniformly distributed in r < 1, with velocities uniform in the range allowed by the condition that their orbits remain in r < 1. In the Schwarzschild potential this initial condition produces a preponderence of box orbits, hence we refer to it as 'box-dominated'. T Particles uniformly distributed in r < 1. Additionally 20% have their velocities chosen as for B, and 80% are given velocities perpendicular to their radius vector with magnitude equal to that of a circular orbit in a spherical potential with the same total force. In the Schwarzschild potential this initial condition produces more tube orbits than B, hence we refer to it as 'tubedominated'. E Particles distributed in r < 1 according to the required mass model. Thus there are extra particles in the inner regions as compared with B or T. Velocities chosen as for T. To summarise the notation by example, simulation PSAB is a Plummer sphere with analytic forces, and box-dominated Table 2. The results of the three-dimensional simulations. Column 1 is the simulation type; column 2 the initial percentage by mass of tube orbits (as defined in the text); column 3 the final percentage by mass; column 4 the final percentage by number; and column 5 the final value of χ. initial condition; simulation STFE is a Schwarzschild triaxial model with Fourier numerical forces, and extra sampling of the inner regions.\\nIn each case we measure the density using a uniform cubical grid with M = 16 elements on a side for a total of 2 12 observables. The core radius of alll the models b = 1 3 , and we restrict particles to orbits with r < 1. Simulation PSAB took about 200 minutes of CPU on the same machine as simulation A, a factor of ∼ 100 longer; most of this factor reflects the factor of 40 increase in the number of particles, while the rest reflects the increased complexity of the differential equations (which must follow 6 phase-space coordinates instead of 2) and increased access time to the much larger array which represents the observables.\\nOrbit classification is carried out by recording the maximum and mininum values of the angular momentum L about the z-axis (the shortest principal axis of the triaxial models). Box orbits tend to reverse direction and 'retrace' their paths, so Lmax ≃ −Lmin. Figure 7 shows the distribution of Lmax and Lmin for simulation PTFT. The box orbits essentially lie in the upper left quadrant. Tube orbits are here defined as those with LmaxLmin > 0. Table 2 gives the percentage of tube orbits in each of the three-dimensional simulations.\\nWith the exception of STFB all simulations converge well: none give final values of χ that are more than a factor of 2 different from the preferred value D = 0.05. Note that the initial condition B, although it works fine for the Plummer models, fails to produce enough tube orbits to support the Schwarzschild potential, and hence the final value of χ is higher than the preferred value D = 0.05. The algorithm converges, but it is forced to substitute smoothness for accuracy because it has too few tube orbits. Also note that the various realisations of the Schwarzschild model have different final weights of tube orbits. This is not surprising as it is expected that the degeneracy in a model which only matches the volume density might lead to a variety of box/tube mixtures.\\nThe behaviour of the weights {wi(t)} and of χ as a function of time look very similar to the one-dimensional case and they are not reproduced here. In Figure 8 we show the density along the principal axes in two of the simulations.\"\n },\n {\n \"title\": \"DISCUSSION\",\n \"content\": \"\"\n },\n {\n \"title\": \"Choosing the parameters\",\n \"content\": \"How should the parameters, (N, M, α, ǫ, µ) be chosen to optimise the investment in CPU? The arguments given in Section 2 lead to the following guidelines.\\nFirst choose D, the value of χ which is as large as can be tolerated-in the simulations above we had χ of a few times 10 −2 . Then choose the resolution required, via the number of resolution elements M d . This informs our choice of N and α through (cf. equation 16)\\nThus, if N is limited by storage requirements, equation (28) determines the desired value of α.\\nOnce we have α then we must choose ǫ smaller than α (equation 19), bearing in mind that the convergence time will be of order ǫ −1 orbital periods. In the simulations we found that ǫ ∼ 0.5α is about right.\\nThe final step is to choose µ. This is largely a matter of taste since µ determines the balance between smoothness and accuracy (Merritt & Tremblay 1992 discuss this issue in another context). The natural choice for µ is the one for which χ 2 reflects the observational errors; and this is the value which will be obtained if µ is allowed to vary according to equation (24). We do not recommend equation (24) for general use however because convergence of µ can be rather slow. One method, which has the advantage of being at least semi-quantitative, is to do a run including equation (24), with D equal to the preferred value of χ. Far from having to converge, it only has to get µ into the right ballpark, and then we go back to constant µ for an extended run.\"\n },\n {\n \"title\": \"The initial condition and prior\",\n \"content\": \"The initial condition of the particles should be chosen to sample phase space as well as possible. Considerable effort is sometimes devoted to this choice in orbit-based calculations. In the simulations presented here we have chosen simply to sample phase-space uniformly (Section 3). With detailed knowledge of the potential and its orbit families, more informed choices could be made.\\nWe can think of the prior {mi}, together with the particle positions and velocities, as a random realisation of some known DF f0(r, v). In the simulations presented here we use {mi} which are all equal. In those where the initial particle positions and velocities sample phase space uniformly, for example, we are effectively using f0 which is initially equal to a constant. If the initial condition is well mixed, as we may reasonably hope, then f0 is independent of time. The smoothness constraint has the effect of driving the DF of the system towards f0.\\nWith a suitable choice of prior {mi}, the final equilibrium should not depend on the initial condition. It should merely reflect the choice of µ, which determines the balance between smoothness and accuracy. The resolution of the final equilibrium in phase space, however, may be affected by the choice of initial condition. The wide range of weights in Figure 1 is a reflection of this fact. However, only perfect knowledge of the target DF would allow one to set up an initial condition which led to an equilibrium with all the {wi} equal. This problem could be alleviated 'on the fly' by methods that kill particles with low weights and split particles with high weights into several daughter particles with similar orbits.\"\n },\n {\n \"title\": \"Comparison with orbit-based methods\",\n \"content\": \"The particle-based method we have described here has several advantages over orbit-based methods.\\n• Particle-based methods use less storage: if there are N particles (or orbits) and J < N observables, orbit-based methods must store O(N J) variables (the contribution of each orbit to each observable), while a particle-based method stores only O(N ) variables (the particle weights at a given time). This advantage is particularly important in systems with a large number of observables (triaxial systems, or systems in which the entire line-ofsight velocity distribution is observed). • Although we have only discussed constructing stellar systems in a fixed potential, it should be possible to generalise particle-based methods so that the potential is determined self-consistently by the particles. Perhaps the best approach would be to expand the potential as a linear combination of a set of basis functions (which can be chosen to preserve any desired symmetries, e.g. spherical symmetry). The coefficient of each basis function, determined from the evolving weights of the particles, could be orbit averaged (cf. equation 14) to reduce the effects of relaxation. • Model construction with orbit-based methods is a multistep process: first compute the luminosity density from the surface brightness; then solve Poisson's equation assuming (say) constant mass-to-light ratio; then integrate orbits in this potential to construct an orbit library (the matrix containing the contribution of each orbit to each observable); then use some inversion/optimization method such as maximum entropy, Lucy's method, or linear or quadratic programming to determine the orbit weights. In a self-consistent particle-based methods all of these steps could be done at the same time.\\nParticle-based methods also have disadvantages compared to orbit-based methods: they use more computing cycles per orbit because the orbits must be followed for a longer time; as in all Monte Carlo methods, accuracy only scales as N 1/2 ; poorly chosen values of parameters such as α and ǫ can cripple the method.\\nThe simple experiments we have described show that particle-based methods might be able to compete with orbitbased methods. Possible improvements include: (i) nonuniform spatial grids so that the resolution is highest near the centre of the galaxy; (ii) methods that kill particles with low weights and split particles with high weights into several daughter particles with similar orbits; (iii) gridless methods based on smooth kernels (Section 2.2); (iv) determining the potential self-consistently from the gravitational field of the particles.\"\n },\n {\n \"title\": \"Figure 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 4 .Figure 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 4 Figure 6 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 7 .Figure 8\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 8\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 1 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"The weights {w i } as a function of time for simulation A.","content":"Figure 1 .\n1The weights {w i } as a function of time for simulation A."},{"id":"","caption":"The density error χ for simulations A (top curve), C (middle curve) and D (bottom curve).","content":"Figure 3 .\n3The density error χ for simulations A (top curve), C (middle curve) and D (bottom curve)."},{"id":"","caption":"A representative sample of 6 of the {w i } for each of the simulations A (top left), B (top right), C (bottom left) and D The density error χ for simulations E and A (with the time coordinate for A shrunk by a factor of ten to facilitate the comparison).","content":"Figure 4 .Figure 5 .\n45A representative sample of 6 of the {w i } for each of the simulations A (top left), B (top right), C (bottom left) and D The density error χ for simulations E and A (with the time coordinate for A shrunk by a factor of ten to facilitate the comparison)."},{"id":"","caption":"shows the evolution of the same 6 particles from simulations A, B, C and D. Comparing A (ǫ = 0.48α) with C and D (ǫ = 95α), we see the effect of excessive temporal smoothing (equation 19): the trajectories wi are noisy As inFigure 4but for simulation E.","content":"Figure 4 Figure 6 .\n46shows the evolution of the same 6 particles from simulations A, B, C and D. Comparing A (ǫ = 0.48α) with C and D (ǫ = 95α), we see the effect of excessive temporal smoothing (equation 19): the trajectories wi are noisy As inFigure 4but for simulation E."},{"id":"","caption":"A scatter plot of Lmax versus L min for simulation PTFT (only 10% of the total number of particles are shown). The values of L plotted are relative to the average magnitude of the angular momentum of each particle. (a). The density along the principal axes in simulation PTFE showing the target mass model (solid) and the final state observed distribution (diamonds).","content":"Figure 7 .Figure 8\n78A scatter plot of Lmax versus L min for simulation PTFT (only 10% of the total number of particles are shown). The values of L plotted are relative to the average magnitude of the angular momentum of each particle. (a). The density along the principal axes in simulation PTFE showing the target mass model (solid) and the final state observed distribution (diamonds)."},{"id":"","caption":"(b). As inFigure 8(a) for simulation STFE.","content":"Figure 8\n8(b). As inFigure 8(a) for simulation STFE."},{"id":"","caption":"The parameter values used in the simulations.","content":"Table 1 .\n1The parameter values used in the simulations.N \n10 2 α \n10 2 ǫ \n10 2 µ \nD \nA \n100 \n5.24 \n2.5 \n1 \n0.05 \nB \n100 \n∞ \n2.5 \n1 \n0.05 \nC \n100 \n5.24 \n500 \n1 \n0.05 \nD \n100 \n5.24 \n500 \n0 \n-\nE \n1000 \n51.9 \n25 \n1 \n0.05 \n3D \n4000 \n5.24 \n2.5 \n0.1 \n0.05 \n\nthe algorithm by examining the time evolution of the {wi} \nand of χ 2 . The parameter values we use are summarised \nin Table 1. In the initial state the particles were uniformly \ndistributed in x ∈ [−1, 1] with equal weights. The veloci-\nties were uniformly distributed in the range allowed by the \ncondition that their orbits are restricted to x ∈ [−1, 1]."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"The weights {w i } as a function of time for simulation A.\",\n \"content\": \"Figure 1 .\\n1The weights {w i } as a function of time for simulation A.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The density error χ for simulations A (top curve), C (middle curve) and D (bottom curve).\",\n \"content\": \"Figure 3 .\\n3The density error χ for simulations A (top curve), C (middle curve) and D (bottom curve).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"A representative sample of 6 of the {w i } for each of the simulations A (top left), B (top right), C (bottom left) and D The density error χ for simulations E and A (with the time coordinate for A shrunk by a factor of ten to facilitate the comparison).\",\n \"content\": \"Figure 4 .Figure 5 .\\n45A representative sample of 6 of the {w i } for each of the simulations A (top left), B (top right), C (bottom left) and D The density error χ for simulations E and A (with the time coordinate for A shrunk by a factor of ten to facilitate the comparison).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"shows the evolution of the same 6 particles from simulations A, B, C and D. Comparing A (ǫ = 0.48α) with C and D (ǫ = 95α), we see the effect of excessive temporal smoothing (equation 19): the trajectories wi are noisy As inFigure 4but for simulation E.\",\n \"content\": \"Figure 4 Figure 6 .\\n46shows the evolution of the same 6 particles from simulations A, B, C and D. Comparing A (ǫ = 0.48α) with C and D (ǫ = 95α), we see the effect of excessive temporal smoothing (equation 19): the trajectories wi are noisy As inFigure 4but for simulation E.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"A scatter plot of Lmax versus L min for simulation PTFT (only 10% of the total number of particles are shown). The values of L plotted are relative to the average magnitude of the angular momentum of each particle. (a). The density along the principal axes in simulation PTFE showing the target mass model (solid) and the final state observed distribution (diamonds).\",\n \"content\": \"Figure 7 .Figure 8\\n78A scatter plot of Lmax versus L min for simulation PTFT (only 10% of the total number of particles are shown). The values of L plotted are relative to the average magnitude of the angular momentum of each particle. (a). The density along the principal axes in simulation PTFE showing the target mass model (solid) and the final state observed distribution (diamonds).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(b). As inFigure 8(a) for simulation STFE.\",\n \"content\": \"Figure 8\\n8(b). As inFigure 8(a) for simulation STFE.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The parameter values used in the simulations.\",\n \"content\": \"Table 1 .\\n1The parameter values used in the simulations.N \\n10 2 α \\n10 2 ǫ \\n10 2 µ \\nD \\nA \\n100 \\n5.24 \\n2.5 \\n1 \\n0.05 \\nB \\n100 \\n∞ \\n2.5 \\n1 \\n0.05 \\nC \\n100 \\n5.24 \\n500 \\n1 \\n0.05 \\nD \\n100 \\n5.24 \\n500 \\n0 \\n-\\nE \\n1000 \\n51.9 \\n25 \\n1 \\n0.05 \\n3D \\n4000 \\n5.24 \\n2.5 \\n0.1 \\n0.05 \\n\\nthe algorithm by examining the time evolution of the {wi} \\nand of χ 2 . The parameter values we use are summarised \\nin Table 1. In the initial state the particles were uniformly \\ndistributed in x ∈ [−1, 1] with equal weights. The veloci-\\nties were uniformly distributed in the range allowed by the \\ncondition that their orbits are restricted to x ∈ [−1, 1].\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". J Binney, G A Mamon, MNRAS. 361Binney, J., & Mamon, G. A. 1982,MNRAS, 200, 361","doi":""},{"text":". J J Binney, R L Davies, G D Illingworth, ApJ. 36178Binney, J. J., Davies, R. L., & Illingworth, G. D. (1990), ApJ, 361, 78","doi":""},{"text":". H Dejonghe, ApJ. 343113Dejonghe, H. (1989),ApJ, 343, 113","doi":""},{"text":". K Kuijken, ApJ. 446194Kuijken, K. (1995), ApJ, 446, 194","doi":""},{"text":". L B Lucy, AJ. 79745Lucy, L. B. 1974, AJ, 79, 745","doi":""},{"text":". J Magorrian, J Binney, MNRAS. 271949Magorrian, J., & Binney, J. (1994), MNRAS, 271, 949","doi":""},{"text":". E E Qian, P T De Zeeuw, R P Van Der Marel, C Hunter, MNRAS. 274602Qian, E. E., de Zeeuw, P. T., van der Marel, R. P., & Hunter, C. (1995), MNRAS, 274, 602","doi":""},{"text":". D Merritt, ApJ. 41379Merritt, D. (1993), ApJ, 413, 79","doi":""},{"text":". D Merritt, B Tremblay, AJ. 108514Merritt, D., & Tremblay, B. (1994), AJ, 108, 514","doi":""},{"text":". A J Newton, J Binney, MNRAS. 210711Newton, A. J., & Binney, J. (1984), MNRAS, 210, 711","doi":""},{"text":". D O Richstone, S Tremaine, ApJ. 32782Richstone, D. O., & Tremaine, S. (1988), ApJ, 327, 82","doi":""},{"text":". D Syer, S Tremaine, MNRAS. 232236ApJSyer, D., & Tremaine, S. (1995), MNRAS, 000, in press Schwarzschild, M. 1979, ApJ, 232, 236","doi":""},{"text":". M Schwarzschild, ApJ. 409563Schwarzschild, M. 1993, ApJ, 409, 563","doi":""}],"string":"[\n {\n \"text\": \". J Binney, G A Mamon, MNRAS. 361Binney, J., & Mamon, G. A. 1982,MNRAS, 200, 361\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J J Binney, R L Davies, G D Illingworth, ApJ. 36178Binney, J. J., Davies, R. L., & Illingworth, G. D. (1990), ApJ, 361, 78\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Dejonghe, ApJ. 343113Dejonghe, H. (1989),ApJ, 343, 113\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Kuijken, ApJ. 446194Kuijken, K. (1995), ApJ, 446, 194\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L B Lucy, AJ. 79745Lucy, L. B. 1974, AJ, 79, 745\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Magorrian, J Binney, MNRAS. 271949Magorrian, J., & Binney, J. (1994), MNRAS, 271, 949\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E E Qian, P T De Zeeuw, R P Van Der Marel, C Hunter, MNRAS. 274602Qian, E. E., de Zeeuw, P. T., van der Marel, R. P., & Hunter, C. (1995), MNRAS, 274, 602\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Merritt, ApJ. 41379Merritt, D. (1993), ApJ, 413, 79\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Merritt, B Tremblay, AJ. 108514Merritt, D., & Tremblay, B. (1994), AJ, 108, 514\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A J Newton, J Binney, MNRAS. 210711Newton, A. J., & Binney, J. (1984), MNRAS, 210, 711\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D O Richstone, S Tremaine, ApJ. 32782Richstone, D. O., & Tremaine, S. (1988), ApJ, 327, 82\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Syer, S Tremaine, MNRAS. 232236ApJSyer, D., & Tremaine, S. (1995), MNRAS, 000, in press Schwarzschild, M. 1979, ApJ, 232, 236\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Schwarzschild, ApJ. 409563Schwarzschild, M. 1993, ApJ, 409, 563\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":497,"cells":{"text":{"kind":"string","value":"\nFrom Arguments to Key Points: Towards Automatic Argument Summarization\nAssociation for Computational LinguisticsCopyright Association for Computational LinguisticsJuly 5 -10, 2020. 2020\n\nRoy Bar roybar@il.ibm.com \nIBM Research\n\n\nHaim Lilach lilache@il.ibm.com \nIBM Research\n\n\nEden Roni Friedman \nIBM Research\n\n\nYoav Kantor yoavka@il.ibm.com \nIBM Research\n\n\nDan Lahav dan.lahav@ibm.com \nIBM Research\n\n\nNoam Slonim noams@il.ibm.com \nIBM Research\n\n\nFrom Arguments to Key Points: Towards Automatic Argument Summarization\n\nProceedings of the 58th Annual Meeting of the Association for Computational Linguistics\nthe 58th Annual Meeting of the Association for Computational LinguisticsAssociation for Computational LinguisticsJuly 5 -10, 2020. 2020\nGenerating a concise summary from a large collection of arguments on a given topic is an intriguing yet understudied problem. We propose to represent such summaries as a small set of talking points, termed key points, each scored according to its salience. We show, by analyzing a large dataset of crowd-contributed arguments, that a small number of key points per topic is typically sufficient for covering the vast majority of the arguments. Furthermore, we found that a domain expert can often predict these key points in advance. We study the task of argument-to-key point mapping, and introduce a novel large-scale dataset for this task. We report empirical results for an extensive set of experiments with this dataset, showing promising performance.\n\nIntroduction\n\nGovernments, businesses and individuals, all need to make decisions on a daily basis: \"Should cannabis be legalized?\",\"Should we develop this product?\", \"Should I become a vegetarian?\". When making an important decision, the process typically comprises several steps: first, we gather as much information as we can about the pros and cons of the proposal under consideration. We may then summarize the collected information as a short list of the main arguments for each side. Lastly, we aim to weigh the pro and con arguments against each other to make the final decision.\n\nWhere can we find relevant arguments for a given topic? In recent years, significant progress was made in the field of argument mining, automatic identification and extraction of argumentative structures in text (Lawrence and Reed, 2020). Specifically, several works focused on topic-related argument mining from the Web or other massive corpora (Levy et al., 2017(Levy et al., , 2018, * All authors equally contributed to this work. 2017; Stab et al., 2018a,b;Ein-Dor et al., 2020). Policy makers in governments or businesses may also conduct surveys to collect from large audiences arguments supporting or contesting some proposal.\n\nEach of the above methods may result in hundreds or thousands of arguments per topic, making it impossible for the decision maker to read and digest such large amounts of information. Several works aimed to alleviate this problem by clustering together related arguments, based on different notions of relatedness, such as similarity (Reimers et al., 2019), frames (Ajjour et al., 2019), and argument facets (Misra et al., 2016). These works, however, did not attempt to create a concise textual summary from the resulting clusters.\n\nIn this work we propose to summarize the arguments supporting each side of the debate by mapping them to a short list of talking points, termed key points. The salience of each key point can be represented by the number of its matching arguments. An example for such summary is shown in Table 1. Key points may be viewed as highlevel arguments. They should be general enough to match a significant portion of the arguments, yet informative enough to make a useful summary.\n\nThe proposed method raises a fundamental question: can a small number of key points effectively summarize massive amount of arguments collected from a large population? In this work we give a positive answer to this question, based on extensive analysis over 28 controversial topics and 7,000 crowd-contributed pro and con arguments for these topics. Furthermore, we found that, given a controversial topic, a domain expert can compose a short, comprehensive list of key points even without looking at the arguments themselves.\n\nMotivated by the above findings, we assume in this work that the key points for each topic are given, and focus on the task of automatically map- ping arguments to these key points. This setting may be viewed as an intermediate step towards fully automatic argument summarization, but also as a valuable setting by itself: argument-to-key point mapping allows measuring the distribution of key points in a massive collection of arguments. It also allows interactive exploration of large argument collections, where key points serve as queries for retrieving matching arguments. In addition, it can be used for novelty detection -identifying unexpected arguments that do not match presupposed key points. We develop the ArgKP dataset for the argumentto-keypoint mapping task, comprising about 24,000 (argument, key point) pairs labeled as matching/non matching. 1 To the best of our knowledge, this is the first dataset for this task. As discussed in the next section in more detail, our dataset is also much larger and far more comprehensive than datasets developed for related tasks such as mapping posts or comments in online debates to reasons or arguments (Hasan and Ng, 2014;Boltužić andŠnajder, 2014).\n\nWe report empirical results for an extensive set of supervised and unsupervised configurations, achieving promising results.\n\nThe main contributions of this work are:\n\n1. We demonstrate, through extensive data annotation and analysis over a variety of topics, the feasibility and effectiveness of summarizing a large set of arguments collected from a large audience by mapping them to a small set of key points.\n\n2. We develop the first large-scale dataset for the task of argument-to-key point mapping.\n\n3. We perform empirical evaluation and analysis of a variety of classification methods for the above task.\n\n2 Related Work\n\n\nArgument Mining\n\nThe starting point for the current work is a collection of pro and con arguments for a given topic. As previously mentioned, these arguments may be collected from a large audience by conducting a survey, or mined automatically from text. Some of the previous work on argument mining focused on specific domains such as legal documents (Moens et al., 2007;Wyner et al., 2010), student essays (Stab and Gurevych, 2017;Persing and Ng, 2016), and user comments on proposed regulations (Park and Cardie, 2014).\n\nMining arguments and argument components for a given topic (also known as context) has been a prominent line of research in argument mining. Levy et al. (2014) introduced the task of context-dependent claim detection in a collection of Wikipedia articles, and Rinott et al. (2015) did the same for context-dependent evidence detection. More recently, several works focused on topic-related argument mining from the Web or other massive corpora (Levy et al., 2017(Levy et al., , 2018Wachsmuth et al., 2017;Stab et al., 2018a,b;Ein-Dor et al., 2020).\n\nStance classification of extracted arguments can be performed as a separate step (Bar-Haim et al., 2017) or jointly with argument detection, as a three-way classification (pro argument/con argument/none), as done by Stab et al. (2018b).\n\n\nArgument Clustering and Summarization\n\nSeveral works have focused on identifying pairs of similar arguments, or clustering similar arguments together. Ajjour et al. (2019) addressed the task of splitting a set of arguments into a set of nonoverlapping frames such as Economics, Environment and Politics. Reimers et al. (2019) classified argument pairs as similar/dissimilar. Misra et al. (2016) aimed to detect argument pairs that are assumed to share the same argument facet, which is similar to our notion of key points. However, they did not attempt to explicitly identify or generate these facets, which remained implicit, but rather focused on detecting similarity between argument pairs. In contrast to these works, we directly map arguments to key points. Egan et al. (2016) proposed to summarize argumentative discussions through the extraction of salient \"points\", where each point is a verb and its syntactic arguments. Applying their unsupervised method to online political debates showed significant improvement over a baseline extractive summarizer, according to human evaluation. While the current work also aims to summarize argumentative content via concise points, our goal is not to extract these points but to accurately map arguments to given points. Our main challenge is to identify the various ways in which the meaning of a point is conveyed in different arguments. The method employed by Egan et al. only matches arguments with the same signature -the same verb, subject and object dependency nodes, hence its ability to capture such variability is limited.\n\nThe line of work that seems most similar to ours is of Hasan and Ng (2014), Boltužić andŠnajder (2014) and Naderi (2016). Hasan and Ng classified posts and individual sentences from online debates into a closed set of reasons, composed manually for each topic. Boltužić andŠnajder mapped comments from one debating website (ProCon.org) to arguments taken from another debating website (iDebate.org). Naderi (2016) addressed a similar task: she used part of the Boltužić andŠnajder corpus as training data for an SVM classifier, which was then tested on sentences and paragraphs from same-sex marriage debates in the Canadian Parliament, annotated with the same set of arguments.\n\nOur work differs from these works in several respects. First, we deal with crowd-contributed arguments, taken from the dataset of Gretz et al. (2020) while these works dealt with posts or comments in debate forums, and parliamentary debates. Second, the dataset developed in this work is far more extensive, covering 28 topics and over 6,500 arguments 2 , as compared to 2-4 topics in the datasets of Boltužić andŠnajder and Hasan and Ng, respectively. This allows us to perform a comprehensive analysis on the feasibility and effectiveness of argument-to-key point mapping over a variety of topics, which has not been possible with previous datasets. Lastly, while Hasan and Ng only perform within-topic classification, where the classifier is trained and tested on the same topic, we address the far more challenging task of cross-topic classification. Boltužić andŠnajder experimented with both within-topic and cross-topic classification, however they used a limited amount of data for training and testing: two topics, with less than 200 comments per topic.\n\nFinally, we point out the similarity between the argument/key point relation and the text/hypothesis relation in textual entailment, also known as natural language inference (NLI) (Dagan et al., 2013). Indeed, Boltužić andŠnajder (2014) used textual entailment as part of their experiments, following the earlier work of Cabrio and Villata (2013), who used textual entailment to detect support/attack relations between arguments.\n\n\nData\n\n\nArguments and Key Points\n\nAs a source of arguments for this work we have used the publicly available IBM-Rank-30k dataset (Gretz et al., 2020). This dataset contains around 30K crowd-sourced arguments, annotated for polarity and point-wise quality. The arguments were collected with strict length limitations, accompanied by extensive quality control measures. Out of the 71 controversial topics in this dataset, we selected the subset of 28 topics for which a corresponding motion exists in the Debatabase repository of the iDebate website 3 . This requirement guaranteed that the selected topics were of high general interest.\n\nWe filtered arguments of low quality (below 0.5) and unclear polarity (below 0.6), to ensure sufficient argument quality in the downstream analysis. We randomly sampled 250 arguments per topic from the set of arguments that passed these filters (7,000 arguments in total for the 28 topics).\n\nDebatabase lists several pro and con points per motion, where each point is typically 1-2 paragraphs long. The headline of each point is a concise sentence that summarizes the point. Initially, we intended to use these point headlines as our key points. However, we found them to be unsuitable for our purpose, due to a large variance in their level of specificity, and their low coverage of the crowd's arguments, as observed in our preliminary analysis.\n\nTo overcome this issue, we let a domain expert who is a professional debater write the key points from scratch. The expert debater received the list of topics and was asked to generate a maximum of 7 key points for each side of the topic, without being exposed to the list of arguments per topic. The maximal number of key points was set according to the typical number of pro and con points in Debatabase motions.\n\nThe process employed by the expert debater to produce the key points comprised several steps:\n\n1. Given a debate topic, generate a list of possible key points in a constrained time frame of 10 minuets per side.\n\n2. Unify related key points that can be expressed as a single key point.\n\n3. Out of the created key points, select a maximum of 7 per side that are estimated to be the most immediate ones, hence the most likely to be chosen by crowd workers.\n\nThe process was completed within two working days. A total of 378 key points were generated, an average of 6.75 per side per topic.\n\n\nMapping Arguments to Key Points\n\n\nAnnotation Process\n\nUsing the Figure Eight crowd labeling platform 4 , we created gold labels for associating the arguments selected as described in Section 3.1 with key points. For each argument, given in the context of its debatable topic, annotators were presented with the key points created for this topic in the relevant stance. They were guided to mark all of the key points this argument can be associated with, and if none are relevant, to select the 'None' option. Each argument was labeled by 8 annotators.\n\nQuality Measures: to ensure the quality of the collected data, the following measures were taken -1. Test questions. Annotators were asked to determine the stance of each argument towards the topic. Similarly to Toledo et al. (2019), this question functioned as a hidden text question 5 . All judgments of annotators failing in more than 10% of the stance questions were discarded.\n\n2. Annotator-κ score. This score, measuring inter annotator agreement, as defined by Toledo et al. (2019), was calculated for each annotator, and all judgments of annotators with annotator-κ < 0.3 were ignored. This score averages all pair-wise Cohen's Kappa (Landis and Koch, 1997) for a given annotator, for any annotator sharing at least 50 judgments with at least 5 other annotators.\n\n3. Selected group of trusted annotators. As in Gretz et al. (2020), the task was only available to a group of annotators which had performed well in previous tasks by our team.\n\nAs described above, the annotation of each key point with respect to a given argument was performed independently, and each annotator could select multiple key points to be associated with each given argument. For the purpose of calculating inter-annotator agreement, we considered (argument, key point) pairs, annotated with a binary label denoting whether the argument was matched to the key point. Fleiss' Kappa for this task was 0.44 (Fleiss, 1971), and Cohen's Kappa was 0.5 (averaging Annotator-κ scores). These scores correspond to \"moderate agreement\" and are comparable to agreement levels previously reported for other annotation tasks in computational argumentation (Boltužić andŠnajder, 2014;Ein-Dor et al., 2020). As for the stance selection question, 98% of the judgments were correct, indicating overall high annotation quality. Data Cleansing: In addition to the above measures, the following annotations were removed from the data: (i) Annotations in which the answer to the stance selection question was wrong; (ii) Annotations in which key point choice was illegalthe 'None' option and one of the key points were both selected. However, the rate of these errors, for each of the annotators, was rather low (< 10% and < 5%, respectively).\n\nArguments left with less than 7 valid judgments after applying the above quality measures and data cleansing were removed from the dataset. 6, 568 labeled arguments remain in the dataset.\n\n\nAnnotation Results\n\nNext, we consolidate the individual annotations as follows. We say that an argument a is mapped to a key point k if at least 60% of the annotators mapped a to k. Recall that an argument can be mapped to more than one key point. Similarly, we say that a has no key point if at least 60% of the annotators mapped a to None (which is equivalent to not selecting any key point for the argument). Otherwise, we say that a is ambiguous, i.e., the annotations were indecisive. Table 2 shows examples for arguments and their matching key points in our dataset.\n\nThe distribution of the arguments in the dataset over the above categories is shown in Table 3. Remarkably, our key points, composed independently of the arguments, were able to cover 72.5% of them, with 5% of the arguments mapped to more than one key point.\n\nWe further investigated the differences between arguments in each category, by comparing their average quality score (taken from the IBM-Rank-30k dataset), number of tokens and number of sentences. The results are shown as additional columns in Table 3. Interestingly, arguments that have no key point tend to be shorter and have lower quality score, comparing to arguments mapped to a single key point; arguments mapped to more than one key point are the longest and have the highest quality. Figure 1 examines the impact of the number of key points on argument coverage. For each topic and stance, we order the key points according to the number of their matched arguments, and add them incrementally. The results indicate that arguments are not trivially mapped to only one or two key points, but a combination of several key points is required to achieve high coverage. The marginal contribution decays for the sixth and seventh key points, suggesting that seven key points indeed suffice for this task.\n\n22.8% of the arguments are ambiguous. Annotations for these arguments are split over several possible key points, none reaching the 60% threshold. For instance, the argument \"homeschooling enables parents with fringe views to push their agenda on their children without allowing exposure to alternative viewpoints.\", had two key points with annotator votes higher than 40%, but below 60%:\n\n1. Homeschools cannot be regulated / standardized.\n\n\nParents are not qualified as teachers.\n\nSuch cases suggest that many arguments are somewhat covered by the key points, but if the judgment is not clear-cut, the different intuitions of the annotators may result in no label receiving the required majority.\n\n\nFinal Dataset Generation\n\nThe ArgKP dataset includes (argument, key point) pairs with binary labels indicating whether the argument is matched to the key point. The dataset was created from the labeled data as follows. We define the label score of a pair as the fraction of annotations that classified the pair as matching . Pairs with label score ≥ 0.6 were labeled as positive (matching). Pairs with label score ≤ 0.15 were labeled as negative (non-matching). Pairs with label score in between these thresholds were removed. We further cleansed our data by discarding key points having less than three matching arguments. This led to the removal of 135 out of the 378 key points and 14,679 out of 38,772 pairs obtained from the previous step.\n\nThe final dataset has 24,093 labeled (argument, key point) pairs, of which 4,998 pairs (20.7%) are positive. It has 6,515 arguments (232.67 per topic), and 243 key points (8.67 key points per topic). For each pair, the dataset also specifies the topic and the stance of the argument towards the topic.\n\nWe assessed the quality of the resulting dataset by having an expert annotator 6 mapping 100 ran- Forcing members of a profession to retire at a certain age creates an experience drain.\n\nA mandatory retirement age decreases institutional knowledge. We should ban the use of child actors.\n\nChild actors are fine to use as long as there is a responsible adult watching them.\n\nChild performers should not be banned as long as there is supervision/regulation. We should close Guantanamo Bay detention camp.\n\nGuantanamo can provide security for accused terrorists who would be hurt in the general prison population.\n\nThe Guantanamo bay detention camp is better for prisoners than the alternatives.\n\nAssisted suicide should be a criminal offence.\n\nPeople have a basic right to bodily autonomy, deciding whether or not to die with minimal suffering and dignity is integral to that right.\n\nPeople should have the freedom to choose to end their life. Assisted suicide gives dignity to the person that wants to commit it. We should ban human cloning.\n\nThe world is already overpopulated, cloning humans will only contribute to this problem.\n\nNo key point domly sampled arguments to key points, and comparing the annotations to the gold labels for all the corresponding pairs in the dataset. We obtained a remarkably high Cohen's Kappa of 0.82 (\"almost perfect agreement\"), validating the high quality of the dataset.\n\n\nExperiments\n\n\nExperimental Setup\n\nWe perform the task of matching arguments to key points in two steps. In the Match Scoring step (Section 4.1.1), we generate a score for each argument and key point. Then, in the Match Classification step (Section 4.1.2), we use these scores to classify the pairs as matching or non-matching. We perform 4-fold cross-validation over the ArgKP dataset. Each fold comprises 7 test topics, 17 train topics and 4 development topics.\n\n\nMatch Scoring\n\nWe experimented with both unsupervised and supervised methods for computing a match score for a given (argument, key point) pair. We also explored transfer learning from the related task of natural language inference (NLI).\n\n\nUnsupervised Methods\n\n• Tf-Idf. In order to assess the role of lexical overlap in the matching task, we represent each argument and key point as tf-idf weighted word vectors and use their cosine similarity as the match score.\n\n• Word Embedding. We examined averaged word embeddings using GloVe (Pennington et al., 2014) and BERT (Devlin et al., 2019). GloVe is a context independent model that computes a single embedding for each word.\n\nBERT is a contextualized embedding model that takes the entire sentence into account. We also experimented with other embedding methods that under-performed BERT and thus their results are not reported here: Universal Sentence Encoder (Cer et al., 2018) and In-ferSent (Conneau et al., 2017). Again, we use cosine similarity to compute the match score.\n\nSupervised Methods. We fine tuned the BERTbase-uncased and BERT-large-uncased models (Devlin et al., 2019) to predict matches between argument and key point pairs. We added a linear fully connected layer of size 1 followed by a sigmoid layer to the special [CLS] token in the BERT model, and trained it for three epochs with a learning rate of 2e-5 and a binary cross entropy loss.\n\nNLI Transfer Learning. We also experimented with transfer learning from NLI to our task of argument-to-key point match classification. This was motivated by the similarity between these tasks (as discussed in Section 2.2), as well as the availability of large-scale NLI labeled datasets. We con-sidered the Stanford (SNLI) and the Multi-Genre (MNLI) datasets (Bowman et al., 2015;Williams et al., 2018), each comprising hundreds of thousands of labeled premise-hypothesis pairs. Pairs labeled as ENTAILMENT were considered positive instances, while the rest of the pairs, labeled as NEUTRAL or CONTRADICTION were considered negative. We trained BERT-base and BERT-large models on each of these datasets, following the procedure described above.\n\n\nMatch Classification\n\nIn the match classification step we select the matching key points for each argument, based on their respective matching scores. The classification can be done locally, treating each pair individually, or globally, by examining all possible key points for each argument. We compared the following policies for selecting matching key points for a given argument.\n\nThreshold. For each fold, we find the threshold on the match score that maximizes the F1 score for the positive (matching) class. Pairs whose score exceeds the learned threshold are considered matched.\n\nBest Match (BM). Using a threshold is not optimal for our data, where most arguments have at most one matched key point. A natural solution is to select the best matching key point. For each argument, we consider all key points for the same topic and stance as candidates and predict only the candidate with the highest match score as matched to the argument and the rest as unmatched. Note that this is the only fully unsupervised selection policy, as it does not require labeled data for learning a threshold.\n\nBM+Threshold. The BM policy always assigns exactly one key point for each argument, while 27.5% of the arguments in our data are not matched to any key point. To address this, we combine the two former policies. The top matching key point is considered a match only if its match score exceeds the learned threshold.\n\nDual Threshold. In order to account for arguments with more than one matching key point, two thresholds are learned. If two key points exceed the lower threshold and at least one of them exceeds the upper threshold, both will be matched. Otherwise, it works the same as the BM+Threshold policy using only the lower threshold. This allows for zero to two matches per argument. Thresholds are learned from the development set for supervised match scoring methods, and from both train and development set for unsupervised match scoring methods. Table 4 compares the various match scoring methods, all using the Threshold key point selection policy. Results are obtained by micro-averaging over the argument-key point pairs in each fold, and averaging over the different folds. We consider Precision, Recall and F1 of the positive class, as well as the overall accuracy. We also list for reference the majority class baseline that always predicts \"no match\", and the random baseline, which randomly predicts the positive class according to its probability in the training data.\n\n\nResults\n\n\nMatch Scoring Methods\n\nThe unsupervised models fail to capture the relation between the argument and the key points. Tf-Idf and Glove perform the worst, showing that simple lexical similarity is insufficient for this task. BERT embedding does better but still reaches a relatively low F1 score of 0.4.\n\nIn contrast to the unsupervised models, supervised models are shown to perform well. BERT with fine tuning leads to a substantial improvement, reaching F1 score of 0.657 with the BERT-base model, and 0.684 with the BERT-large model.\n\nBERT Models trained on NLI data are considerably better than the unsupervised methods, with the best model reaching F1 of 0.526, yet their performance is still far below the supervised models trained on our ArgKP dataset. This may reflect both the similarities and the differences between NLI and the current task. We have also experimented with combining these two types of data in cascade: BERT was first trained on a large NLI dataset (SNLI, MNLI or their union), and was then fine-tuned on the smaller ArgKP data. However, it did not improve the supervised results.\n\nError Analysis. By analyzing the top errors of the supervised classifier (BERT-large), we found several systematic patterns of errors. In most cases, non-matching arguments and key points received a high match score in one of the following cases:\n\n• They share some key phrases. For example:\n\n\"It is unfair to only subsidize vocational education. Achieving a more advanced education is very expensive and it would also need to be subsidized.\" and \"Subsidizing vocational education is expensive\".\n\n• They share a large portion of the sentence, but not the main point, for example: \"Women should be able to fight if they are strong enough\" and \"Women should be able to serve in combat if they choose to\".\n\n• They are at least partially related, but labeled as non-matching due to a better fitting key point for the same argument. For example: \"We should subsidize space exploration because it increases the knowledge of the universe we are in\" and \"Space exploration improves science/technology\" can be considered matched, but were labeled as unmatched due to the key point \"Space exploration unravels information about the universe\". Using the Best Match policy helps in these cases.\n\nFor arguments and key points that were labeled as matched but received a low match score, the relation was in many cases implied or required some further knowledge, for examples: \"Journalism is an essential part of democracy and freedom of expression and should not be subsidized by the state.\" and \"government intervention has the risk of inserting bias/harming objectivity\". Table 5 compares different key point selection policies, all using the best performing match scoring method: BERT-large fine-tuned on ArgKP. We report the results over the whole dataset (\"all arguments\"), as well as the subsets of arguments having none, single or multiple matching key points according to the labeled data. In case of no matches we present the accuracy, as recall and F1 scores are undefined. When considering all the arguments, the Dual Threshold policy achieves the best F1 score of 0.73. The Threshold method performs well for arguments with no matches or multiple matches. When there is exactly one match (the common case in our data), it has lower precision. The Best Match policy performs well when there is a single match, but is not able to cope with arguments that have no matches or have multiple matches. The BM+Threshold method combines the two and is useful when there are no matching key points or a single matching key point, but still have lower recall when there are multiple matching key points. The Dual Threshold method improves the recall and therefore the F1 score for multiple matches while maintaining good performance for arguments with single or no matches. Figure 2 shows Precision-Recall trade-off for the various policies, using the different possible thresholds, computed for one of the folds. For each policy, we specify the best F1 score, as well as the F1 score obtained for the selected threshold, which was optimized over the development set. The Threshold policy allows to control recall, up to one (where the threshold is zero), at the price of low precision. The BM+Threshold policy generates the highest precision, but low recall, since at most one candidate is selected. Note that when the threshold is zero, the BM+Threshold policy is equivalent to the BM policy. The Dual Threshold policy offers the best trade-off, for mid-range precision and recall.\n\n\nKey Point Selection Policies\n\n\nConclusion\n\nThis work addressed the practical problem of summarizing a large collection of arguments on a given topic. We proposed to represent such summaries as a set of key points scored according to their relative salience. Such summary aims to provide both textual and quantitative views of the argument data in a concise form. We demonstrated the feasibility and effectiveness of the proposed approach through extensive data annotation and analysis. We showed that a domain expert can quickly come up with a short list of pro and con key points per topic, that would capture the gist of crowd-contributed arguments, even without being exposed to the arguments themselves. We studied the problem of automatically matching arguments to key points, and developed the first large-scale dataset for this task, which we make publicly available.\n\nOur experimental results demonstrate that the problem is far from trivial, and cannot be effectively solved using unsupervised methods based on word or sentence-level embedding. However, by using state of the art supervised learning methods for match scoring, together with an appropriate key point selection policy for match classification, we were able to achieve promising results on this task. The natural next step for this work is the challenging task of automatic key point generation. In addition, we plan to apply the methods presented in this work also to automatically-mined arguments. Finally, detecting the more implicit relations between the argument and the key point, as seen in our error analysis, is another intriguing direction for future work.\n\nFigure 1 :\n1Argument coverage per number of key points.\n\nFigure 2 :\n2Precision/Recall trade-off for different key point selection policies. For each method, the highest F1 score, as well as the F1 score for the chosen threshold are specified. For the Best Match + Threshold policy, these two scores coincide.\n\n\nHomeschooling should be banned #Args Pro Mainstream schools are essential to develop social skills. 61 Parents are not qualified as teachers. 20 Homeschools cannot be regulated/standardized. 15 Mainstream schools are of higher educational quality. 9 Con Parents should be permitted to choose the education of their children.28 Homeschooling is often the best option for catering for the needs of exceptional/religious/ill/disabled students.Homeschools can be personalized to the child's pace/needs. 21 Mainstream schools have a lot of violence/bullying. 21 The home is a good learning environment. 13 Parents will have more ability to pay-attention/educate their child. 7Table 1: A sample key point-based summary, extracted from our ArgKP dataset.25 \n\n\n\nTable 2 :\n2Examples for key point association to arguments.% Arguments Quality # Tokens # Sentences \nNo key point \n4.7% \n0.75 \n16.35 \n1.09 \nAmbiguous \n22.8% \n0.80 \n18.97 \n1.15 \nSingle key point \n67.5% \n0.84 \n18.54 \n1.15 \nMultiple key points \n5.0% \n0.91 \n23.66 \n1.33 \n\n\n\nTable 3 :\n3Argument statistics by key point matches.\n\nTable 4 :\n4Comparison of match scoring methods, using the Threshold selection policy. P, R and F1 refer to the positive class. Acc is the accuracy.All \nSingle \nMultiple \nNo \nArguments \nKey Point \nKey Points \nKey Points \nAcc \nP \nR \nF1 \nP \nR \nF1 \nP \nR \nF1 \nAcc \nThreshold \n.868 .685 .688 .684 \n.720 .686 .701 \n.904 .690 .782 \n.933 \nBest Match \n.876 .696 .711 .703 \n.836 .747 .789 \n.936 .448 .606 \n.839 \nBM+Threshold \n.890 .772 .665 .713 \n.856 .699 .769 \n.941 .421 .580 \n.915 \nDual Threshold \n.887 .721 .740 .730 \n.784 .752 .767 \n.945 .656 .773 \n.908 \n\n\n\nTable 5 :\n5Comparing key point selection policies, using BERT-large trained on the ArgKP dataset for match scoring.\nThe dataset is available at https://www.research. ibm.com/haifa/dept/vst/debating_data. shtml\nAs detailed in the next section, a few hundreds of arguments out of the initial 7,000 were filtered in the process of constructing the dataset.3 https://idebate.org/debatabase\nhttp://figure-eight.com\n Unlike Toledo et al., the results were analyzed after the task was completed, and the annotators were not aware of their success/failure.\nA professional debater who was not involved in the development of the dataset.\nAcknowledgmentsWe would like to thank the anonymous reviewers for their helpful comments and suggestions.\nAssociation for Computational Linguistics. Yamen Ajjour, Milad Alshomary, Henning Wachsmuth, Benno Stein, 10.18653/v1/D19-1290Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP). the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)Hong Kong, ChinaModeling frames in argumentationYamen Ajjour, Milad Alshomary, Henning Wachsmuth, and Benno Stein. 2019. Modeling frames in ar- gumentation. In Proceedings of the 2019 Confer- ence on Empirical Methods in Natural Language Processing and the 9th International Joint Confer- ence on Natural Language Processing (EMNLP- IJCNLP), pages 2922-2932, Hong Kong, China. As- sociation for Computational Linguistics.\n\nStance classification of context-dependent claims. Roy Bar-Haim, Indrajit Bhattacharya, Francesco Dinuzzo, Amrita Saha, Noam Slonim, Proceedings of the 15th Conference of the European Chapter of the Association for Computational Linguistics. the 15th Conference of the European Chapter of the Association for Computational LinguisticsAssociation for Computational Linguistics1Roy Bar-Haim, Indrajit Bhattacharya, Francesco Din- uzzo, Amrita Saha, and Noam Slonim. 2017. Stance classification of context-dependent claims. In Pro- ceedings of the 15th Conference of the European Chapter of the Association for Computational Lin- guistics: Volume 1, Long Papers, pages 251-261. Association for Computational Linguistics.\n\nBack up your stance: Recognizing arguments in online discussions. Filip Boltužić, Janšnajder, 10.3115/v1/W14-2107Proceedings of the First Workshop on Argumentation Mining. the First Workshop on Argumentation MiningBaltimore, MarylandAssociation for Computational LinguisticsFilip Boltužić and JanŠnajder. 2014. Back up your stance: Recognizing arguments in online discus- sions. In Proceedings of the First Workshop on Ar- gumentation Mining, pages 49-58, Baltimore, Mary- land. Association for Computational Linguistics.\n\nA large annotated corpus for learning natural language inference. R Samuel, Gabor Bowman, Christopher Angeli, Christopher D Potts, Manning, 10.18653/v1/D15-1075Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing. the 2015 Conference on Empirical Methods in Natural Language ProcessingLisbon, PortugalAssociation for Computational LinguisticsSamuel R. Bowman, Gabor Angeli, Christopher Potts, and Christopher D. Manning. 2015. A large anno- tated corpus for learning natural language inference. In Proceedings of the 2015 Conference on Empiri- cal Methods in Natural Language Processing, pages 632-642, Lisbon, Portugal. Association for Compu- tational Linguistics.\n\nA natural language bipolar argumentation approach to support users in online debate interactions. Elena Cabrio, Serena Villata, 10.1080/19462166.2013.862303Argument & Computation. 43Elena Cabrio and Serena Villata. 2013. A natural lan- guage bipolar argumentation approach to support users in online debate interactions. Argument & Computation, 4(3):209-230.\n\nUniversal sentence encoder. Daniel Cer, Yinfei Yang, Sheng-Yi Kong, Nan Hua, Nicole Limtiaco, Rhomni St John, Noah Constant, Mario Guajardo-Cespedes, Steve Yuan, Chris Tar, arXiv:1803.11175arXiv preprintDaniel Cer, Yinfei Yang, Sheng-yi Kong, Nan Hua, Nicole Limtiaco, Rhomni St John, Noah Constant, Mario Guajardo-Cespedes, Steve Yuan, Chris Tar, et al. 2018. Universal sentence encoder. arXiv preprint arXiv:1803.11175.\n\nSupervised learning of universal sentence representations from natural language inference data. Alexis Conneau, Douwe Kiela, Holger Schwenk, Loic Barrault, Antoine Bordes, arXiv:1705.02364arXiv preprintAlexis Conneau, Douwe Kiela, Holger Schwenk, Loic Barrault, and Antoine Bordes. 2017. Supervised learning of universal sentence representations from natural language inference data. arXiv preprint arXiv:1705.02364.\n\nRecognizing Textual Entailment: Models and Applications. Synthesis Lectures on Human Language Technologies. Dan Ido Dagan, Mark Roth, Fabio Massimo Sammons, Zanzotto, 10.2200/S00509ED1V01Y201305HLT023Morgan & Claypool PublishersIdo Dagan, Dan Roth, Mark Sammons, and Fabio Mas- simo Zanzotto. 2013. Recognizing Textual Entail- ment: Models and Applications. Synthesis Lectures on Human Language Technologies. Morgan & Clay- pool Publishers.\n\nBERT: Pre-training of deep bidirectional transformers for language understanding. Jacob Devlin, Ming-Wei Chang, Kenton Lee, Kristina Toutanova, 10.18653/v1/N19-1423Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies. the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language TechnologiesMinneapolis, MinnesotaLong and Short Papers1Association for Computational LinguisticsJacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019. BERT: Pre-training of deep bidirectional transformers for language under- standing. In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pages 4171-4186, Minneapolis, Minnesota. Associ- ation for Computational Linguistics.\n\nSummarising the points made in online political debates. Charlie Egan, Advaith Siddharthan, Adam Wyner, 10.18653/v1/W16-2816Proceedings of the Third Workshop on Argument Mining (ArgMining2016). the Third Workshop on Argument Mining (ArgMining2016)Berlin, GermanyAssociation for Computational LinguisticsCharlie Egan, Advaith Siddharthan, and Adam Wyner. 2016. Summarising the points made in online po- litical debates. In Proceedings of the Third Work- shop on Argument Mining (ArgMining2016), pages 134-143, Berlin, Germany. Association for Compu- tational Linguistics.\n\nRanit Aharonov, and Noam Slonim. 2020. Corpus wide argument mining -a working solution. Eyal Liat Ein-Dor, Lena Shnarch, Alon Dankin, Benjamin Halfon, Ariel Sznajder, Carlos Gera, Martin Alzate, Leshem Gleize, Yufang Choshen, Yonatan Hou, Bilu, AAAI. Liat Ein-Dor, Eyal Shnarch, Lena Dankin, Alon Hal- fon, Benjamin Sznajder, Ariel Gera, Carlos Alzate, Martin Gleize, Leshem Choshen, Yufang Hou, Yonatan Bilu, Ranit Aharonov, and Noam Slonim. 2020. Corpus wide argument mining -a working solution. In AAAI.\n\nMeasuring nominal scale agreement among many raters. Joseph L Fleiss, Psychological Bulletin. 765Joseph L. Fleiss. 1971. Measuring nominal scale agree- ment among many raters. Psychological Bulletin, 76(5):378-382.\n\nRanit Aharonov, and Noam Slonim. 2020. A large-scale dataset for argument quality ranking: Construction and analysis. Shai Gretz, Roni Friedman, Edo Cohen-Karlik, Assaf Toledo, Dan Lahav, AAAI. Shai Gretz, Roni Friedman, Edo Cohen-Karlik, As- saf Toledo, Dan Lahav, Ranit Aharonov, and Noam Slonim. 2020. A large-scale dataset for argument quality ranking: Construction and analysis. In AAAI.\n\nWhy are you taking this stance? identifying and classifying reasons in ideological debates. Saidul Kazi, Vincent Hasan, Ng, 10.3115/v1/D14-1083Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP). the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP)Doha, QatarAssociation for Computational LinguisticsKazi Saidul Hasan and Vincent Ng. 2014. Why are you taking this stance? identifying and classify- ing reasons in ideological debates. In Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 751- 762, Doha, Qatar. Association for Computational Linguistics.\n\nThe measurements of observer agreement for categorical data. J R Landis, G G Koch, Biometrics. 33J. R. Landis and G. G. Koch. 1997. The measurements of observer agreement for categorical data. Biomet- rics, 33:159-174.\n\nArgument mining: A survey. John Lawrence, Chris Reed, 10.1162/coli_a_00364Computational Linguistics. 454John Lawrence and Chris Reed. 2020. Argument mining: A survey. Computational Linguistics, 45(4):765-818.\n\nContext dependent claim detection. Ran Levy, Yonatan Bilu, Daniel Hershcovich, Ehud Aharoni, Noam Slonim, Proceedings of COLING 2014, the 25th International Conference on Computational Linguistics: Technical Papers. COLING 2014, the 25th International Conference on Computational Linguistics: Technical PapersDublin, IrelandDublin City University and Association for Computational LinguisticsRan Levy, Yonatan Bilu, Daniel Hershcovich, Ehud Aharoni, and Noam Slonim. 2014. Context depen- dent claim detection. In Proceedings of COLING 2014, the 25th International Conference on Compu- tational Linguistics: Technical Papers, pages 1489- 1500, Dublin, Ireland. Dublin City University and Association for Computational Linguistics.\n\nTowards an argumentative content search engine using weak supervision. Ran Levy, Ben Bogin, Shai Gretz, Ranit Aharonov, Noam Slonim, Proceedings of the 27th International Conference on Computational Linguistics. the 27th International Conference on Computational LinguisticsAssociation for Computational LinguisticsRan Levy, Ben Bogin, Shai Gretz, Ranit Aharonov, and Noam Slonim. 2018. Towards an argumentative con- tent search engine using weak supervision. In Pro- ceedings of the 27th International Conference on Computational Linguistics, pages 2066-2081. Asso- ciation for Computational Linguistics.\n\nUnsupervised corpus-wide claim detection. Ran Levy, Shai Gretz, Benjamin Sznajder, Shay Hummel, Ranit Aharonov, Noam Slonim, 10.18653/v1/W17-5110Proceedings of the 4th Workshop on Argument Mining. the 4th Workshop on Argument MiningAssociation for Computational LinguisticsRan Levy, Shai Gretz, Benjamin Sznajder, Shay Hum- mel, Ranit Aharonov, and Noam Slonim. 2017. Un- supervised corpus-wide claim detection. In Pro- ceedings of the 4th Workshop on Argument Mining, pages 79-84. Association for Computational Lin- guistics.\n\nMeasuring the similarity of sentential arguments in dialogue. Amita Misra, Brian Ecker, Marilyn Walker, 10.18653/v1/W16-3636Proceedings of the 17th Annual Meeting of the Special Interest Group on Discourse and Dialogue. the 17th Annual Meeting of the Special Interest Group on Discourse and DialogueLos AngelesAssociation for Computational LinguisticsAmita Misra, Brian Ecker, and Marilyn Walker. 2016. Measuring the similarity of sentential arguments in dialogue. In Proceedings of the 17th Annual Meet- ing of the Special Interest Group on Discourse and Dialogue, pages 276-287, Los Angeles. Association for Computational Linguistics.\n\nAutomatic detection of arguments in legal texts. Marie-Francine Moens, Erik Boiy, 10.1145/1276318.1276362Proceedings of the 11th International Conference on Artificial Intelligence and Law, ICAIL '07. the 11th International Conference on Artificial Intelligence and Law, ICAIL '07New York, NY, USAACMRaquel Mochales Palau, and Chris ReedMarie-Francine Moens, Erik Boiy, Raquel Mochales Palau, and Chris Reed. 2007. Automatic detection of arguments in legal texts. In Proceedings of the 11th International Conference on Artificial Intelli- gence and Law, ICAIL '07, pages 225-230, New York, NY, USA. ACM.\n\nArgumentation mining in parliamentary discourse. Nona Naderi, Proceedings of the 11th International Conference of the Ontario Society for the Study of Argumentation. the 11th International Conference of the Ontario Society for the Study of ArgumentationNona Naderi. 2016. Argumentation mining in parlia- mentary discourse. In Proceedings of the 11th Inter- national Conference of the Ontario Society for the Study of Argumentation, pages 1-9.\n\nIdentifying appropriate support for propositions in online user comments. Joonsuk Park, Claire Cardie, 10.3115/v1/W14-2105Proceedings of the First Workshop on Argumentation Mining. the First Workshop on Argumentation MiningBaltimore, MarylandAssociation for Computational LinguisticsJoonsuk Park and Claire Cardie. 2014. Identifying appropriate support for propositions in online user comments. In Proceedings of the First Workshop on Argumentation Mining, pages 29-38, Baltimore, Maryland. Association for Computational Linguis- tics.\n\nGlove: Global vectors for word representation. Jeffrey Pennington, Richard Socher, Christopher Manning, Proceedings of the 2014 conference on empirical methods in natural language processing (EMNLP). the 2014 conference on empirical methods in natural language processing (EMNLP)Jeffrey Pennington, Richard Socher, and Christopher Manning. 2014. Glove: Global vectors for word rep- resentation. In Proceedings of the 2014 conference on empirical methods in natural language process- ing (EMNLP), pages 1532-1543.\n\nEnd-to-end argumentation mining in student essays. Isaac Persing, Vincent Ng, 10.18653/v1/N16-1164Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies. the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language TechnologiesSan Diego, CaliforniaAssociation for Computational LinguisticsIsaac Persing and Vincent Ng. 2016. End-to-end ar- gumentation mining in student essays. In Proceed- ings of the 2016 Conference of the North Ameri- can Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 1384-1394, San Diego, California. Association for Computational Linguistics.\n\nClassification and clustering of arguments with contextualized word embeddings. Nils Reimers, Benjamin Schiller, Tilman Beck, Johannes Daxenberger, Christian Stab, Iryna Gurevych, 10.18653/v1/P19-1054Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics. the 57th Annual Meeting of the Association for Computational LinguisticsFlorence, ItalyAssociation for Computational LinguisticsNils Reimers, Benjamin Schiller, Tilman Beck, Jo- hannes Daxenberger, Christian Stab, and Iryna Gurevych. 2019. Classification and clustering of ar- guments with contextualized word embeddings. In Proceedings of the 57th Annual Meeting of the As- sociation for Computational Linguistics, pages 567- 578, Florence, Italy. Association for Computational Linguistics.\n\nShow me your evidence -an automatic method for context dependent evidence detection. Ruty Rinott, Lena Dankin, Carlos Alzate Perez, Mitesh M Khapra, Ehud Aharoni, Noam Slonim, 10.18653/v1/D15-1050Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing. the 2015 Conference on Empirical Methods in Natural Language ProcessingLisbon, PortugalAssociation for Computational LinguisticsRuty Rinott, Lena Dankin, Carlos Alzate Perez, Mitesh M. Khapra, Ehud Aharoni, and Noam Slonim. 2015. Show me your evidence -an au- tomatic method for context dependent evidence de- tection. In Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing, pages 440-450, Lisbon, Portugal. Association for Computational Linguistics.\n\nArgumentext: Searching for arguments in heterogeneous sources. Christian Stab, Johannes Daxenberger, Chris Stahlhut, Tristan Miller, Benjamin Schiller, Christopher Tauchmann, Steffen Eger, Iryna Gurevych, 10.18653/v1/N18-5005Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Demonstrations. the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: DemonstrationsAssociation for Computational LinguisticsChristian Stab, Johannes Daxenberger, Chris Stahlhut, Tristan Miller, Benjamin Schiller, Christopher Tauchmann, Steffen Eger, and Iryna Gurevych. 2018a. Argumentext: Searching for arguments in heterogeneous sources. In Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Demon- strations, pages 21-25. Association for Computa- tional Linguistics.\n\nParsing argumentation structures in persuasive essays. Christian Stab, Iryna Gurevych, 10.1162/COLI_a_00295Computational Linguistics. 433Christian Stab and Iryna Gurevych. 2017. Parsing ar- gumentation structures in persuasive essays. Com- putational Linguistics, 43(3):619-659.\n\nCrosstopic argument mining from heterogeneous sources. Christian Stab, Tristan Miller, Benjamin Schiller, Pranav Rai, Iryna Gurevych, Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing. the 2018 Conference on Empirical Methods in Natural Language ProcessingAssociation for Computational LinguisticsChristian Stab, Tristan Miller, Benjamin Schiller, Pranav Rai, and Iryna Gurevych. 2018b. Cross- topic argument mining from heterogeneous sources. In Proceedings of the 2018 Conference on Empiri- cal Methods in Natural Language Processing, pages 3664-3674. Association for Computational Linguis- tics.\n\nAutomatic argument quality assessment -new datasets and methods. Assaf Toledo, Shai Gretz, Edo Cohen-Karlik, Roni Friedman, Elad Venezian, Dan Lahav, Michal Jacovi, Ranit Aharonov, Noam Slonim, 10.18653/v1/D19-1564Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP). the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)Hong Kong, ChinaAssociation for Computational LinguisticsAssaf Toledo, Shai Gretz, Edo Cohen-Karlik, Roni Friedman, Elad Venezian, Dan Lahav, Michal Ja- covi, Ranit Aharonov, and Noam Slonim. 2019. Au- tomatic argument quality assessment -new datasets and methods. In Proceedings of the 2019 Confer- ence on Empirical Methods in Natural Language Processing and the 9th International Joint Confer- ence on Natural Language Processing (EMNLP- IJCNLP), pages 5624-5634, Hong Kong, China. As- sociation for Computational Linguistics.\n\nBuilding an argument search engine for the web. Henning Wachsmuth, Martin Potthast, Khalid Al Khatib, Yamen Ajjour, Jana Puschmann, Jiani Qu, Jonas Dorsch, Viorel Morari, Janek Bevendorff, Benno Stein, 10.18653/v1/W17-5106Proceedings of the 4th Workshop on Argument Mining. the 4th Workshop on Argument MiningAssociation for Computational LinguisticsHenning Wachsmuth, Martin Potthast, Khalid Al Khatib, Yamen Ajjour, Jana Puschmann, Jiani Qu, Jonas Dorsch, Viorel Morari, Janek Bevendorff, and Benno Stein. 2017. Building an argument search engine for the web. In Proceedings of the 4th Workshop on Argument Mining, pages 49-59. Association for Computational Linguistics.\n\nA broad-coverage challenge corpus for sentence understanding through inference. Adina Williams, Nikita Nangia, Samuel Bowman, 10.18653/v1/N18-1101Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies. the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language TechnologiesNew Orleans, LouisianaLong Papers1Association for Computational LinguisticsAdina Williams, Nikita Nangia, and Samuel Bowman. 2018. A broad-coverage challenge corpus for sen- tence understanding through inference. In Proceed- ings of the 2018 Conference of the North American Chapter of the Association for Computational Lin- guistics: Human Language Technologies, Volume 1 (Long Papers), pages 1112-1122, New Orleans, Louisiana. Association for Computational Linguis- tics.\n\nApproaches to text mining arguments from legal cases. Adam Wyner, Raquel Mochales-Palau, Marie-Francine Moens, David Milward, Semantic Processing of Legal Texts: Where the Language of Law Meets the Law of Language. Berlin, HeidelbergSpringer-VerlagAdam Wyner, Raquel Mochales-Palau, Marie-Francine Moens, and David Milward. 2010. Approaches to text mining arguments from legal cases. In Seman- tic Processing of Legal Texts: Where the Language of Law Meets the Law of Language, page 60-79, Berlin, Heidelberg. Springer-Verlag.\n"},"corpus_id":{"kind":"number","value":218487506,"string":"218,487,506"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2005.01619\",\n \"mag\": 3035554416,\n \"acl\": \"2020.acl-main.371\",\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"conf/acl/Bar-HaimEFKLS20\",\n \"doi\": \"10.18653/v1/2020.acl-main.371\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://www.aclweb.org/anthology/2020.acl-main.371.pdf\"\n ],\n \"pdf_sha\": \"eb3718caf089629c90b58775019e74b41b4028ca\",\n \"open_access\": {\n \"license\": \"CCBY\",\n \"url\": \"https://www.aclweb.org/anthology/2020.acl-main.371.pdf\",\n \"status\": \"HYBRID\"\n }\n}"},"title":{"kind":"string","value":"From Arguments to Key Points: Towards Automatic Argument Summarization"},"authors":{"kind":"list like","value":["Roy Bar roybar@il.ibm.com \nIBM Research","Haim Lilach lilache@il.ibm.com \nIBM Research","Eden Roni Friedman \nIBM Research","Yoav Kantor yoavka@il.ibm.com \nIBM Research","Dan Lahav dan.lahav@ibm.com \nIBM Research","Noam Slonim noams@il.ibm.com \nIBM Research"],"string":"[\n \"Roy Bar roybar@il.ibm.com \\nIBM Research\",\n \"Haim Lilach lilache@il.ibm.com \\nIBM Research\",\n \"Eden Roni Friedman \\nIBM Research\",\n \"Yoav Kantor yoavka@il.ibm.com \\nIBM Research\",\n \"Dan Lahav dan.lahav@ibm.com \\nIBM Research\",\n \"Noam Slonim noams@il.ibm.com \\nIBM Research\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Generating a concise summary from a large collection of arguments on a given topic is an intriguing yet understudied problem. We propose to represent such summaries as a small set of talking points, termed key points, each scored according to its salience. We show, by analyzing a large dataset of crowd-contributed arguments, that a small number of key points per topic is typically sufficient for covering the vast majority of the arguments. Furthermore, we found that a domain expert can often predict these key points in advance. We study the task of argument-to-key point mapping, and introduce a novel large-scale dataset for this task. We report empirical results for an extensive set of experiments with this dataset, showing promising performance."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Governments, businesses and individuals, all need to make decisions on a daily basis: \"Should cannabis be legalized?\",\"Should we develop this product?\", \"Should I become a vegetarian?\". When making an important decision, the process typically comprises several steps: first, we gather as much information as we can about the pros and cons of the proposal under consideration. We may then summarize the collected information as a short list of the main arguments for each side. Lastly, we aim to weigh the pro and con arguments against each other to make the final decision.\nWhere can we find relevant arguments for a given topic? In recent years, significant progress was made in the field of argument mining, automatic identification and extraction of argumentative structures in text (Lawrence and Reed, 2020). Specifically, several works focused on topic-related argument mining from the Web or other massive corpora (Levy et al., 2017(Levy et al., , 2018, * All authors equally contributed to this work. 2017; Stab et al., 2018a,b;Ein-Dor et al., 2020). Policy makers in governments or businesses may also conduct surveys to collect from large audiences arguments supporting or contesting some proposal.\nEach of the above methods may result in hundreds or thousands of arguments per topic, making it impossible for the decision maker to read and digest such large amounts of information. Several works aimed to alleviate this problem by clustering together related arguments, based on different notions of relatedness, such as similarity (Reimers et al., 2019), frames (Ajjour et al., 2019), and argument facets (Misra et al., 2016). These works, however, did not attempt to create a concise textual summary from the resulting clusters.\nIn this work we propose to summarize the arguments supporting each side of the debate by mapping them to a short list of talking points, termed key points. The salience of each key point can be represented by the number of its matching arguments. An example for such summary is shown in Table 1. Key points may be viewed as highlevel arguments. They should be general enough to match a significant portion of the arguments, yet informative enough to make a useful summary.\nThe proposed method raises a fundamental question: can a small number of key points effectively summarize massive amount of arguments collected from a large population? In this work we give a positive answer to this question, based on extensive analysis over 28 controversial topics and 7,000 crowd-contributed pro and con arguments for these topics. Furthermore, we found that, given a controversial topic, a domain expert can compose a short, comprehensive list of key points even without looking at the arguments themselves.\nMotivated by the above findings, we assume in this work that the key points for each topic are given, and focus on the task of automatically map- ping arguments to these key points. This setting may be viewed as an intermediate step towards fully automatic argument summarization, but also as a valuable setting by itself: argument-to-key point mapping allows measuring the distribution of key points in a massive collection of arguments. It also allows interactive exploration of large argument collections, where key points serve as queries for retrieving matching arguments. In addition, it can be used for novelty detection -identifying unexpected arguments that do not match presupposed key points. We develop the ArgKP dataset for the argumentto-keypoint mapping task, comprising about 24,000 (argument, key point) pairs labeled as matching/non matching. 1 To the best of our knowledge, this is the first dataset for this task. As discussed in the next section in more detail, our dataset is also much larger and far more comprehensive than datasets developed for related tasks such as mapping posts or comments in online debates to reasons or arguments (Hasan and Ng, 2014;Boltužić andŠnajder, 2014).\nWe report empirical results for an extensive set of supervised and unsupervised configurations, achieving promising results.\nThe main contributions of this work are:\n1. We demonstrate, through extensive data annotation and analysis over a variety of topics, the feasibility and effectiveness of summarizing a large set of arguments collected from a large audience by mapping them to a small set of key points.\n2. We develop the first large-scale dataset for the task of argument-to-key point mapping.\n3. We perform empirical evaluation and analysis of a variety of classification methods for the above task.\n2 Related Work"},{"title":"Argument Mining","content":"The starting point for the current work is a collection of pro and con arguments for a given topic. As previously mentioned, these arguments may be collected from a large audience by conducting a survey, or mined automatically from text. Some of the previous work on argument mining focused on specific domains such as legal documents (Moens et al., 2007;Wyner et al., 2010), student essays (Stab and Gurevych, 2017;Persing and Ng, 2016), and user comments on proposed regulations (Park and Cardie, 2014).\nMining arguments and argument components for a given topic (also known as context) has been a prominent line of research in argument mining. Levy et al. (2014) introduced the task of context-dependent claim detection in a collection of Wikipedia articles, and Rinott et al. (2015) did the same for context-dependent evidence detection. More recently, several works focused on topic-related argument mining from the Web or other massive corpora (Levy et al., 2017(Levy et al., , 2018Wachsmuth et al., 2017;Stab et al., 2018a,b;Ein-Dor et al., 2020).\nStance classification of extracted arguments can be performed as a separate step (Bar-Haim et al., 2017) or jointly with argument detection, as a three-way classification (pro argument/con argument/none), as done by Stab et al. (2018b)."},{"title":"Argument Clustering and Summarization","content":"Several works have focused on identifying pairs of similar arguments, or clustering similar arguments together. Ajjour et al. (2019) addressed the task of splitting a set of arguments into a set of nonoverlapping frames such as Economics, Environment and Politics. Reimers et al. (2019) classified argument pairs as similar/dissimilar. Misra et al. (2016) aimed to detect argument pairs that are assumed to share the same argument facet, which is similar to our notion of key points. However, they did not attempt to explicitly identify or generate these facets, which remained implicit, but rather focused on detecting similarity between argument pairs. In contrast to these works, we directly map arguments to key points. Egan et al. (2016) proposed to summarize argumentative discussions through the extraction of salient \"points\", where each point is a verb and its syntactic arguments. Applying their unsupervised method to online political debates showed significant improvement over a baseline extractive summarizer, according to human evaluation. While the current work also aims to summarize argumentative content via concise points, our goal is not to extract these points but to accurately map arguments to given points. Our main challenge is to identify the various ways in which the meaning of a point is conveyed in different arguments. The method employed by Egan et al. only matches arguments with the same signature -the same verb, subject and object dependency nodes, hence its ability to capture such variability is limited.\nThe line of work that seems most similar to ours is of Hasan and Ng (2014), Boltužić andŠnajder (2014) and Naderi (2016). Hasan and Ng classified posts and individual sentences from online debates into a closed set of reasons, composed manually for each topic. Boltužić andŠnajder mapped comments from one debating website (ProCon.org) to arguments taken from another debating website (iDebate.org). Naderi (2016) addressed a similar task: she used part of the Boltužić andŠnajder corpus as training data for an SVM classifier, which was then tested on sentences and paragraphs from same-sex marriage debates in the Canadian Parliament, annotated with the same set of arguments.\nOur work differs from these works in several respects. First, we deal with crowd-contributed arguments, taken from the dataset of Gretz et al. (2020) while these works dealt with posts or comments in debate forums, and parliamentary debates. Second, the dataset developed in this work is far more extensive, covering 28 topics and over 6,500 arguments 2 , as compared to 2-4 topics in the datasets of Boltužić andŠnajder and Hasan and Ng, respectively. This allows us to perform a comprehensive analysis on the feasibility and effectiveness of argument-to-key point mapping over a variety of topics, which has not been possible with previous datasets. Lastly, while Hasan and Ng only perform within-topic classification, where the classifier is trained and tested on the same topic, we address the far more challenging task of cross-topic classification. Boltužić andŠnajder experimented with both within-topic and cross-topic classification, however they used a limited amount of data for training and testing: two topics, with less than 200 comments per topic.\nFinally, we point out the similarity between the argument/key point relation and the text/hypothesis relation in textual entailment, also known as natural language inference (NLI) (Dagan et al., 2013). Indeed, Boltužić andŠnajder (2014) used textual entailment as part of their experiments, following the earlier work of Cabrio and Villata (2013), who used textual entailment to detect support/attack relations between arguments."},{"title":"Data","content":""},{"title":"Arguments and Key Points","content":"As a source of arguments for this work we have used the publicly available IBM-Rank-30k dataset (Gretz et al., 2020). This dataset contains around 30K crowd-sourced arguments, annotated for polarity and point-wise quality. The arguments were collected with strict length limitations, accompanied by extensive quality control measures. Out of the 71 controversial topics in this dataset, we selected the subset of 28 topics for which a corresponding motion exists in the Debatabase repository of the iDebate website 3 . This requirement guaranteed that the selected topics were of high general interest.\nWe filtered arguments of low quality (below 0.5) and unclear polarity (below 0.6), to ensure sufficient argument quality in the downstream analysis. We randomly sampled 250 arguments per topic from the set of arguments that passed these filters (7,000 arguments in total for the 28 topics).\nDebatabase lists several pro and con points per motion, where each point is typically 1-2 paragraphs long. The headline of each point is a concise sentence that summarizes the point. Initially, we intended to use these point headlines as our key points. However, we found them to be unsuitable for our purpose, due to a large variance in their level of specificity, and their low coverage of the crowd's arguments, as observed in our preliminary analysis.\nTo overcome this issue, we let a domain expert who is a professional debater write the key points from scratch. The expert debater received the list of topics and was asked to generate a maximum of 7 key points for each side of the topic, without being exposed to the list of arguments per topic. The maximal number of key points was set according to the typical number of pro and con points in Debatabase motions.\nThe process employed by the expert debater to produce the key points comprised several steps:\n1. Given a debate topic, generate a list of possible key points in a constrained time frame of 10 minuets per side.\n2. Unify related key points that can be expressed as a single key point.\n3. Out of the created key points, select a maximum of 7 per side that are estimated to be the most immediate ones, hence the most likely to be chosen by crowd workers.\nThe process was completed within two working days. A total of 378 key points were generated, an average of 6.75 per side per topic."},{"title":"Mapping Arguments to Key Points","content":""},{"title":"Annotation Process","content":"Using the Figure Eight crowd labeling platform 4 , we created gold labels for associating the arguments selected as described in Section 3.1 with key points. For each argument, given in the context of its debatable topic, annotators were presented with the key points created for this topic in the relevant stance. They were guided to mark all of the key points this argument can be associated with, and if none are relevant, to select the 'None' option. Each argument was labeled by 8 annotators.\nQuality Measures: to ensure the quality of the collected data, the following measures were taken -1. Test questions. Annotators were asked to determine the stance of each argument towards the topic. Similarly to Toledo et al. (2019), this question functioned as a hidden text question 5 . All judgments of annotators failing in more than 10% of the stance questions were discarded.\n2. Annotator-κ score. This score, measuring inter annotator agreement, as defined by Toledo et al. (2019), was calculated for each annotator, and all judgments of annotators with annotator-κ < 0.3 were ignored. This score averages all pair-wise Cohen's Kappa (Landis and Koch, 1997) for a given annotator, for any annotator sharing at least 50 judgments with at least 5 other annotators.\n3. Selected group of trusted annotators. As in Gretz et al. (2020), the task was only available to a group of annotators which had performed well in previous tasks by our team.\nAs described above, the annotation of each key point with respect to a given argument was performed independently, and each annotator could select multiple key points to be associated with each given argument. For the purpose of calculating inter-annotator agreement, we considered (argument, key point) pairs, annotated with a binary label denoting whether the argument was matched to the key point. Fleiss' Kappa for this task was 0.44 (Fleiss, 1971), and Cohen's Kappa was 0.5 (averaging Annotator-κ scores). These scores correspond to \"moderate agreement\" and are comparable to agreement levels previously reported for other annotation tasks in computational argumentation (Boltužić andŠnajder, 2014;Ein-Dor et al., 2020). As for the stance selection question, 98% of the judgments were correct, indicating overall high annotation quality. Data Cleansing: In addition to the above measures, the following annotations were removed from the data: (i) Annotations in which the answer to the stance selection question was wrong; (ii) Annotations in which key point choice was illegalthe 'None' option and one of the key points were both selected. However, the rate of these errors, for each of the annotators, was rather low (< 10% and < 5%, respectively).\nArguments left with less than 7 valid judgments after applying the above quality measures and data cleansing were removed from the dataset. 6, 568 labeled arguments remain in the dataset."},{"title":"Annotation Results","content":"Next, we consolidate the individual annotations as follows. We say that an argument a is mapped to a key point k if at least 60% of the annotators mapped a to k. Recall that an argument can be mapped to more than one key point. Similarly, we say that a has no key point if at least 60% of the annotators mapped a to None (which is equivalent to not selecting any key point for the argument). Otherwise, we say that a is ambiguous, i.e., the annotations were indecisive. Table 2 shows examples for arguments and their matching key points in our dataset.\nThe distribution of the arguments in the dataset over the above categories is shown in Table 3. Remarkably, our key points, composed independently of the arguments, were able to cover 72.5% of them, with 5% of the arguments mapped to more than one key point.\nWe further investigated the differences between arguments in each category, by comparing their average quality score (taken from the IBM-Rank-30k dataset), number of tokens and number of sentences. The results are shown as additional columns in Table 3. Interestingly, arguments that have no key point tend to be shorter and have lower quality score, comparing to arguments mapped to a single key point; arguments mapped to more than one key point are the longest and have the highest quality. Figure 1 examines the impact of the number of key points on argument coverage. For each topic and stance, we order the key points according to the number of their matched arguments, and add them incrementally. The results indicate that arguments are not trivially mapped to only one or two key points, but a combination of several key points is required to achieve high coverage. The marginal contribution decays for the sixth and seventh key points, suggesting that seven key points indeed suffice for this task.\n22.8% of the arguments are ambiguous. Annotations for these arguments are split over several possible key points, none reaching the 60% threshold. For instance, the argument \"homeschooling enables parents with fringe views to push their agenda on their children without allowing exposure to alternative viewpoints.\", had two key points with annotator votes higher than 40%, but below 60%:\n1. Homeschools cannot be regulated / standardized."},{"title":"Parents are not qualified as teachers.","content":"Such cases suggest that many arguments are somewhat covered by the key points, but if the judgment is not clear-cut, the different intuitions of the annotators may result in no label receiving the required majority."},{"title":"Final Dataset Generation","content":"The ArgKP dataset includes (argument, key point) pairs with binary labels indicating whether the argument is matched to the key point. The dataset was created from the labeled data as follows. We define the label score of a pair as the fraction of annotations that classified the pair as matching . Pairs with label score ≥ 0.6 were labeled as positive (matching). Pairs with label score ≤ 0.15 were labeled as negative (non-matching). Pairs with label score in between these thresholds were removed. We further cleansed our data by discarding key points having less than three matching arguments. This led to the removal of 135 out of the 378 key points and 14,679 out of 38,772 pairs obtained from the previous step.\nThe final dataset has 24,093 labeled (argument, key point) pairs, of which 4,998 pairs (20.7%) are positive. It has 6,515 arguments (232.67 per topic), and 243 key points (8.67 key points per topic). For each pair, the dataset also specifies the topic and the stance of the argument towards the topic.\nWe assessed the quality of the resulting dataset by having an expert annotator 6 mapping 100 ran- Forcing members of a profession to retire at a certain age creates an experience drain.\nA mandatory retirement age decreases institutional knowledge. We should ban the use of child actors.\nChild actors are fine to use as long as there is a responsible adult watching them.\nChild performers should not be banned as long as there is supervision/regulation. We should close Guantanamo Bay detention camp.\nGuantanamo can provide security for accused terrorists who would be hurt in the general prison population.\nThe Guantanamo bay detention camp is better for prisoners than the alternatives.\nAssisted suicide should be a criminal offence.\nPeople have a basic right to bodily autonomy, deciding whether or not to die with minimal suffering and dignity is integral to that right.\nPeople should have the freedom to choose to end their life. Assisted suicide gives dignity to the person that wants to commit it. We should ban human cloning.\nThe world is already overpopulated, cloning humans will only contribute to this problem.\nNo key point domly sampled arguments to key points, and comparing the annotations to the gold labels for all the corresponding pairs in the dataset. We obtained a remarkably high Cohen's Kappa of 0.82 (\"almost perfect agreement\"), validating the high quality of the dataset."},{"title":"Experiments","content":""},{"title":"Experimental Setup","content":"We perform the task of matching arguments to key points in two steps. In the Match Scoring step (Section 4.1.1), we generate a score for each argument and key point. Then, in the Match Classification step (Section 4.1.2), we use these scores to classify the pairs as matching or non-matching. We perform 4-fold cross-validation over the ArgKP dataset. Each fold comprises 7 test topics, 17 train topics and 4 development topics."},{"title":"Match Scoring","content":"We experimented with both unsupervised and supervised methods for computing a match score for a given (argument, key point) pair. We also explored transfer learning from the related task of natural language inference (NLI)."},{"title":"Unsupervised Methods","content":"• Tf-Idf. In order to assess the role of lexical overlap in the matching task, we represent each argument and key point as tf-idf weighted word vectors and use their cosine similarity as the match score.\n• Word Embedding. We examined averaged word embeddings using GloVe (Pennington et al., 2014) and BERT (Devlin et al., 2019). GloVe is a context independent model that computes a single embedding for each word.\nBERT is a contextualized embedding model that takes the entire sentence into account. We also experimented with other embedding methods that under-performed BERT and thus their results are not reported here: Universal Sentence Encoder (Cer et al., 2018) and In-ferSent (Conneau et al., 2017). Again, we use cosine similarity to compute the match score.\nSupervised Methods. We fine tuned the BERTbase-uncased and BERT-large-uncased models (Devlin et al., 2019) to predict matches between argument and key point pairs. We added a linear fully connected layer of size 1 followed by a sigmoid layer to the special [CLS] token in the BERT model, and trained it for three epochs with a learning rate of 2e-5 and a binary cross entropy loss.\nNLI Transfer Learning. We also experimented with transfer learning from NLI to our task of argument-to-key point match classification. This was motivated by the similarity between these tasks (as discussed in Section 2.2), as well as the availability of large-scale NLI labeled datasets. We con-sidered the Stanford (SNLI) and the Multi-Genre (MNLI) datasets (Bowman et al., 2015;Williams et al., 2018), each comprising hundreds of thousands of labeled premise-hypothesis pairs. Pairs labeled as ENTAILMENT were considered positive instances, while the rest of the pairs, labeled as NEUTRAL or CONTRADICTION were considered negative. We trained BERT-base and BERT-large models on each of these datasets, following the procedure described above."},{"title":"Match Classification","content":"In the match classification step we select the matching key points for each argument, based on their respective matching scores. The classification can be done locally, treating each pair individually, or globally, by examining all possible key points for each argument. We compared the following policies for selecting matching key points for a given argument.\nThreshold. For each fold, we find the threshold on the match score that maximizes the F1 score for the positive (matching) class. Pairs whose score exceeds the learned threshold are considered matched.\nBest Match (BM). Using a threshold is not optimal for our data, where most arguments have at most one matched key point. A natural solution is to select the best matching key point. For each argument, we consider all key points for the same topic and stance as candidates and predict only the candidate with the highest match score as matched to the argument and the rest as unmatched. Note that this is the only fully unsupervised selection policy, as it does not require labeled data for learning a threshold.\nBM+Threshold. The BM policy always assigns exactly one key point for each argument, while 27.5% of the arguments in our data are not matched to any key point. To address this, we combine the two former policies. The top matching key point is considered a match only if its match score exceeds the learned threshold.\nDual Threshold. In order to account for arguments with more than one matching key point, two thresholds are learned. If two key points exceed the lower threshold and at least one of them exceeds the upper threshold, both will be matched. Otherwise, it works the same as the BM+Threshold policy using only the lower threshold. This allows for zero to two matches per argument. Thresholds are learned from the development set for supervised match scoring methods, and from both train and development set for unsupervised match scoring methods. Table 4 compares the various match scoring methods, all using the Threshold key point selection policy. Results are obtained by micro-averaging over the argument-key point pairs in each fold, and averaging over the different folds. We consider Precision, Recall and F1 of the positive class, as well as the overall accuracy. We also list for reference the majority class baseline that always predicts \"no match\", and the random baseline, which randomly predicts the positive class according to its probability in the training data."},{"title":"Results","content":""},{"title":"Match Scoring Methods","content":"The unsupervised models fail to capture the relation between the argument and the key points. Tf-Idf and Glove perform the worst, showing that simple lexical similarity is insufficient for this task. BERT embedding does better but still reaches a relatively low F1 score of 0.4.\nIn contrast to the unsupervised models, supervised models are shown to perform well. BERT with fine tuning leads to a substantial improvement, reaching F1 score of 0.657 with the BERT-base model, and 0.684 with the BERT-large model.\nBERT Models trained on NLI data are considerably better than the unsupervised methods, with the best model reaching F1 of 0.526, yet their performance is still far below the supervised models trained on our ArgKP dataset. This may reflect both the similarities and the differences between NLI and the current task. We have also experimented with combining these two types of data in cascade: BERT was first trained on a large NLI dataset (SNLI, MNLI or their union), and was then fine-tuned on the smaller ArgKP data. However, it did not improve the supervised results.\nError Analysis. By analyzing the top errors of the supervised classifier (BERT-large), we found several systematic patterns of errors. In most cases, non-matching arguments and key points received a high match score in one of the following cases:\n• They share some key phrases. For example:\n\"It is unfair to only subsidize vocational education. Achieving a more advanced education is very expensive and it would also need to be subsidized.\" and \"Subsidizing vocational education is expensive\".\n• They share a large portion of the sentence, but not the main point, for example: \"Women should be able to fight if they are strong enough\" and \"Women should be able to serve in combat if they choose to\".\n• They are at least partially related, but labeled as non-matching due to a better fitting key point for the same argument. For example: \"We should subsidize space exploration because it increases the knowledge of the universe we are in\" and \"Space exploration improves science/technology\" can be considered matched, but were labeled as unmatched due to the key point \"Space exploration unravels information about the universe\". Using the Best Match policy helps in these cases.\nFor arguments and key points that were labeled as matched but received a low match score, the relation was in many cases implied or required some further knowledge, for examples: \"Journalism is an essential part of democracy and freedom of expression and should not be subsidized by the state.\" and \"government intervention has the risk of inserting bias/harming objectivity\". Table 5 compares different key point selection policies, all using the best performing match scoring method: BERT-large fine-tuned on ArgKP. We report the results over the whole dataset (\"all arguments\"), as well as the subsets of arguments having none, single or multiple matching key points according to the labeled data. In case of no matches we present the accuracy, as recall and F1 scores are undefined. When considering all the arguments, the Dual Threshold policy achieves the best F1 score of 0.73. The Threshold method performs well for arguments with no matches or multiple matches. When there is exactly one match (the common case in our data), it has lower precision. The Best Match policy performs well when there is a single match, but is not able to cope with arguments that have no matches or have multiple matches. The BM+Threshold method combines the two and is useful when there are no matching key points or a single matching key point, but still have lower recall when there are multiple matching key points. The Dual Threshold method improves the recall and therefore the F1 score for multiple matches while maintaining good performance for arguments with single or no matches. Figure 2 shows Precision-Recall trade-off for the various policies, using the different possible thresholds, computed for one of the folds. For each policy, we specify the best F1 score, as well as the F1 score obtained for the selected threshold, which was optimized over the development set. The Threshold policy allows to control recall, up to one (where the threshold is zero), at the price of low precision. The BM+Threshold policy generates the highest precision, but low recall, since at most one candidate is selected. Note that when the threshold is zero, the BM+Threshold policy is equivalent to the BM policy. The Dual Threshold policy offers the best trade-off, for mid-range precision and recall."},{"title":"Key Point Selection Policies","content":""},{"title":"Conclusion","content":"This work addressed the practical problem of summarizing a large collection of arguments on a given topic. We proposed to represent such summaries as a set of key points scored according to their relative salience. Such summary aims to provide both textual and quantitative views of the argument data in a concise form. We demonstrated the feasibility and effectiveness of the proposed approach through extensive data annotation and analysis. We showed that a domain expert can quickly come up with a short list of pro and con key points per topic, that would capture the gist of crowd-contributed arguments, even without being exposed to the arguments themselves. We studied the problem of automatically matching arguments to key points, and developed the first large-scale dataset for this task, which we make publicly available.\nOur experimental results demonstrate that the problem is far from trivial, and cannot be effectively solved using unsupervised methods based on word or sentence-level embedding. However, by using state of the art supervised learning methods for match scoring, together with an appropriate key point selection policy for match classification, we were able to achieve promising results on this task. The natural next step for this work is the challenging task of automatic key point generation. In addition, we plan to apply the methods presented in this work also to automatically-mined arguments. Finally, detecting the more implicit relations between the argument and the key point, as seen in our error analysis, is another intriguing direction for future work."},{"title":"Figure 1 :","content":""},{"title":"Figure 2 :","content":""},{"title":"Table 2 :","content":""},{"title":"Table 3 :","content":""},{"title":"Table 4 :","content":""},{"title":"Table 5 :","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Governments, businesses and individuals, all need to make decisions on a daily basis: \\\"Should cannabis be legalized?\\\",\\\"Should we develop this product?\\\", \\\"Should I become a vegetarian?\\\". When making an important decision, the process typically comprises several steps: first, we gather as much information as we can about the pros and cons of the proposal under consideration. We may then summarize the collected information as a short list of the main arguments for each side. Lastly, we aim to weigh the pro and con arguments against each other to make the final decision.\\nWhere can we find relevant arguments for a given topic? In recent years, significant progress was made in the field of argument mining, automatic identification and extraction of argumentative structures in text (Lawrence and Reed, 2020). Specifically, several works focused on topic-related argument mining from the Web or other massive corpora (Levy et al., 2017(Levy et al., , 2018, * All authors equally contributed to this work. 2017; Stab et al., 2018a,b;Ein-Dor et al., 2020). Policy makers in governments or businesses may also conduct surveys to collect from large audiences arguments supporting or contesting some proposal.\\nEach of the above methods may result in hundreds or thousands of arguments per topic, making it impossible for the decision maker to read and digest such large amounts of information. Several works aimed to alleviate this problem by clustering together related arguments, based on different notions of relatedness, such as similarity (Reimers et al., 2019), frames (Ajjour et al., 2019), and argument facets (Misra et al., 2016). These works, however, did not attempt to create a concise textual summary from the resulting clusters.\\nIn this work we propose to summarize the arguments supporting each side of the debate by mapping them to a short list of talking points, termed key points. The salience of each key point can be represented by the number of its matching arguments. An example for such summary is shown in Table 1. Key points may be viewed as highlevel arguments. They should be general enough to match a significant portion of the arguments, yet informative enough to make a useful summary.\\nThe proposed method raises a fundamental question: can a small number of key points effectively summarize massive amount of arguments collected from a large population? In this work we give a positive answer to this question, based on extensive analysis over 28 controversial topics and 7,000 crowd-contributed pro and con arguments for these topics. Furthermore, we found that, given a controversial topic, a domain expert can compose a short, comprehensive list of key points even without looking at the arguments themselves.\\nMotivated by the above findings, we assume in this work that the key points for each topic are given, and focus on the task of automatically map- ping arguments to these key points. This setting may be viewed as an intermediate step towards fully automatic argument summarization, but also as a valuable setting by itself: argument-to-key point mapping allows measuring the distribution of key points in a massive collection of arguments. It also allows interactive exploration of large argument collections, where key points serve as queries for retrieving matching arguments. In addition, it can be used for novelty detection -identifying unexpected arguments that do not match presupposed key points. We develop the ArgKP dataset for the argumentto-keypoint mapping task, comprising about 24,000 (argument, key point) pairs labeled as matching/non matching. 1 To the best of our knowledge, this is the first dataset for this task. As discussed in the next section in more detail, our dataset is also much larger and far more comprehensive than datasets developed for related tasks such as mapping posts or comments in online debates to reasons or arguments (Hasan and Ng, 2014;Boltužić andŠnajder, 2014).\\nWe report empirical results for an extensive set of supervised and unsupervised configurations, achieving promising results.\\nThe main contributions of this work are:\\n1. We demonstrate, through extensive data annotation and analysis over a variety of topics, the feasibility and effectiveness of summarizing a large set of arguments collected from a large audience by mapping them to a small set of key points.\\n2. We develop the first large-scale dataset for the task of argument-to-key point mapping.\\n3. We perform empirical evaluation and analysis of a variety of classification methods for the above task.\\n2 Related Work\"\n },\n {\n \"title\": \"Argument Mining\",\n \"content\": \"The starting point for the current work is a collection of pro and con arguments for a given topic. As previously mentioned, these arguments may be collected from a large audience by conducting a survey, or mined automatically from text. Some of the previous work on argument mining focused on specific domains such as legal documents (Moens et al., 2007;Wyner et al., 2010), student essays (Stab and Gurevych, 2017;Persing and Ng, 2016), and user comments on proposed regulations (Park and Cardie, 2014).\\nMining arguments and argument components for a given topic (also known as context) has been a prominent line of research in argument mining. Levy et al. (2014) introduced the task of context-dependent claim detection in a collection of Wikipedia articles, and Rinott et al. (2015) did the same for context-dependent evidence detection. More recently, several works focused on topic-related argument mining from the Web or other massive corpora (Levy et al., 2017(Levy et al., , 2018Wachsmuth et al., 2017;Stab et al., 2018a,b;Ein-Dor et al., 2020).\\nStance classification of extracted arguments can be performed as a separate step (Bar-Haim et al., 2017) or jointly with argument detection, as a three-way classification (pro argument/con argument/none), as done by Stab et al. (2018b).\"\n },\n {\n \"title\": \"Argument Clustering and Summarization\",\n \"content\": \"Several works have focused on identifying pairs of similar arguments, or clustering similar arguments together. Ajjour et al. (2019) addressed the task of splitting a set of arguments into a set of nonoverlapping frames such as Economics, Environment and Politics. Reimers et al. (2019) classified argument pairs as similar/dissimilar. Misra et al. (2016) aimed to detect argument pairs that are assumed to share the same argument facet, which is similar to our notion of key points. However, they did not attempt to explicitly identify or generate these facets, which remained implicit, but rather focused on detecting similarity between argument pairs. In contrast to these works, we directly map arguments to key points. Egan et al. (2016) proposed to summarize argumentative discussions through the extraction of salient \\\"points\\\", where each point is a verb and its syntactic arguments. Applying their unsupervised method to online political debates showed significant improvement over a baseline extractive summarizer, according to human evaluation. While the current work also aims to summarize argumentative content via concise points, our goal is not to extract these points but to accurately map arguments to given points. Our main challenge is to identify the various ways in which the meaning of a point is conveyed in different arguments. The method employed by Egan et al. only matches arguments with the same signature -the same verb, subject and object dependency nodes, hence its ability to capture such variability is limited.\\nThe line of work that seems most similar to ours is of Hasan and Ng (2014), Boltužić andŠnajder (2014) and Naderi (2016). Hasan and Ng classified posts and individual sentences from online debates into a closed set of reasons, composed manually for each topic. Boltužić andŠnajder mapped comments from one debating website (ProCon.org) to arguments taken from another debating website (iDebate.org). Naderi (2016) addressed a similar task: she used part of the Boltužić andŠnajder corpus as training data for an SVM classifier, which was then tested on sentences and paragraphs from same-sex marriage debates in the Canadian Parliament, annotated with the same set of arguments.\\nOur work differs from these works in several respects. First, we deal with crowd-contributed arguments, taken from the dataset of Gretz et al. (2020) while these works dealt with posts or comments in debate forums, and parliamentary debates. Second, the dataset developed in this work is far more extensive, covering 28 topics and over 6,500 arguments 2 , as compared to 2-4 topics in the datasets of Boltužić andŠnajder and Hasan and Ng, respectively. This allows us to perform a comprehensive analysis on the feasibility and effectiveness of argument-to-key point mapping over a variety of topics, which has not been possible with previous datasets. Lastly, while Hasan and Ng only perform within-topic classification, where the classifier is trained and tested on the same topic, we address the far more challenging task of cross-topic classification. Boltužić andŠnajder experimented with both within-topic and cross-topic classification, however they used a limited amount of data for training and testing: two topics, with less than 200 comments per topic.\\nFinally, we point out the similarity between the argument/key point relation and the text/hypothesis relation in textual entailment, also known as natural language inference (NLI) (Dagan et al., 2013). Indeed, Boltužić andŠnajder (2014) used textual entailment as part of their experiments, following the earlier work of Cabrio and Villata (2013), who used textual entailment to detect support/attack relations between arguments.\"\n },\n {\n \"title\": \"Data\",\n \"content\": \"\"\n },\n {\n \"title\": \"Arguments and Key Points\",\n \"content\": \"As a source of arguments for this work we have used the publicly available IBM-Rank-30k dataset (Gretz et al., 2020). This dataset contains around 30K crowd-sourced arguments, annotated for polarity and point-wise quality. The arguments were collected with strict length limitations, accompanied by extensive quality control measures. Out of the 71 controversial topics in this dataset, we selected the subset of 28 topics for which a corresponding motion exists in the Debatabase repository of the iDebate website 3 . This requirement guaranteed that the selected topics were of high general interest.\\nWe filtered arguments of low quality (below 0.5) and unclear polarity (below 0.6), to ensure sufficient argument quality in the downstream analysis. We randomly sampled 250 arguments per topic from the set of arguments that passed these filters (7,000 arguments in total for the 28 topics).\\nDebatabase lists several pro and con points per motion, where each point is typically 1-2 paragraphs long. The headline of each point is a concise sentence that summarizes the point. Initially, we intended to use these point headlines as our key points. However, we found them to be unsuitable for our purpose, due to a large variance in their level of specificity, and their low coverage of the crowd's arguments, as observed in our preliminary analysis.\\nTo overcome this issue, we let a domain expert who is a professional debater write the key points from scratch. The expert debater received the list of topics and was asked to generate a maximum of 7 key points for each side of the topic, without being exposed to the list of arguments per topic. The maximal number of key points was set according to the typical number of pro and con points in Debatabase motions.\\nThe process employed by the expert debater to produce the key points comprised several steps:\\n1. Given a debate topic, generate a list of possible key points in a constrained time frame of 10 minuets per side.\\n2. Unify related key points that can be expressed as a single key point.\\n3. Out of the created key points, select a maximum of 7 per side that are estimated to be the most immediate ones, hence the most likely to be chosen by crowd workers.\\nThe process was completed within two working days. A total of 378 key points were generated, an average of 6.75 per side per topic.\"\n },\n {\n \"title\": \"Mapping Arguments to Key Points\",\n \"content\": \"\"\n },\n {\n \"title\": \"Annotation Process\",\n \"content\": \"Using the Figure Eight crowd labeling platform 4 , we created gold labels for associating the arguments selected as described in Section 3.1 with key points. For each argument, given in the context of its debatable topic, annotators were presented with the key points created for this topic in the relevant stance. They were guided to mark all of the key points this argument can be associated with, and if none are relevant, to select the 'None' option. Each argument was labeled by 8 annotators.\\nQuality Measures: to ensure the quality of the collected data, the following measures were taken -1. Test questions. Annotators were asked to determine the stance of each argument towards the topic. Similarly to Toledo et al. (2019), this question functioned as a hidden text question 5 . All judgments of annotators failing in more than 10% of the stance questions were discarded.\\n2. Annotator-κ score. This score, measuring inter annotator agreement, as defined by Toledo et al. (2019), was calculated for each annotator, and all judgments of annotators with annotator-κ < 0.3 were ignored. This score averages all pair-wise Cohen's Kappa (Landis and Koch, 1997) for a given annotator, for any annotator sharing at least 50 judgments with at least 5 other annotators.\\n3. Selected group of trusted annotators. As in Gretz et al. (2020), the task was only available to a group of annotators which had performed well in previous tasks by our team.\\nAs described above, the annotation of each key point with respect to a given argument was performed independently, and each annotator could select multiple key points to be associated with each given argument. For the purpose of calculating inter-annotator agreement, we considered (argument, key point) pairs, annotated with a binary label denoting whether the argument was matched to the key point. Fleiss' Kappa for this task was 0.44 (Fleiss, 1971), and Cohen's Kappa was 0.5 (averaging Annotator-κ scores). These scores correspond to \\\"moderate agreement\\\" and are comparable to agreement levels previously reported for other annotation tasks in computational argumentation (Boltužić andŠnajder, 2014;Ein-Dor et al., 2020). As for the stance selection question, 98% of the judgments were correct, indicating overall high annotation quality. Data Cleansing: In addition to the above measures, the following annotations were removed from the data: (i) Annotations in which the answer to the stance selection question was wrong; (ii) Annotations in which key point choice was illegalthe 'None' option and one of the key points were both selected. However, the rate of these errors, for each of the annotators, was rather low (< 10% and < 5%, respectively).\\nArguments left with less than 7 valid judgments after applying the above quality measures and data cleansing were removed from the dataset. 6, 568 labeled arguments remain in the dataset.\"\n },\n {\n \"title\": \"Annotation Results\",\n \"content\": \"Next, we consolidate the individual annotations as follows. We say that an argument a is mapped to a key point k if at least 60% of the annotators mapped a to k. Recall that an argument can be mapped to more than one key point. Similarly, we say that a has no key point if at least 60% of the annotators mapped a to None (which is equivalent to not selecting any key point for the argument). Otherwise, we say that a is ambiguous, i.e., the annotations were indecisive. Table 2 shows examples for arguments and their matching key points in our dataset.\\nThe distribution of the arguments in the dataset over the above categories is shown in Table 3. Remarkably, our key points, composed independently of the arguments, were able to cover 72.5% of them, with 5% of the arguments mapped to more than one key point.\\nWe further investigated the differences between arguments in each category, by comparing their average quality score (taken from the IBM-Rank-30k dataset), number of tokens and number of sentences. The results are shown as additional columns in Table 3. Interestingly, arguments that have no key point tend to be shorter and have lower quality score, comparing to arguments mapped to a single key point; arguments mapped to more than one key point are the longest and have the highest quality. Figure 1 examines the impact of the number of key points on argument coverage. For each topic and stance, we order the key points according to the number of their matched arguments, and add them incrementally. The results indicate that arguments are not trivially mapped to only one or two key points, but a combination of several key points is required to achieve high coverage. The marginal contribution decays for the sixth and seventh key points, suggesting that seven key points indeed suffice for this task.\\n22.8% of the arguments are ambiguous. Annotations for these arguments are split over several possible key points, none reaching the 60% threshold. For instance, the argument \\\"homeschooling enables parents with fringe views to push their agenda on their children without allowing exposure to alternative viewpoints.\\\", had two key points with annotator votes higher than 40%, but below 60%:\\n1. Homeschools cannot be regulated / standardized.\"\n },\n {\n \"title\": \"Parents are not qualified as teachers.\",\n \"content\": \"Such cases suggest that many arguments are somewhat covered by the key points, but if the judgment is not clear-cut, the different intuitions of the annotators may result in no label receiving the required majority.\"\n },\n {\n \"title\": \"Final Dataset Generation\",\n \"content\": \"The ArgKP dataset includes (argument, key point) pairs with binary labels indicating whether the argument is matched to the key point. The dataset was created from the labeled data as follows. We define the label score of a pair as the fraction of annotations that classified the pair as matching . Pairs with label score ≥ 0.6 were labeled as positive (matching). Pairs with label score ≤ 0.15 were labeled as negative (non-matching). Pairs with label score in between these thresholds were removed. We further cleansed our data by discarding key points having less than three matching arguments. This led to the removal of 135 out of the 378 key points and 14,679 out of 38,772 pairs obtained from the previous step.\\nThe final dataset has 24,093 labeled (argument, key point) pairs, of which 4,998 pairs (20.7%) are positive. It has 6,515 arguments (232.67 per topic), and 243 key points (8.67 key points per topic). For each pair, the dataset also specifies the topic and the stance of the argument towards the topic.\\nWe assessed the quality of the resulting dataset by having an expert annotator 6 mapping 100 ran- Forcing members of a profession to retire at a certain age creates an experience drain.\\nA mandatory retirement age decreases institutional knowledge. We should ban the use of child actors.\\nChild actors are fine to use as long as there is a responsible adult watching them.\\nChild performers should not be banned as long as there is supervision/regulation. We should close Guantanamo Bay detention camp.\\nGuantanamo can provide security for accused terrorists who would be hurt in the general prison population.\\nThe Guantanamo bay detention camp is better for prisoners than the alternatives.\\nAssisted suicide should be a criminal offence.\\nPeople have a basic right to bodily autonomy, deciding whether or not to die with minimal suffering and dignity is integral to that right.\\nPeople should have the freedom to choose to end their life. Assisted suicide gives dignity to the person that wants to commit it. We should ban human cloning.\\nThe world is already overpopulated, cloning humans will only contribute to this problem.\\nNo key point domly sampled arguments to key points, and comparing the annotations to the gold labels for all the corresponding pairs in the dataset. We obtained a remarkably high Cohen's Kappa of 0.82 (\\\"almost perfect agreement\\\"), validating the high quality of the dataset.\"\n },\n {\n \"title\": \"Experiments\",\n \"content\": \"\"\n },\n {\n \"title\": \"Experimental Setup\",\n \"content\": \"We perform the task of matching arguments to key points in two steps. In the Match Scoring step (Section 4.1.1), we generate a score for each argument and key point. Then, in the Match Classification step (Section 4.1.2), we use these scores to classify the pairs as matching or non-matching. We perform 4-fold cross-validation over the ArgKP dataset. Each fold comprises 7 test topics, 17 train topics and 4 development topics.\"\n },\n {\n \"title\": \"Match Scoring\",\n \"content\": \"We experimented with both unsupervised and supervised methods for computing a match score for a given (argument, key point) pair. We also explored transfer learning from the related task of natural language inference (NLI).\"\n },\n {\n \"title\": \"Unsupervised Methods\",\n \"content\": \"• Tf-Idf. In order to assess the role of lexical overlap in the matching task, we represent each argument and key point as tf-idf weighted word vectors and use their cosine similarity as the match score.\\n• Word Embedding. We examined averaged word embeddings using GloVe (Pennington et al., 2014) and BERT (Devlin et al., 2019). GloVe is a context independent model that computes a single embedding for each word.\\nBERT is a contextualized embedding model that takes the entire sentence into account. We also experimented with other embedding methods that under-performed BERT and thus their results are not reported here: Universal Sentence Encoder (Cer et al., 2018) and In-ferSent (Conneau et al., 2017). Again, we use cosine similarity to compute the match score.\\nSupervised Methods. We fine tuned the BERTbase-uncased and BERT-large-uncased models (Devlin et al., 2019) to predict matches between argument and key point pairs. We added a linear fully connected layer of size 1 followed by a sigmoid layer to the special [CLS] token in the BERT model, and trained it for three epochs with a learning rate of 2e-5 and a binary cross entropy loss.\\nNLI Transfer Learning. We also experimented with transfer learning from NLI to our task of argument-to-key point match classification. This was motivated by the similarity between these tasks (as discussed in Section 2.2), as well as the availability of large-scale NLI labeled datasets. We con-sidered the Stanford (SNLI) and the Multi-Genre (MNLI) datasets (Bowman et al., 2015;Williams et al., 2018), each comprising hundreds of thousands of labeled premise-hypothesis pairs. Pairs labeled as ENTAILMENT were considered positive instances, while the rest of the pairs, labeled as NEUTRAL or CONTRADICTION were considered negative. We trained BERT-base and BERT-large models on each of these datasets, following the procedure described above.\"\n },\n {\n \"title\": \"Match Classification\",\n \"content\": \"In the match classification step we select the matching key points for each argument, based on their respective matching scores. The classification can be done locally, treating each pair individually, or globally, by examining all possible key points for each argument. We compared the following policies for selecting matching key points for a given argument.\\nThreshold. For each fold, we find the threshold on the match score that maximizes the F1 score for the positive (matching) class. Pairs whose score exceeds the learned threshold are considered matched.\\nBest Match (BM). Using a threshold is not optimal for our data, where most arguments have at most one matched key point. A natural solution is to select the best matching key point. For each argument, we consider all key points for the same topic and stance as candidates and predict only the candidate with the highest match score as matched to the argument and the rest as unmatched. Note that this is the only fully unsupervised selection policy, as it does not require labeled data for learning a threshold.\\nBM+Threshold. The BM policy always assigns exactly one key point for each argument, while 27.5% of the arguments in our data are not matched to any key point. To address this, we combine the two former policies. The top matching key point is considered a match only if its match score exceeds the learned threshold.\\nDual Threshold. In order to account for arguments with more than one matching key point, two thresholds are learned. If two key points exceed the lower threshold and at least one of them exceeds the upper threshold, both will be matched. Otherwise, it works the same as the BM+Threshold policy using only the lower threshold. This allows for zero to two matches per argument. Thresholds are learned from the development set for supervised match scoring methods, and from both train and development set for unsupervised match scoring methods. Table 4 compares the various match scoring methods, all using the Threshold key point selection policy. Results are obtained by micro-averaging over the argument-key point pairs in each fold, and averaging over the different folds. We consider Precision, Recall and F1 of the positive class, as well as the overall accuracy. We also list for reference the majority class baseline that always predicts \\\"no match\\\", and the random baseline, which randomly predicts the positive class according to its probability in the training data.\"\n },\n {\n \"title\": \"Results\",\n \"content\": \"\"\n },\n {\n \"title\": \"Match Scoring Methods\",\n \"content\": \"The unsupervised models fail to capture the relation between the argument and the key points. Tf-Idf and Glove perform the worst, showing that simple lexical similarity is insufficient for this task. BERT embedding does better but still reaches a relatively low F1 score of 0.4.\\nIn contrast to the unsupervised models, supervised models are shown to perform well. BERT with fine tuning leads to a substantial improvement, reaching F1 score of 0.657 with the BERT-base model, and 0.684 with the BERT-large model.\\nBERT Models trained on NLI data are considerably better than the unsupervised methods, with the best model reaching F1 of 0.526, yet their performance is still far below the supervised models trained on our ArgKP dataset. This may reflect both the similarities and the differences between NLI and the current task. We have also experimented with combining these two types of data in cascade: BERT was first trained on a large NLI dataset (SNLI, MNLI or their union), and was then fine-tuned on the smaller ArgKP data. However, it did not improve the supervised results.\\nError Analysis. By analyzing the top errors of the supervised classifier (BERT-large), we found several systematic patterns of errors. In most cases, non-matching arguments and key points received a high match score in one of the following cases:\\n• They share some key phrases. For example:\\n\\\"It is unfair to only subsidize vocational education. Achieving a more advanced education is very expensive and it would also need to be subsidized.\\\" and \\\"Subsidizing vocational education is expensive\\\".\\n• They share a large portion of the sentence, but not the main point, for example: \\\"Women should be able to fight if they are strong enough\\\" and \\\"Women should be able to serve in combat if they choose to\\\".\\n• They are at least partially related, but labeled as non-matching due to a better fitting key point for the same argument. For example: \\\"We should subsidize space exploration because it increases the knowledge of the universe we are in\\\" and \\\"Space exploration improves science/technology\\\" can be considered matched, but were labeled as unmatched due to the key point \\\"Space exploration unravels information about the universe\\\". Using the Best Match policy helps in these cases.\\nFor arguments and key points that were labeled as matched but received a low match score, the relation was in many cases implied or required some further knowledge, for examples: \\\"Journalism is an essential part of democracy and freedom of expression and should not be subsidized by the state.\\\" and \\\"government intervention has the risk of inserting bias/harming objectivity\\\". Table 5 compares different key point selection policies, all using the best performing match scoring method: BERT-large fine-tuned on ArgKP. We report the results over the whole dataset (\\\"all arguments\\\"), as well as the subsets of arguments having none, single or multiple matching key points according to the labeled data. In case of no matches we present the accuracy, as recall and F1 scores are undefined. When considering all the arguments, the Dual Threshold policy achieves the best F1 score of 0.73. The Threshold method performs well for arguments with no matches or multiple matches. When there is exactly one match (the common case in our data), it has lower precision. The Best Match policy performs well when there is a single match, but is not able to cope with arguments that have no matches or have multiple matches. The BM+Threshold method combines the two and is useful when there are no matching key points or a single matching key point, but still have lower recall when there are multiple matching key points. The Dual Threshold method improves the recall and therefore the F1 score for multiple matches while maintaining good performance for arguments with single or no matches. Figure 2 shows Precision-Recall trade-off for the various policies, using the different possible thresholds, computed for one of the folds. For each policy, we specify the best F1 score, as well as the F1 score obtained for the selected threshold, which was optimized over the development set. The Threshold policy allows to control recall, up to one (where the threshold is zero), at the price of low precision. The BM+Threshold policy generates the highest precision, but low recall, since at most one candidate is selected. Note that when the threshold is zero, the BM+Threshold policy is equivalent to the BM policy. The Dual Threshold policy offers the best trade-off, for mid-range precision and recall.\"\n },\n {\n \"title\": \"Key Point Selection Policies\",\n \"content\": \"\"\n },\n {\n \"title\": \"Conclusion\",\n \"content\": \"This work addressed the practical problem of summarizing a large collection of arguments on a given topic. We proposed to represent such summaries as a set of key points scored according to their relative salience. Such summary aims to provide both textual and quantitative views of the argument data in a concise form. We demonstrated the feasibility and effectiveness of the proposed approach through extensive data annotation and analysis. We showed that a domain expert can quickly come up with a short list of pro and con key points per topic, that would capture the gist of crowd-contributed arguments, even without being exposed to the arguments themselves. We studied the problem of automatically matching arguments to key points, and developed the first large-scale dataset for this task, which we make publicly available.\\nOur experimental results demonstrate that the problem is far from trivial, and cannot be effectively solved using unsupervised methods based on word or sentence-level embedding. However, by using state of the art supervised learning methods for match scoring, together with an appropriate key point selection policy for match classification, we were able to achieve promising results on this task. The natural next step for this work is the challenging task of automatic key point generation. In addition, we plan to apply the methods presented in this work also to automatically-mined arguments. Finally, detecting the more implicit relations between the argument and the key point, as seen in our error analysis, is another intriguing direction for future work.\"\n },\n {\n \"title\": \"Figure 1 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 2 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 2 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 3 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 4 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 5 :\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Argument coverage per number of key points.","content":"Figure 1 :\n1Argument coverage per number of key points."},{"id":"","caption":"Precision/Recall trade-off for different key point selection policies. For each method, the highest F1 score, as well as the F1 score for the chosen threshold are specified. For the Best Match + Threshold policy, these two scores coincide.","content":"Figure 2 :\n2Precision/Recall trade-off for different key point selection policies. For each method, the highest F1 score, as well as the F1 score for the chosen threshold are specified. For the Best Match + Threshold policy, these two scores coincide."},{"id":"","caption":"Homeschooling should be banned #Args Pro Mainstream schools are essential to develop social skills. 61 Parents are not qualified as teachers. 20 Homeschools cannot be regulated/standardized. 15 Mainstream schools are of higher educational quality. 9 Con Parents should be permitted to choose the education of their children.28 Homeschooling is often the best option for catering for the needs of exceptional/religious/ill/disabled students.Homeschools can be personalized to the child's pace/needs. 21 Mainstream schools have a lot of violence/bullying. 21 The home is a good learning environment. 13 Parents will have more ability to pay-attention/educate their child. 7Table 1: A sample key point-based summary, extracted from our ArgKP dataset.","content":"Homeschooling should be banned #Args Pro Mainstream schools are essential to develop social skills. 61 Parents are not qualified as teachers. 20 Homeschools cannot be regulated/standardized. 15 Mainstream schools are of higher educational quality. 9 Con Parents should be permitted to choose the education of their children.28 Homeschooling is often the best option for catering for the needs of exceptional/religious/ill/disabled students.Homeschools can be personalized to the child's pace/needs. 21 Mainstream schools have a lot of violence/bullying. 21 The home is a good learning environment. 13 Parents will have more ability to pay-attention/educate their child. 7Table 1: A sample key point-based summary, extracted from our ArgKP dataset.25"},{"id":"","caption":"Examples for key point association to arguments.","content":"Table 2 :\n2Examples for key point association to arguments.% Arguments Quality # Tokens # Sentences \nNo key point \n4.7% \n0.75 \n16.35 \n1.09 \nAmbiguous \n22.8% \n0.80 \n18.97 \n1.15 \nSingle key point \n67.5% \n0.84 \n18.54 \n1.15 \nMultiple key points \n5.0% \n0.91 \n23.66 \n1.33"},{"id":"","caption":"Argument statistics by key point matches.","content":"Table 3 :\n3Argument statistics by key point matches."},{"id":"","caption":"Comparison of match scoring methods, using the Threshold selection policy. P, R and F1 refer to the positive class. Acc is the accuracy.","content":"Table 4 :\n4Comparison of match scoring methods, using the Threshold selection policy. P, R and F1 refer to the positive class. Acc is the accuracy.All \nSingle \nMultiple \nNo \nArguments \nKey Point \nKey Points \nKey Points \nAcc \nP \nR \nF1 \nP \nR \nF1 \nP \nR \nF1 \nAcc \nThreshold \n.868 .685 .688 .684 \n.720 .686 .701 \n.904 .690 .782 \n.933 \nBest Match \n.876 .696 .711 .703 \n.836 .747 .789 \n.936 .448 .606 \n.839 \nBM+Threshold \n.890 .772 .665 .713 \n.856 .699 .769 \n.941 .421 .580 \n.915 \nDual Threshold \n.887 .721 .740 .730 \n.784 .752 .767 \n.945 .656 .773 \n.908"},{"id":"","caption":"Comparing key point selection policies, using BERT-large trained on the ArgKP dataset for match scoring.","content":"Table 5 :\n5Comparing key point selection policies, using BERT-large trained on the ArgKP dataset for match scoring."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Argument coverage per number of key points.\",\n \"content\": \"Figure 1 :\\n1Argument coverage per number of key points.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Precision/Recall trade-off for different key point selection policies. For each method, the highest F1 score, as well as the F1 score for the chosen threshold are specified. For the Best Match + Threshold policy, these two scores coincide.\",\n \"content\": \"Figure 2 :\\n2Precision/Recall trade-off for different key point selection policies. For each method, the highest F1 score, as well as the F1 score for the chosen threshold are specified. For the Best Match + Threshold policy, these two scores coincide.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Homeschooling should be banned #Args Pro Mainstream schools are essential to develop social skills. 61 Parents are not qualified as teachers. 20 Homeschools cannot be regulated/standardized. 15 Mainstream schools are of higher educational quality. 9 Con Parents should be permitted to choose the education of their children.28 Homeschooling is often the best option for catering for the needs of exceptional/religious/ill/disabled students.Homeschools can be personalized to the child's pace/needs. 21 Mainstream schools have a lot of violence/bullying. 21 The home is a good learning environment. 13 Parents will have more ability to pay-attention/educate their child. 7Table 1: A sample key point-based summary, extracted from our ArgKP dataset.\",\n \"content\": \"Homeschooling should be banned #Args Pro Mainstream schools are essential to develop social skills. 61 Parents are not qualified as teachers. 20 Homeschools cannot be regulated/standardized. 15 Mainstream schools are of higher educational quality. 9 Con Parents should be permitted to choose the education of their children.28 Homeschooling is often the best option for catering for the needs of exceptional/religious/ill/disabled students.Homeschools can be personalized to the child's pace/needs. 21 Mainstream schools have a lot of violence/bullying. 21 The home is a good learning environment. 13 Parents will have more ability to pay-attention/educate their child. 7Table 1: A sample key point-based summary, extracted from our ArgKP dataset.25\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Examples for key point association to arguments.\",\n \"content\": \"Table 2 :\\n2Examples for key point association to arguments.% Arguments Quality # Tokens # Sentences \\nNo key point \\n4.7% \\n0.75 \\n16.35 \\n1.09 \\nAmbiguous \\n22.8% \\n0.80 \\n18.97 \\n1.15 \\nSingle key point \\n67.5% \\n0.84 \\n18.54 \\n1.15 \\nMultiple key points \\n5.0% \\n0.91 \\n23.66 \\n1.33\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Argument statistics by key point matches.\",\n \"content\": \"Table 3 :\\n3Argument statistics by key point matches.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Comparison of match scoring methods, using the Threshold selection policy. P, R and F1 refer to the positive class. Acc is the accuracy.\",\n \"content\": \"Table 4 :\\n4Comparison of match scoring methods, using the Threshold selection policy. P, R and F1 refer to the positive class. Acc is the accuracy.All \\nSingle \\nMultiple \\nNo \\nArguments \\nKey Point \\nKey Points \\nKey Points \\nAcc \\nP \\nR \\nF1 \\nP \\nR \\nF1 \\nP \\nR \\nF1 \\nAcc \\nThreshold \\n.868 .685 .688 .684 \\n.720 .686 .701 \\n.904 .690 .782 \\n.933 \\nBest Match \\n.876 .696 .711 .703 \\n.836 .747 .789 \\n.936 .448 .606 \\n.839 \\nBM+Threshold \\n.890 .772 .665 .713 \\n.856 .699 .769 \\n.941 .421 .580 \\n.915 \\nDual Threshold \\n.887 .721 .740 .730 \\n.784 .752 .767 \\n.945 .656 .773 \\n.908\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Comparing key point selection policies, using BERT-large trained on the ArgKP dataset for match scoring.\",\n \"content\": \"Table 5 :\\n5Comparing key point selection policies, using BERT-large trained on the ArgKP dataset for match scoring.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Association for Computational Linguistics. Yamen Ajjour, Milad Alshomary, Henning Wachsmuth, Benno Stein, 10.18653/v1/D19-1290Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP). the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)Hong Kong, ChinaModeling frames in argumentationYamen Ajjour, Milad Alshomary, Henning Wachsmuth, and Benno Stein. 2019. Modeling frames in ar- gumentation. In Proceedings of the 2019 Confer- ence on Empirical Methods in Natural Language Processing and the 9th International Joint Confer- ence on Natural Language Processing (EMNLP- IJCNLP), pages 2922-2932, Hong Kong, China. As- sociation for Computational Linguistics.","doi":"10.18653/v1/D19-1290"},{"text":"Stance classification of context-dependent claims. Roy Bar-Haim, Indrajit Bhattacharya, Francesco Dinuzzo, Amrita Saha, Noam Slonim, Proceedings of the 15th Conference of the European Chapter of the Association for Computational Linguistics. the 15th Conference of the European Chapter of the Association for Computational LinguisticsAssociation for Computational Linguistics1Roy Bar-Haim, Indrajit Bhattacharya, Francesco Din- uzzo, Amrita Saha, and Noam Slonim. 2017. Stance classification of context-dependent claims. In Pro- ceedings of the 15th Conference of the European Chapter of the Association for Computational Lin- guistics: Volume 1, Long Papers, pages 251-261. Association for Computational Linguistics.","doi":""},{"text":"Back up your stance: Recognizing arguments in online discussions. Filip Boltužić, Janšnajder, 10.3115/v1/W14-2107Proceedings of the First Workshop on Argumentation Mining. the First Workshop on Argumentation MiningBaltimore, MarylandAssociation for Computational LinguisticsFilip Boltužić and JanŠnajder. 2014. Back up your stance: Recognizing arguments in online discus- sions. In Proceedings of the First Workshop on Ar- gumentation Mining, pages 49-58, Baltimore, Mary- land. Association for Computational Linguistics.","doi":"10.3115/v1/W14-2107"},{"text":"A large annotated corpus for learning natural language inference. R Samuel, Gabor Bowman, Christopher Angeli, Christopher D Potts, Manning, 10.18653/v1/D15-1075Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing. the 2015 Conference on Empirical Methods in Natural Language ProcessingLisbon, PortugalAssociation for Computational LinguisticsSamuel R. Bowman, Gabor Angeli, Christopher Potts, and Christopher D. Manning. 2015. A large anno- tated corpus for learning natural language inference. In Proceedings of the 2015 Conference on Empiri- cal Methods in Natural Language Processing, pages 632-642, Lisbon, Portugal. Association for Compu- tational Linguistics.","doi":"10.18653/v1/D15-1075"},{"text":"A natural language bipolar argumentation approach to support users in online debate interactions. Elena Cabrio, Serena Villata, 10.1080/19462166.2013.862303Argument & Computation. 43Elena Cabrio and Serena Villata. 2013. A natural lan- guage bipolar argumentation approach to support users in online debate interactions. Argument & Computation, 4(3):209-230.","doi":"10.1080/19462166.2013.862303"},{"text":"Universal sentence encoder. Daniel Cer, Yinfei Yang, Sheng-Yi Kong, Nan Hua, Nicole Limtiaco, Rhomni St John, Noah Constant, Mario Guajardo-Cespedes, Steve Yuan, Chris Tar, arXiv:1803.11175arXiv preprintDaniel Cer, Yinfei Yang, Sheng-yi Kong, Nan Hua, Nicole Limtiaco, Rhomni St John, Noah Constant, Mario Guajardo-Cespedes, Steve Yuan, Chris Tar, et al. 2018. Universal sentence encoder. arXiv preprint arXiv:1803.11175.","doi":"arXiv:1803.11175"},{"text":"Supervised learning of universal sentence representations from natural language inference data. Alexis Conneau, Douwe Kiela, Holger Schwenk, Loic Barrault, Antoine Bordes, arXiv:1705.02364arXiv preprintAlexis Conneau, Douwe Kiela, Holger Schwenk, Loic Barrault, and Antoine Bordes. 2017. Supervised learning of universal sentence representations from natural language inference data. arXiv preprint arXiv:1705.02364.","doi":"arXiv:1705.02364"},{"text":"Recognizing Textual Entailment: Models and Applications. Synthesis Lectures on Human Language Technologies. Dan Ido Dagan, Mark Roth, Fabio Massimo Sammons, Zanzotto, 10.2200/S00509ED1V01Y201305HLT023Morgan & Claypool PublishersIdo Dagan, Dan Roth, Mark Sammons, and Fabio Mas- simo Zanzotto. 2013. Recognizing Textual Entail- ment: Models and Applications. Synthesis Lectures on Human Language Technologies. Morgan & Clay- pool Publishers.","doi":"10.2200/S00509ED1V01Y201305HLT023"},{"text":"BERT: Pre-training of deep bidirectional transformers for language understanding. Jacob Devlin, Ming-Wei Chang, Kenton Lee, Kristina Toutanova, 10.18653/v1/N19-1423Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies. the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language TechnologiesMinneapolis, MinnesotaLong and Short Papers1Association for Computational LinguisticsJacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019. BERT: Pre-training of deep bidirectional transformers for language under- standing. In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pages 4171-4186, Minneapolis, Minnesota. Associ- ation for Computational Linguistics.","doi":"10.18653/v1/N19-1423"},{"text":"Summarising the points made in online political debates. Charlie Egan, Advaith Siddharthan, Adam Wyner, 10.18653/v1/W16-2816Proceedings of the Third Workshop on Argument Mining (ArgMining2016). the Third Workshop on Argument Mining (ArgMining2016)Berlin, GermanyAssociation for Computational LinguisticsCharlie Egan, Advaith Siddharthan, and Adam Wyner. 2016. Summarising the points made in online po- litical debates. In Proceedings of the Third Work- shop on Argument Mining (ArgMining2016), pages 134-143, Berlin, Germany. Association for Compu- tational Linguistics.","doi":"10.18653/v1/W16-2816"},{"text":"Ranit Aharonov, and Noam Slonim. 2020. Corpus wide argument mining -a working solution. Eyal Liat Ein-Dor, Lena Shnarch, Alon Dankin, Benjamin Halfon, Ariel Sznajder, Carlos Gera, Martin Alzate, Leshem Gleize, Yufang Choshen, Yonatan Hou, Bilu, AAAI. Liat Ein-Dor, Eyal Shnarch, Lena Dankin, Alon Hal- fon, Benjamin Sznajder, Ariel Gera, Carlos Alzate, Martin Gleize, Leshem Choshen, Yufang Hou, Yonatan Bilu, Ranit Aharonov, and Noam Slonim. 2020. Corpus wide argument mining -a working solution. In AAAI.","doi":""},{"text":"Measuring nominal scale agreement among many raters. Joseph L Fleiss, Psychological Bulletin. 765Joseph L. Fleiss. 1971. Measuring nominal scale agree- ment among many raters. Psychological Bulletin, 76(5):378-382.","doi":""},{"text":"Ranit Aharonov, and Noam Slonim. 2020. A large-scale dataset for argument quality ranking: Construction and analysis. Shai Gretz, Roni Friedman, Edo Cohen-Karlik, Assaf Toledo, Dan Lahav, AAAI. Shai Gretz, Roni Friedman, Edo Cohen-Karlik, As- saf Toledo, Dan Lahav, Ranit Aharonov, and Noam Slonim. 2020. A large-scale dataset for argument quality ranking: Construction and analysis. In AAAI.","doi":""},{"text":"Why are you taking this stance? identifying and classifying reasons in ideological debates. Saidul Kazi, Vincent Hasan, Ng, 10.3115/v1/D14-1083Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP). the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP)Doha, QatarAssociation for Computational LinguisticsKazi Saidul Hasan and Vincent Ng. 2014. Why are you taking this stance? identifying and classify- ing reasons in ideological debates. In Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 751- 762, Doha, Qatar. Association for Computational Linguistics.","doi":"10.3115/v1/D14-1083"},{"text":"The measurements of observer agreement for categorical data. J R Landis, G G Koch, Biometrics. 33J. R. Landis and G. G. Koch. 1997. The measurements of observer agreement for categorical data. Biomet- rics, 33:159-174.","doi":""},{"text":"Argument mining: A survey. John Lawrence, Chris Reed, 10.1162/coli_a_00364Computational Linguistics. 454John Lawrence and Chris Reed. 2020. Argument mining: A survey. Computational Linguistics, 45(4):765-818.","doi":"10.1162/coli_a_00364"},{"text":"Context dependent claim detection. Ran Levy, Yonatan Bilu, Daniel Hershcovich, Ehud Aharoni, Noam Slonim, Proceedings of COLING 2014, the 25th International Conference on Computational Linguistics: Technical Papers. COLING 2014, the 25th International Conference on Computational Linguistics: Technical PapersDublin, IrelandDublin City University and Association for Computational LinguisticsRan Levy, Yonatan Bilu, Daniel Hershcovich, Ehud Aharoni, and Noam Slonim. 2014. Context depen- dent claim detection. In Proceedings of COLING 2014, the 25th International Conference on Compu- tational Linguistics: Technical Papers, pages 1489- 1500, Dublin, Ireland. Dublin City University and Association for Computational Linguistics.","doi":""},{"text":"Towards an argumentative content search engine using weak supervision. Ran Levy, Ben Bogin, Shai Gretz, Ranit Aharonov, Noam Slonim, Proceedings of the 27th International Conference on Computational Linguistics. the 27th International Conference on Computational LinguisticsAssociation for Computational LinguisticsRan Levy, Ben Bogin, Shai Gretz, Ranit Aharonov, and Noam Slonim. 2018. Towards an argumentative con- tent search engine using weak supervision. In Pro- ceedings of the 27th International Conference on Computational Linguistics, pages 2066-2081. Asso- ciation for Computational Linguistics.","doi":""},{"text":"Unsupervised corpus-wide claim detection. Ran Levy, Shai Gretz, Benjamin Sznajder, Shay Hummel, Ranit Aharonov, Noam Slonim, 10.18653/v1/W17-5110Proceedings of the 4th Workshop on Argument Mining. the 4th Workshop on Argument MiningAssociation for Computational LinguisticsRan Levy, Shai Gretz, Benjamin Sznajder, Shay Hum- mel, Ranit Aharonov, and Noam Slonim. 2017. Un- supervised corpus-wide claim detection. In Pro- ceedings of the 4th Workshop on Argument Mining, pages 79-84. Association for Computational Lin- guistics.","doi":"10.18653/v1/W17-5110"},{"text":"Measuring the similarity of sentential arguments in dialogue. Amita Misra, Brian Ecker, Marilyn Walker, 10.18653/v1/W16-3636Proceedings of the 17th Annual Meeting of the Special Interest Group on Discourse and Dialogue. the 17th Annual Meeting of the Special Interest Group on Discourse and DialogueLos AngelesAssociation for Computational LinguisticsAmita Misra, Brian Ecker, and Marilyn Walker. 2016. Measuring the similarity of sentential arguments in dialogue. In Proceedings of the 17th Annual Meet- ing of the Special Interest Group on Discourse and Dialogue, pages 276-287, Los Angeles. Association for Computational Linguistics.","doi":"10.18653/v1/W16-3636"},{"text":"Automatic detection of arguments in legal texts. Marie-Francine Moens, Erik Boiy, 10.1145/1276318.1276362Proceedings of the 11th International Conference on Artificial Intelligence and Law, ICAIL '07. the 11th International Conference on Artificial Intelligence and Law, ICAIL '07New York, NY, USAACMRaquel Mochales Palau, and Chris ReedMarie-Francine Moens, Erik Boiy, Raquel Mochales Palau, and Chris Reed. 2007. Automatic detection of arguments in legal texts. In Proceedings of the 11th International Conference on Artificial Intelli- gence and Law, ICAIL '07, pages 225-230, New York, NY, USA. ACM.","doi":"10.1145/1276318.1276362"},{"text":"Argumentation mining in parliamentary discourse. Nona Naderi, Proceedings of the 11th International Conference of the Ontario Society for the Study of Argumentation. the 11th International Conference of the Ontario Society for the Study of ArgumentationNona Naderi. 2016. Argumentation mining in parlia- mentary discourse. In Proceedings of the 11th Inter- national Conference of the Ontario Society for the Study of Argumentation, pages 1-9.","doi":""},{"text":"Identifying appropriate support for propositions in online user comments. Joonsuk Park, Claire Cardie, 10.3115/v1/W14-2105Proceedings of the First Workshop on Argumentation Mining. the First Workshop on Argumentation MiningBaltimore, MarylandAssociation for Computational LinguisticsJoonsuk Park and Claire Cardie. 2014. Identifying appropriate support for propositions in online user comments. In Proceedings of the First Workshop on Argumentation Mining, pages 29-38, Baltimore, Maryland. Association for Computational Linguis- tics.","doi":"10.3115/v1/W14-2105"},{"text":"Glove: Global vectors for word representation. Jeffrey Pennington, Richard Socher, Christopher Manning, Proceedings of the 2014 conference on empirical methods in natural language processing (EMNLP). the 2014 conference on empirical methods in natural language processing (EMNLP)Jeffrey Pennington, Richard Socher, and Christopher Manning. 2014. Glove: Global vectors for word rep- resentation. In Proceedings of the 2014 conference on empirical methods in natural language process- ing (EMNLP), pages 1532-1543.","doi":""},{"text":"End-to-end argumentation mining in student essays. Isaac Persing, Vincent Ng, 10.18653/v1/N16-1164Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies. the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language TechnologiesSan Diego, CaliforniaAssociation for Computational LinguisticsIsaac Persing and Vincent Ng. 2016. End-to-end ar- gumentation mining in student essays. In Proceed- ings of the 2016 Conference of the North Ameri- can Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 1384-1394, San Diego, California. Association for Computational Linguistics.","doi":"10.18653/v1/N16-1164"},{"text":"Classification and clustering of arguments with contextualized word embeddings. Nils Reimers, Benjamin Schiller, Tilman Beck, Johannes Daxenberger, Christian Stab, Iryna Gurevych, 10.18653/v1/P19-1054Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics. the 57th Annual Meeting of the Association for Computational LinguisticsFlorence, ItalyAssociation for Computational LinguisticsNils Reimers, Benjamin Schiller, Tilman Beck, Jo- hannes Daxenberger, Christian Stab, and Iryna Gurevych. 2019. Classification and clustering of ar- guments with contextualized word embeddings. In Proceedings of the 57th Annual Meeting of the As- sociation for Computational Linguistics, pages 567- 578, Florence, Italy. Association for Computational Linguistics.","doi":"10.18653/v1/P19-1054"},{"text":"Show me your evidence -an automatic method for context dependent evidence detection. Ruty Rinott, Lena Dankin, Carlos Alzate Perez, Mitesh M Khapra, Ehud Aharoni, Noam Slonim, 10.18653/v1/D15-1050Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing. the 2015 Conference on Empirical Methods in Natural Language ProcessingLisbon, PortugalAssociation for Computational LinguisticsRuty Rinott, Lena Dankin, Carlos Alzate Perez, Mitesh M. Khapra, Ehud Aharoni, and Noam Slonim. 2015. Show me your evidence -an au- tomatic method for context dependent evidence de- tection. In Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing, pages 440-450, Lisbon, Portugal. Association for Computational Linguistics.","doi":"10.18653/v1/D15-1050"},{"text":"Argumentext: Searching for arguments in heterogeneous sources. Christian Stab, Johannes Daxenberger, Chris Stahlhut, Tristan Miller, Benjamin Schiller, Christopher Tauchmann, Steffen Eger, Iryna Gurevych, 10.18653/v1/N18-5005Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Demonstrations. the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: DemonstrationsAssociation for Computational LinguisticsChristian Stab, Johannes Daxenberger, Chris Stahlhut, Tristan Miller, Benjamin Schiller, Christopher Tauchmann, Steffen Eger, and Iryna Gurevych. 2018a. Argumentext: Searching for arguments in heterogeneous sources. In Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Demon- strations, pages 21-25. Association for Computa- tional Linguistics.","doi":"10.18653/v1/N18-5005"},{"text":"Parsing argumentation structures in persuasive essays. Christian Stab, Iryna Gurevych, 10.1162/COLI_a_00295Computational Linguistics. 433Christian Stab and Iryna Gurevych. 2017. Parsing ar- gumentation structures in persuasive essays. Com- putational Linguistics, 43(3):619-659.","doi":"10.1162/COLI_a_00295"},{"text":"Crosstopic argument mining from heterogeneous sources. Christian Stab, Tristan Miller, Benjamin Schiller, Pranav Rai, Iryna Gurevych, Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing. the 2018 Conference on Empirical Methods in Natural Language ProcessingAssociation for Computational LinguisticsChristian Stab, Tristan Miller, Benjamin Schiller, Pranav Rai, and Iryna Gurevych. 2018b. Cross- topic argument mining from heterogeneous sources. In Proceedings of the 2018 Conference on Empiri- cal Methods in Natural Language Processing, pages 3664-3674. Association for Computational Linguis- tics.","doi":""},{"text":"Automatic argument quality assessment -new datasets and methods. Assaf Toledo, Shai Gretz, Edo Cohen-Karlik, Roni Friedman, Elad Venezian, Dan Lahav, Michal Jacovi, Ranit Aharonov, Noam Slonim, 10.18653/v1/D19-1564Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP). the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)Hong Kong, ChinaAssociation for Computational LinguisticsAssaf Toledo, Shai Gretz, Edo Cohen-Karlik, Roni Friedman, Elad Venezian, Dan Lahav, Michal Ja- covi, Ranit Aharonov, and Noam Slonim. 2019. Au- tomatic argument quality assessment -new datasets and methods. In Proceedings of the 2019 Confer- ence on Empirical Methods in Natural Language Processing and the 9th International Joint Confer- ence on Natural Language Processing (EMNLP- IJCNLP), pages 5624-5634, Hong Kong, China. As- sociation for Computational Linguistics.","doi":"10.18653/v1/D19-1564"},{"text":"Building an argument search engine for the web. Henning Wachsmuth, Martin Potthast, Khalid Al Khatib, Yamen Ajjour, Jana Puschmann, Jiani Qu, Jonas Dorsch, Viorel Morari, Janek Bevendorff, Benno Stein, 10.18653/v1/W17-5106Proceedings of the 4th Workshop on Argument Mining. the 4th Workshop on Argument MiningAssociation for Computational LinguisticsHenning Wachsmuth, Martin Potthast, Khalid Al Khatib, Yamen Ajjour, Jana Puschmann, Jiani Qu, Jonas Dorsch, Viorel Morari, Janek Bevendorff, and Benno Stein. 2017. Building an argument search engine for the web. In Proceedings of the 4th Workshop on Argument Mining, pages 49-59. Association for Computational Linguistics.","doi":"10.18653/v1/W17-5106"},{"text":"A broad-coverage challenge corpus for sentence understanding through inference. Adina Williams, Nikita Nangia, Samuel Bowman, 10.18653/v1/N18-1101Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies. the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language TechnologiesNew Orleans, LouisianaLong Papers1Association for Computational LinguisticsAdina Williams, Nikita Nangia, and Samuel Bowman. 2018. A broad-coverage challenge corpus for sen- tence understanding through inference. In Proceed- ings of the 2018 Conference of the North American Chapter of the Association for Computational Lin- guistics: Human Language Technologies, Volume 1 (Long Papers), pages 1112-1122, New Orleans, Louisiana. Association for Computational Linguis- tics.","doi":"10.18653/v1/N18-1101"},{"text":"Approaches to text mining arguments from legal cases. Adam Wyner, Raquel Mochales-Palau, Marie-Francine Moens, David Milward, Semantic Processing of Legal Texts: Where the Language of Law Meets the Law of Language. Berlin, HeidelbergSpringer-VerlagAdam Wyner, Raquel Mochales-Palau, Marie-Francine Moens, and David Milward. 2010. Approaches to text mining arguments from legal cases. In Seman- tic Processing of Legal Texts: Where the Language of Law Meets the Law of Language, page 60-79, Berlin, Heidelberg. Springer-Verlag.","doi":""}],"string":"[\n {\n \"text\": \"Association for Computational Linguistics. Yamen Ajjour, Milad Alshomary, Henning Wachsmuth, Benno Stein, 10.18653/v1/D19-1290Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP). the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)Hong Kong, ChinaModeling frames in argumentationYamen Ajjour, Milad Alshomary, Henning Wachsmuth, and Benno Stein. 2019. Modeling frames in ar- gumentation. In Proceedings of the 2019 Confer- ence on Empirical Methods in Natural Language Processing and the 9th International Joint Confer- ence on Natural Language Processing (EMNLP- IJCNLP), pages 2922-2932, Hong Kong, China. As- sociation for Computational Linguistics.\",\n \"doi\": \"10.18653/v1/D19-1290\"\n },\n {\n \"text\": \"Stance classification of context-dependent claims. Roy Bar-Haim, Indrajit Bhattacharya, Francesco Dinuzzo, Amrita Saha, Noam Slonim, Proceedings of the 15th Conference of the European Chapter of the Association for Computational Linguistics. the 15th Conference of the European Chapter of the Association for Computational LinguisticsAssociation for Computational Linguistics1Roy Bar-Haim, Indrajit Bhattacharya, Francesco Din- uzzo, Amrita Saha, and Noam Slonim. 2017. Stance classification of context-dependent claims. In Pro- ceedings of the 15th Conference of the European Chapter of the Association for Computational Lin- guistics: Volume 1, Long Papers, pages 251-261. Association for Computational Linguistics.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Back up your stance: Recognizing arguments in online discussions. Filip Boltužić, Janšnajder, 10.3115/v1/W14-2107Proceedings of the First Workshop on Argumentation Mining. the First Workshop on Argumentation MiningBaltimore, MarylandAssociation for Computational LinguisticsFilip Boltužić and JanŠnajder. 2014. Back up your stance: Recognizing arguments in online discus- sions. In Proceedings of the First Workshop on Ar- gumentation Mining, pages 49-58, Baltimore, Mary- land. Association for Computational Linguistics.\",\n \"doi\": \"10.3115/v1/W14-2107\"\n },\n {\n \"text\": \"A large annotated corpus for learning natural language inference. R Samuel, Gabor Bowman, Christopher Angeli, Christopher D Potts, Manning, 10.18653/v1/D15-1075Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing. the 2015 Conference on Empirical Methods in Natural Language ProcessingLisbon, PortugalAssociation for Computational LinguisticsSamuel R. Bowman, Gabor Angeli, Christopher Potts, and Christopher D. Manning. 2015. A large anno- tated corpus for learning natural language inference. In Proceedings of the 2015 Conference on Empiri- cal Methods in Natural Language Processing, pages 632-642, Lisbon, Portugal. Association for Compu- tational Linguistics.\",\n \"doi\": \"10.18653/v1/D15-1075\"\n },\n {\n \"text\": \"A natural language bipolar argumentation approach to support users in online debate interactions. Elena Cabrio, Serena Villata, 10.1080/19462166.2013.862303Argument & Computation. 43Elena Cabrio and Serena Villata. 2013. A natural lan- guage bipolar argumentation approach to support users in online debate interactions. Argument & Computation, 4(3):209-230.\",\n \"doi\": \"10.1080/19462166.2013.862303\"\n },\n {\n \"text\": \"Universal sentence encoder. Daniel Cer, Yinfei Yang, Sheng-Yi Kong, Nan Hua, Nicole Limtiaco, Rhomni St John, Noah Constant, Mario Guajardo-Cespedes, Steve Yuan, Chris Tar, arXiv:1803.11175arXiv preprintDaniel Cer, Yinfei Yang, Sheng-yi Kong, Nan Hua, Nicole Limtiaco, Rhomni St John, Noah Constant, Mario Guajardo-Cespedes, Steve Yuan, Chris Tar, et al. 2018. Universal sentence encoder. arXiv preprint arXiv:1803.11175.\",\n \"doi\": \"arXiv:1803.11175\"\n },\n {\n \"text\": \"Supervised learning of universal sentence representations from natural language inference data. Alexis Conneau, Douwe Kiela, Holger Schwenk, Loic Barrault, Antoine Bordes, arXiv:1705.02364arXiv preprintAlexis Conneau, Douwe Kiela, Holger Schwenk, Loic Barrault, and Antoine Bordes. 2017. Supervised learning of universal sentence representations from natural language inference data. arXiv preprint arXiv:1705.02364.\",\n \"doi\": \"arXiv:1705.02364\"\n },\n {\n \"text\": \"Recognizing Textual Entailment: Models and Applications. Synthesis Lectures on Human Language Technologies. Dan Ido Dagan, Mark Roth, Fabio Massimo Sammons, Zanzotto, 10.2200/S00509ED1V01Y201305HLT023Morgan & Claypool PublishersIdo Dagan, Dan Roth, Mark Sammons, and Fabio Mas- simo Zanzotto. 2013. Recognizing Textual Entail- ment: Models and Applications. Synthesis Lectures on Human Language Technologies. Morgan & Clay- pool Publishers.\",\n \"doi\": \"10.2200/S00509ED1V01Y201305HLT023\"\n },\n {\n \"text\": \"BERT: Pre-training of deep bidirectional transformers for language understanding. Jacob Devlin, Ming-Wei Chang, Kenton Lee, Kristina Toutanova, 10.18653/v1/N19-1423Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies. the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language TechnologiesMinneapolis, MinnesotaLong and Short Papers1Association for Computational LinguisticsJacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019. BERT: Pre-training of deep bidirectional transformers for language under- standing. In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pages 4171-4186, Minneapolis, Minnesota. Associ- ation for Computational Linguistics.\",\n \"doi\": \"10.18653/v1/N19-1423\"\n },\n {\n \"text\": \"Summarising the points made in online political debates. Charlie Egan, Advaith Siddharthan, Adam Wyner, 10.18653/v1/W16-2816Proceedings of the Third Workshop on Argument Mining (ArgMining2016). the Third Workshop on Argument Mining (ArgMining2016)Berlin, GermanyAssociation for Computational LinguisticsCharlie Egan, Advaith Siddharthan, and Adam Wyner. 2016. Summarising the points made in online po- litical debates. In Proceedings of the Third Work- shop on Argument Mining (ArgMining2016), pages 134-143, Berlin, Germany. Association for Compu- tational Linguistics.\",\n \"doi\": \"10.18653/v1/W16-2816\"\n },\n {\n \"text\": \"Ranit Aharonov, and Noam Slonim. 2020. Corpus wide argument mining -a working solution. Eyal Liat Ein-Dor, Lena Shnarch, Alon Dankin, Benjamin Halfon, Ariel Sznajder, Carlos Gera, Martin Alzate, Leshem Gleize, Yufang Choshen, Yonatan Hou, Bilu, AAAI. Liat Ein-Dor, Eyal Shnarch, Lena Dankin, Alon Hal- fon, Benjamin Sznajder, Ariel Gera, Carlos Alzate, Martin Gleize, Leshem Choshen, Yufang Hou, Yonatan Bilu, Ranit Aharonov, and Noam Slonim. 2020. Corpus wide argument mining -a working solution. In AAAI.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Measuring nominal scale agreement among many raters. Joseph L Fleiss, Psychological Bulletin. 765Joseph L. Fleiss. 1971. Measuring nominal scale agree- ment among many raters. Psychological Bulletin, 76(5):378-382.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Ranit Aharonov, and Noam Slonim. 2020. A large-scale dataset for argument quality ranking: Construction and analysis. Shai Gretz, Roni Friedman, Edo Cohen-Karlik, Assaf Toledo, Dan Lahav, AAAI. Shai Gretz, Roni Friedman, Edo Cohen-Karlik, As- saf Toledo, Dan Lahav, Ranit Aharonov, and Noam Slonim. 2020. A large-scale dataset for argument quality ranking: Construction and analysis. In AAAI.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Why are you taking this stance? identifying and classifying reasons in ideological debates. Saidul Kazi, Vincent Hasan, Ng, 10.3115/v1/D14-1083Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP). the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP)Doha, QatarAssociation for Computational LinguisticsKazi Saidul Hasan and Vincent Ng. 2014. Why are you taking this stance? identifying and classify- ing reasons in ideological debates. In Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 751- 762, Doha, Qatar. Association for Computational Linguistics.\",\n \"doi\": \"10.3115/v1/D14-1083\"\n },\n {\n \"text\": \"The measurements of observer agreement for categorical data. J R Landis, G G Koch, Biometrics. 33J. R. Landis and G. G. Koch. 1997. The measurements of observer agreement for categorical data. Biomet- rics, 33:159-174.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Argument mining: A survey. John Lawrence, Chris Reed, 10.1162/coli_a_00364Computational Linguistics. 454John Lawrence and Chris Reed. 2020. Argument mining: A survey. Computational Linguistics, 45(4):765-818.\",\n \"doi\": \"10.1162/coli_a_00364\"\n },\n {\n \"text\": \"Context dependent claim detection. Ran Levy, Yonatan Bilu, Daniel Hershcovich, Ehud Aharoni, Noam Slonim, Proceedings of COLING 2014, the 25th International Conference on Computational Linguistics: Technical Papers. COLING 2014, the 25th International Conference on Computational Linguistics: Technical PapersDublin, IrelandDublin City University and Association for Computational LinguisticsRan Levy, Yonatan Bilu, Daniel Hershcovich, Ehud Aharoni, and Noam Slonim. 2014. Context depen- dent claim detection. In Proceedings of COLING 2014, the 25th International Conference on Compu- tational Linguistics: Technical Papers, pages 1489- 1500, Dublin, Ireland. Dublin City University and Association for Computational Linguistics.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Towards an argumentative content search engine using weak supervision. Ran Levy, Ben Bogin, Shai Gretz, Ranit Aharonov, Noam Slonim, Proceedings of the 27th International Conference on Computational Linguistics. the 27th International Conference on Computational LinguisticsAssociation for Computational LinguisticsRan Levy, Ben Bogin, Shai Gretz, Ranit Aharonov, and Noam Slonim. 2018. Towards an argumentative con- tent search engine using weak supervision. In Pro- ceedings of the 27th International Conference on Computational Linguistics, pages 2066-2081. Asso- ciation for Computational Linguistics.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Unsupervised corpus-wide claim detection. Ran Levy, Shai Gretz, Benjamin Sznajder, Shay Hummel, Ranit Aharonov, Noam Slonim, 10.18653/v1/W17-5110Proceedings of the 4th Workshop on Argument Mining. the 4th Workshop on Argument MiningAssociation for Computational LinguisticsRan Levy, Shai Gretz, Benjamin Sznajder, Shay Hum- mel, Ranit Aharonov, and Noam Slonim. 2017. Un- supervised corpus-wide claim detection. In Pro- ceedings of the 4th Workshop on Argument Mining, pages 79-84. Association for Computational Lin- guistics.\",\n \"doi\": \"10.18653/v1/W17-5110\"\n },\n {\n \"text\": \"Measuring the similarity of sentential arguments in dialogue. Amita Misra, Brian Ecker, Marilyn Walker, 10.18653/v1/W16-3636Proceedings of the 17th Annual Meeting of the Special Interest Group on Discourse and Dialogue. the 17th Annual Meeting of the Special Interest Group on Discourse and DialogueLos AngelesAssociation for Computational LinguisticsAmita Misra, Brian Ecker, and Marilyn Walker. 2016. Measuring the similarity of sentential arguments in dialogue. In Proceedings of the 17th Annual Meet- ing of the Special Interest Group on Discourse and Dialogue, pages 276-287, Los Angeles. Association for Computational Linguistics.\",\n \"doi\": \"10.18653/v1/W16-3636\"\n },\n {\n \"text\": \"Automatic detection of arguments in legal texts. Marie-Francine Moens, Erik Boiy, 10.1145/1276318.1276362Proceedings of the 11th International Conference on Artificial Intelligence and Law, ICAIL '07. the 11th International Conference on Artificial Intelligence and Law, ICAIL '07New York, NY, USAACMRaquel Mochales Palau, and Chris ReedMarie-Francine Moens, Erik Boiy, Raquel Mochales Palau, and Chris Reed. 2007. Automatic detection of arguments in legal texts. In Proceedings of the 11th International Conference on Artificial Intelli- gence and Law, ICAIL '07, pages 225-230, New York, NY, USA. ACM.\",\n \"doi\": \"10.1145/1276318.1276362\"\n },\n {\n \"text\": \"Argumentation mining in parliamentary discourse. Nona Naderi, Proceedings of the 11th International Conference of the Ontario Society for the Study of Argumentation. the 11th International Conference of the Ontario Society for the Study of ArgumentationNona Naderi. 2016. Argumentation mining in parlia- mentary discourse. In Proceedings of the 11th Inter- national Conference of the Ontario Society for the Study of Argumentation, pages 1-9.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Identifying appropriate support for propositions in online user comments. Joonsuk Park, Claire Cardie, 10.3115/v1/W14-2105Proceedings of the First Workshop on Argumentation Mining. the First Workshop on Argumentation MiningBaltimore, MarylandAssociation for Computational LinguisticsJoonsuk Park and Claire Cardie. 2014. Identifying appropriate support for propositions in online user comments. In Proceedings of the First Workshop on Argumentation Mining, pages 29-38, Baltimore, Maryland. Association for Computational Linguis- tics.\",\n \"doi\": \"10.3115/v1/W14-2105\"\n },\n {\n \"text\": \"Glove: Global vectors for word representation. Jeffrey Pennington, Richard Socher, Christopher Manning, Proceedings of the 2014 conference on empirical methods in natural language processing (EMNLP). the 2014 conference on empirical methods in natural language processing (EMNLP)Jeffrey Pennington, Richard Socher, and Christopher Manning. 2014. Glove: Global vectors for word rep- resentation. In Proceedings of the 2014 conference on empirical methods in natural language process- ing (EMNLP), pages 1532-1543.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"End-to-end argumentation mining in student essays. Isaac Persing, Vincent Ng, 10.18653/v1/N16-1164Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies. the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language TechnologiesSan Diego, CaliforniaAssociation for Computational LinguisticsIsaac Persing and Vincent Ng. 2016. End-to-end ar- gumentation mining in student essays. In Proceed- ings of the 2016 Conference of the North Ameri- can Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 1384-1394, San Diego, California. Association for Computational Linguistics.\",\n \"doi\": \"10.18653/v1/N16-1164\"\n },\n {\n \"text\": \"Classification and clustering of arguments with contextualized word embeddings. Nils Reimers, Benjamin Schiller, Tilman Beck, Johannes Daxenberger, Christian Stab, Iryna Gurevych, 10.18653/v1/P19-1054Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics. the 57th Annual Meeting of the Association for Computational LinguisticsFlorence, ItalyAssociation for Computational LinguisticsNils Reimers, Benjamin Schiller, Tilman Beck, Jo- hannes Daxenberger, Christian Stab, and Iryna Gurevych. 2019. Classification and clustering of ar- guments with contextualized word embeddings. In Proceedings of the 57th Annual Meeting of the As- sociation for Computational Linguistics, pages 567- 578, Florence, Italy. Association for Computational Linguistics.\",\n \"doi\": \"10.18653/v1/P19-1054\"\n },\n {\n \"text\": \"Show me your evidence -an automatic method for context dependent evidence detection. Ruty Rinott, Lena Dankin, Carlos Alzate Perez, Mitesh M Khapra, Ehud Aharoni, Noam Slonim, 10.18653/v1/D15-1050Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing. the 2015 Conference on Empirical Methods in Natural Language ProcessingLisbon, PortugalAssociation for Computational LinguisticsRuty Rinott, Lena Dankin, Carlos Alzate Perez, Mitesh M. Khapra, Ehud Aharoni, and Noam Slonim. 2015. Show me your evidence -an au- tomatic method for context dependent evidence de- tection. In Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing, pages 440-450, Lisbon, Portugal. Association for Computational Linguistics.\",\n \"doi\": \"10.18653/v1/D15-1050\"\n },\n {\n \"text\": \"Argumentext: Searching for arguments in heterogeneous sources. Christian Stab, Johannes Daxenberger, Chris Stahlhut, Tristan Miller, Benjamin Schiller, Christopher Tauchmann, Steffen Eger, Iryna Gurevych, 10.18653/v1/N18-5005Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Demonstrations. the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: DemonstrationsAssociation for Computational LinguisticsChristian Stab, Johannes Daxenberger, Chris Stahlhut, Tristan Miller, Benjamin Schiller, Christopher Tauchmann, Steffen Eger, and Iryna Gurevych. 2018a. Argumentext: Searching for arguments in heterogeneous sources. In Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Demon- strations, pages 21-25. Association for Computa- tional Linguistics.\",\n \"doi\": \"10.18653/v1/N18-5005\"\n },\n {\n \"text\": \"Parsing argumentation structures in persuasive essays. Christian Stab, Iryna Gurevych, 10.1162/COLI_a_00295Computational Linguistics. 433Christian Stab and Iryna Gurevych. 2017. Parsing ar- gumentation structures in persuasive essays. Com- putational Linguistics, 43(3):619-659.\",\n \"doi\": \"10.1162/COLI_a_00295\"\n },\n {\n \"text\": \"Crosstopic argument mining from heterogeneous sources. Christian Stab, Tristan Miller, Benjamin Schiller, Pranav Rai, Iryna Gurevych, Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing. the 2018 Conference on Empirical Methods in Natural Language ProcessingAssociation for Computational LinguisticsChristian Stab, Tristan Miller, Benjamin Schiller, Pranav Rai, and Iryna Gurevych. 2018b. Cross- topic argument mining from heterogeneous sources. In Proceedings of the 2018 Conference on Empiri- cal Methods in Natural Language Processing, pages 3664-3674. Association for Computational Linguis- tics.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Automatic argument quality assessment -new datasets and methods. Assaf Toledo, Shai Gretz, Edo Cohen-Karlik, Roni Friedman, Elad Venezian, Dan Lahav, Michal Jacovi, Ranit Aharonov, Noam Slonim, 10.18653/v1/D19-1564Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP). the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)Hong Kong, ChinaAssociation for Computational LinguisticsAssaf Toledo, Shai Gretz, Edo Cohen-Karlik, Roni Friedman, Elad Venezian, Dan Lahav, Michal Ja- covi, Ranit Aharonov, and Noam Slonim. 2019. Au- tomatic argument quality assessment -new datasets and methods. In Proceedings of the 2019 Confer- ence on Empirical Methods in Natural Language Processing and the 9th International Joint Confer- ence on Natural Language Processing (EMNLP- IJCNLP), pages 5624-5634, Hong Kong, China. As- sociation for Computational Linguistics.\",\n \"doi\": \"10.18653/v1/D19-1564\"\n },\n {\n \"text\": \"Building an argument search engine for the web. Henning Wachsmuth, Martin Potthast, Khalid Al Khatib, Yamen Ajjour, Jana Puschmann, Jiani Qu, Jonas Dorsch, Viorel Morari, Janek Bevendorff, Benno Stein, 10.18653/v1/W17-5106Proceedings of the 4th Workshop on Argument Mining. the 4th Workshop on Argument MiningAssociation for Computational LinguisticsHenning Wachsmuth, Martin Potthast, Khalid Al Khatib, Yamen Ajjour, Jana Puschmann, Jiani Qu, Jonas Dorsch, Viorel Morari, Janek Bevendorff, and Benno Stein. 2017. Building an argument search engine for the web. In Proceedings of the 4th Workshop on Argument Mining, pages 49-59. Association for Computational Linguistics.\",\n \"doi\": \"10.18653/v1/W17-5106\"\n },\n {\n \"text\": \"A broad-coverage challenge corpus for sentence understanding through inference. Adina Williams, Nikita Nangia, Samuel Bowman, 10.18653/v1/N18-1101Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies. the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language TechnologiesNew Orleans, LouisianaLong Papers1Association for Computational LinguisticsAdina Williams, Nikita Nangia, and Samuel Bowman. 2018. A broad-coverage challenge corpus for sen- tence understanding through inference. In Proceed- ings of the 2018 Conference of the North American Chapter of the Association for Computational Lin- guistics: Human Language Technologies, Volume 1 (Long Papers), pages 1112-1122, New Orleans, Louisiana. Association for Computational Linguis- tics.\",\n \"doi\": \"10.18653/v1/N18-1101\"\n },\n {\n \"text\": \"Approaches to text mining arguments from legal cases. Adam Wyner, Raquel Mochales-Palau, Marie-Francine Moens, David Milward, Semantic Processing of Legal Texts: Where the Language of Law Meets the Law of Language. Berlin, HeidelbergSpringer-VerlagAdam Wyner, Raquel Mochales-Palau, Marie-Francine Moens, and David Milward. 2010. Approaches to text mining arguments from legal cases. In Seman- tic Processing of Legal Texts: Where the Language of Law Meets the Law of Language, page 60-79, Berlin, Heidelberg. Springer-Verlag.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":498,"cells":{"text":{"kind":"string","value":"\nNonlocal scalar field in an external potential: WKB approximation\n\n\nValeri P Frolov \nTheoretical Physics Institute\nUniversity of Alberta\nT6G 2E1EdmontonAlbertaCanada\n\nNonlocal scalar field in an external potential: WKB approximation\n\nWe consider a nonlocal theory of a scalar massive field in a flat spacetime background in the presence of an external potential and construct WKB solutions for this theory. We use a model in which the kinetic part of the scalar field action is modified by changing to f ( ) operator. We discuss conditions when the corresponding form factor f is chosen so that the theory does not contain new unphysical degrees of freedom. We applied the obtained WKB solutions for study energy levels of the field trapped by a one-dimensional potential and the probability of the barrier penetration. This allows us to illustrate how the effects of the nonlocality change the known results obtained for the local field theory.\n\nI. INTRODUCTION\n\nThe idea of nonlocality is quite old in the theoretical physics. Nonlocal modifications of the field theory were discussed already in the publications [1][2][3]. Even if one works with a local quantum field theory, already at the one loop level its effective action is nonlocal. This nonlocality is connected with the vacuum polarization and particle creation effects and it reflects a fact that a vacuum is a physical medium and in this sense it behaves as a condensed matter with very special properties. A nonlocality that we consider in this paper is of a different origin. We assume that a field theory is described by a nonlocal action already at the tree level. In particular this means that its interaction with external sources as well as a self-interaction is nonlocal.\n\nUsually such a nonlocal modification contains a scale parameter which determines either the energy at which the effects of the nonlocality become important or the corresponding spacetime length when it happens. Nonlocal theories were discussed in the beginning of seventies of the past century by Efimov [4][5][6][7]. The nonlocality is often introduced to modify high energy properties of the theory and to improve its ultraviolet behavior. More recently the interest to nonlocal theories increased. This was mainly stimulated by the development of the string theory. Nonlocal fields naturally arise in the string theory and in the theories with a noncommutive geometry (see e.g. [8][9][10][11][12][13][14][15] and references therein). Nonlocal effects in quantum gravity were widely discussed recently. A comprehensive review of modifications of gravity involving a minimal length scale and related references can be found in [16].\n\nThere is a subclass of the non-local field theories sometimes called ghost-free theories. An effective action in these models contains an infinite number of derivatives so that the corresponding field equations are effectively nonlocal. These models have been widely discussed recently and they have rather \"nice properties\". In these theories the nonlocality is introduced in such a way that * vfrolov@ualberta.ca it preserves local Lorentz invariance and it does not introduce new unphysical degrees of freedom (ghosts). In these models the ultraviolet (UV) behaviour of the theory at short distances is improved, while in the infrared (IR) regime (at large scales) they reproduce results of a corresponding local theory [17][18][19][20][21][22][23]. Main motivation for study such infinite derivative modifications of the gravity equations is connected with attempts to solve longstanding problems of cosmological and black hole singularities [18,[24][25][26][27][28][29][30][31][32][33][34].\n\nComplete equations of a modified gravity which include both nonlocality and nonlinearity are quite complicated. Much easier for study is a linearized version of the theory. However already study of linearized models allows one to obtain several stimulating results: (i) The nonlocality removes singularities of the field produced by a point-like sources [18,[35][36][37][38]; (ii) There exists a mass gap for mini black hole formation [39,40]; (iii) It allows one to demonstrate a formation of the inner horizon for black hole creation in the scattering of ultrarelativistic particles [41]. A standard technique for solving these problems is usage of nonlocal Green functions which for the flat spacetime background can be found by means of the Fourier transform.\n\nStudy of the nonlocal field in the presence of an external potential is a much more complicated problem. There exist very special cases when it is possible to find an explicit solution. For example, such a solution was obtained for scattering of a nonlocal scalar field by a delta-like potential [42][43][44]. However in a general case in the presence of an arbitrary potential one needs to solve a nonlocal linear equation with the space-dependent coefficients which is a very non-trivial problem.\n\nThe purpose of this paper is to demonstrate that one can obtain asymptotic solutions for this problem by using a standard WKB method. Its main idea is to search for a solution Φ(x) of the field equation in the form Φ(x) ≈ u(x) exp [iS(x)/ ]. After substitution of this ansatz into the field equation one collects terms of the expansion into the powers of which are of the same order of . In the leading order one gets a first-order partial differential equation of the form H(∇S, x) = 0 known as an eikonal equation. A sub-leading equation determines evolution of a slowly changing field ampli-tude u(x). The eikonal equation can be identified with the Hamilton-Jacobi equation for the Hamiltonian obtained by the substitution ∇S = p into it. Initial data for the Hamilton-Jacobi equation specify a beam of trajectories in the phase space which forms a Lagrangian submanifold (for details see e.g. a remarkable book [45]). Knowledge of this Lagrangian submanifold allows one not only to construct the eikonal function S(x) but also to find a solution of the transport equation for the amplitude u(x) by using the Liouville theorem. This WKB method is widely used in the standard quantum mechanics and field theory where the corresponding equations are second order partial differential equations. However, it can be applied to a wider class of so called quasilinear differential equations. A comprehensive presentation of these results can be found in the book [46]. In these paper we apply this method for study quasiclassical solutions of the linear nonlocal scalar field equations in the presence of an external potential. A similar approach for other higher and infinite order equations can be found in [47].\n\nThis paper is organized as follows. In section II we describe a model of a nonlocal scalar field with infinite number of derivatives which is analysed in the paper. In section III we construct a WKB solution for such a field in the presence of an external potential in any number of spacetime dimensions. In section IV we consider a special case when the potential depends on only one spatial Cartesian coordinate. Energy levels for the nonlocal scalar field confined by the one-dimensional parabolic potential are calculated in section V. Under-barrier propagation of the nonlocal field and barrier penetration effect are discussed in section VI. Section VII contains discussion of the obtained results. Additional technical details are collected in two appendices.\n\n\nII. NONLOCAL SCALAR FIELD EQUATION\n\nWe consider N -dimensional flat spacetime. Its metric in Cartesian coordinates is\nds 2 = η µν dx µ dx ν , µ, ν = 0, 1, . . . , N − 1 ,(1)\nwhere η µν = diag(−1, 1, . . . , 1). Let us consider a scalar massive field ϕ obeying the Klein-Gordon equation\n( 2 − m 2 − V )ϕ = 0 .(2)\nHere = η µν ∂ µ ∂ ν , m is the mass of the field and V (x) is an external potential. In what follows we shall study solutions of this equation and its nonlocal generalization in the WKB approximation. For this reason we keep the Planck constant , while as usual put c = 1 1 . The\n\nKlein-Gordon equation follows from the action\nW [ϕ] = 1 2 d N xϕ(F ( ) − V )ϕ , F ( ) = 2 − m 2 .\n(3) We consider a generalization of the Klein-Gordon equation in which the operator F ( ) is modified. Namely, we assume that it is a scalar operator which may contain an arbitrary (finite or infinite) number of partial derivatives ∂/∂x µ . It is easy to check that the covariance of the action requires that such an operator can be written as a scalar function of the -operator. In order to keep a proper dimensionality of the action we introduce a parameter µ which has a dimension of the mass and we write the operator F ( ) in the form\nF ( ) = µ 2F ( 2 /µ 2 ) .(4)\nThe corresponding generalized Klein-Gordon equation is\n[−F (z) +V ]ϕ = 0 ,(5)\nwhere z = 2 /µ 2 andV = V /µ 2 .\n\nWe consider a class of theories for which the function F (z) has the following properties:\n\n1.F (z) does not vanish anywhere on the complex plane of z besides the point z =m 2 ;\n\n2.F | z=m 2 = 0 and dF (z)/dz| z=m 2 = 1;\n\n3. For a real value of its argument z the functionF (z) is real.\n\nThese conditions guarantee that the inverse of the op-eratorF has a single pole with its residue equal to 1. In other words, there is no new unphysical (ghost) degrees of freedom in this theory. To satisfy the required properties it is sufficient to chooseF in the form\nF (z) = F(ẑ), F(ẑ) =ẑf (ẑ),ẑ = z −m 2 ,(6)\nwhere f (ẑ) = exp(g(ẑ)) and g(ẑ) is an entire function of the complex variable satisfying the condition g(ẑ = 0) = 0. We call f (ẑ) a form factor. In order to illustrate the results we shall use a special example of the form factor. Namely we put g(ẑ) =ẑ 2 then one has\nF(ẑ) = X(ẑ) ≡ẑ exp(ẑ 2 ) .(7)\nThis function obeys all the conditions listed above. Besides this for realẑ it is a monotonically increasing from −∞ atẑ = −∞ to +∞ atẑ = +∞ and is positive for z > 0. One also has F(−ẑ) = −F(ẑ). The function X(x) is shown at figure 1. Other properties of this function which are used later in the paper can be found in appendix B. One can expand the function F(ẑ) into powers ofẑ. This gives an infinite series of the powers of for the field operatorF ( 2 /µ 2 ). This means that this operator is nonlocal. Let us also mention that the mass parameter µ which enters this operator determines a characteristic energy scale at which the effects of the nonlocality become important. In the limit µ → ∞ the operator F becomes 2 − m 2 and the theory is local.\n\n\nIII. WKB APPROXIMATION\n\n\nA. A solution of the nonlocal field equation in the WKB approximation\n\nLet us write the nonlocal field equation (5) in the form 2\nOϕ = 0,Ô = 1 2 (−µF(ẑ) +V ),ẑ = 2 /µ 2 −m 2 . (8)\nWe include a factor 1/2 in the operatorÔ which evidently does not change solutions but this simplifies the form of some relations. The operatorÔ belongs to the class of equations (A1) discussed in the appendix. We are looking for a solution of the field equation (8) in the WKB approximation and write it in the following form\nϕ(x) = exp iS(x) ∞ j=0 i j u j (x) .(9)\nHere S(x) is a fast changing phase, while u j (x) are slowly changing functions. Let us emphasize that besides standard conditions of the validity of the WKB approximation formulated in textbooks on quantum mechanics we also assume that the parameter of nonlocality µ does not 2 In the nonlocal theory it is convenient to use the parameter µ to define dimensionless quantities. We specify such quantities by using a hat over them. For example, we havê\nm = m/µ ,η = η/µ ,η 0 = η 0 /µ ,V = V /µ ,ẑ =η 2 0 −η 2 .\ndepend on . We keep this parameter fixed in the limit → 0. This allows us to discuss effects of the nonlocality in the quasiclassical approximation.\n\nIn order to find the eikonal function S(x) and the amplitudes u j (x) we follow the steps described in appendix A. First of all we define a symbol H(x, p) of the operatorÔ. Let p µ be a covector and denote p 2 = p µ p µ . Then one should substitute − 2 → p 2 into (8). The corresponding symbol for the operatorÔ is\nH(x, p) = 1 2 −µ 2 F(ẑ) + V (x) ,ẑ = − 1 µ 2 (p 2 + m 2 ) .(10)\nThe eikonal equation (A8) implies that the following constraint H(x, p) = 0 is valid.\n\nThe next step is to study a dynamical system in 2Ndimensional phase space satisfying the Hamilton's equations\ndx µ dτ = ∂H ∂p µ , dp µ dτ = − ∂H ∂x µ .(11)\nSince the Hamiltonian H(x, p) is an integral of motion it is sufficient to choose H(x, p) = 0 at the initial \"time\" τ = 0, then the constraint H(x, p) = 0 is valid for any later \"time\". We define a (N − 1)-dimensional surface Σ by conditions\nx µ = f µ (y i ), i = 1, . . . , N − 1 ,(12)\nand choose an initial state for the field ϕ in the form\nϕ(x)| Σ = u 0 (y) exp i S 0 (y) .(13)\nAfter solving equations (A13) we impose the initial conditions (A14) for the Hamiltonian equations (11). Solving this system we find\nx µ = x µ (τ, y i ), p µ = p µ (τ, y i ) .(14)\nA sought solution ϕ(x) of the nonlocal equation (8) in the leading order of the WKB approximation is\nϕ(x) = u 0 (y) J(0, y) J(τ, y) exp i S 0 (y) + τ 0 p µ dx µ .(15)\nThis expression is obtained from (A19) by using the relation\n∂ 2 H(x, p) ∂x µ ∂p µ = 0(16)\nvalid for the Hamiltonian (10). The integral in the exponent in (15) is taken over phase trajectories which belong to the Lagrangian submanifold determied by given initial conditions. Here the function J which enters into the prefactor in (15) is defined by (A18).\n\n\nB. Remarks on the Hamilton's equations\n\nLet us discuss now the Hamilton's equations in more detail. One has\n∂H ∂p µ = F p µ .(17)\nHere a prime denotes a derivative of F(ẑ) with respect to its argumentẑ. Hence the Hamilton's equations (11) take the formẋ\nµ ≡ dx µ dτ = F p µ ,(18)p µ ≡ dp µ dτ = − 1 2 V ,µ .(19)\nThe first of these equations implieṡ\nx 2 ≡ η µνẋ µẋν = (F ) 2 p 2 .(20)\nSinceẑ = −(p 2 + m 2 )/µ 2 the right-hand side of this relation is a function of p 2 . We assume that one can solve equation (20) and find p 2 as a function ofẋ 2 . Then the Lagrangian of this dynamical system is\nL(x,ẋ) =ẋ µ p µ − H = 1 2 [µ 2 (F − 2ẑF ) − 2m 2 F ] − 1 2 V (x) .(21)\nIt is understood that in this expression for L one should expressẑ as a function ofẋ 2 . Since L does not contain explicitly the parameter τ the corresponding \"energy\" for this Lagrangian is conserved. This \"energy\" is nothing but the Hamiltonian in which p 2 is a function ofẋ 2 defined by (20). The constraint equation in the dimensionless form is (see footnote 2) )\n− F(ẑ) +V (x) = 0 .(22)\nIfV (x) ≥ 0 then F(ẑ) =ẑf (ẑ) should be real and positive. The argument of this functionẑ = −(p 2 + m 2 )/µ 2 is real. Since f (0) = 1 and f (ẑ) does not vanish this function is positive for realẑ. This means that the sign of F(ẑ) coincides with the sign ofẑ. The equation (22) implies that the classical motion is possible only in a domain whereẑ ≥ 0.\n\nIn the absence of the potential the constraint equation takes the form\nH 0 (p) = − 1 2 µ 2ẑ f (ẑ) = 0 .(23)\nSince f (ẑ) does not vanish this equation gives\np 2 = −m 2 . The condition µ 2 F (ẑ = 0) = 1 implies that ∂H 0 (p) ∂p µ p 2 +m 2 =0 = p µ .(24)\nThis means that in the absence of the potential the Hamilton's equations (18)- (19) coincide with the equations for a free particle with the Hamiltonian (1/2)(p 2 + m 2 ). In other words, the effects of the nonlocality are important only off-shell, that is in the presence on the external potential.\n\n\nIV. ONE-DIMENSIONAL CASE\n\nTo illustrate an application of the general approach described in the previous section we consider now a simple model. Namely, we assume that the external potential V (x) depends on only one spatial Cartesian coordinate. We denote this coordinate by q and use the following notations\nx µ = (t, q, x ⊥ ), x ⊥ = (x 2 , . . . , x N ) .(25)\nWe use index a = 2, . . . , N to enumerate transverse coordinates x ⊥ . We write the eikonal function S which enters the WKB solution (9) in the form\nS = −εt + (p ⊥ , x ⊥ ) + S(q), (p ⊥ , x ⊥ ) = N a=2 p a x a .(26)\nWe also assume that the functions u j depend only on the coordinate q.\n\nLet us assume first that the potential V (q) vanishes. In order to solve the corresponding field equation (8) one can put S(q) = ηq where η =const, u 0 =const and u j≥1 = 0. This is a plane wave solution of the field equation (8). Denote p µ = (−ε, η, p ⊥ ), where the components of the covector p ⊥ are p a with a = 2, . . . , N . Hence S = (p, x) = p µ x µ . Other solutions of the field equation can be obtained by superimposing the plane-wave solutions, which in fact is just by using the Fourier representation. The constraint equation\np 2 + m 2 = 0 gives − ε 2 + p 2 ⊥ + m 2 + η 2 = 0 .(27)\nIn the presence of the potential we look for a solution of the nonlocal field equation (8) in the form\nϕ(x) = exp i (−εt + (p ⊥ , x ⊥ )) Φ(q) .(28)\nSubstituting this expression into (8) one obtains the following reduced equation\nO 1 Φ = 0 ,(29)O 1 = 1 2 [−µ 2D f (D) + V (q)] ,(30)D = 1 µ 2 η 2 0 + 2 ∂ 2 ∂q 2 , η 2 0 = ε 2 − p 2 ⊥ − m 2 . (31)\nTo obtain the symbol H 1 (q, η) of the operatorÔ 1 we substitute\ni ∂ ∂q → η .(32)\nThe result is\nH 1 (q, η) = 1 2 −µ 2 F(ẑ) + V (q) ,(33)F(ẑ) =ẑf (ẑ),ẑ =η 2 0 −η 2 .(34)\nFor real η the variableẑ is also real as well as the Hamiltonian itself. We also require that for realẑ one has F (ẑ) > 0. For f (ẑ) = exp(g(ẑ)) this condition is satisfied when\n1 +ẑ dg(ẑ) dẑ > 0.(35)\nIn particular, this is valid for the choice (7). The Hamilton-Jacobi equations (18)- (19) in (q, η)phase space reduce tȯ\nq = ∂H 1 ∂η = µ 2 F η,η = − ∂H 1 ∂q = − dV dq ,(36)\nwhile the constraint equation is H 1 (q, η) = 0. Let us denote by (q(t), η(t)) a solution of these equations with the initial condition q(0) = q 0 and η(0) = η 0 . A solution of the field equation (29) in the leading order of the WKB approximation is\nΦ(q) = 1 √ J exp[ i S(q)]Φ 0 (q 0 ) ,(37)S(q) = q q0 ηdq = τ 0 η(τ )q(τ )dτ .(38)\nIn the expression for S(q) the integral is taken along the path q(τ ) connecting the initial point q 0 with a final point\nq. A function Φ 0 (q 0 ) = exp[ i S(q 0 )]u 0 (q 0 ) is the initial value for Φ(q). The factor J is J = dx dτ = µF ẑ 0 −ẑ,ẑ 0 =η 2 0 .(39)\nThe second equality in the expression for J is obtained by using the first Hamilton's equation (36).\n\nSince the Hamiltonian H 1 (q, η) is invariant under the reflection η → −η one can write another WKB solution by a simple change of the sign in the exponent of (37). Taking a superposition of these two WKB solutions one can write Φ(q) in the following form\nΦ(q) = C 1 √ J exp[ i S(q)] + C 2 √ J exp[− i S(q)] .(40)\nFor\nC 2 =C 1 the WKB solution (40) is real. V. BOUND MOTION. ENERGY LEVELS A. A general case\nLet us assume that the potential V (q) is nonnegative and at some point a the following condition is valid\nF(η 0 ) =V (a) .(41)\nThen the constraint equation\nF(ẑ) =V (q)(42)\nimplies that the momentum η vanishes at a. This means that a is a turning point. If dV /dq(a) > 0 then the \"particle\" comes to this point from the left, that is from the domain where q < a and η > 0. After reaching this turning point η changes its sign and the \"particle\" moves to the left. If the potential V (q) has another turning point b where dV /dq(b) < 0, then the \"particle\" comes to it from the right with η < 0 changes its direction of motion at b to the opposite and moves to the right with η > 0. Let us assume that there exist two turning points a < b such that the motion is restricted by the interval a ≤ q ≤ b. Such a motion is called bound. Let us consider a nonnegative potential V (q) which has its minimum at q 0 and V (q 0 ) = 0. At this point the absolute value of the momentum |η(q)| reaches its maximum which is equal to η 0 . Thus for such a bound motion the phase trajectory lies within a rectangle [a, b] × [−η 0 , η 0 ]. Let us emphasize that the Hamiltonian (33) contains a free parameter η 0 which determines a size and a shape of the phase trajectory. Since the parameterẑ depends on the square of the momentum η, a phase trajectory on the plane (q, η) is symmetric with respect to a reflection η → −η. For a given value η 0 > 0 constraint equation (42) \n\n\nFIG. 2. A phase trajectory\n\nThe energy levels of the states corresponding to the bound motion are quantized. In the quasiclassical approximation these levels can be found by using standard Bohr-Sommerfeld quantization condition [46] J = ηdq = π (2n + 1)\n\nHere the integral is taken over a complete period. For a given η 0 > 0 this integral is equal to the surface area inside the phase-trajectory curve corresponding to this parameter. Because of the reflection symmetry η → −η this integral can be written as\nJ = 2 b a η(q)dq .(44)\nThe integral is taken between the turning points a and b.\n\nSometimes it is convenient to rewrite J in another form. Using constraint equation (42) one can express the coordinate q as a function of η. This function has two branches which we denote by q ± (η) (see figure 2). Then, integrating by parts (44) and taking into account that η(a) = η(b) = 0 one finds\nJ = 2 η0 0 (q + (η) − q − (η))dη .(45)\nIf the potential V (x) is a symmetric function of q, V (−q) = V (q), then one has q 0 = 0 and q − (η) = −q + (η).\n\nIn this case the quantity J can be written in the form\nJ = 4 η0 0 q + (η)dη .(46)\nB. Parabolic potential: Local theory\n\nLet us consider a case when the potential V (q) is parabolic\nV (q) = V 2 1 q 2 ,(47)\nwhere V 1 is a positive constant which has the dimension [V 1 ] = M/L. For the local theory the form factor f (ẑ) = 1 and the constraint equation takes the form\nη 2 + V 2 1 q 2 = η 2 0 .(48)\nSolving (48) we find\nq = ±V −1 1 η 2 0 − η 2 .(49)\nThe integral (46) can be easily calculated. We denote the result by J 0 . Then one has\nJ 0 = π η 2 0 V 1 .(50)\nThis gives the following expression for the energy levels\nε 2 n = m 2 + p 2 ⊥ + V 1 (2n + 1) .(51)\nIt is easy to see that the number of levels ∆n within the interval ∆(ε 2 ) is\n∆n = 1 2 V 1 ∆(ε 2 ) .(52)\nThe coefficient 1/(2 V 1 ) does not depend on n. In this sense, the corresponding distribution of energy levels, ∆n/∆(ε 2 ), is equidistant.\n\nC. Parabolic potential: Nonlocal theory\n\n\nPhase trajectories\n\nIn order to discuss the effect of the nonlocality on the distribution of the energy levels for the trapped nonlocal field we consider the same parabolic potential (47) as in the previous subsection, but modify the kinetic part of the effective Hamiltonian. We use dimensionless units defined in the footnote 2 and denotê\nq = V 1 q/µ,ẑ 0 =η 2 0 = 1 µ 2 (ε 2 − m 2 − p 2 ⊥ ) .(53)\nThen the constraint (42) giveŝ\nq = ±F(ẑ) 1/2 .(54)\nLet us denoteη =η 0 sin φ,ẑ =ẑ 0 cos 2 φ .\n\nThen relations (54) and (55) allow one to write the equation for phase trajectories in the parametric form (q(φ), η(φ)). The phase trajectories for the special choice of the form factor (7) are shown at figure 3.\n\n\nEnergy levels\n\nUsing dimensionless variables the action integral (46), which enters the Bohr-Sommerfeld relation (43) for the energy levels, can be written as follows\nJ = 2µ 2 V 1 ẑ0 0 dẑ √ z √ẑ 0 −ẑ f 1/2 (ẑ) .(56)\nWe use here the expression forF in terms of the form factor f ,F =ẑf (ẑ). After change of variables,ẑ =ẑ 0 ξ, relation (56) takes the form\nJ = σ(ẑ 0 )J 0 .(57)\nHere J 0 is the value of the action integral for the local theory (50) and the factor σ(ẑ 0 ) is\nσ(ẑ 0 ) = 2 π 1 0 dξ √ ξ √ 1 − ξ f 1/2 (ẑ 0 ξ) .(58)\nFor a fixed value of η 0 and µ → ∞ the parameterẑ 0 → 0. Since f (0) = 1 the integral (58) simplifies. It can be calculated analytically and the result is σ(0) = 1. Hence in this limit J coincides with J 0 . This is not surprising since in the limit µ → ∞ the theory becomes local. For a numerical calculation of σ(ẑ 0 ) it is convenient to put ξ = cos 2 φ and rewrite (58) in the form σ(ẑ 0 ) = 4 π π/2 0 dφ cos 2 φf 1/2 (ẑ 0 cos 2 φ) . The parameterẑ 0 which enters J depends on the energy ε. The condition of quantization J = π (2n + 1)\n\nallows one to find the corresponding quasiclassical energy levels ε n . Since for smallẑ 0 the coefficient σ(ẑ 0 ) is close to 1, the number of the energy levels n in the interval ∆ε 2 has the form (52). However for large n this distribution of the number of levels is quite different. For the form factor f (ẑ) = exp(ẑ 2 ) one can find it by using the asymptotic form of the coefficient σ(ẑ 0 )\nJ = 1.13 πµ 2 V 1 exp(ẑ 2 0 /2) .(62)\nThe condition of quantization (61) gives\nexp(ẑ 2 0 /2) = V 1 µ 2 (2n + 1) .(63)\nFor n 1 one getsẑ 2 0 ≈ 2 ln n .\n\nSinceẑ 0 = η 2 0 /µ 2 , one finds\nε 2 ≈ m 2 + p 2 ⊥ + √ 2µ 2 √ ln n .(65)\nThis relation implies\n∆n ≈ √ 2 µ 2 n √ ln n ∆(ε 2 ) .(66)\nThe obtained result means that the equidistance of the energy level distribution (52) of the local theory is broken by nonlocality. In the limit of large n and given interval ∆(ε 2 ) the number of the corresponding levels ∆n grows with n as n √ ln n.\n\n\nVI. BARRIER PENETRATION\n\nA. WKB approximation for under-barrier \"motion\"\n\nLet us consider a potential which has the form shown at figure 6 and assume that the effective energy of the \"particle\" is such that its trajectory has a turning point at q = a. At this point one has\nF(ẑ 0 ) =V (a),ẑ 0 =η 2 0 .(67)\nLet us write the constraint equation (42) in the form\nF(ẑ 0 ) − F(ẑ) =V (a) −V (q) .(68)\nIn the domain to the left from the turning point a one hasV (q) <V (a). We assume that for realẑ the function F(ẑ) is monotonically growing. Then one hasẑ <ẑ 0 . In the region q > a whereV (q) >V (a) one should havê z >ẑ 0 . This condition implies that η 2 < 0 which is impossible for a real value of η.\n\nIt is well known that in the local theory a quasiclassical solution for a wave function in the classically forbidden domain can be obtained by considering complex trajectories. Namely, one should change η → iη. The same trick does work for the non-local theories considered in this paper. After the changeẑ →η 2 0 +η 2 the value of z can be larger thanẑ 0 for realη and the equation (68) has a solution in the region where q > a, that is in the classically forbidden domain.\n\nA quasiclassical solution in the underbarrier domain can be obtained by the change η → iη in (40) and it is of the form\nΦ(q) = C − √ J exp[− 1 S(q)] + C + √ J exp[ 1 S(q)] , (69) S(q) = q q0 ηdq = τ 0 η(τ )q(τ )dτ .(70)\nHere\nJ = µF ẑ −ẑ 0 ,ẑ 0 =η 2 0 .(71)\nFor real coefficients C ± the WKB solution (69) is real.\n\n\nB. Nonlocal field in a linear potential\n\nFor a linear potential\nV (q) = U 0 + U 1 (q − a) ,(72)\none-dimensional nonlocal field equation (29) can be solved exactly. For this purpose we denote q = a + x and write Φ(x) in the form\nΦ(x) = ∞ −∞ dη e iηx/ Φ (η) .(73)\nUsing the inverse Fourier transform one gets\nΦ(η) = 1 2π ∞ −∞ dx e −iηx/ Φ(x) .(74)\nThe field equation (29) implies the following differential equation forΦ(η)\nU 1 i ∂ ∂η + U 0 + µ 2 F(ẑ) Φ (η) = 0 .(75)\nA solution of this equation is\nΦ(η) = C exp − i U 1 S(η) ,(76)S(η) = U 0 η + µ 2 η 0 dηF(ẑ) .(77)\nSubstitution of this expression into (73) gives\nΦ(x) = C ∞ −∞ dη exp i ηx − 1 U 1 S(η) .(78)\nThis is a desired solution of the nonlocal field equation (29) C. Connection of WKB solutions at a turning point Consider a potential of the form shown at figure 6 and denote by a a coordinate of a left turning point. One has V ,q (a) > 0. To the left of a and at some distance from it one can use WKB solution (40). For the real field Φ(q) we denote\nC 1 = 1 2 C exp(−iψ), C 2 = 1 2 C exp(iψ) ,(79)\nwhere C and ψ is real and write the WKB solution (40) in the form\nΦ(q) = C √ J cos (S/ − ψ) ,(80)S = q a η(q)dq, J = µF ẑ 0 −ẑ .(81)\nTo the right of the turning point a and at some distance from it we use a decreasing in the under-barrier domain WKB solution\nΦ(q) = C − √ J exp(−S/ ) ,(82)S = q a η(q)dq, J = µF ẑ −ẑ 0 .(83)\nAt the turning point the WKB approximation is not valid. In order to establish a relation between (80) and (82) solutions one can use the following trick [48] which can be easily adapted to our problem. For this purpose we consider formally Φ(q) as a function of a complex variable q and find it along a path on the complex plane of q connecting positive and negative values of q − a. We choose this path to be a half of the circle of constant radius ρ with the center at q = a and take ρ so that the conditions of the validity of the WKB approximation are satisfied along the path. Let us denote\nq − a = ρe ±φ .(84)\nWe choose a sign + for a semi-circle in the upper half of the complex q-plane, and − for a semi-circle in the lower half of the complex q-plane. The angle φ changes from 0 to π. At the turning pointẑ =ẑ 0 =η 2 0 and q = a. We assume that ρ is small so that |ẑ −ẑ 0 | and |q − a| are also small and use the following linearized form of the constraint equation\nF 0 (ẑ −ẑ 0 ) =V ,q (a)(q − a) , F 0 = (dF/dẑ)|ẑ 0 . (85)\nLet us denote B = V ,q (a)/F 0 . One also hasẑ−ẑ 0 =η 2 . Then (84) and (85) imply\nη = µBρ 1/2 e ±iφ/2 .(86)\nFor the analytically continued WKB solution (82) we get\nS ± = 2 3 µBρ 3/2 exp(±3iφ/2) ,(87)J ± = µBF 0 ρ 1/2 exp(±iφ/2) .(88)\nHere ± stands for the quantities calculated along a semicircle in the upper (+) and lower (−) half of the q-plane.\n\nUsing these relations at φ = π where a − q = ρ one obtains\nC = 1 2 C − , ψ = −π/4 .(89)\nThis establishes a relation between the WKB solution (82) in the domain to the right of the turning point and the WKB solution (80) to the left from it. This gives a required connection formula. It should be emphasized that for the operator F ( ) with the properties specified in section II this connection relation is the same as for a standard Schrödinger equation.\n\nWe assume now that the potential has two turning points a < b and the domain between these points is forbidden for the classical motion. We suppose that in the region q < a the field Φ(q) has the form (80) and its amplitude is C. Then in the region q > b it has the same form with a different amplitudeC. Similarly to the local case the ratio |C/C| 2 can be found by using the connection formula (89) twice [48] \nR = C C 2 = exp(− 2 P ) ,(90)P = µ b a ẑ −ẑ 0 dq .(91)\nHereẑ 0 =η 2 0 andẑ =ẑ 2 0 +η 2 . In the expression for P the quantitiesẑ and q satisfy the constraint (42). The quantity R gives a probability of the barrier penetration. It should be emphasized that these relations are obtained in the WKB approximation which is valid when R 1.\n\n\nD. Barrier penetration\n\n\nInverse parabolic potential: Local theory\n\nFor the local theory the effective Hamiltonian is\nH(q, η) = 1 2 [−(η 2 0 − η 2 ) + V (q)] ,(92)\nwhere η 2 0 = ε 2 − p 2 ⊥ − m 2 . Let the potential V (q) be of the form\nV (q) = V 0 − V 2 1 q 2 , if |q| ≤ q * , 0 , if |q| > q * .(93)\nwhere V 0 > 0 and q * = √ V 0 /V 1 . This potential is shown at figure 7. It is called an inverse parabolic potential.\n\nWe assume that η 2 0 < V 0 so that there exit two turning points at q = ±a, where\na = V 0 − η 2 0 V 1 .(94)\nFor the motion in the under-barrier domain we put η = iη whereη\n= ± V − η 2 0 = ±V 1 a 2 − q 2 .(95)\n\nFIG. 7. The inverse parabolic potential\n\nWe denote by P 0 the following integral\nP 0 = a −a |η|dq = V 1 a −a dq a 2 − q 2 = 1 2 πa 2 V 1 .(96)\nUsing relation (94) we can also write P 0 in the form\nP 0 = π(V 0 − η 2 0 ) 2V 1 .(97)\nLet us emphasize that P 0 depends only on the difference V 0 −η 2 0 but it does not depend on the value of η 0 itself. In other words, the penetration probability is sensitive only to the fact how close the effective energy η 2 0 is to the top of the potential V 0 . Let us also remind that the adopted WKB approximation is valid only when the dimensionless quantity P 0 is large, that is when the probability of the penetration R 0 = exp(−2P 0 / ) is small (see e,g, [48]).\n\n\nInverse parabolic potential: Nonlocal theory\n\nLet us assume again that the potential V (q) has the form (93). We denote byẑ 0 andẑ 1 solutions of the following equations\nF(ẑ 0 ) =V 0 −V 2 1 a 2 , F(ẑ 1 ) =V 0 .(98)\nThen the WKB probability of the barrier penetration is\nR = exp − 2 P ,(99)P = μ V 1 ẑ1 z0 dẑ √ẑ −ẑ 0 F(ẑ 1 ) − F(ẑ) .(100)\nLet us calculate the value of P for the case when F is of the form (7). When both parameters V 0 and η 2 0 are much smaller than µ 2 , the function X(ẑ) ≈ẑ and the integral in (100) can be easily taken with the following result\nP = πµ(ẑ 1 −ẑ 0 ) 2V 1 = π(V 0 − η 2 0 ) 2V 1 .(101)\nIn this regime the effects of the nonlocality are not important and we reproduce the expression for P 0 obtained in the local case (see equation (97)). The effects of the nonlocality play an important role in the different regime, whenV 0 1 andη 2 0 1. To demonstrate this we consider again the theory with the form factor f (ẑ) = exp(ẑ 2 ). In order to compare the expression for P with P 0 we fix as earlier the value of the difference V 0 − η 2 0 , but now we assume thatV 0 1. Then one hasẑ 1 −ẑ 0 1. In the appendix B it is shown that one can write the following approximate expression for P in this case\nP = κ(V 0 )P 0 , κ(V 0 ) = 1 X 1 .(102)\nHere X 1 = dX/dx| x=ẑ1 and X(ẑ 1 ) =V 0 . Using (B2) one can write\nX 1 (ẑ 1 ) = (1 + 2ẑ 2 1 ) exp(ẑ 2 1 ) ,(103)V 0 =ẑ 2 1 exp(ẑ 2 1 ) .(104)\nThese relations give a parametric representation of X 1 (ẑ 1 ) as a function ofV 0 . The plot of the function κ(V 0 ) is shown at figure 8. It shows that this function is equal to 1 atV 0 = 0 and it is less than 1 and monotonically decreases for positiveV 0 . Since R = exp(−2P/ ) the relation (102) can be rewritten as follows\nR = (R 0 ) κ(V0) .(105)\nSince R 0 < 1 the value of R is always greater that R 0 . This means that in the nonlocal theory the probability of the barrier penetration is larger that in the local case.\n\nIn other words nonlocality makes the barrier more transparent in the high-energy regime.\n\n\nVII. DISCISSION\n\nLet us discuss the obtained results. In this paper we study quasiclassical solutions of the nonlocal massive scalar field equations. The kinetic part of the action for such a theory contains a function of the -operator, which is chosen so that the theory does not contain new unphysical degrees of freedom. In section II we formulated conditions imposed on the form factor of such a theory. We also described a special model which is used later in the paper for the illustration. In section III we described an ansatz used in the WKB approximation for a solution of the nonlocal field equation. In the leading order of the 1/ -expansion one obtains the equation for the eikonal function S(x) which is of the form H(x, ∇S) = 0. This is a first order partial differential equation which has the form of the Hamilton-Jacobi equation for the Hamiltonian H(x, p), obtained by substitution ∇S = p. To calculate the eikonal function S(x) it is sufficixent to construct the Lagrangian submanifold in the phase space (x, p) by using the initial conditions for S. Such a Lagrangian submanifold is formed by phase trajectories which are solutions of the Hamiltonian equations of motion. Since the system is conservative this submanifold belongs to the subspace H(x, p) = 0. Let us emphasize that this procedure is very similar to the standard one adopted for the quantum mechanics and local field theory. When the field operator is of the second order the corresponding effective Hamiltonian H(x, p) is a quadratic polynomial in momentum p. For the nonlocal theory H(x, p) is a non-polynomial function of p and this is a main difference with the local case. As a result the construction of the WKB solutions becomes technically more complicated.\n\nWKB solutions presented in section III are valid for a wide class of nonlocal scalar field theories with an arbitrary external potential V (x) in a flat spacetime with any number of spatial dimensions. These relations are greatly simplified for the case when the potential V depends only on one spatial Cartesian coordinate. For this case the phase space is two-dimensional and the Hamilton's equations are completely integrable. The corresponding WKB solutions are described in section IV. In section V we used the WKB approximation for the calculation of the energy levels of the nonlocal field trapped by a one-dimensional potential. To illustrate general formulas we considered the case when the form factor of the nonlocality is of the special form (7) and the potential is parabolic. The energy levels ε n of the field depend on a discrete integer parameter n. For the local case the number ∆n of the levels in the interval ∆(ε 2 ) is ∆n = ∆(ε 2 )/(2 V 1 ) and it does not depend on the value of n. In this sense the levels are equidistant. In the nonlocal case for large n this dependence is different\n∆n ≈ √ 2 µ 2 n √ ln n ∆(ε 2 ) .(106)\nThis means that at very high energy for a given interval of ∆(ε 2 ) there are much more energy levels than in the local case. Finally, we discussed the under-barrier motion and obtained an expression for the probability of the barrier penetration. This probability has the form ∼ exp(−2P/ ). We calculated the factor P for the the special form of the nonlocality (7) in the presence of the inverse parabolic potential. For the local theory this factor is P 0 = π(V 0 − η 2 0 )/(2V 1 ). In other words, it depends on how close is the energy of the state η 2 0 to the top of the potential V 0 , but it does not depend on the height V 0 of the potential itself. The situation in the nonlocal case is quite different. Namely, one has P = κP 0 , where the factor κ depends on the height V 0 of the potential. This factor is always less than 1 and it decreases when V 0 grows. This means that the nonlocality \"helps\" the field to penetrate the potential barrier.\n\nThe examples considered in this paper are rather simple. It would be interesting to apply the developed in the paper method for study the effects of the nonlocality in the early cosmology and in the black holes interior.\n\n\nAppendix A: WKB approximation\n\nWe collected here useful information concerning construction of solutions of the nonlocal field equations in the WKB approximation. Further details and required mathematical foundations can be found in the book [46].\n\nLet (x 1 , . . . , x N ) be coordinates of a point x in R N . We denote by α a set of N non-negative integer numbers α = (α 1 , . . . , α N ). We consider the following operator in this spacê\nO = α a α i ∂ ∂x 1 α1 . . . i ∂ ∂x N α N . (A1)\nThe sum is taken over all different combinations of α.\n\nWe do not assume that the number of terms in the sum is finite, so that this expression can be used as a formal series in derivatives representation of nonlocal operators. Let us make a substitution\ni ∂ ∂x µ → p µ (A2)\nin the expression (A1). Then we obtain a function of 2N variables which we denote by H = H(x, p)\nH = H(x, p) = α a α p α1 1 . . . p α N N .(A3)\nHere p = (p 1 , . . . , p N ). This object is called a symbol of the operatorD. We are looking for a solution ϕ of the equation\nOϕ = 0 (A4)\nin the form of the following formal series\nϕ(x) = exp iS(x) ∞ j=0 i j u j (x) .(A5)\nUsually it is sufficient to consider only first few terms in the series. This gives what is known as the WKB approximation for the solution ϕ. To obtain the functions S(x) and u j (x) that enter (A5) the following procedure is used (see e.g. [45,46] and references therein). If one substitute expression (A5) into the equation (A4) then one finds that in the leading order j = 0 the following condition is to be satisfied\nH x, ∂S ∂x u 0 = 0 ,(A6)\nwhere\n∂S ∂x = ∂S ∂x 1 . . . , ∂S ∂x N .(A7)\nFor a nonvanishing function u 0 (x) the following Hamilton-Jacobi equation is valid\nH x, ∂S ∂x = 0 .(A8)\nLet us consider a (2N )-dimension space Γ 2N with coordinates (x, p) = (x 1 , . . . , x N , p 1 , . . . , P N ) and denote by Ω a symplectic two-form in it\nΩ = N µ=1 dp µ ∧ dx µ .(A9)\nAn N -dimensional surface Λ N in Γ 2N is called a Lagrangian submanifold if the Ω vanishes on Λ N . If a Lagrangian submanifold can be uniquely projected onto the x-space, then there exists such a function S(x) that p µ = ∂S/∂ µ on Λ N . The Hamiltonian H(x, p) by the relation (A3) implies the following Hamilton's equations are\ndx µ dτ = ∂H(x, p) ∂p µ , dp µ dτ = − ∂H(x, p) ∂x µ .(A10)\nThese equations determine the Hamiltonian flow in the phase space Γ 2N preserving the symplectic form Ω and the Hamiltonian itself is an integral of motion. Let Σ N −1 be a (N −1)-dimensional surface in the coordinate space. Its embedding is defined by the equations\nx µ = f µ (y i ), i = 1, . . . , N − 1 .(A11)\nOne can consider y i as coordinates on Σ N −1 . We associate with Σ N −1 a (2N − 2)-dimensional phase space Γ 2N −2 and denote its canonical coordinates by (y, v) = (y 1 , . . . , y N −1 , v 1 , . . . , v N −1 ). The symplectic 2-form ω in\nΓ 2N −2 is ω = N −1 i=1 dv i ∧ dy i .(A12)\nDenote by S 0 = S 0 (y) a function on Σ N −1 and ∇S 0 = (S 0 ,y 1 , . . . , S 0 ,y N −1 ), then a (N − 1)-dimensional surface (y, ∇S) of Γ 2N −2 is a Lagrangian submanifold Λ N −1 , so that ω| Λ N −1 = 0.\n\nLet us define a covector p(y) = p µ (y) on Σ N −1 by two conditions p µ (y)dx µ = p µ (y) ∂f µ ∂y i = 0 , H(x(y), p(y)) = 0 . (A13)\n\nWe assume that this system of N equations has a solution which determines p µ (y)| Σ N −1 . Let us consider the following initial conditions for the Hamilton's equations (A10)\nx µ | τ =0 = f µ (y i ), p µ | τ =0 = p µ (y) .(A14)\nA family of solutions of the Hamilton's equations (A10) with these initial conditions, (x(τ, y), p(τ, y)), forms a Ndimensional surface in Γ 2N which we denote by B. It is possible to show that B is a Lagrangian submanifold in Γ 2N and that H(x, p) = 0 on it. This implies that there exists such a function S(x) that\np µ | B = ∂S ∂x µ .(A15)\nThis function is a solution of the Hamilton-Jacobi equation (A8) satisfying the initial condition S(x)| Σ N −1 = S 0 (y). This function S(x) can be found as follows. Denote a 1-form θ = µ p µ dx µ , then Ω = dθ. Since the symplectic form Ω vanishes on B, the integral\np µ dx µ (A16)\ncalculated along a path on B between its two points does not depend on the choice of the path. The required solution S(x) of the Hamilton-Jacobi equation satisfying the initial condition S| Σ N −1 = S(y) can be written in the form\nS(x) = S 0 (y) + τ 0 p µ dx µ .(A17)\nHere x µ = x µ (τ, y) and p µ = p µ (τ, y). Let us note that the map x µ = x µ (τ, y) may be considered as a transformation between the coordinates (τ, y and x µ . Let us denote by J a Jacobian of this transformation J(τ, y) = det ∂x(τ, y) ∂(τ, y) .\n\nThen the leading term of the asymptotic solution (A5) is [46] ϕ(x) = u 0 (y) J(0, y) J(τ, y) exp i S 0 (y) + For τ = 0 the right-hand side of this expression reduces to u 0 (y) exp i S 0 (y) and it should be found from the initial condition for the function ϕ(x) at Σ.\n\nAppendix B: Function X(x) = x exp(x 2 ) and its properties\n\nIn this appendix we discuss some properties of the function\nX(x) = x exp(x 2 ) .(B1)\nThe derivatives of this function are X = (1 + 2x 2 ) exp(x 2 ) ,\nX = 6x(1 + 2 3 x 2 ) exp(x 2 ) , (B2) α = 1 2 X X = x 1 + 2 2x 2 + 1 .\nWe denote by Y an inverse function of X such that the following relations are valid y = X(x), x = Y (y) .\n\nThe inverse function can be written as follows Y (y) = y exp − 1 2 W (2y 2 ) .\n\nHere W is Lambert W function defined by the relation W (z) exp(W (z)) = z .\n\nFor a real positive argument W is a real single-values function. Consider a following integral\nI = x1 x0 dx √ x − x 0 X(x 1 ) − X(x) .(B6)\nLet us assume that x 1 − x 0 is small. To obtain an approximate value of the integral I in this case we use the Taylor expansion\nX(x) = X 1 − X 1 (x 1 − x) + 1 2 X 1 (x 1 − x) 2 + . . . . (B7)\nHere X 1 = X(x 1 ), X 1 = X (x 1 ), X 1 = X (x 1 ). Skipping the higher order terms denoted in (B7) by dots one has\nI = X 1 x1 x0 dx √ x − x 0 (x 1 − x) − α(x 1 − x) 2 ,(B8)\nwhere α = 1 2 (X 1 /X 1 ). This integral can be calculated exactly. Let us denote β = x 1 − x 0 , then for αβ < 1 the answer is I = 2 X 1 3α (2αβ − 1)E( αβ) − (αβ − 1)K( αβ) .\n\nHere E and K are complete elliptic integrals. Forthe small value of αβ one has\nI = πβ 2 X 1 1 − 3 8 αβ .(B9)\nFor αβ 1 one can neglect the second term. Keeping the leading term one gets\nI = πβ 2 X 1 = π 2 X 1 − X 0 X 1 . (B10)\nFor small x 1 one has X 1 ≈ 1 and\nI ≈ π 2 (X 1 − X 0 ) . (B11) ACKNOWLEDGMENTS\nThe author thanks the Natural Sciences and Engineering Research Council of Canada and the Killam Trust for their financial support.\n\nFIG. 1 .\n1Function X(x).\n\n\ndefines a closed curve on (q, η) phase plane (see figure 2).\n\nFIG. 3 .\n3Phase trajectories for the form factor f = exp(ẑ 2 ). Different curves correspond to the different values ofẑ0 parameter: 0.14 for line 1; 0.7 for line 2; 1.0 for line 3 and 1.1 for line 4.\n\nFIG. 4 .Figure 5\n45Function σ(ẑ0) for the form factor f (ẑ) = exp(ẑ 2 ).The function σ(ẑ 0 ) for the choice of the form factor f (ẑ) = exp(ẑ 2 ) is shown at figure 4. For this form factor it is possible to show that for largeẑ 0 the function σ(ẑ 0 ) has the follows asymptotic form which plots the ratio σ(ẑ 0 )/σ 0 (ẑ 0 ) demonstrates this property.\n\nFIG. 5 .\n5Ratio σ/σ0 as a function ofẑ0.\n\nFIG. 6 .\n6Potential V (q).\n\nFIG. 8 .\n8Parameter κ as a function ofV0.\nIn these units one has [ ] = M L, [ ] = L −2 , and [m 2 ] = [V ] = M 2 .\n\n. G Wataghin, 10.1007/BF01352311Z. Phys. 8892G. Wataghin, Z. Phys. 88, 92 (1934).\n\n. H Yukawa, 10.1103/PhysRev.77.219Phys. Rev. 77219H. Yukawa, Phys. Rev. 77, 219 (1950).\n\n. H Yukawa, 10.1103/PhysRev.80.1047Phys. Rev. 801047H. Yukawa, Phys. Rev. 80, 1047 (1950).\n\n. G V Efimov, 10.1007/BF01648331Comm. Math. Phys. 7138G. V. Efimov, Comm. Math. Phys. 7, 138 (1968).\n\n. G V Efimov, 10.1016/0003-4916(72)90127-3Annals Phys. (NY). 71466G. V. Efimov, Annals Phys. (NY) 71, 466 (1972).\n\n. G V Efimov, 10.1007/BF01808314Int. J. Theor. Phys. 1019G. V. Efimov, Int. J. Theor. Phys. 10, 19 (1974).\n\n. V A Alebastrov, G V Efimov, 10.1007/BF01651546Commun. Math. Phys. 3811V. A. Alebastrov and G. V. Efimov, Commun. Math. Phys. 38, 11 (1974).\n\n. E Witten, 10.1016/0550-3213(86)90155-0Nucl. Phys. 268253E. Witten, Nucl. Phys. B268, 253 (1986).\n\n. P H Frampton, Y Okada, 10.1103/PhysRevD.37.3077Phys. Rev. 373077P. H. Frampton and Y. Okada, Phys. Rev. D37, 3077 (1988).\n\n. D A Eliezer, R P Woodard, 10.1016/0550-3213(89)90461-6Nucl. Phys. 325389D. A. Eliezer and R. P. Woodard, Nucl. Phys. B325, 389 (1989).\n\n. V A Kostelecky, S Samuel, 10.1103/PhysRevD.42.1289Phys. Rev. 421289V. A. Kostelecky and S. Samuel, Phys. Rev. D42, 1289 (1990).\n\n. B Harms, Y Leblanc, 10.1006/aphy.1995.1080arXiv:hep-th/9307042Annals Phys. 242hep-thB. Harms and Y. Leblanc, Annals Phys. 242, 265 (1995), arXiv:hep-th/9307042 [hep-th].\n\n. A A Tseytlin, 10.1016/0370-2693(95)01228-7arXiv:hep-th/9509050Phys. Lett. 363223hep-thA. A. Tseytlin, Phys. Lett. B363, 223 (1995), arXiv:hep- th/9509050 [hep-th].\n\n. G Calcagni, L Modesto, Journal of Physics A: Mathematical and Theoretical. 47355G. Calcagni and L. Modesto, Journal of Physics A: Math- ematical and Theoretical 47, 355 (2014).\n\n. G Calcagni, L Modesto, Physical Review D. 91124059G. Calcagni and L. Modesto, Physical Review D 91, 124059 (2015).\n\n. S Hossenfelder, 10.12942/lrr-2013-2arXiv:1203.6191Living Rev. Rel. 16gr-qcS. Hossenfelder, Living Rev. Rel. 16, 2 (2013), arXiv:1203.6191 [gr-qc].\n\n. E Tomboulis, arXiv:hep-th/9702146hep-thE. Tomboulis, (1997), arXiv:hep-th/9702146 [hep-th].\n\n. T Biswas, E Gerwick, T Koivisto, A Mazumdar, 10.1103/PhysRevLett.108.031101arXiv:1110.5249Phys. Rev. Lett. 10831101gr-qcT. Biswas, E. Gerwick, T. Koivisto, and A. Mazumdar, Phys. Rev. Lett. 108, 031101 (2012), arXiv:1110.5249 [gr-qc].\n\n. L Modesto, 10.1103/PhysRevD.86.044005arXiv:1107.2403Phys. Rev. 8644005hep-thL. Modesto, Phys. Rev. D86, 044005 (2012), arXiv:1107.2403 [hep-th].\n\n. T Biswas, A Conroy, A S Koshelev, A Mazumdar, 10.1088/0264-9381/31/1/015022,10.1088/0264-9381/31/15/159501arXiv:1308.2319Class.Quant.Grav. 3115022hep-thT. Biswas, A. Conroy, A. S. Koshelev, and A. Mazumdar, Class.Quant.Grav. 31, 015022 (2014), arXiv:1308.2319 [hep-th].\n\n. I L Shapiro, 10.1016/j.physletb.2015.03.037arXiv:1502.00106Phys. Lett. 74467hep-thI. L. Shapiro, Phys. Lett. B744, 67 (2015), arXiv:1502.00106 [hep-th].\n\n. T Biswas, T Koivisto, A Mazumdar, 10.1088/1475-7516/2010/11/008arXiv:1005.0590JCAP. 10118hep-thT. Biswas, T. Koivisto, and A. Mazumdar, JCAP 1011, 008 (2010), arXiv:1005.0590 [hep-th].\n\nT Biswas, T Koivisto, A Mazumdar, arXiv:1302.0532Proceedings. nullgr-qcT. Biswas, T. Koivisto, and A. Mazumdar, in Pro- ceedings, Barcelona Postgrad Encounters on Fundamen- tal Physics (2013) pp. 13-24, arXiv:1302.0532 [gr-qc].\n\n. T Biswas, A Mazumdar, W Siegel, Journal of Cosmology and Astroparticle Physics. 06039T. Biswas, A. Mazumdar, and W. Siegel, Journal of Cosmology and Astroparticle Physics 0603, 009 (2006).\n\n. A S Koshelev, Journal of High Energy Physics. 071029A. S. Koshelev, Journal of High Energy Physics 0710, 029 (2007).\n\n. T Biswas, T Koivisto, A Mazumdar, Journal of Cosmology and Astroparticle Physics. 10118T. Biswas, T. Koivisto, and A. Mazumdar, Journal of Cosmology and Astroparticle Physics 1011, 008 (2010).\n\n. L Modesto, J W Moffat, P Nicolini, Physics Letters B. 695397L. Modesto, J. W. Moffat, and P. Nicolini, Physics Let- ters B 695, 397 (2011).\n\n. A Conroy, A Mazumdar, S Talaganis, A Teimouri, 10.1103/PhysRevD.92.124051arXiv:1509.01247Phys. Rev. 92124051hepthA. Conroy, A. Mazumdar, S. Talaganis, and A. Teimouri, Phys. Rev. D92, 124051 (2015), arXiv:1509.01247 [hep- th].\n\n. L Modesto, L , 10.1142/S0218271817300208Int. J. Mod. Phys. 261730020L. Modesto and L. Rachwa l, Int. J. Mod. Phys. D26, 1730020 (2017).\n\n. L Buoninfante, A S Koshelev, G Lambiase, A Mazumdar, 10.1088/1475-7516/2018/09/034arXiv:1802.00399JCAP. 180934gr-qcL. Buoninfante, A. S. Koshelev, G. Lambiase, and A. Mazumdar, JCAP 1809, 034 (2018), arXiv:1802.00399 [gr-qc].\n\n. A S Koshelev, J Marto, A Mazumdar, 10.1103/PhysRevD.98.064023arXiv:1803.00309Phys. Rev. 9864023gr-qcA. S. Koshelev, J. Marto, and A. Mazumdar, Phys. Rev. D98, 064023 (2018), arXiv:1803.00309 [gr-qc].\n\n. E Kilicarslan, Turk, 10.3906/fiz-1811-2arXiv:1811.00843J. Phys. 43gr-qcE. Kilicarslan, Turk. J. Phys. 43, 126 (2019), arXiv:1811.00843 [gr-qc].\n\n. A S Koshelev, J Marto, A Mazumdar, Journal of Cosmology and Astroparticle Physics. 20A. S. Koshelev, J. Marto, and A. Mazumdar, Journal of Cosmology and Astroparticle Physics 2019, 020 (2019).\n\n. E Kilicarslan, 10.1103/PhysRevD.99.124048arXiv:1903.04283Phys. Rev. 99gr-qcE. Kilicarslan, Phys. Rev. D99, 124048 (2019), arXiv:1903.04283 [gr-qc].\n\n. F J Cruz-Dombriz, A Maldonado Torralba, Mazumdar, 10.1103/PhysRevD.99.104021arXiv:1812.04037Phys. Rev. 99gr-qcl. de la Cruz-Dombriz, F. J. Maldonado Torralba, and A. Mazumdar, Phys. Rev. D99, 104021 (2019), arXiv:1812.04037 [gr-qc].\n\n. L Buoninfante, A Mazumdar, 10.1103/PhysRevD.100.024031arXiv:1903.01542Phys. Rev. 10024031gr-qcL. Buoninfante and A. Mazumdar, Phys. Rev. D100, 024031 (2019), arXiv:1903.01542 [gr-qc].\n\n. J Boos, V P Frolov, A Zelnikov, 10.1103/PhysRevD.97.084021arXiv:1802.09573Phys. Rev. 9784021gr-qcJ. Boos, V. P. Frolov, and A. Zelnikov, Phys. Rev. D97, 084021 (2018), arXiv:1802.09573 [gr-qc].\n\n. J Boos, J Pinedo Soto, V P Frolov, 10.1103/PhysRevD.101.124065arXiv:2004.07420Phys. Rev. 101124065gr-qcJ. Boos, J. Pinedo Soto, and V. P. Frolov, Phys. Rev. D101, 124065 (2020), arXiv:2004.07420 [gr-qc].\n\n. V P Frolov, 10.1103/PhysRevLett.115.051102arXiv:1505.00492Phys. Rev. Lett. 11551102hep-thV. P. Frolov, Phys. Rev. Lett. 115, 051102 (2015), arXiv:1505.00492 [hep-th].\n\n. V P Frolov, A Zelnikov, T De Paula, Netto, 10.1007/JHEP06(2015)107arXiv:1504.00412JHEP. 06107hep-thV. P. Frolov, A. Zelnikov, and T. de Paula Netto, JHEP 06, 107 (2015), arXiv:1504.00412 [hep-th].\n\n. V P Frolov, A Zelnikov, 10.1103/PhysRevD.93.064048arXiv:1509.03336Phys. Rev. 9364048hep-thV. P. Frolov and A. Zelnikov, Phys. Rev. D93, 064048 (2016), arXiv:1509.03336 [hep-th].\n\n. J Boos, V P Frolov, A Zelnikov, 10.1016/j.physletb.2018.06.018arXiv:1805.01875Phys. Lett. 782688hep-thJ. Boos, V. P. Frolov, and A. Zelnikov, Phys. Lett. B782, 688 (2018), arXiv:1805.01875 [hep-th].\n\n. V P Frolov, A Zelnikov, 10.1103/PhysRevD.98.084035arXiv:1809.00417Phys. Rev. 9884035hep-thV. P. Frolov and A. Zelnikov, Phys. Rev. D98, 084035 (2018), arXiv:1809.00417 [hep-th].\n\n. L Buoninfante, A Mazumdar, J Peng, 10.1103/PhysRevD.100.104059arXiv:1906.03624Phys. Rev. D. 100104059gr-qcL. Buoninfante, A. Mazumdar, and J. Peng, Phys. Rev. D 100, 104059 (2019), arXiv:1906.03624 [gr-qc].\n\n. V Arnold, K Vogtmann, A Weinstein, Mathematical Methods of Classical Mechanics. SpringerV. Arnold, K. Vogtmann, and A. Weinstein, Mathemati- cal Methods of Classical Mechanics (Springer New York, 2013).\n\nV Maslov, M Fedoriuk, Semi-Classical Approximation in Quantum Mechanics, Mathematical Physics and Applied Mathematics. NetherlandsSpringerV. Maslov and M. Fedoriuk, Semi-Classical Approxima- tion in Quantum Mechanics, Mathematical Physics and Applied Mathematics (Springer Netherlands, 1981).\n\n. F Lu, B Lv, P Wang, H Yang, 10.1016/j.nuclphysb.2018.06.002arXiv:1701.00395Nucl. Phys. 937gr-qcF. Lu, B. Lv, P. Wang, and H. Yang, Nucl. Phys. B937, 502 (2018), arXiv:1701.00395 [gr-qc].\n\nL D Landau, E M Lifshitz, Quantum Mechanics. Bristol, UKPergamon Press2nd ed.L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 2nd ed. (Pergamon Press, Bristol, UK, 1965).\n"},"corpus_id":{"kind":"number","value":227152308,"string":"227,152,308"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2011.11700\",\n \"mag\": 3124882961,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1103/physrevd.103.025020\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/2011.11700v1.pdf\"\n ],\n \"pdf_sha\": \"cc3e7ff94da27a10534aa0979b31d6b6f12fefa3\",\n \"open_access\": {\n \"license\": \"CCBY\",\n \"url\": \"http://link.aps.org/pdf/10.1103/PhysRevD.103.025020\",\n \"status\": \"HYBRID\"\n }\n}"},"title":{"kind":"string","value":"Nonlocal scalar field in an external potential: WKB approximation"},"authors":{"kind":"list like","value":["Valeri P Frolov \nTheoretical Physics Institute\nUniversity of Alberta\nT6G 2E1EdmontonAlbertaCanada"],"string":"[\n \"Valeri P Frolov \\nTheoretical Physics Institute\\nUniversity of Alberta\\nT6G 2E1EdmontonAlbertaCanada\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We consider a nonlocal theory of a scalar massive field in a flat spacetime background in the presence of an external potential and construct WKB solutions for this theory. We use a model in which the kinetic part of the scalar field action is modified by changing to f ( ) operator. We discuss conditions when the corresponding form factor f is chosen so that the theory does not contain new unphysical degrees of freedom. We applied the obtained WKB solutions for study energy levels of the field trapped by a one-dimensional potential and the probability of the barrier penetration. This allows us to illustrate how the effects of the nonlocality change the known results obtained for the local field theory."},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":"The idea of nonlocality is quite old in the theoretical physics. Nonlocal modifications of the field theory were discussed already in the publications [1][2][3]. Even if one works with a local quantum field theory, already at the one loop level its effective action is nonlocal. This nonlocality is connected with the vacuum polarization and particle creation effects and it reflects a fact that a vacuum is a physical medium and in this sense it behaves as a condensed matter with very special properties. A nonlocality that we consider in this paper is of a different origin. We assume that a field theory is described by a nonlocal action already at the tree level. In particular this means that its interaction with external sources as well as a self-interaction is nonlocal.\nUsually such a nonlocal modification contains a scale parameter which determines either the energy at which the effects of the nonlocality become important or the corresponding spacetime length when it happens. Nonlocal theories were discussed in the beginning of seventies of the past century by Efimov [4][5][6][7]. The nonlocality is often introduced to modify high energy properties of the theory and to improve its ultraviolet behavior. More recently the interest to nonlocal theories increased. This was mainly stimulated by the development of the string theory. Nonlocal fields naturally arise in the string theory and in the theories with a noncommutive geometry (see e.g. [8][9][10][11][12][13][14][15] and references therein). Nonlocal effects in quantum gravity were widely discussed recently. A comprehensive review of modifications of gravity involving a minimal length scale and related references can be found in [16].\nThere is a subclass of the non-local field theories sometimes called ghost-free theories. An effective action in these models contains an infinite number of derivatives so that the corresponding field equations are effectively nonlocal. These models have been widely discussed recently and they have rather \"nice properties\". In these theories the nonlocality is introduced in such a way that * vfrolov@ualberta.ca it preserves local Lorentz invariance and it does not introduce new unphysical degrees of freedom (ghosts). In these models the ultraviolet (UV) behaviour of the theory at short distances is improved, while in the infrared (IR) regime (at large scales) they reproduce results of a corresponding local theory [17][18][19][20][21][22][23]. Main motivation for study such infinite derivative modifications of the gravity equations is connected with attempts to solve longstanding problems of cosmological and black hole singularities [18,[24][25][26][27][28][29][30][31][32][33][34].\nComplete equations of a modified gravity which include both nonlocality and nonlinearity are quite complicated. Much easier for study is a linearized version of the theory. However already study of linearized models allows one to obtain several stimulating results: (i) The nonlocality removes singularities of the field produced by a point-like sources [18,[35][36][37][38]; (ii) There exists a mass gap for mini black hole formation [39,40]; (iii) It allows one to demonstrate a formation of the inner horizon for black hole creation in the scattering of ultrarelativistic particles [41]. A standard technique for solving these problems is usage of nonlocal Green functions which for the flat spacetime background can be found by means of the Fourier transform.\nStudy of the nonlocal field in the presence of an external potential is a much more complicated problem. There exist very special cases when it is possible to find an explicit solution. For example, such a solution was obtained for scattering of a nonlocal scalar field by a delta-like potential [42][43][44]. However in a general case in the presence of an arbitrary potential one needs to solve a nonlocal linear equation with the space-dependent coefficients which is a very non-trivial problem.\nThe purpose of this paper is to demonstrate that one can obtain asymptotic solutions for this problem by using a standard WKB method. Its main idea is to search for a solution Φ(x) of the field equation in the form Φ(x) ≈ u(x) exp [iS(x)/ ]. After substitution of this ansatz into the field equation one collects terms of the expansion into the powers of which are of the same order of . In the leading order one gets a first-order partial differential equation of the form H(∇S, x) = 0 known as an eikonal equation. A sub-leading equation determines evolution of a slowly changing field ampli-tude u(x). The eikonal equation can be identified with the Hamilton-Jacobi equation for the Hamiltonian obtained by the substitution ∇S = p into it. Initial data for the Hamilton-Jacobi equation specify a beam of trajectories in the phase space which forms a Lagrangian submanifold (for details see e.g. a remarkable book [45]). Knowledge of this Lagrangian submanifold allows one not only to construct the eikonal function S(x) but also to find a solution of the transport equation for the amplitude u(x) by using the Liouville theorem. This WKB method is widely used in the standard quantum mechanics and field theory where the corresponding equations are second order partial differential equations. However, it can be applied to a wider class of so called quasilinear differential equations. A comprehensive presentation of these results can be found in the book [46]. In these paper we apply this method for study quasiclassical solutions of the linear nonlocal scalar field equations in the presence of an external potential. A similar approach for other higher and infinite order equations can be found in [47].\nThis paper is organized as follows. In section II we describe a model of a nonlocal scalar field with infinite number of derivatives which is analysed in the paper. In section III we construct a WKB solution for such a field in the presence of an external potential in any number of spacetime dimensions. In section IV we consider a special case when the potential depends on only one spatial Cartesian coordinate. Energy levels for the nonlocal scalar field confined by the one-dimensional parabolic potential are calculated in section V. Under-barrier propagation of the nonlocal field and barrier penetration effect are discussed in section VI. Section VII contains discussion of the obtained results. Additional technical details are collected in two appendices."},{"title":"II. NONLOCAL SCALAR FIELD EQUATION","content":"We consider N -dimensional flat spacetime. Its metric in Cartesian coordinates is\nwhere η µν = diag(−1, 1, . . . , 1). Let us consider a scalar massive field ϕ obeying the Klein-Gordon equation\nHere = η µν ∂ µ ∂ ν , m is the mass of the field and V (x) is an external potential. In what follows we shall study solutions of this equation and its nonlocal generalization in the WKB approximation. For this reason we keep the Planck constant , while as usual put c = 1 1 . The\nKlein-Gordon equation follows from the action\n(3) We consider a generalization of the Klein-Gordon equation in which the operator F ( ) is modified. Namely, we assume that it is a scalar operator which may contain an arbitrary (finite or infinite) number of partial derivatives ∂/∂x µ . It is easy to check that the covariance of the action requires that such an operator can be written as a scalar function of the -operator. In order to keep a proper dimensionality of the action we introduce a parameter µ which has a dimension of the mass and we write the operator F ( ) in the form\nThe corresponding generalized Klein-Gordon equation is\nwhere z = 2 /µ 2 andV = V /µ 2 .\nWe consider a class of theories for which the function F (z) has the following properties:\n1.F (z) does not vanish anywhere on the complex plane of z besides the point z =m 2 ;\n2.F | z=m 2 = 0 and dF (z)/dz| z=m 2 = 1;\n3. For a real value of its argument z the functionF (z) is real.\nThese conditions guarantee that the inverse of the op-eratorF has a single pole with its residue equal to 1. In other words, there is no new unphysical (ghost) degrees of freedom in this theory. To satisfy the required properties it is sufficient to chooseF in the form\nwhere f (ẑ) = exp(g(ẑ)) and g(ẑ) is an entire function of the complex variable satisfying the condition g(ẑ = 0) = 0. We call f (ẑ) a form factor. In order to illustrate the results we shall use a special example of the form factor. Namely we put g(ẑ) =ẑ 2 then one has\nThis function obeys all the conditions listed above. Besides this for realẑ it is a monotonically increasing from −∞ atẑ = −∞ to +∞ atẑ = +∞ and is positive for z > 0. One also has F(−ẑ) = −F(ẑ). The function X(x) is shown at figure 1. Other properties of this function which are used later in the paper can be found in appendix B. One can expand the function F(ẑ) into powers ofẑ. This gives an infinite series of the powers of for the field operatorF ( 2 /µ 2 ). This means that this operator is nonlocal. Let us also mention that the mass parameter µ which enters this operator determines a characteristic energy scale at which the effects of the nonlocality become important. In the limit µ → ∞ the operator F becomes 2 − m 2 and the theory is local."},{"title":"III. WKB APPROXIMATION","content":""},{"title":"A. A solution of the nonlocal field equation in the WKB approximation","content":"Let us write the nonlocal field equation (5) in the form 2\nWe include a factor 1/2 in the operatorÔ which evidently does not change solutions but this simplifies the form of some relations. The operatorÔ belongs to the class of equations (A1) discussed in the appendix. We are looking for a solution of the field equation (8) in the WKB approximation and write it in the following form\nHere S(x) is a fast changing phase, while u j (x) are slowly changing functions. Let us emphasize that besides standard conditions of the validity of the WKB approximation formulated in textbooks on quantum mechanics we also assume that the parameter of nonlocality µ does not 2 In the nonlocal theory it is convenient to use the parameter µ to define dimensionless quantities. We specify such quantities by using a hat over them. For example, we havê\ndepend on . We keep this parameter fixed in the limit → 0. This allows us to discuss effects of the nonlocality in the quasiclassical approximation.\nIn order to find the eikonal function S(x) and the amplitudes u j (x) we follow the steps described in appendix A. First of all we define a symbol H(x, p) of the operatorÔ. Let p µ be a covector and denote p 2 = p µ p µ . Then one should substitute − 2 → p 2 into (8). The corresponding symbol for the operatorÔ is\nThe eikonal equation (A8) implies that the following constraint H(x, p) = 0 is valid.\nThe next step is to study a dynamical system in 2Ndimensional phase space satisfying the Hamilton's equations\nSince the Hamiltonian H(x, p) is an integral of motion it is sufficient to choose H(x, p) = 0 at the initial \"time\" τ = 0, then the constraint H(x, p) = 0 is valid for any later \"time\". We define a (N − 1)-dimensional surface Σ by conditions\nand choose an initial state for the field ϕ in the form\nAfter solving equations (A13) we impose the initial conditions (A14) for the Hamiltonian equations (11). Solving this system we find\nA sought solution ϕ(x) of the nonlocal equation (8) in the leading order of the WKB approximation is\nThis expression is obtained from (A19) by using the relation\nvalid for the Hamiltonian (10). The integral in the exponent in (15) is taken over phase trajectories which belong to the Lagrangian submanifold determied by given initial conditions. Here the function J which enters into the prefactor in (15) is defined by (A18)."},{"title":"B. Remarks on the Hamilton's equations","content":"Let us discuss now the Hamilton's equations in more detail. One has\nHere a prime denotes a derivative of F(ẑ) with respect to its argumentẑ. Hence the Hamilton's equations (11) take the formẋ\nThe first of these equations implieṡ\nSinceẑ = −(p 2 + m 2 )/µ 2 the right-hand side of this relation is a function of p 2 . We assume that one can solve equation (20) and find p 2 as a function ofẋ 2 . Then the Lagrangian of this dynamical system is\nIt is understood that in this expression for L one should expressẑ as a function ofẋ 2 . Since L does not contain explicitly the parameter τ the corresponding \"energy\" for this Lagrangian is conserved. This \"energy\" is nothing but the Hamiltonian in which p 2 is a function ofẋ 2 defined by (20). The constraint equation in the dimensionless form is (see footnote 2) )\nIfV (x) ≥ 0 then F(ẑ) =ẑf (ẑ) should be real and positive. The argument of this functionẑ = −(p 2 + m 2 )/µ 2 is real. Since f (0) = 1 and f (ẑ) does not vanish this function is positive for realẑ. This means that the sign of F(ẑ) coincides with the sign ofẑ. The equation (22) implies that the classical motion is possible only in a domain whereẑ ≥ 0.\nIn the absence of the potential the constraint equation takes the form\nSince f (ẑ) does not vanish this equation gives\nThis means that in the absence of the potential the Hamilton's equations (18)- (19) coincide with the equations for a free particle with the Hamiltonian (1/2)(p 2 + m 2 ). In other words, the effects of the nonlocality are important only off-shell, that is in the presence on the external potential."},{"title":"IV. ONE-DIMENSIONAL CASE","content":"To illustrate an application of the general approach described in the previous section we consider now a simple model. Namely, we assume that the external potential V (x) depends on only one spatial Cartesian coordinate. We denote this coordinate by q and use the following notations\nWe use index a = 2, . . . , N to enumerate transverse coordinates x ⊥ . We write the eikonal function S which enters the WKB solution (9) in the form\nWe also assume that the functions u j depend only on the coordinate q.\nLet us assume first that the potential V (q) vanishes. In order to solve the corresponding field equation (8) one can put S(q) = ηq where η =const, u 0 =const and u j≥1 = 0. This is a plane wave solution of the field equation (8). Denote p µ = (−ε, η, p ⊥ ), where the components of the covector p ⊥ are p a with a = 2, . . . , N . Hence S = (p, x) = p µ x µ . Other solutions of the field equation can be obtained by superimposing the plane-wave solutions, which in fact is just by using the Fourier representation. The constraint equation\nIn the presence of the potential we look for a solution of the nonlocal field equation (8) in the form\nSubstituting this expression into (8) one obtains the following reduced equation\nTo obtain the symbol H 1 (q, η) of the operatorÔ 1 we substitute\nThe result is\nFor real η the variableẑ is also real as well as the Hamiltonian itself. We also require that for realẑ one has F (ẑ) > 0. For f (ẑ) = exp(g(ẑ)) this condition is satisfied when\nIn particular, this is valid for the choice (7). The Hamilton-Jacobi equations (18)- (19) in (q, η)phase space reduce tȯ\nwhile the constraint equation is H 1 (q, η) = 0. Let us denote by (q(t), η(t)) a solution of these equations with the initial condition q(0) = q 0 and η(0) = η 0 . A solution of the field equation (29) in the leading order of the WKB approximation is\nIn the expression for S(q) the integral is taken along the path q(τ ) connecting the initial point q 0 with a final point\nThe second equality in the expression for J is obtained by using the first Hamilton's equation (36).\nSince the Hamiltonian H 1 (q, η) is invariant under the reflection η → −η one can write another WKB solution by a simple change of the sign in the exponent of (37). Taking a superposition of these two WKB solutions one can write Φ(q) in the following form\nFor\nLet us assume that the potential V (q) is nonnegative and at some point a the following condition is valid\nThen the constraint equation\nimplies that the momentum η vanishes at a. This means that a is a turning point. If dV /dq(a) > 0 then the \"particle\" comes to this point from the left, that is from the domain where q < a and η > 0. After reaching this turning point η changes its sign and the \"particle\" moves to the left. If the potential V (q) has another turning point b where dV /dq(b) < 0, then the \"particle\" comes to it from the right with η < 0 changes its direction of motion at b to the opposite and moves to the right with η > 0. Let us assume that there exist two turning points a < b such that the motion is restricted by the interval a ≤ q ≤ b. Such a motion is called bound. Let us consider a nonnegative potential V (q) which has its minimum at q 0 and V (q 0 ) = 0. At this point the absolute value of the momentum |η(q)| reaches its maximum which is equal to η 0 . Thus for such a bound motion the phase trajectory lies within a rectangle [a, b] × [−η 0 , η 0 ]. Let us emphasize that the Hamiltonian (33) contains a free parameter η 0 which determines a size and a shape of the phase trajectory. Since the parameterẑ depends on the square of the momentum η, a phase trajectory on the plane (q, η) is symmetric with respect to a reflection η → −η. For a given value η 0 > 0 constraint equation (42)"},{"title":"FIG. 2. A phase trajectory","content":"The energy levels of the states corresponding to the bound motion are quantized. In the quasiclassical approximation these levels can be found by using standard Bohr-Sommerfeld quantization condition [46] J = ηdq = π (2n + 1)\nHere the integral is taken over a complete period. For a given η 0 > 0 this integral is equal to the surface area inside the phase-trajectory curve corresponding to this parameter. Because of the reflection symmetry η → −η this integral can be written as\nThe integral is taken between the turning points a and b.\nSometimes it is convenient to rewrite J in another form. Using constraint equation (42) one can express the coordinate q as a function of η. This function has two branches which we denote by q ± (η) (see figure 2). Then, integrating by parts (44) and taking into account that η(a) = η(b) = 0 one finds\nIf the potential V (x) is a symmetric function of q, V (−q) = V (q), then one has q 0 = 0 and q − (η) = −q + (η).\nIn this case the quantity J can be written in the form\nB. Parabolic potential: Local theory\nLet us consider a case when the potential V (q) is parabolic\nwhere V 1 is a positive constant which has the dimension [V 1 ] = M/L. For the local theory the form factor f (ẑ) = 1 and the constraint equation takes the form\nSolving (48) we find\nThe integral (46) can be easily calculated. We denote the result by J 0 . Then one has\nThis gives the following expression for the energy levels\nIt is easy to see that the number of levels ∆n within the interval ∆(ε 2 ) is\nThe coefficient 1/(2 V 1 ) does not depend on n. In this sense, the corresponding distribution of energy levels, ∆n/∆(ε 2 ), is equidistant.\nC. Parabolic potential: Nonlocal theory"},{"title":"Phase trajectories","content":"In order to discuss the effect of the nonlocality on the distribution of the energy levels for the trapped nonlocal field we consider the same parabolic potential (47) as in the previous subsection, but modify the kinetic part of the effective Hamiltonian. We use dimensionless units defined in the footnote 2 and denotê\nThen the constraint (42) giveŝ\nLet us denoteη =η 0 sin φ,ẑ =ẑ 0 cos 2 φ .\nThen relations (54) and (55) allow one to write the equation for phase trajectories in the parametric form (q(φ), η(φ)). The phase trajectories for the special choice of the form factor (7) are shown at figure 3."},{"title":"Energy levels","content":"Using dimensionless variables the action integral (46), which enters the Bohr-Sommerfeld relation (43) for the energy levels, can be written as follows\nWe use here the expression forF in terms of the form factor f ,F =ẑf (ẑ). After change of variables,ẑ =ẑ 0 ξ, relation (56) takes the form\nHere J 0 is the value of the action integral for the local theory (50) and the factor σ(ẑ 0 ) is\nFor a fixed value of η 0 and µ → ∞ the parameterẑ 0 → 0. Since f (0) = 1 the integral (58) simplifies. It can be calculated analytically and the result is σ(0) = 1. Hence in this limit J coincides with J 0 . This is not surprising since in the limit µ → ∞ the theory becomes local. For a numerical calculation of σ(ẑ 0 ) it is convenient to put ξ = cos 2 φ and rewrite (58) in the form σ(ẑ 0 ) = 4 π π/2 0 dφ cos 2 φf 1/2 (ẑ 0 cos 2 φ) . The parameterẑ 0 which enters J depends on the energy ε. The condition of quantization J = π (2n + 1)\nallows one to find the corresponding quasiclassical energy levels ε n . Since for smallẑ 0 the coefficient σ(ẑ 0 ) is close to 1, the number of the energy levels n in the interval ∆ε 2 has the form (52). However for large n this distribution of the number of levels is quite different. For the form factor f (ẑ) = exp(ẑ 2 ) one can find it by using the asymptotic form of the coefficient σ(ẑ 0 )\nThe condition of quantization (61) gives\nFor n 1 one getsẑ 2 0 ≈ 2 ln n .\nSinceẑ 0 = η 2 0 /µ 2 , one finds\nThis relation implies\nThe obtained result means that the equidistance of the energy level distribution (52) of the local theory is broken by nonlocality. In the limit of large n and given interval ∆(ε 2 ) the number of the corresponding levels ∆n grows with n as n √ ln n."},{"title":"VI. BARRIER PENETRATION","content":"A. WKB approximation for under-barrier \"motion\"\nLet us consider a potential which has the form shown at figure 6 and assume that the effective energy of the \"particle\" is such that its trajectory has a turning point at q = a. At this point one has\nLet us write the constraint equation (42) in the form\nIn the domain to the left from the turning point a one hasV (q) <V (a). We assume that for realẑ the function F(ẑ) is monotonically growing. Then one hasẑ <ẑ 0 . In the region q > a whereV (q) >V (a) one should havê z >ẑ 0 . This condition implies that η 2 < 0 which is impossible for a real value of η.\nIt is well known that in the local theory a quasiclassical solution for a wave function in the classically forbidden domain can be obtained by considering complex trajectories. Namely, one should change η → iη. The same trick does work for the non-local theories considered in this paper. After the changeẑ →η 2 0 +η 2 the value of z can be larger thanẑ 0 for realη and the equation (68) has a solution in the region where q > a, that is in the classically forbidden domain.\nA quasiclassical solution in the underbarrier domain can be obtained by the change η → iη in (40) and it is of the form\nHere\nFor real coefficients C ± the WKB solution (69) is real."},{"title":"B. Nonlocal field in a linear potential","content":"For a linear potential\none-dimensional nonlocal field equation (29) can be solved exactly. For this purpose we denote q = a + x and write Φ(x) in the form\nUsing the inverse Fourier transform one gets\nThe field equation (29) implies the following differential equation forΦ(η)\nA solution of this equation is\nSubstitution of this expression into (73) gives\nThis is a desired solution of the nonlocal field equation (29) C. Connection of WKB solutions at a turning point Consider a potential of the form shown at figure 6 and denote by a a coordinate of a left turning point. One has V ,q (a) > 0. To the left of a and at some distance from it one can use WKB solution (40). For the real field Φ(q) we denote\nwhere C and ψ is real and write the WKB solution (40) in the form\nTo the right of the turning point a and at some distance from it we use a decreasing in the under-barrier domain WKB solution\nAt the turning point the WKB approximation is not valid. In order to establish a relation between (80) and (82) solutions one can use the following trick [48] which can be easily adapted to our problem. For this purpose we consider formally Φ(q) as a function of a complex variable q and find it along a path on the complex plane of q connecting positive and negative values of q − a. We choose this path to be a half of the circle of constant radius ρ with the center at q = a and take ρ so that the conditions of the validity of the WKB approximation are satisfied along the path. Let us denote\nWe choose a sign + for a semi-circle in the upper half of the complex q-plane, and − for a semi-circle in the lower half of the complex q-plane. The angle φ changes from 0 to π. At the turning pointẑ =ẑ 0 =η 2 0 and q = a. We assume that ρ is small so that |ẑ −ẑ 0 | and |q − a| are also small and use the following linearized form of the constraint equation\nLet us denote B = V ,q (a)/F 0 . One also hasẑ−ẑ 0 =η 2 . Then (84) and (85) imply\nFor the analytically continued WKB solution (82) we get\nHere ± stands for the quantities calculated along a semicircle in the upper (+) and lower (−) half of the q-plane.\nUsing these relations at φ = π where a − q = ρ one obtains\nThis establishes a relation between the WKB solution (82) in the domain to the right of the turning point and the WKB solution (80) to the left from it. This gives a required connection formula. It should be emphasized that for the operator F ( ) with the properties specified in section II this connection relation is the same as for a standard Schrödinger equation.\nWe assume now that the potential has two turning points a < b and the domain between these points is forbidden for the classical motion. We suppose that in the region q < a the field Φ(q) has the form (80) and its amplitude is C. Then in the region q > b it has the same form with a different amplitudeC. Similarly to the local case the ratio |C/C| 2 can be found by using the connection formula (89) twice [48]\nHereẑ 0 =η 2 0 andẑ =ẑ 2 0 +η 2 . In the expression for P the quantitiesẑ and q satisfy the constraint (42). The quantity R gives a probability of the barrier penetration. It should be emphasized that these relations are obtained in the WKB approximation which is valid when R 1."},{"title":"D. Barrier penetration","content":""},{"title":"Inverse parabolic potential: Local theory","content":"For the local theory the effective Hamiltonian is\nwhere η 2 0 = ε 2 − p 2 ⊥ − m 2 . Let the potential V (q) be of the form\nwhere V 0 > 0 and q * = √ V 0 /V 1 . This potential is shown at figure 7. It is called an inverse parabolic potential.\nWe assume that η 2 0 < V 0 so that there exit two turning points at q = ±a, where\nFor the motion in the under-barrier domain we put η = iη whereη"},{"title":"FIG. 7. The inverse parabolic potential","content":"We denote by P 0 the following integral\nUsing relation (94) we can also write P 0 in the form\nLet us emphasize that P 0 depends only on the difference V 0 −η 2 0 but it does not depend on the value of η 0 itself. In other words, the penetration probability is sensitive only to the fact how close the effective energy η 2 0 is to the top of the potential V 0 . Let us also remind that the adopted WKB approximation is valid only when the dimensionless quantity P 0 is large, that is when the probability of the penetration R 0 = exp(−2P 0 / ) is small (see e,g, [48])."},{"title":"Inverse parabolic potential: Nonlocal theory","content":"Let us assume again that the potential V (q) has the form (93). We denote byẑ 0 andẑ 1 solutions of the following equations\nThen the WKB probability of the barrier penetration is\nLet us calculate the value of P for the case when F is of the form (7). When both parameters V 0 and η 2 0 are much smaller than µ 2 , the function X(ẑ) ≈ẑ and the integral in (100) can be easily taken with the following result\nIn this regime the effects of the nonlocality are not important and we reproduce the expression for P 0 obtained in the local case (see equation (97)). The effects of the nonlocality play an important role in the different regime, whenV 0 1 andη 2 0 1. To demonstrate this we consider again the theory with the form factor f (ẑ) = exp(ẑ 2 ). In order to compare the expression for P with P 0 we fix as earlier the value of the difference V 0 − η 2 0 , but now we assume thatV 0 1. Then one hasẑ 1 −ẑ 0 1. In the appendix B it is shown that one can write the following approximate expression for P in this case\nHere X 1 = dX/dx| x=ẑ1 and X(ẑ 1 ) =V 0 . Using (B2) one can write\nThese relations give a parametric representation of X 1 (ẑ 1 ) as a function ofV 0 . The plot of the function κ(V 0 ) is shown at figure 8. It shows that this function is equal to 1 atV 0 = 0 and it is less than 1 and monotonically decreases for positiveV 0 . Since R = exp(−2P/ ) the relation (102) can be rewritten as follows\nSince R 0 < 1 the value of R is always greater that R 0 . This means that in the nonlocal theory the probability of the barrier penetration is larger that in the local case.\nIn other words nonlocality makes the barrier more transparent in the high-energy regime."},{"title":"VII. DISCISSION","content":"Let us discuss the obtained results. In this paper we study quasiclassical solutions of the nonlocal massive scalar field equations. The kinetic part of the action for such a theory contains a function of the -operator, which is chosen so that the theory does not contain new unphysical degrees of freedom. In section II we formulated conditions imposed on the form factor of such a theory. We also described a special model which is used later in the paper for the illustration. In section III we described an ansatz used in the WKB approximation for a solution of the nonlocal field equation. In the leading order of the 1/ -expansion one obtains the equation for the eikonal function S(x) which is of the form H(x, ∇S) = 0. This is a first order partial differential equation which has the form of the Hamilton-Jacobi equation for the Hamiltonian H(x, p), obtained by substitution ∇S = p. To calculate the eikonal function S(x) it is sufficixent to construct the Lagrangian submanifold in the phase space (x, p) by using the initial conditions for S. Such a Lagrangian submanifold is formed by phase trajectories which are solutions of the Hamiltonian equations of motion. Since the system is conservative this submanifold belongs to the subspace H(x, p) = 0. Let us emphasize that this procedure is very similar to the standard one adopted for the quantum mechanics and local field theory. When the field operator is of the second order the corresponding effective Hamiltonian H(x, p) is a quadratic polynomial in momentum p. For the nonlocal theory H(x, p) is a non-polynomial function of p and this is a main difference with the local case. As a result the construction of the WKB solutions becomes technically more complicated.\nWKB solutions presented in section III are valid for a wide class of nonlocal scalar field theories with an arbitrary external potential V (x) in a flat spacetime with any number of spatial dimensions. These relations are greatly simplified for the case when the potential V depends only on one spatial Cartesian coordinate. For this case the phase space is two-dimensional and the Hamilton's equations are completely integrable. The corresponding WKB solutions are described in section IV. In section V we used the WKB approximation for the calculation of the energy levels of the nonlocal field trapped by a one-dimensional potential. To illustrate general formulas we considered the case when the form factor of the nonlocality is of the special form (7) and the potential is parabolic. The energy levels ε n of the field depend on a discrete integer parameter n. For the local case the number ∆n of the levels in the interval ∆(ε 2 ) is ∆n = ∆(ε 2 )/(2 V 1 ) and it does not depend on the value of n. In this sense the levels are equidistant. In the nonlocal case for large n this dependence is different\nThis means that at very high energy for a given interval of ∆(ε 2 ) there are much more energy levels than in the local case. Finally, we discussed the under-barrier motion and obtained an expression for the probability of the barrier penetration. This probability has the form ∼ exp(−2P/ ). We calculated the factor P for the the special form of the nonlocality (7) in the presence of the inverse parabolic potential. For the local theory this factor is P 0 = π(V 0 − η 2 0 )/(2V 1 ). In other words, it depends on how close is the energy of the state η 2 0 to the top of the potential V 0 , but it does not depend on the height V 0 of the potential itself. The situation in the nonlocal case is quite different. Namely, one has P = κP 0 , where the factor κ depends on the height V 0 of the potential. This factor is always less than 1 and it decreases when V 0 grows. This means that the nonlocality \"helps\" the field to penetrate the potential barrier.\nThe examples considered in this paper are rather simple. It would be interesting to apply the developed in the paper method for study the effects of the nonlocality in the early cosmology and in the black holes interior."},{"title":"Appendix A: WKB approximation","content":"We collected here useful information concerning construction of solutions of the nonlocal field equations in the WKB approximation. Further details and required mathematical foundations can be found in the book [46].\nLet (x 1 , . . . , x N ) be coordinates of a point x in R N . We denote by α a set of N non-negative integer numbers α = (α 1 , . . . , α N ). We consider the following operator in this spacê\nThe sum is taken over all different combinations of α.\nWe do not assume that the number of terms in the sum is finite, so that this expression can be used as a formal series in derivatives representation of nonlocal operators. Let us make a substitution\nin the expression (A1). Then we obtain a function of 2N variables which we denote by H = H(x, p)\nHere p = (p 1 , . . . , p N ). This object is called a symbol of the operatorD. We are looking for a solution ϕ of the equation\nin the form of the following formal series\nUsually it is sufficient to consider only first few terms in the series. This gives what is known as the WKB approximation for the solution ϕ. To obtain the functions S(x) and u j (x) that enter (A5) the following procedure is used (see e.g. [45,46] and references therein). If one substitute expression (A5) into the equation (A4) then one finds that in the leading order j = 0 the following condition is to be satisfied\nwhere\nFor a nonvanishing function u 0 (x) the following Hamilton-Jacobi equation is valid\nLet us consider a (2N )-dimension space Γ 2N with coordinates (x, p) = (x 1 , . . . , x N , p 1 , . . . , P N ) and denote by Ω a symplectic two-form in it\nAn N -dimensional surface Λ N in Γ 2N is called a Lagrangian submanifold if the Ω vanishes on Λ N . If a Lagrangian submanifold can be uniquely projected onto the x-space, then there exists such a function S(x) that p µ = ∂S/∂ µ on Λ N . The Hamiltonian H(x, p) by the relation (A3) implies the following Hamilton's equations are\nThese equations determine the Hamiltonian flow in the phase space Γ 2N preserving the symplectic form Ω and the Hamiltonian itself is an integral of motion. Let Σ N −1 be a (N −1)-dimensional surface in the coordinate space. Its embedding is defined by the equations\nOne can consider y i as coordinates on Σ N −1 . We associate with Σ N −1 a (2N − 2)-dimensional phase space Γ 2N −2 and denote its canonical coordinates by (y, v) = (y 1 , . . . , y N −1 , v 1 , . . . , v N −1 ). The symplectic 2-form ω in\nDenote by S 0 = S 0 (y) a function on Σ N −1 and ∇S 0 = (S 0 ,y 1 , . . . , S 0 ,y N −1 ), then a (N − 1)-dimensional surface (y, ∇S) of Γ 2N −2 is a Lagrangian submanifold Λ N −1 , so that ω| Λ N −1 = 0.\nLet us define a covector p(y) = p µ (y) on Σ N −1 by two conditions p µ (y)dx µ = p µ (y) ∂f µ ∂y i = 0 , H(x(y), p(y)) = 0 . (A13)\nWe assume that this system of N equations has a solution which determines p µ (y)| Σ N −1 . Let us consider the following initial conditions for the Hamilton's equations (A10)\nA family of solutions of the Hamilton's equations (A10) with these initial conditions, (x(τ, y), p(τ, y)), forms a Ndimensional surface in Γ 2N which we denote by B. It is possible to show that B is a Lagrangian submanifold in Γ 2N and that H(x, p) = 0 on it. This implies that there exists such a function S(x) that\nThis function is a solution of the Hamilton-Jacobi equation (A8) satisfying the initial condition S(x)| Σ N −1 = S 0 (y). This function S(x) can be found as follows. Denote a 1-form θ = µ p µ dx µ , then Ω = dθ. Since the symplectic form Ω vanishes on B, the integral\ncalculated along a path on B between its two points does not depend on the choice of the path. The required solution S(x) of the Hamilton-Jacobi equation satisfying the initial condition S| Σ N −1 = S(y) can be written in the form\nHere x µ = x µ (τ, y) and p µ = p µ (τ, y). Let us note that the map x µ = x µ (τ, y) may be considered as a transformation between the coordinates (τ, y and x µ . Let us denote by J a Jacobian of this transformation J(τ, y) = det ∂x(τ, y) ∂(τ, y) .\nThen the leading term of the asymptotic solution (A5) is [46] ϕ(x) = u 0 (y) J(0, y) J(τ, y) exp i S 0 (y) + For τ = 0 the right-hand side of this expression reduces to u 0 (y) exp i S 0 (y) and it should be found from the initial condition for the function ϕ(x) at Σ.\nAppendix B: Function X(x) = x exp(x 2 ) and its properties\nIn this appendix we discuss some properties of the function\nThe derivatives of this function are X = (1 + 2x 2 ) exp(x 2 ) ,\nWe denote by Y an inverse function of X such that the following relations are valid y = X(x), x = Y (y) .\nThe inverse function can be written as follows Y (y) = y exp − 1 2 W (2y 2 ) .\nHere W is Lambert W function defined by the relation W (z) exp(W (z)) = z .\nFor a real positive argument W is a real single-values function. Consider a following integral\nLet us assume that x 1 − x 0 is small. To obtain an approximate value of the integral I in this case we use the Taylor expansion\nHere X 1 = X(x 1 ), X 1 = X (x 1 ), X 1 = X (x 1 ). Skipping the higher order terms denoted in (B7) by dots one has\nwhere α = 1 2 (X 1 /X 1 ). This integral can be calculated exactly. Let us denote β = x 1 − x 0 , then for αβ < 1 the answer is I = 2 X 1 3α (2αβ − 1)E( αβ) − (αβ − 1)K( αβ) .\nHere E and K are complete elliptic integrals. Forthe small value of αβ one has\nFor αβ 1 one can neglect the second term. Keeping the leading term one gets\nFor small x 1 one has X 1 ≈ 1 and\nThe author thanks the Natural Sciences and Engineering Research Council of Canada and the Killam Trust for their financial support."},{"title":"FIG. 1 .","content":""},{"title":"FIG. 3 .","content":""},{"title":"FIG. 4 .Figure 5","content":""},{"title":"FIG. 5 .","content":""},{"title":"FIG. 6 .","content":""},{"title":"FIG. 8 .","content":""}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"The idea of nonlocality is quite old in the theoretical physics. Nonlocal modifications of the field theory were discussed already in the publications [1][2][3]. Even if one works with a local quantum field theory, already at the one loop level its effective action is nonlocal. This nonlocality is connected with the vacuum polarization and particle creation effects and it reflects a fact that a vacuum is a physical medium and in this sense it behaves as a condensed matter with very special properties. A nonlocality that we consider in this paper is of a different origin. We assume that a field theory is described by a nonlocal action already at the tree level. In particular this means that its interaction with external sources as well as a self-interaction is nonlocal.\\nUsually such a nonlocal modification contains a scale parameter which determines either the energy at which the effects of the nonlocality become important or the corresponding spacetime length when it happens. Nonlocal theories were discussed in the beginning of seventies of the past century by Efimov [4][5][6][7]. The nonlocality is often introduced to modify high energy properties of the theory and to improve its ultraviolet behavior. More recently the interest to nonlocal theories increased. This was mainly stimulated by the development of the string theory. Nonlocal fields naturally arise in the string theory and in the theories with a noncommutive geometry (see e.g. [8][9][10][11][12][13][14][15] and references therein). Nonlocal effects in quantum gravity were widely discussed recently. A comprehensive review of modifications of gravity involving a minimal length scale and related references can be found in [16].\\nThere is a subclass of the non-local field theories sometimes called ghost-free theories. An effective action in these models contains an infinite number of derivatives so that the corresponding field equations are effectively nonlocal. These models have been widely discussed recently and they have rather \\\"nice properties\\\". In these theories the nonlocality is introduced in such a way that * vfrolov@ualberta.ca it preserves local Lorentz invariance and it does not introduce new unphysical degrees of freedom (ghosts). In these models the ultraviolet (UV) behaviour of the theory at short distances is improved, while in the infrared (IR) regime (at large scales) they reproduce results of a corresponding local theory [17][18][19][20][21][22][23]. Main motivation for study such infinite derivative modifications of the gravity equations is connected with attempts to solve longstanding problems of cosmological and black hole singularities [18,[24][25][26][27][28][29][30][31][32][33][34].\\nComplete equations of a modified gravity which include both nonlocality and nonlinearity are quite complicated. Much easier for study is a linearized version of the theory. However already study of linearized models allows one to obtain several stimulating results: (i) The nonlocality removes singularities of the field produced by a point-like sources [18,[35][36][37][38]; (ii) There exists a mass gap for mini black hole formation [39,40]; (iii) It allows one to demonstrate a formation of the inner horizon for black hole creation in the scattering of ultrarelativistic particles [41]. A standard technique for solving these problems is usage of nonlocal Green functions which for the flat spacetime background can be found by means of the Fourier transform.\\nStudy of the nonlocal field in the presence of an external potential is a much more complicated problem. There exist very special cases when it is possible to find an explicit solution. For example, such a solution was obtained for scattering of a nonlocal scalar field by a delta-like potential [42][43][44]. However in a general case in the presence of an arbitrary potential one needs to solve a nonlocal linear equation with the space-dependent coefficients which is a very non-trivial problem.\\nThe purpose of this paper is to demonstrate that one can obtain asymptotic solutions for this problem by using a standard WKB method. Its main idea is to search for a solution Φ(x) of the field equation in the form Φ(x) ≈ u(x) exp [iS(x)/ ]. After substitution of this ansatz into the field equation one collects terms of the expansion into the powers of which are of the same order of . In the leading order one gets a first-order partial differential equation of the form H(∇S, x) = 0 known as an eikonal equation. A sub-leading equation determines evolution of a slowly changing field ampli-tude u(x). The eikonal equation can be identified with the Hamilton-Jacobi equation for the Hamiltonian obtained by the substitution ∇S = p into it. Initial data for the Hamilton-Jacobi equation specify a beam of trajectories in the phase space which forms a Lagrangian submanifold (for details see e.g. a remarkable book [45]). Knowledge of this Lagrangian submanifold allows one not only to construct the eikonal function S(x) but also to find a solution of the transport equation for the amplitude u(x) by using the Liouville theorem. This WKB method is widely used in the standard quantum mechanics and field theory where the corresponding equations are second order partial differential equations. However, it can be applied to a wider class of so called quasilinear differential equations. A comprehensive presentation of these results can be found in the book [46]. In these paper we apply this method for study quasiclassical solutions of the linear nonlocal scalar field equations in the presence of an external potential. A similar approach for other higher and infinite order equations can be found in [47].\\nThis paper is organized as follows. In section II we describe a model of a nonlocal scalar field with infinite number of derivatives which is analysed in the paper. In section III we construct a WKB solution for such a field in the presence of an external potential in any number of spacetime dimensions. In section IV we consider a special case when the potential depends on only one spatial Cartesian coordinate. Energy levels for the nonlocal scalar field confined by the one-dimensional parabolic potential are calculated in section V. Under-barrier propagation of the nonlocal field and barrier penetration effect are discussed in section VI. Section VII contains discussion of the obtained results. Additional technical details are collected in two appendices.\"\n },\n {\n \"title\": \"II. NONLOCAL SCALAR FIELD EQUATION\",\n \"content\": \"We consider N -dimensional flat spacetime. Its metric in Cartesian coordinates is\\nwhere η µν = diag(−1, 1, . . . , 1). Let us consider a scalar massive field ϕ obeying the Klein-Gordon equation\\nHere = η µν ∂ µ ∂ ν , m is the mass of the field and V (x) is an external potential. In what follows we shall study solutions of this equation and its nonlocal generalization in the WKB approximation. For this reason we keep the Planck constant , while as usual put c = 1 1 . The\\nKlein-Gordon equation follows from the action\\n(3) We consider a generalization of the Klein-Gordon equation in which the operator F ( ) is modified. Namely, we assume that it is a scalar operator which may contain an arbitrary (finite or infinite) number of partial derivatives ∂/∂x µ . It is easy to check that the covariance of the action requires that such an operator can be written as a scalar function of the -operator. In order to keep a proper dimensionality of the action we introduce a parameter µ which has a dimension of the mass and we write the operator F ( ) in the form\\nThe corresponding generalized Klein-Gordon equation is\\nwhere z = 2 /µ 2 andV = V /µ 2 .\\nWe consider a class of theories for which the function F (z) has the following properties:\\n1.F (z) does not vanish anywhere on the complex plane of z besides the point z =m 2 ;\\n2.F | z=m 2 = 0 and dF (z)/dz| z=m 2 = 1;\\n3. For a real value of its argument z the functionF (z) is real.\\nThese conditions guarantee that the inverse of the op-eratorF has a single pole with its residue equal to 1. In other words, there is no new unphysical (ghost) degrees of freedom in this theory. To satisfy the required properties it is sufficient to chooseF in the form\\nwhere f (ẑ) = exp(g(ẑ)) and g(ẑ) is an entire function of the complex variable satisfying the condition g(ẑ = 0) = 0. We call f (ẑ) a form factor. In order to illustrate the results we shall use a special example of the form factor. Namely we put g(ẑ) =ẑ 2 then one has\\nThis function obeys all the conditions listed above. Besides this for realẑ it is a monotonically increasing from −∞ atẑ = −∞ to +∞ atẑ = +∞ and is positive for z > 0. One also has F(−ẑ) = −F(ẑ). The function X(x) is shown at figure 1. Other properties of this function which are used later in the paper can be found in appendix B. One can expand the function F(ẑ) into powers ofẑ. This gives an infinite series of the powers of for the field operatorF ( 2 /µ 2 ). This means that this operator is nonlocal. Let us also mention that the mass parameter µ which enters this operator determines a characteristic energy scale at which the effects of the nonlocality become important. In the limit µ → ∞ the operator F becomes 2 − m 2 and the theory is local.\"\n },\n {\n \"title\": \"III. WKB APPROXIMATION\",\n \"content\": \"\"\n },\n {\n \"title\": \"A. A solution of the nonlocal field equation in the WKB approximation\",\n \"content\": \"Let us write the nonlocal field equation (5) in the form 2\\nWe include a factor 1/2 in the operatorÔ which evidently does not change solutions but this simplifies the form of some relations. The operatorÔ belongs to the class of equations (A1) discussed in the appendix. We are looking for a solution of the field equation (8) in the WKB approximation and write it in the following form\\nHere S(x) is a fast changing phase, while u j (x) are slowly changing functions. Let us emphasize that besides standard conditions of the validity of the WKB approximation formulated in textbooks on quantum mechanics we also assume that the parameter of nonlocality µ does not 2 In the nonlocal theory it is convenient to use the parameter µ to define dimensionless quantities. We specify such quantities by using a hat over them. For example, we havê\\ndepend on . We keep this parameter fixed in the limit → 0. This allows us to discuss effects of the nonlocality in the quasiclassical approximation.\\nIn order to find the eikonal function S(x) and the amplitudes u j (x) we follow the steps described in appendix A. First of all we define a symbol H(x, p) of the operatorÔ. Let p µ be a covector and denote p 2 = p µ p µ . Then one should substitute − 2 → p 2 into (8). The corresponding symbol for the operatorÔ is\\nThe eikonal equation (A8) implies that the following constraint H(x, p) = 0 is valid.\\nThe next step is to study a dynamical system in 2Ndimensional phase space satisfying the Hamilton's equations\\nSince the Hamiltonian H(x, p) is an integral of motion it is sufficient to choose H(x, p) = 0 at the initial \\\"time\\\" τ = 0, then the constraint H(x, p) = 0 is valid for any later \\\"time\\\". We define a (N − 1)-dimensional surface Σ by conditions\\nand choose an initial state for the field ϕ in the form\\nAfter solving equations (A13) we impose the initial conditions (A14) for the Hamiltonian equations (11). Solving this system we find\\nA sought solution ϕ(x) of the nonlocal equation (8) in the leading order of the WKB approximation is\\nThis expression is obtained from (A19) by using the relation\\nvalid for the Hamiltonian (10). The integral in the exponent in (15) is taken over phase trajectories which belong to the Lagrangian submanifold determied by given initial conditions. Here the function J which enters into the prefactor in (15) is defined by (A18).\"\n },\n {\n \"title\": \"B. Remarks on the Hamilton's equations\",\n \"content\": \"Let us discuss now the Hamilton's equations in more detail. One has\\nHere a prime denotes a derivative of F(ẑ) with respect to its argumentẑ. Hence the Hamilton's equations (11) take the formẋ\\nThe first of these equations implieṡ\\nSinceẑ = −(p 2 + m 2 )/µ 2 the right-hand side of this relation is a function of p 2 . We assume that one can solve equation (20) and find p 2 as a function ofẋ 2 . Then the Lagrangian of this dynamical system is\\nIt is understood that in this expression for L one should expressẑ as a function ofẋ 2 . Since L does not contain explicitly the parameter τ the corresponding \\\"energy\\\" for this Lagrangian is conserved. This \\\"energy\\\" is nothing but the Hamiltonian in which p 2 is a function ofẋ 2 defined by (20). The constraint equation in the dimensionless form is (see footnote 2) )\\nIfV (x) ≥ 0 then F(ẑ) =ẑf (ẑ) should be real and positive. The argument of this functionẑ = −(p 2 + m 2 )/µ 2 is real. Since f (0) = 1 and f (ẑ) does not vanish this function is positive for realẑ. This means that the sign of F(ẑ) coincides with the sign ofẑ. The equation (22) implies that the classical motion is possible only in a domain whereẑ ≥ 0.\\nIn the absence of the potential the constraint equation takes the form\\nSince f (ẑ) does not vanish this equation gives\\nThis means that in the absence of the potential the Hamilton's equations (18)- (19) coincide with the equations for a free particle with the Hamiltonian (1/2)(p 2 + m 2 ). In other words, the effects of the nonlocality are important only off-shell, that is in the presence on the external potential.\"\n },\n {\n \"title\": \"IV. ONE-DIMENSIONAL CASE\",\n \"content\": \"To illustrate an application of the general approach described in the previous section we consider now a simple model. Namely, we assume that the external potential V (x) depends on only one spatial Cartesian coordinate. We denote this coordinate by q and use the following notations\\nWe use index a = 2, . . . , N to enumerate transverse coordinates x ⊥ . We write the eikonal function S which enters the WKB solution (9) in the form\\nWe also assume that the functions u j depend only on the coordinate q.\\nLet us assume first that the potential V (q) vanishes. In order to solve the corresponding field equation (8) one can put S(q) = ηq where η =const, u 0 =const and u j≥1 = 0. This is a plane wave solution of the field equation (8). Denote p µ = (−ε, η, p ⊥ ), where the components of the covector p ⊥ are p a with a = 2, . . . , N . Hence S = (p, x) = p µ x µ . Other solutions of the field equation can be obtained by superimposing the plane-wave solutions, which in fact is just by using the Fourier representation. The constraint equation\\nIn the presence of the potential we look for a solution of the nonlocal field equation (8) in the form\\nSubstituting this expression into (8) one obtains the following reduced equation\\nTo obtain the symbol H 1 (q, η) of the operatorÔ 1 we substitute\\nThe result is\\nFor real η the variableẑ is also real as well as the Hamiltonian itself. We also require that for realẑ one has F (ẑ) > 0. For f (ẑ) = exp(g(ẑ)) this condition is satisfied when\\nIn particular, this is valid for the choice (7). The Hamilton-Jacobi equations (18)- (19) in (q, η)phase space reduce tȯ\\nwhile the constraint equation is H 1 (q, η) = 0. Let us denote by (q(t), η(t)) a solution of these equations with the initial condition q(0) = q 0 and η(0) = η 0 . A solution of the field equation (29) in the leading order of the WKB approximation is\\nIn the expression for S(q) the integral is taken along the path q(τ ) connecting the initial point q 0 with a final point\\nThe second equality in the expression for J is obtained by using the first Hamilton's equation (36).\\nSince the Hamiltonian H 1 (q, η) is invariant under the reflection η → −η one can write another WKB solution by a simple change of the sign in the exponent of (37). Taking a superposition of these two WKB solutions one can write Φ(q) in the following form\\nFor\\nLet us assume that the potential V (q) is nonnegative and at some point a the following condition is valid\\nThen the constraint equation\\nimplies that the momentum η vanishes at a. This means that a is a turning point. If dV /dq(a) > 0 then the \\\"particle\\\" comes to this point from the left, that is from the domain where q < a and η > 0. After reaching this turning point η changes its sign and the \\\"particle\\\" moves to the left. If the potential V (q) has another turning point b where dV /dq(b) < 0, then the \\\"particle\\\" comes to it from the right with η < 0 changes its direction of motion at b to the opposite and moves to the right with η > 0. Let us assume that there exist two turning points a < b such that the motion is restricted by the interval a ≤ q ≤ b. Such a motion is called bound. Let us consider a nonnegative potential V (q) which has its minimum at q 0 and V (q 0 ) = 0. At this point the absolute value of the momentum |η(q)| reaches its maximum which is equal to η 0 . Thus for such a bound motion the phase trajectory lies within a rectangle [a, b] × [−η 0 , η 0 ]. Let us emphasize that the Hamiltonian (33) contains a free parameter η 0 which determines a size and a shape of the phase trajectory. Since the parameterẑ depends on the square of the momentum η, a phase trajectory on the plane (q, η) is symmetric with respect to a reflection η → −η. For a given value η 0 > 0 constraint equation (42)\"\n },\n {\n \"title\": \"FIG. 2. A phase trajectory\",\n \"content\": \"The energy levels of the states corresponding to the bound motion are quantized. In the quasiclassical approximation these levels can be found by using standard Bohr-Sommerfeld quantization condition [46] J = ηdq = π (2n + 1)\\nHere the integral is taken over a complete period. For a given η 0 > 0 this integral is equal to the surface area inside the phase-trajectory curve corresponding to this parameter. Because of the reflection symmetry η → −η this integral can be written as\\nThe integral is taken between the turning points a and b.\\nSometimes it is convenient to rewrite J in another form. Using constraint equation (42) one can express the coordinate q as a function of η. This function has two branches which we denote by q ± (η) (see figure 2). Then, integrating by parts (44) and taking into account that η(a) = η(b) = 0 one finds\\nIf the potential V (x) is a symmetric function of q, V (−q) = V (q), then one has q 0 = 0 and q − (η) = −q + (η).\\nIn this case the quantity J can be written in the form\\nB. Parabolic potential: Local theory\\nLet us consider a case when the potential V (q) is parabolic\\nwhere V 1 is a positive constant which has the dimension [V 1 ] = M/L. For the local theory the form factor f (ẑ) = 1 and the constraint equation takes the form\\nSolving (48) we find\\nThe integral (46) can be easily calculated. We denote the result by J 0 . Then one has\\nThis gives the following expression for the energy levels\\nIt is easy to see that the number of levels ∆n within the interval ∆(ε 2 ) is\\nThe coefficient 1/(2 V 1 ) does not depend on n. In this sense, the corresponding distribution of energy levels, ∆n/∆(ε 2 ), is equidistant.\\nC. Parabolic potential: Nonlocal theory\"\n },\n {\n \"title\": \"Phase trajectories\",\n \"content\": \"In order to discuss the effect of the nonlocality on the distribution of the energy levels for the trapped nonlocal field we consider the same parabolic potential (47) as in the previous subsection, but modify the kinetic part of the effective Hamiltonian. We use dimensionless units defined in the footnote 2 and denotê\\nThen the constraint (42) giveŝ\\nLet us denoteη =η 0 sin φ,ẑ =ẑ 0 cos 2 φ .\\nThen relations (54) and (55) allow one to write the equation for phase trajectories in the parametric form (q(φ), η(φ)). The phase trajectories for the special choice of the form factor (7) are shown at figure 3.\"\n },\n {\n \"title\": \"Energy levels\",\n \"content\": \"Using dimensionless variables the action integral (46), which enters the Bohr-Sommerfeld relation (43) for the energy levels, can be written as follows\\nWe use here the expression forF in terms of the form factor f ,F =ẑf (ẑ). After change of variables,ẑ =ẑ 0 ξ, relation (56) takes the form\\nHere J 0 is the value of the action integral for the local theory (50) and the factor σ(ẑ 0 ) is\\nFor a fixed value of η 0 and µ → ∞ the parameterẑ 0 → 0. Since f (0) = 1 the integral (58) simplifies. It can be calculated analytically and the result is σ(0) = 1. Hence in this limit J coincides with J 0 . This is not surprising since in the limit µ → ∞ the theory becomes local. For a numerical calculation of σ(ẑ 0 ) it is convenient to put ξ = cos 2 φ and rewrite (58) in the form σ(ẑ 0 ) = 4 π π/2 0 dφ cos 2 φf 1/2 (ẑ 0 cos 2 φ) . The parameterẑ 0 which enters J depends on the energy ε. The condition of quantization J = π (2n + 1)\\nallows one to find the corresponding quasiclassical energy levels ε n . Since for smallẑ 0 the coefficient σ(ẑ 0 ) is close to 1, the number of the energy levels n in the interval ∆ε 2 has the form (52). However for large n this distribution of the number of levels is quite different. For the form factor f (ẑ) = exp(ẑ 2 ) one can find it by using the asymptotic form of the coefficient σ(ẑ 0 )\\nThe condition of quantization (61) gives\\nFor n 1 one getsẑ 2 0 ≈ 2 ln n .\\nSinceẑ 0 = η 2 0 /µ 2 , one finds\\nThis relation implies\\nThe obtained result means that the equidistance of the energy level distribution (52) of the local theory is broken by nonlocality. In the limit of large n and given interval ∆(ε 2 ) the number of the corresponding levels ∆n grows with n as n √ ln n.\"\n },\n {\n \"title\": \"VI. BARRIER PENETRATION\",\n \"content\": \"A. WKB approximation for under-barrier \\\"motion\\\"\\nLet us consider a potential which has the form shown at figure 6 and assume that the effective energy of the \\\"particle\\\" is such that its trajectory has a turning point at q = a. At this point one has\\nLet us write the constraint equation (42) in the form\\nIn the domain to the left from the turning point a one hasV (q) <V (a). We assume that for realẑ the function F(ẑ) is monotonically growing. Then one hasẑ <ẑ 0 . In the region q > a whereV (q) >V (a) one should havê z >ẑ 0 . This condition implies that η 2 < 0 which is impossible for a real value of η.\\nIt is well known that in the local theory a quasiclassical solution for a wave function in the classically forbidden domain can be obtained by considering complex trajectories. Namely, one should change η → iη. The same trick does work for the non-local theories considered in this paper. After the changeẑ →η 2 0 +η 2 the value of z can be larger thanẑ 0 for realη and the equation (68) has a solution in the region where q > a, that is in the classically forbidden domain.\\nA quasiclassical solution in the underbarrier domain can be obtained by the change η → iη in (40) and it is of the form\\nHere\\nFor real coefficients C ± the WKB solution (69) is real.\"\n },\n {\n \"title\": \"B. Nonlocal field in a linear potential\",\n \"content\": \"For a linear potential\\none-dimensional nonlocal field equation (29) can be solved exactly. For this purpose we denote q = a + x and write Φ(x) in the form\\nUsing the inverse Fourier transform one gets\\nThe field equation (29) implies the following differential equation forΦ(η)\\nA solution of this equation is\\nSubstitution of this expression into (73) gives\\nThis is a desired solution of the nonlocal field equation (29) C. Connection of WKB solutions at a turning point Consider a potential of the form shown at figure 6 and denote by a a coordinate of a left turning point. One has V ,q (a) > 0. To the left of a and at some distance from it one can use WKB solution (40). For the real field Φ(q) we denote\\nwhere C and ψ is real and write the WKB solution (40) in the form\\nTo the right of the turning point a and at some distance from it we use a decreasing in the under-barrier domain WKB solution\\nAt the turning point the WKB approximation is not valid. In order to establish a relation between (80) and (82) solutions one can use the following trick [48] which can be easily adapted to our problem. For this purpose we consider formally Φ(q) as a function of a complex variable q and find it along a path on the complex plane of q connecting positive and negative values of q − a. We choose this path to be a half of the circle of constant radius ρ with the center at q = a and take ρ so that the conditions of the validity of the WKB approximation are satisfied along the path. Let us denote\\nWe choose a sign + for a semi-circle in the upper half of the complex q-plane, and − for a semi-circle in the lower half of the complex q-plane. The angle φ changes from 0 to π. At the turning pointẑ =ẑ 0 =η 2 0 and q = a. We assume that ρ is small so that |ẑ −ẑ 0 | and |q − a| are also small and use the following linearized form of the constraint equation\\nLet us denote B = V ,q (a)/F 0 . One also hasẑ−ẑ 0 =η 2 . Then (84) and (85) imply\\nFor the analytically continued WKB solution (82) we get\\nHere ± stands for the quantities calculated along a semicircle in the upper (+) and lower (−) half of the q-plane.\\nUsing these relations at φ = π where a − q = ρ one obtains\\nThis establishes a relation between the WKB solution (82) in the domain to the right of the turning point and the WKB solution (80) to the left from it. This gives a required connection formula. It should be emphasized that for the operator F ( ) with the properties specified in section II this connection relation is the same as for a standard Schrödinger equation.\\nWe assume now that the potential has two turning points a < b and the domain between these points is forbidden for the classical motion. We suppose that in the region q < a the field Φ(q) has the form (80) and its amplitude is C. Then in the region q > b it has the same form with a different amplitudeC. Similarly to the local case the ratio |C/C| 2 can be found by using the connection formula (89) twice [48]\\nHereẑ 0 =η 2 0 andẑ =ẑ 2 0 +η 2 . In the expression for P the quantitiesẑ and q satisfy the constraint (42). The quantity R gives a probability of the barrier penetration. It should be emphasized that these relations are obtained in the WKB approximation which is valid when R 1.\"\n },\n {\n \"title\": \"D. Barrier penetration\",\n \"content\": \"\"\n },\n {\n \"title\": \"Inverse parabolic potential: Local theory\",\n \"content\": \"For the local theory the effective Hamiltonian is\\nwhere η 2 0 = ε 2 − p 2 ⊥ − m 2 . Let the potential V (q) be of the form\\nwhere V 0 > 0 and q * = √ V 0 /V 1 . This potential is shown at figure 7. It is called an inverse parabolic potential.\\nWe assume that η 2 0 < V 0 so that there exit two turning points at q = ±a, where\\nFor the motion in the under-barrier domain we put η = iη whereη\"\n },\n {\n \"title\": \"FIG. 7. The inverse parabolic potential\",\n \"content\": \"We denote by P 0 the following integral\\nUsing relation (94) we can also write P 0 in the form\\nLet us emphasize that P 0 depends only on the difference V 0 −η 2 0 but it does not depend on the value of η 0 itself. In other words, the penetration probability is sensitive only to the fact how close the effective energy η 2 0 is to the top of the potential V 0 . Let us also remind that the adopted WKB approximation is valid only when the dimensionless quantity P 0 is large, that is when the probability of the penetration R 0 = exp(−2P 0 / ) is small (see e,g, [48]).\"\n },\n {\n \"title\": \"Inverse parabolic potential: Nonlocal theory\",\n \"content\": \"Let us assume again that the potential V (q) has the form (93). We denote byẑ 0 andẑ 1 solutions of the following equations\\nThen the WKB probability of the barrier penetration is\\nLet us calculate the value of P for the case when F is of the form (7). When both parameters V 0 and η 2 0 are much smaller than µ 2 , the function X(ẑ) ≈ẑ and the integral in (100) can be easily taken with the following result\\nIn this regime the effects of the nonlocality are not important and we reproduce the expression for P 0 obtained in the local case (see equation (97)). The effects of the nonlocality play an important role in the different regime, whenV 0 1 andη 2 0 1. To demonstrate this we consider again the theory with the form factor f (ẑ) = exp(ẑ 2 ). In order to compare the expression for P with P 0 we fix as earlier the value of the difference V 0 − η 2 0 , but now we assume thatV 0 1. Then one hasẑ 1 −ẑ 0 1. In the appendix B it is shown that one can write the following approximate expression for P in this case\\nHere X 1 = dX/dx| x=ẑ1 and X(ẑ 1 ) =V 0 . Using (B2) one can write\\nThese relations give a parametric representation of X 1 (ẑ 1 ) as a function ofV 0 . The plot of the function κ(V 0 ) is shown at figure 8. It shows that this function is equal to 1 atV 0 = 0 and it is less than 1 and monotonically decreases for positiveV 0 . Since R = exp(−2P/ ) the relation (102) can be rewritten as follows\\nSince R 0 < 1 the value of R is always greater that R 0 . This means that in the nonlocal theory the probability of the barrier penetration is larger that in the local case.\\nIn other words nonlocality makes the barrier more transparent in the high-energy regime.\"\n },\n {\n \"title\": \"VII. DISCISSION\",\n \"content\": \"Let us discuss the obtained results. In this paper we study quasiclassical solutions of the nonlocal massive scalar field equations. The kinetic part of the action for such a theory contains a function of the -operator, which is chosen so that the theory does not contain new unphysical degrees of freedom. In section II we formulated conditions imposed on the form factor of such a theory. We also described a special model which is used later in the paper for the illustration. In section III we described an ansatz used in the WKB approximation for a solution of the nonlocal field equation. In the leading order of the 1/ -expansion one obtains the equation for the eikonal function S(x) which is of the form H(x, ∇S) = 0. This is a first order partial differential equation which has the form of the Hamilton-Jacobi equation for the Hamiltonian H(x, p), obtained by substitution ∇S = p. To calculate the eikonal function S(x) it is sufficixent to construct the Lagrangian submanifold in the phase space (x, p) by using the initial conditions for S. Such a Lagrangian submanifold is formed by phase trajectories which are solutions of the Hamiltonian equations of motion. Since the system is conservative this submanifold belongs to the subspace H(x, p) = 0. Let us emphasize that this procedure is very similar to the standard one adopted for the quantum mechanics and local field theory. When the field operator is of the second order the corresponding effective Hamiltonian H(x, p) is a quadratic polynomial in momentum p. For the nonlocal theory H(x, p) is a non-polynomial function of p and this is a main difference with the local case. As a result the construction of the WKB solutions becomes technically more complicated.\\nWKB solutions presented in section III are valid for a wide class of nonlocal scalar field theories with an arbitrary external potential V (x) in a flat spacetime with any number of spatial dimensions. These relations are greatly simplified for the case when the potential V depends only on one spatial Cartesian coordinate. For this case the phase space is two-dimensional and the Hamilton's equations are completely integrable. The corresponding WKB solutions are described in section IV. In section V we used the WKB approximation for the calculation of the energy levels of the nonlocal field trapped by a one-dimensional potential. To illustrate general formulas we considered the case when the form factor of the nonlocality is of the special form (7) and the potential is parabolic. The energy levels ε n of the field depend on a discrete integer parameter n. For the local case the number ∆n of the levels in the interval ∆(ε 2 ) is ∆n = ∆(ε 2 )/(2 V 1 ) and it does not depend on the value of n. In this sense the levels are equidistant. In the nonlocal case for large n this dependence is different\\nThis means that at very high energy for a given interval of ∆(ε 2 ) there are much more energy levels than in the local case. Finally, we discussed the under-barrier motion and obtained an expression for the probability of the barrier penetration. This probability has the form ∼ exp(−2P/ ). We calculated the factor P for the the special form of the nonlocality (7) in the presence of the inverse parabolic potential. For the local theory this factor is P 0 = π(V 0 − η 2 0 )/(2V 1 ). In other words, it depends on how close is the energy of the state η 2 0 to the top of the potential V 0 , but it does not depend on the height V 0 of the potential itself. The situation in the nonlocal case is quite different. Namely, one has P = κP 0 , where the factor κ depends on the height V 0 of the potential. This factor is always less than 1 and it decreases when V 0 grows. This means that the nonlocality \\\"helps\\\" the field to penetrate the potential barrier.\\nThe examples considered in this paper are rather simple. It would be interesting to apply the developed in the paper method for study the effects of the nonlocality in the early cosmology and in the black holes interior.\"\n },\n {\n \"title\": \"Appendix A: WKB approximation\",\n \"content\": \"We collected here useful information concerning construction of solutions of the nonlocal field equations in the WKB approximation. Further details and required mathematical foundations can be found in the book [46].\\nLet (x 1 , . . . , x N ) be coordinates of a point x in R N . We denote by α a set of N non-negative integer numbers α = (α 1 , . . . , α N ). We consider the following operator in this spacê\\nThe sum is taken over all different combinations of α.\\nWe do not assume that the number of terms in the sum is finite, so that this expression can be used as a formal series in derivatives representation of nonlocal operators. Let us make a substitution\\nin the expression (A1). Then we obtain a function of 2N variables which we denote by H = H(x, p)\\nHere p = (p 1 , . . . , p N ). This object is called a symbol of the operatorD. We are looking for a solution ϕ of the equation\\nin the form of the following formal series\\nUsually it is sufficient to consider only first few terms in the series. This gives what is known as the WKB approximation for the solution ϕ. To obtain the functions S(x) and u j (x) that enter (A5) the following procedure is used (see e.g. [45,46] and references therein). If one substitute expression (A5) into the equation (A4) then one finds that in the leading order j = 0 the following condition is to be satisfied\\nwhere\\nFor a nonvanishing function u 0 (x) the following Hamilton-Jacobi equation is valid\\nLet us consider a (2N )-dimension space Γ 2N with coordinates (x, p) = (x 1 , . . . , x N , p 1 , . . . , P N ) and denote by Ω a symplectic two-form in it\\nAn N -dimensional surface Λ N in Γ 2N is called a Lagrangian submanifold if the Ω vanishes on Λ N . If a Lagrangian submanifold can be uniquely projected onto the x-space, then there exists such a function S(x) that p µ = ∂S/∂ µ on Λ N . The Hamiltonian H(x, p) by the relation (A3) implies the following Hamilton's equations are\\nThese equations determine the Hamiltonian flow in the phase space Γ 2N preserving the symplectic form Ω and the Hamiltonian itself is an integral of motion. Let Σ N −1 be a (N −1)-dimensional surface in the coordinate space. Its embedding is defined by the equations\\nOne can consider y i as coordinates on Σ N −1 . We associate with Σ N −1 a (2N − 2)-dimensional phase space Γ 2N −2 and denote its canonical coordinates by (y, v) = (y 1 , . . . , y N −1 , v 1 , . . . , v N −1 ). The symplectic 2-form ω in\\nDenote by S 0 = S 0 (y) a function on Σ N −1 and ∇S 0 = (S 0 ,y 1 , . . . , S 0 ,y N −1 ), then a (N − 1)-dimensional surface (y, ∇S) of Γ 2N −2 is a Lagrangian submanifold Λ N −1 , so that ω| Λ N −1 = 0.\\nLet us define a covector p(y) = p µ (y) on Σ N −1 by two conditions p µ (y)dx µ = p µ (y) ∂f µ ∂y i = 0 , H(x(y), p(y)) = 0 . (A13)\\nWe assume that this system of N equations has a solution which determines p µ (y)| Σ N −1 . Let us consider the following initial conditions for the Hamilton's equations (A10)\\nA family of solutions of the Hamilton's equations (A10) with these initial conditions, (x(τ, y), p(τ, y)), forms a Ndimensional surface in Γ 2N which we denote by B. It is possible to show that B is a Lagrangian submanifold in Γ 2N and that H(x, p) = 0 on it. This implies that there exists such a function S(x) that\\nThis function is a solution of the Hamilton-Jacobi equation (A8) satisfying the initial condition S(x)| Σ N −1 = S 0 (y). This function S(x) can be found as follows. Denote a 1-form θ = µ p µ dx µ , then Ω = dθ. Since the symplectic form Ω vanishes on B, the integral\\ncalculated along a path on B between its two points does not depend on the choice of the path. The required solution S(x) of the Hamilton-Jacobi equation satisfying the initial condition S| Σ N −1 = S(y) can be written in the form\\nHere x µ = x µ (τ, y) and p µ = p µ (τ, y). Let us note that the map x µ = x µ (τ, y) may be considered as a transformation between the coordinates (τ, y and x µ . Let us denote by J a Jacobian of this transformation J(τ, y) = det ∂x(τ, y) ∂(τ, y) .\\nThen the leading term of the asymptotic solution (A5) is [46] ϕ(x) = u 0 (y) J(0, y) J(τ, y) exp i S 0 (y) + For τ = 0 the right-hand side of this expression reduces to u 0 (y) exp i S 0 (y) and it should be found from the initial condition for the function ϕ(x) at Σ.\\nAppendix B: Function X(x) = x exp(x 2 ) and its properties\\nIn this appendix we discuss some properties of the function\\nThe derivatives of this function are X = (1 + 2x 2 ) exp(x 2 ) ,\\nWe denote by Y an inverse function of X such that the following relations are valid y = X(x), x = Y (y) .\\nThe inverse function can be written as follows Y (y) = y exp − 1 2 W (2y 2 ) .\\nHere W is Lambert W function defined by the relation W (z) exp(W (z)) = z .\\nFor a real positive argument W is a real single-values function. Consider a following integral\\nLet us assume that x 1 − x 0 is small. To obtain an approximate value of the integral I in this case we use the Taylor expansion\\nHere X 1 = X(x 1 ), X 1 = X (x 1 ), X 1 = X (x 1 ). Skipping the higher order terms denoted in (B7) by dots one has\\nwhere α = 1 2 (X 1 /X 1 ). This integral can be calculated exactly. Let us denote β = x 1 − x 0 , then for αβ < 1 the answer is I = 2 X 1 3α (2αβ − 1)E( αβ) − (αβ − 1)K( αβ) .\\nHere E and K are complete elliptic integrals. Forthe small value of αβ one has\\nFor αβ 1 one can neglect the second term. Keeping the leading term one gets\\nFor small x 1 one has X 1 ≈ 1 and\\nThe author thanks the Natural Sciences and Engineering Research Council of Canada and the Killam Trust for their financial support.\"\n },\n {\n \"title\": \"FIG. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 4 .Figure 5\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 6 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 8 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Function X(x).","content":"FIG. 1 .\n1Function X(x)."},{"id":"","caption":"defines a closed curve on (q, η) phase plane (see figure 2).","content":"defines a closed curve on (q, η) phase plane (see figure 2)."},{"id":"","caption":"Phase trajectories for the form factor f = exp(ẑ 2 ). Different curves correspond to the different values ofẑ0 parameter: 0.14 for line 1; 0.7 for line 2; 1.0 for line 3 and 1.1 for line 4.","content":"FIG. 3 .\n3Phase trajectories for the form factor f = exp(ẑ 2 ). Different curves correspond to the different values ofẑ0 parameter: 0.14 for line 1; 0.7 for line 2; 1.0 for line 3 and 1.1 for line 4."},{"id":"","caption":"Function σ(ẑ0) for the form factor f (ẑ) = exp(ẑ 2 ).The function σ(ẑ 0 ) for the choice of the form factor f (ẑ) = exp(ẑ 2 ) is shown at figure 4. For this form factor it is possible to show that for largeẑ 0 the function σ(ẑ 0 ) has the follows asymptotic form which plots the ratio σ(ẑ 0 )/σ 0 (ẑ 0 ) demonstrates this property.","content":"FIG. 4 .Figure 5\n45Function σ(ẑ0) for the form factor f (ẑ) = exp(ẑ 2 ).The function σ(ẑ 0 ) for the choice of the form factor f (ẑ) = exp(ẑ 2 ) is shown at figure 4. For this form factor it is possible to show that for largeẑ 0 the function σ(ẑ 0 ) has the follows asymptotic form which plots the ratio σ(ẑ 0 )/σ 0 (ẑ 0 ) demonstrates this property."},{"id":"","caption":"Ratio σ/σ0 as a function ofẑ0.","content":"FIG. 5 .\n5Ratio σ/σ0 as a function ofẑ0."},{"id":"","caption":"Potential V (q).","content":"FIG. 6 .\n6Potential V (q)."},{"id":"","caption":"Parameter κ as a function ofV0.","content":"FIG. 8 .\n8Parameter κ as a function ofV0."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Function X(x).\",\n \"content\": \"FIG. 1 .\\n1Function X(x).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"defines a closed curve on (q, η) phase plane (see figure 2).\",\n \"content\": \"defines a closed curve on (q, η) phase plane (see figure 2).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Phase trajectories for the form factor f = exp(ẑ 2 ). Different curves correspond to the different values ofẑ0 parameter: 0.14 for line 1; 0.7 for line 2; 1.0 for line 3 and 1.1 for line 4.\",\n \"content\": \"FIG. 3 .\\n3Phase trajectories for the form factor f = exp(ẑ 2 ). Different curves correspond to the different values ofẑ0 parameter: 0.14 for line 1; 0.7 for line 2; 1.0 for line 3 and 1.1 for line 4.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Function σ(ẑ0) for the form factor f (ẑ) = exp(ẑ 2 ).The function σ(ẑ 0 ) for the choice of the form factor f (ẑ) = exp(ẑ 2 ) is shown at figure 4. For this form factor it is possible to show that for largeẑ 0 the function σ(ẑ 0 ) has the follows asymptotic form which plots the ratio σ(ẑ 0 )/σ 0 (ẑ 0 ) demonstrates this property.\",\n \"content\": \"FIG. 4 .Figure 5\\n45Function σ(ẑ0) for the form factor f (ẑ) = exp(ẑ 2 ).The function σ(ẑ 0 ) for the choice of the form factor f (ẑ) = exp(ẑ 2 ) is shown at figure 4. For this form factor it is possible to show that for largeẑ 0 the function σ(ẑ 0 ) has the follows asymptotic form which plots the ratio σ(ẑ 0 )/σ 0 (ẑ 0 ) demonstrates this property.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Ratio σ/σ0 as a function ofẑ0.\",\n \"content\": \"FIG. 5 .\\n5Ratio σ/σ0 as a function ofẑ0.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Potential V (q).\",\n \"content\": \"FIG. 6 .\\n6Potential V (q).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Parameter κ as a function ofV0.\",\n \"content\": \"FIG. 8 .\\n8Parameter κ as a function ofV0.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". G Wataghin, 10.1007/BF01352311Z. Phys. 8892G. Wataghin, Z. Phys. 88, 92 (1934).","doi":""},{"text":". H Yukawa, 10.1103/PhysRev.77.219Phys. Rev. 77219H. Yukawa, Phys. Rev. 77, 219 (1950).","doi":""},{"text":". H Yukawa, 10.1103/PhysRev.80.1047Phys. Rev. 801047H. Yukawa, Phys. Rev. 80, 1047 (1950).","doi":""},{"text":". G V Efimov, 10.1007/BF01648331Comm. Math. Phys. 7138G. V. Efimov, Comm. Math. Phys. 7, 138 (1968).","doi":""},{"text":". G V Efimov, 10.1016/0003-4916(72)90127-3Annals Phys. (NY). 71466G. V. Efimov, Annals Phys. (NY) 71, 466 (1972).","doi":""},{"text":". G V Efimov, 10.1007/BF01808314Int. J. Theor. Phys. 1019G. V. Efimov, Int. J. Theor. Phys. 10, 19 (1974).","doi":""},{"text":". V A Alebastrov, G V Efimov, 10.1007/BF01651546Commun. Math. Phys. 3811V. A. Alebastrov and G. V. Efimov, Commun. Math. Phys. 38, 11 (1974).","doi":""},{"text":". E Witten, 10.1016/0550-3213(86)90155-0Nucl. Phys. 268253E. Witten, Nucl. Phys. B268, 253 (1986).","doi":""},{"text":". P H Frampton, Y Okada, 10.1103/PhysRevD.37.3077Phys. Rev. 373077P. H. Frampton and Y. Okada, Phys. Rev. D37, 3077 (1988).","doi":""},{"text":". D A Eliezer, R P Woodard, 10.1016/0550-3213(89)90461-6Nucl. Phys. 325389D. A. Eliezer and R. P. Woodard, Nucl. Phys. B325, 389 (1989).","doi":""},{"text":". V A Kostelecky, S Samuel, 10.1103/PhysRevD.42.1289Phys. Rev. 421289V. A. Kostelecky and S. Samuel, Phys. Rev. D42, 1289 (1990).","doi":""},{"text":". B Harms, Y Leblanc, 10.1006/aphy.1995.1080arXiv:hep-th/9307042Annals Phys. 242hep-thB. Harms and Y. Leblanc, Annals Phys. 242, 265 (1995), arXiv:hep-th/9307042 [hep-th].","doi":""},{"text":". A A Tseytlin, 10.1016/0370-2693(95)01228-7arXiv:hep-th/9509050Phys. Lett. 363223hep-thA. A. Tseytlin, Phys. Lett. B363, 223 (1995), arXiv:hep- th/9509050 [hep-th].","doi":""},{"text":". G Calcagni, L Modesto, Journal of Physics A: Mathematical and Theoretical. 47355G. Calcagni and L. Modesto, Journal of Physics A: Math- ematical and Theoretical 47, 355 (2014).","doi":""},{"text":". G Calcagni, L Modesto, Physical Review D. 91124059G. Calcagni and L. Modesto, Physical Review D 91, 124059 (2015).","doi":""},{"text":". S Hossenfelder, 10.12942/lrr-2013-2arXiv:1203.6191Living Rev. Rel. 16gr-qcS. Hossenfelder, Living Rev. Rel. 16, 2 (2013), arXiv:1203.6191 [gr-qc].","doi":""},{"text":". E Tomboulis, arXiv:hep-th/9702146hep-thE. Tomboulis, (1997), arXiv:hep-th/9702146 [hep-th].","doi":""},{"text":". T Biswas, E Gerwick, T Koivisto, A Mazumdar, 10.1103/PhysRevLett.108.031101arXiv:1110.5249Phys. Rev. Lett. 10831101gr-qcT. Biswas, E. Gerwick, T. Koivisto, and A. Mazumdar, Phys. Rev. Lett. 108, 031101 (2012), arXiv:1110.5249 [gr-qc].","doi":""},{"text":". L Modesto, 10.1103/PhysRevD.86.044005arXiv:1107.2403Phys. Rev. 8644005hep-thL. Modesto, Phys. Rev. D86, 044005 (2012), arXiv:1107.2403 [hep-th].","doi":""},{"text":". T Biswas, A Conroy, A S Koshelev, A Mazumdar, 10.1088/0264-9381/31/1/015022,10.1088/0264-9381/31/15/159501arXiv:1308.2319Class.Quant.Grav. 3115022hep-thT. Biswas, A. Conroy, A. S. Koshelev, and A. Mazumdar, Class.Quant.Grav. 31, 015022 (2014), arXiv:1308.2319 [hep-th].","doi":""},{"text":". I L Shapiro, 10.1016/j.physletb.2015.03.037arXiv:1502.00106Phys. Lett. 74467hep-thI. L. Shapiro, Phys. Lett. B744, 67 (2015), arXiv:1502.00106 [hep-th].","doi":""},{"text":". T Biswas, T Koivisto, A Mazumdar, 10.1088/1475-7516/2010/11/008arXiv:1005.0590JCAP. 10118hep-thT. Biswas, T. Koivisto, and A. Mazumdar, JCAP 1011, 008 (2010), arXiv:1005.0590 [hep-th].","doi":""},{"text":"T Biswas, T Koivisto, A Mazumdar, arXiv:1302.0532Proceedings. nullgr-qcT. Biswas, T. Koivisto, and A. Mazumdar, in Pro- ceedings, Barcelona Postgrad Encounters on Fundamen- tal Physics (2013) pp. 13-24, arXiv:1302.0532 [gr-qc].","doi":""},{"text":". T Biswas, A Mazumdar, W Siegel, Journal of Cosmology and Astroparticle Physics. 06039T. Biswas, A. Mazumdar, and W. Siegel, Journal of Cosmology and Astroparticle Physics 0603, 009 (2006).","doi":""},{"text":". A S Koshelev, Journal of High Energy Physics. 071029A. S. Koshelev, Journal of High Energy Physics 0710, 029 (2007).","doi":""},{"text":". T Biswas, T Koivisto, A Mazumdar, Journal of Cosmology and Astroparticle Physics. 10118T. Biswas, T. Koivisto, and A. Mazumdar, Journal of Cosmology and Astroparticle Physics 1011, 008 (2010).","doi":""},{"text":". L Modesto, J W Moffat, P Nicolini, Physics Letters B. 695397L. Modesto, J. W. Moffat, and P. Nicolini, Physics Let- ters B 695, 397 (2011).","doi":""},{"text":". A Conroy, A Mazumdar, S Talaganis, A Teimouri, 10.1103/PhysRevD.92.124051arXiv:1509.01247Phys. Rev. 92124051hepthA. Conroy, A. Mazumdar, S. Talaganis, and A. Teimouri, Phys. Rev. D92, 124051 (2015), arXiv:1509.01247 [hep- th].","doi":""},{"text":". L Modesto, L , 10.1142/S0218271817300208Int. J. Mod. Phys. 261730020L. Modesto and L. Rachwa l, Int. J. Mod. Phys. D26, 1730020 (2017).","doi":""},{"text":". L Buoninfante, A S Koshelev, G Lambiase, A Mazumdar, 10.1088/1475-7516/2018/09/034arXiv:1802.00399JCAP. 180934gr-qcL. Buoninfante, A. S. Koshelev, G. Lambiase, and A. Mazumdar, JCAP 1809, 034 (2018), arXiv:1802.00399 [gr-qc].","doi":""},{"text":". A S Koshelev, J Marto, A Mazumdar, 10.1103/PhysRevD.98.064023arXiv:1803.00309Phys. Rev. 9864023gr-qcA. S. Koshelev, J. Marto, and A. Mazumdar, Phys. Rev. D98, 064023 (2018), arXiv:1803.00309 [gr-qc].","doi":""},{"text":". E Kilicarslan, Turk, 10.3906/fiz-1811-2arXiv:1811.00843J. Phys. 43gr-qcE. Kilicarslan, Turk. J. Phys. 43, 126 (2019), arXiv:1811.00843 [gr-qc].","doi":""},{"text":". A S Koshelev, J Marto, A Mazumdar, Journal of Cosmology and Astroparticle Physics. 20A. S. Koshelev, J. Marto, and A. Mazumdar, Journal of Cosmology and Astroparticle Physics 2019, 020 (2019).","doi":""},{"text":". E Kilicarslan, 10.1103/PhysRevD.99.124048arXiv:1903.04283Phys. Rev. 99gr-qcE. Kilicarslan, Phys. Rev. D99, 124048 (2019), arXiv:1903.04283 [gr-qc].","doi":""},{"text":". F J Cruz-Dombriz, A Maldonado Torralba, Mazumdar, 10.1103/PhysRevD.99.104021arXiv:1812.04037Phys. Rev. 99gr-qcl. de la Cruz-Dombriz, F. J. Maldonado Torralba, and A. Mazumdar, Phys. Rev. D99, 104021 (2019), arXiv:1812.04037 [gr-qc].","doi":""},{"text":". L Buoninfante, A Mazumdar, 10.1103/PhysRevD.100.024031arXiv:1903.01542Phys. Rev. 10024031gr-qcL. Buoninfante and A. Mazumdar, Phys. Rev. D100, 024031 (2019), arXiv:1903.01542 [gr-qc].","doi":""},{"text":". J Boos, V P Frolov, A Zelnikov, 10.1103/PhysRevD.97.084021arXiv:1802.09573Phys. Rev. 9784021gr-qcJ. Boos, V. P. Frolov, and A. Zelnikov, Phys. Rev. D97, 084021 (2018), arXiv:1802.09573 [gr-qc].","doi":""},{"text":". J Boos, J Pinedo Soto, V P Frolov, 10.1103/PhysRevD.101.124065arXiv:2004.07420Phys. Rev. 101124065gr-qcJ. Boos, J. Pinedo Soto, and V. P. Frolov, Phys. Rev. D101, 124065 (2020), arXiv:2004.07420 [gr-qc].","doi":""},{"text":". V P Frolov, 10.1103/PhysRevLett.115.051102arXiv:1505.00492Phys. Rev. Lett. 11551102hep-thV. P. Frolov, Phys. Rev. Lett. 115, 051102 (2015), arXiv:1505.00492 [hep-th].","doi":""},{"text":". V P Frolov, A Zelnikov, T De Paula, Netto, 10.1007/JHEP06(2015)107arXiv:1504.00412JHEP. 06107hep-thV. P. Frolov, A. Zelnikov, and T. de Paula Netto, JHEP 06, 107 (2015), arXiv:1504.00412 [hep-th].","doi":""},{"text":". V P Frolov, A Zelnikov, 10.1103/PhysRevD.93.064048arXiv:1509.03336Phys. Rev. 9364048hep-thV. P. Frolov and A. Zelnikov, Phys. Rev. D93, 064048 (2016), arXiv:1509.03336 [hep-th].","doi":""},{"text":". J Boos, V P Frolov, A Zelnikov, 10.1016/j.physletb.2018.06.018arXiv:1805.01875Phys. Lett. 782688hep-thJ. Boos, V. P. Frolov, and A. Zelnikov, Phys. Lett. B782, 688 (2018), arXiv:1805.01875 [hep-th].","doi":""},{"text":". V P Frolov, A Zelnikov, 10.1103/PhysRevD.98.084035arXiv:1809.00417Phys. Rev. 9884035hep-thV. P. Frolov and A. Zelnikov, Phys. Rev. D98, 084035 (2018), arXiv:1809.00417 [hep-th].","doi":""},{"text":". L Buoninfante, A Mazumdar, J Peng, 10.1103/PhysRevD.100.104059arXiv:1906.03624Phys. Rev. D. 100104059gr-qcL. Buoninfante, A. Mazumdar, and J. Peng, Phys. Rev. D 100, 104059 (2019), arXiv:1906.03624 [gr-qc].","doi":""},{"text":". V Arnold, K Vogtmann, A Weinstein, Mathematical Methods of Classical Mechanics. SpringerV. Arnold, K. Vogtmann, and A. Weinstein, Mathemati- cal Methods of Classical Mechanics (Springer New York, 2013).","doi":""},{"text":"V Maslov, M Fedoriuk, Semi-Classical Approximation in Quantum Mechanics, Mathematical Physics and Applied Mathematics. NetherlandsSpringerV. Maslov and M. Fedoriuk, Semi-Classical Approxima- tion in Quantum Mechanics, Mathematical Physics and Applied Mathematics (Springer Netherlands, 1981).","doi":""},{"text":". F Lu, B Lv, P Wang, H Yang, 10.1016/j.nuclphysb.2018.06.002arXiv:1701.00395Nucl. Phys. 937gr-qcF. Lu, B. Lv, P. Wang, and H. Yang, Nucl. Phys. B937, 502 (2018), arXiv:1701.00395 [gr-qc].","doi":""},{"text":"L D Landau, E M Lifshitz, Quantum Mechanics. Bristol, UKPergamon Press2nd ed.L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 2nd ed. (Pergamon Press, Bristol, UK, 1965).","doi":""}],"string":"[\n {\n \"text\": \". G Wataghin, 10.1007/BF01352311Z. Phys. 8892G. Wataghin, Z. Phys. 88, 92 (1934).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Yukawa, 10.1103/PhysRev.77.219Phys. Rev. 77219H. Yukawa, Phys. Rev. 77, 219 (1950).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Yukawa, 10.1103/PhysRev.80.1047Phys. Rev. 801047H. Yukawa, Phys. Rev. 80, 1047 (1950).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G V Efimov, 10.1007/BF01648331Comm. Math. Phys. 7138G. V. Efimov, Comm. Math. Phys. 7, 138 (1968).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G V Efimov, 10.1016/0003-4916(72)90127-3Annals Phys. (NY). 71466G. V. Efimov, Annals Phys. (NY) 71, 466 (1972).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G V Efimov, 10.1007/BF01808314Int. J. Theor. Phys. 1019G. V. Efimov, Int. J. Theor. Phys. 10, 19 (1974).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V A Alebastrov, G V Efimov, 10.1007/BF01651546Commun. Math. Phys. 3811V. A. Alebastrov and G. V. Efimov, Commun. Math. Phys. 38, 11 (1974).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Witten, 10.1016/0550-3213(86)90155-0Nucl. Phys. 268253E. Witten, Nucl. Phys. B268, 253 (1986).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P H Frampton, Y Okada, 10.1103/PhysRevD.37.3077Phys. Rev. 373077P. H. Frampton and Y. Okada, Phys. Rev. D37, 3077 (1988).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D A Eliezer, R P Woodard, 10.1016/0550-3213(89)90461-6Nucl. Phys. 325389D. A. Eliezer and R. P. Woodard, Nucl. Phys. B325, 389 (1989).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V A Kostelecky, S Samuel, 10.1103/PhysRevD.42.1289Phys. Rev. 421289V. A. Kostelecky and S. Samuel, Phys. Rev. D42, 1289 (1990).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Harms, Y Leblanc, 10.1006/aphy.1995.1080arXiv:hep-th/9307042Annals Phys. 242hep-thB. Harms and Y. Leblanc, Annals Phys. 242, 265 (1995), arXiv:hep-th/9307042 [hep-th].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A A Tseytlin, 10.1016/0370-2693(95)01228-7arXiv:hep-th/9509050Phys. Lett. 363223hep-thA. A. Tseytlin, Phys. Lett. B363, 223 (1995), arXiv:hep- th/9509050 [hep-th].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Calcagni, L Modesto, Journal of Physics A: Mathematical and Theoretical. 47355G. Calcagni and L. Modesto, Journal of Physics A: Math- ematical and Theoretical 47, 355 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Calcagni, L Modesto, Physical Review D. 91124059G. Calcagni and L. Modesto, Physical Review D 91, 124059 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Hossenfelder, 10.12942/lrr-2013-2arXiv:1203.6191Living Rev. Rel. 16gr-qcS. Hossenfelder, Living Rev. Rel. 16, 2 (2013), arXiv:1203.6191 [gr-qc].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Tomboulis, arXiv:hep-th/9702146hep-thE. Tomboulis, (1997), arXiv:hep-th/9702146 [hep-th].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Biswas, E Gerwick, T Koivisto, A Mazumdar, 10.1103/PhysRevLett.108.031101arXiv:1110.5249Phys. Rev. Lett. 10831101gr-qcT. Biswas, E. Gerwick, T. Koivisto, and A. Mazumdar, Phys. Rev. Lett. 108, 031101 (2012), arXiv:1110.5249 [gr-qc].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Modesto, 10.1103/PhysRevD.86.044005arXiv:1107.2403Phys. Rev. 8644005hep-thL. Modesto, Phys. Rev. D86, 044005 (2012), arXiv:1107.2403 [hep-th].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Biswas, A Conroy, A S Koshelev, A Mazumdar, 10.1088/0264-9381/31/1/015022,10.1088/0264-9381/31/15/159501arXiv:1308.2319Class.Quant.Grav. 3115022hep-thT. Biswas, A. Conroy, A. S. Koshelev, and A. Mazumdar, Class.Quant.Grav. 31, 015022 (2014), arXiv:1308.2319 [hep-th].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I L Shapiro, 10.1016/j.physletb.2015.03.037arXiv:1502.00106Phys. Lett. 74467hep-thI. L. Shapiro, Phys. Lett. B744, 67 (2015), arXiv:1502.00106 [hep-th].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Biswas, T Koivisto, A Mazumdar, 10.1088/1475-7516/2010/11/008arXiv:1005.0590JCAP. 10118hep-thT. Biswas, T. Koivisto, and A. Mazumdar, JCAP 1011, 008 (2010), arXiv:1005.0590 [hep-th].\",\n \"doi\": \"\"\n },\n {\n \"text\": \"T Biswas, T Koivisto, A Mazumdar, arXiv:1302.0532Proceedings. nullgr-qcT. Biswas, T. Koivisto, and A. Mazumdar, in Pro- ceedings, Barcelona Postgrad Encounters on Fundamen- tal Physics (2013) pp. 13-24, arXiv:1302.0532 [gr-qc].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Biswas, A Mazumdar, W Siegel, Journal of Cosmology and Astroparticle Physics. 06039T. Biswas, A. Mazumdar, and W. Siegel, Journal of Cosmology and Astroparticle Physics 0603, 009 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A S Koshelev, Journal of High Energy Physics. 071029A. S. Koshelev, Journal of High Energy Physics 0710, 029 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Biswas, T Koivisto, A Mazumdar, Journal of Cosmology and Astroparticle Physics. 10118T. Biswas, T. Koivisto, and A. Mazumdar, Journal of Cosmology and Astroparticle Physics 1011, 008 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Modesto, J W Moffat, P Nicolini, Physics Letters B. 695397L. Modesto, J. W. Moffat, and P. Nicolini, Physics Let- ters B 695, 397 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Conroy, A Mazumdar, S Talaganis, A Teimouri, 10.1103/PhysRevD.92.124051arXiv:1509.01247Phys. Rev. 92124051hepthA. Conroy, A. Mazumdar, S. Talaganis, and A. Teimouri, Phys. Rev. D92, 124051 (2015), arXiv:1509.01247 [hep- th].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Modesto, L , 10.1142/S0218271817300208Int. J. Mod. Phys. 261730020L. Modesto and L. Rachwa l, Int. J. Mod. Phys. D26, 1730020 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Buoninfante, A S Koshelev, G Lambiase, A Mazumdar, 10.1088/1475-7516/2018/09/034arXiv:1802.00399JCAP. 180934gr-qcL. Buoninfante, A. S. Koshelev, G. Lambiase, and A. Mazumdar, JCAP 1809, 034 (2018), arXiv:1802.00399 [gr-qc].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A S Koshelev, J Marto, A Mazumdar, 10.1103/PhysRevD.98.064023arXiv:1803.00309Phys. Rev. 9864023gr-qcA. S. Koshelev, J. Marto, and A. Mazumdar, Phys. Rev. D98, 064023 (2018), arXiv:1803.00309 [gr-qc].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Kilicarslan, Turk, 10.3906/fiz-1811-2arXiv:1811.00843J. Phys. 43gr-qcE. Kilicarslan, Turk. J. Phys. 43, 126 (2019), arXiv:1811.00843 [gr-qc].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A S Koshelev, J Marto, A Mazumdar, Journal of Cosmology and Astroparticle Physics. 20A. S. Koshelev, J. Marto, and A. Mazumdar, Journal of Cosmology and Astroparticle Physics 2019, 020 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Kilicarslan, 10.1103/PhysRevD.99.124048arXiv:1903.04283Phys. Rev. 99gr-qcE. Kilicarslan, Phys. Rev. D99, 124048 (2019), arXiv:1903.04283 [gr-qc].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F J Cruz-Dombriz, A Maldonado Torralba, Mazumdar, 10.1103/PhysRevD.99.104021arXiv:1812.04037Phys. Rev. 99gr-qcl. de la Cruz-Dombriz, F. J. Maldonado Torralba, and A. Mazumdar, Phys. Rev. D99, 104021 (2019), arXiv:1812.04037 [gr-qc].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Buoninfante, A Mazumdar, 10.1103/PhysRevD.100.024031arXiv:1903.01542Phys. Rev. 10024031gr-qcL. Buoninfante and A. Mazumdar, Phys. Rev. D100, 024031 (2019), arXiv:1903.01542 [gr-qc].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Boos, V P Frolov, A Zelnikov, 10.1103/PhysRevD.97.084021arXiv:1802.09573Phys. Rev. 9784021gr-qcJ. Boos, V. P. Frolov, and A. Zelnikov, Phys. Rev. D97, 084021 (2018), arXiv:1802.09573 [gr-qc].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Boos, J Pinedo Soto, V P Frolov, 10.1103/PhysRevD.101.124065arXiv:2004.07420Phys. Rev. 101124065gr-qcJ. Boos, J. Pinedo Soto, and V. P. Frolov, Phys. Rev. D101, 124065 (2020), arXiv:2004.07420 [gr-qc].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V P Frolov, 10.1103/PhysRevLett.115.051102arXiv:1505.00492Phys. Rev. Lett. 11551102hep-thV. P. Frolov, Phys. Rev. Lett. 115, 051102 (2015), arXiv:1505.00492 [hep-th].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V P Frolov, A Zelnikov, T De Paula, Netto, 10.1007/JHEP06(2015)107arXiv:1504.00412JHEP. 06107hep-thV. P. Frolov, A. Zelnikov, and T. de Paula Netto, JHEP 06, 107 (2015), arXiv:1504.00412 [hep-th].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V P Frolov, A Zelnikov, 10.1103/PhysRevD.93.064048arXiv:1509.03336Phys. Rev. 9364048hep-thV. P. Frolov and A. Zelnikov, Phys. Rev. D93, 064048 (2016), arXiv:1509.03336 [hep-th].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Boos, V P Frolov, A Zelnikov, 10.1016/j.physletb.2018.06.018arXiv:1805.01875Phys. Lett. 782688hep-thJ. Boos, V. P. Frolov, and A. Zelnikov, Phys. Lett. B782, 688 (2018), arXiv:1805.01875 [hep-th].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V P Frolov, A Zelnikov, 10.1103/PhysRevD.98.084035arXiv:1809.00417Phys. Rev. 9884035hep-thV. P. Frolov and A. Zelnikov, Phys. Rev. D98, 084035 (2018), arXiv:1809.00417 [hep-th].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Buoninfante, A Mazumdar, J Peng, 10.1103/PhysRevD.100.104059arXiv:1906.03624Phys. Rev. D. 100104059gr-qcL. Buoninfante, A. Mazumdar, and J. Peng, Phys. Rev. D 100, 104059 (2019), arXiv:1906.03624 [gr-qc].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Arnold, K Vogtmann, A Weinstein, Mathematical Methods of Classical Mechanics. SpringerV. Arnold, K. Vogtmann, and A. Weinstein, Mathemati- cal Methods of Classical Mechanics (Springer New York, 2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"V Maslov, M Fedoriuk, Semi-Classical Approximation in Quantum Mechanics, Mathematical Physics and Applied Mathematics. NetherlandsSpringerV. Maslov and M. Fedoriuk, Semi-Classical Approxima- tion in Quantum Mechanics, Mathematical Physics and Applied Mathematics (Springer Netherlands, 1981).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Lu, B Lv, P Wang, H Yang, 10.1016/j.nuclphysb.2018.06.002arXiv:1701.00395Nucl. Phys. 937gr-qcF. Lu, B. Lv, P. Wang, and H. Yang, Nucl. Phys. B937, 502 (2018), arXiv:1701.00395 [gr-qc].\",\n \"doi\": \"\"\n },\n {\n \"text\": \"L D Landau, E M Lifshitz, Quantum Mechanics. Bristol, UKPergamon Press2nd ed.L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 2nd ed. (Pergamon Press, Bristol, UK, 1965).\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":499,"cells":{"text":{"kind":"string","value":"\nPolynomial equations for additive functions I. The inner parameter case\nAugust 31, 2023\n\nEszter Gselmann gselmann@science.unideb.hu \nUniversity of Debrecen\n\n\nGergely Kiss \nUniversity of Debrecen\n\n\nPolynomial equations for additive functions I. The inner parameter case\nAugust 31, 2023EA85966ED7279EBC9B8F373053D85867arXiv:2211.03605v2[math.CA]homomorphismderivationhigher order derivationexponential polynomialdecomposable function. 2020 MSC classes. 43A4513N1516W2039B3239B72\nThe aim of this sequence of work is to investigate polynomial equations satisfied by additive functions.As a result of this, new characterization theorems for homomorphisms and derivations can be given.More exactly, in this paper the following type of equation is consideredwhere n is a positive integer, F Ă C is a field, f i , g i : F Ñ C are additive functions and p i , q i are positive integers for all i \" 1, . . ., n.\n\nIntroduction and preliminaries\n\nEquations satisfied by additive functions play an important role not only in the theory of commutative algebra, but also in the theory of functional equations.It is an important and challenging question how special morphisms (such as homomorphisms and derivations) can be characterized among additive mappings in general.In this paper classes of multivariable algebraic equations are introduced with appropriate solutions as field homomorphisms and derivations.\n\nConcerning all the cases we consider here, the involved additive functions are defined on a field F Ă C and have values in the complex field, therefore we introduce the preliminaries in this setting.\n\nWe adopt the standard notations, that is, N and C denote the set of positive integers and the set of complex numbers, respectively.\n\nHenceforth we assume F Ă C to be a field.\n\nDefinition 1.We say that a function f : F Ñ C is additive if it fulfills the so-called Cauchy functional equation, that is, f px `yq \" f pxq `f pyq px, y P Fq .\n\nAn additive function d : F Ñ C is termed to be a derivation (of order 1) if it also fulfills the Leibniz equation, i.e., dpxyq \" dpxqy `xdpyq px, y P Fq .\n\nAn additive function ϕ : F Ñ C is said to be a homomorphism if it is multiplicative as well, in other words, besides additivity we also have ϕpxyq \" ϕpxqϕpyq px, y P Fq .\n\nIf F \" C and ϕ is an isomorphism, then ϕ is called a complex automorphism.\n\nCertain well-known equations are especially important.For instance, the additive solutions of following equation on a ring R with charR 2 f px 2 q \" 2x f pxq px P q are derivations, under some assumptions, where the additive mapping f acts.As an extension of such type of results in [1,2,5] the additive solutions of the equations n ÿ i\"0\n\nx i f n`1´i px n`1´i q \" 0 px P Rq and n ÿ i\"0 f px p i qx q i \" 0 px P Fq were described, here R denotes a ring with charpRq ě n, while F Ă C is a field.By a polynomial equation of additive functions we mean an equation of the form Pp f r 1 1 px s 1 q, . . ., f r n n px s n qq \" 0, where P : C n Ñ C is a n-variable polynomial, r i , s i are positive integers and f i denote the unknown additive functions.Without further restrictions (e.g., on the polynomial P or on the parameters r i , s i ), the above equation is unfortunately too general for its solutions to be fully determined.Indeed, it is not too hard to specify a polynomial P and parameters r i , s i so that the above equation is satisfied by all additive functions.Therefore, in this series, we will focus on the following classes of equations.\nn ÿ i\"1\nf i px p i qg i px q i q \" 0, n ÿ i\"1 f i px p i qg i pxq q i \" 0, n ÿ i\"1 f i pxq p i g i pxq q i \" 0,\n\n\npx P Fq\n\nIn this paper the most impressive equation, namely\nn ÿ i\"1\nf i px p i qg i px q i q \" 0, px P Fq (1) will be studied from this list with a fruitful theoretical description.(This we call 'inner case', as the parameters p i , q i are exponents of the variable x, so they act on the domain of the functions f i , g i , respectively.)Under some natural conditions equation ( 1) is satisfied by compositions of (higher order) derivations and homomorphisms.The purpose of this paper is about the converse by showing proper characterizations for the solutions of (1) in the class of additive functions.\n\nThe diagonalization or trace of an n-additive function A : G n Ñ S is defined as A ˚pxq \" A px, . . ., xq px P Gq .\n\nAs a direct consequence of the definition each n-additive function A : G n Ñ S satisfies Apx 1 , . . ., x i´1 , kx i , x i`1 , . . ., x n q \" kApx 1 , . . ., x i´1 , x i , x i`1 , . . ., x n q px 1 , . . ., x n P Gq for all i \" 1, . . ., n, where k P N is arbitrary.The same identity holds for any k P Z provided that G and S are groups, and for k P Q, provided that G and S are linear spaces over the rationals.For the diagonalization of A we have A ˚pk xq \" k n A ˚pxq px P Gq .\n\nThe above notion can also be extended for the case n \" 0 by letting G 0 \" G and by calling 0-additive any constant function from G to S .\n\nOne of the most important theoretical results concerning multiadditive functions is the so-called Polarization formula, that briefly expresses that every n-additive symmetric function is uniquely determined by its diagonalization under some conditions on the domain as well as on the range.Suppose that G is a commutative semigroup and S is a commutative group.The action of the difference operator ∆ on a function f : G Ñ S is defined by the formula ∆ y f pxq \" f px `yq ´f pxq px, y P Gq .\n\nNote that the addition in the argument of the function is the operation of the semigroup G and the subtraction means the inverse of the operation of the group S .\n\nTheorem 1 (Polarization formula).Suppose that G is a commutative semigroup, S is a commutative group, n P N. If A : G n Ñ S is a symmetric, n-additive function, then for all x, y 1 , . . ., y m P G we have ∆ y 1 ,...,y m A ˚pxq \" \" 0 if m ą n n!Apy 1 , . . ., y m q if m \" n.\n\nCorollary 1. Suppose that G is a commutative semigroup, S is a commutative group, n P N. If A : G n Ñ S is a symmetric, n-additive function, then for all x, y P G ∆ n y A ˚pxq \" n!A ˚pyq.Lemma 1.Let n P N and suppose that the multiplication by n! is surjective in the commutative semigroup G or injective in the commutative group S .Then for any symmetric, n-additive function A : G n Ñ S , A ˚\" 0 implies that A is identically zero, as well.Definition 3. Let G and S be commutative semigroups, a function p : G Ñ S is called a generalized polynomial from G to S , if it has a representation as the sum of diagonalizations of symmetric multi-additive functions from G to S .In other words, a function p : G Ñ S is a generalized polynomial if and only if, it has a representation\np \" n ÿ k\"0 A k ,\nwhere n is a nonnegative integer and A k : G k Ñ S is a symmetric, k-additive function for each k \" 0, 1, . . ., n.In this case we also say that p is a generalized polynomial of degree at most n.\n\nLet n be a nonnegative integer, functions p n : G Ñ S of the form\np n \" A n ,\nwhere A n : G n Ñ S is a symmetric and n-additive mapping, are the so-called generalized monomials of degree n.\n\nIn this subsection pG, ¨q is assumed to be a commutative group (written multiplicatively).\n\nDefinition 4. Polynomials are elements of the algebra generated by additive functions over G.More exactly, a mapping f : G Ñ C is called a polynomial if there is a positive integer n, there exists a (classical) complex polynomial P : C n Ñ C in n variables and there are additive functions a k : G Ñ C pk \" 1, . . ., nq such that f pxq \" Ppa 1 pxq, . . ., a n pxqq px P Gq .\n\nRemark 1.We recall that the elements of N n for any positive integer n are called (n-dimensional) multi-indices.Addition, multiplication and inequalities between multi-indices of the same dimension are defined componentwise.Further, we define x α for any n-dimensional multi-index α and for any x \" px 1 , . . ., x n q in C n by\nx α \" n ź i\"1 x α i i\nwhere we always adopt the convention 0 0 \" 0. We also use the notation |α| \" α 1 `¨¨¨`α n .With these notations any polynomial of degree at most N on the commutative semigroup G has the form ppxq \"\nÿ |α|ďN c α apxq α px P Gq ,\nwhere c α P C and a \" pa 1 , . . ., a n q : G Ñ C n is an additive function.Furthermore, the homogeneous term of degree k of p is ÿ\n|α|\"k c α apxq α .\nLemma 2 (Lemma 2.7 of [10]).Let G be a commutative group, n be a positive integer and let a \" pa 1 , . . ., a n q , where a 1 , . . ., a n are linearly independent complex valued additive functions defined on G. Then the monomials ta α u for different multi-indices are linearly independent.\nDefinition 5. A function m : G Ñ C is called an exponential function if it satisfies mpxyq \" mpxqmpyq px, y P Gq .\nFurthermore, on a(n) (generalized) exponential polynomial we mean a linear combination of functions of the form p ¨m, where p is a (generalized) polynomial and m is an exponential function.\n\nThe following lemma shows that generalized exponential polynomial functions are linearly independent.Although it can be stated in a more general way (see [10]), we adopt it to our situation, when the functions are complex valued.Lemma 3 (Lemma 4.3 of [10]).Let G be a commutative group, n a positive integer, m 1 , . . ., m n : G Ñ C pi \" 1, . . ., nq be distinct nonzero exponentials and p 1 , . . ., p n : G Ñ K pi \" 1, . . ., nq be generalized polynomials.\n\n\nIf n ÿ\n\ni\"1 p i ¨mi is identically zero, then for all i \" 1, . . ., n the generalized polynomial p i is identically zero.\n\nAdditionally, we will need the analogous statement for polynomial expressions of generalized exponential polynomials which was proved in [5].\n\nTheorem 2. Let K be a field of characteristic 0 and k, l, N be positive integers such that k, l ď N. Let m 1 , . . ., m k : K ˆÑ C be distinct exponential functions that are additive on K, let a 1 , . . ., a l : K ˆÑ C be additive functions that are linearly independent over C and for all |s| ď N let P s : C l Ñ C be classical complex polynomials of l variables.If ÿ |s|ďN P s pa 1 , . . ., a l qm s 1 1 ¨¨¨m s k k \" 0 then for all |s| ď N, the polynomials P s vanish identically.\n\nDefinition 6.Let G be a commutative group and V Ď C G a set of functions.We say that V is translation invariant if for every f P V the function τ g f P V also holds for all g P G, where τ g f phq \" f phgq ph P Gq .\n\nIn view of Theorem 10.1 of Székelyhidi [10], any finite dimensional translation invariant linear space of complex valued functions on a commutative group consists of exponential polynomials.This implies that if G is a commutative group, then any function f : G Ñ C, satisfying the functional equation\nf pxyq \" n ÿ i\"1 g i pxqh i pyq px, y P Gq\nfor some positive integer n and functions g i , h i : G Ñ C (i \" 1, . . ., n), is an exponential polynomial of degree at most n.This enlightens the connection between generalized polynomials and polynomials.It is easy to see that each polynomial, that is, any function of the form\nx Þ ÝÑ Ppa 1 pxq, . . . , a n pxqq,\nwhere n is a positive integer, P : C n Ñ C is a (classical) complex polynomial in n variables and a k : G Ñ C pk \" 1, . . ., nq are additive functions, is a generalized polynomial.The converse however is in general not true.A complex-valued generalized polynomial p defined on a commutative group G is a polynomial if and only if its variety (the linear space spanned by its translates) is of finite dimension.\n\nHenceforth, not only the notion of (exponential) polynomials, but also that of decomposable functions will be used.The basics of this concept are due to Shulman [9], besides this we heavily rely on the work of Laczkovich [7].Definition 7. Let G be a group and n P N, n ě 2. A function F : G n Ñ C is said to be decomposable if it can be written as a finite sum of products F 1 ¨¨¨F k , where all F i depend on disjoint sets of variables.\n\nRemark 2. Without loss of generality we can suppose that k \" 2 in the above definition, that is, decomposable functions are those mappings that can be written in the form\nFpx 1 , . . . , x n q \" ÿ E ÿ j A E j B E j\nwhere E runs through all non-void proper subsets of t1, . . ., nu and for each E and j the function A E j depends only on variables x i with i P E, while B E j depends only on the variables x i with i E. Theorem 3. Let G be a commutative topological semigroup with unit.A continuous function f : G Ñ C is a generalized exponential polynomial if and only if there is a positive integer n ě 2 such that the mapping\nG n Q px 1 , . . . , x n q Þ ÝÑ f px 1 `¨¨¨`x n q is decomposable.\nThe notion of derivations can be extended in several ways.We will employ the concept of higher order derivations according to Reich [8] and Unger-Reich [11].For further results on characterization theorems on higher order derivations consult e.g.[1,2,3] and [5].Definition 8. Let F Ă C be a field.The identically zero map is the only derivation of order zero.For each n P N, an additive mapping f : F Ñ C is termed to be a derivation of order n, if there exists B : F ˆF Ñ C such that B is a bi-derivation of order n ´1 (that is, B is a derivation of order n ´1 in each variable) and f pxyq ´x f pyq ´f pxqy \" Bpx, yq px, y P Fq .\n\nThe set of derivations of order n of the ring R will be denoted by D n pFq.For our future purposes the notion of differential operators will also be important, see [6].\n\nDefinition 9. Let F Ă C be a field.We say that the map D : is a derivation on K for every i \" 1, . . ., k.Therefore, every differential operator\nF Ñ C is a differentialD \" ÿ i 1 `¨¨¨`i k ďn c i 1 ,...,i k ¨Bi 1 `¨¨¨`i k Bt i 1 1 ¨¨¨Bt i k k ,\nwhere the coefficients c i 1 ,...,i k belong to K, is a differential operator of order at most n, and also conversely, if D is a differential operator of order at most n on the field K \" Qpt 1 , . . ., t k q, then D is of the above form.\n\nThe main result of [6] is Theorem 1.1 that reads in our settings as follows.\n\nTheorem 4. Let F Ă C be a field and let n be a positive integer.Then, for every function D : F Ñ C, the following are equivalent.\n\n(i) D P D n pFq (ii) D P cl pO n pFqq (iii) D is additive on F, Dp1q \" 0, and D{ j, as a map from the group F ˆto C, is a generalized polynomial of degree at most n.Here j stands for the identity map defined on F.\n\n\nResults\n\n\nElementary observations: reduction of the problem\n\nThis part begins with some elementary, yet fundamental observations.As the following lemmata show, the original problem can be reduced to a more simpler equation.\n\nLemma 4 (Homogenization).Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers.Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation (1), that is,\nn ÿ i\"1\nf i px p i qg i px q i q \" 0 for each x P F. If the set tp 1 , . . ., p n u has a partition P 1 , . . ., P k with the property if p α , p β P P j for a certain index j, then p α `qα \" p β `qβ , then the system of equations ÿ p α PP j f α px p α qg α px q α q \" 0 px P F, j \" 1, . . ., kq is satisfied.\n\nProof.Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers.Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation (1) for each x P F. Assume further that the set tp 1 , . . ., p n u has a partition P 1 , . . ., P k with the property if p α , p β P P j for a certain index j, then p α `qα \" p β `qβ .\n\nObserve that for all i \" 1, . . ., n, the mapping\nF Q x Þ ÝÑ f i px p i qg i px q i q\nis a generalized monomial of degree p i `qi .Indeed, it is the diagonalization of the symmetric pp i `qi q-additive mapping\nF p i `qi Q px 1 , . . . , x p i `qi q Þ ÝÑ f i px σp1q ¨¨¨x σpp i q qg i px σpp i `1q ¨¨¨x σpp i `qi q q.\nSince F Ă C, we necessarily have Q Ă F. Let now r P Q be arbitrary and substitute rx in place of x in equation (1) to get\nn ÿ i\"1 f i pprxq p i qg i pprxq q i q \" 0 pr P Q, x P Fq .\nUsing the Q-homogeneity of the additive functions f 1 , . . ., f n and g 1 , . . ., g n , we deduce 0 \"\nn ÿ i\"1 f i pprxq p i qg i pprxq q i q \" n ÿ i\"1 f i pr p i x p i qg i pr q i x q i q \" n ÿ i\"1 r p i `qi f i px p i qg i px q i q \" k ÿ j\"1 ÿ p α PP j r p α `qα f α px p α qg α px q α q pr P Q, x P Fq .\nNote that the right hand side of this equation is a (classical) polynomial in r which is identically zero.Thus all of its coefficients should be (identically) zero, yielding that the system of equations ÿ p α PP j f α px p α qg α px q α q \" 0 px P F, j \" 1, . . ., kq is fulfilled.\n\nRemark 5.The above lemma guarantees that ab initio p i `qi \" N pi \" 1, . . ., nq can be assumed.Otherwise, after using the above homogenization, we get a system of functional equations in which this condition is already fulfilled.For instance, due to the above lemma, if the additive functions f 1 , . . ., f 5 : F Ñ C and g 1 , . . ., g 5 : F Ñ C satisfy equation f 1 px 24 qg 1 px 5 q `f2 px 20 qg 2 px 9 q `f3 px 19 qg 3 px 10 q `f4 px 13 qg 4 px 7 q `f5 px 12 qg 4 px 8 q \" 0 px P Fq then the equations f 1 px 24 qg 1 px 5 q `f2 px 20 qg 2 px 9 q `f3 px 19 qg 3 px 10 q \" 0 px P Fq and f 4 px 13 qg 4 px 7 q `f5 px 12 qg 4 px 8 q \" 0 px P Fq are also fulfilled (separately).\n\nRemark 6.At first glance the assumption that p 1 , . . ., p n are different seems a reasonable and sufficient supposition.Clearly, if the parameters are not necessarily different then we cannot expect anything special for the form of the involved additive functions.Indeed, let L Ă C n be a linear subspace and let f 1 , . . ., f n : F Ñ C and g 1 , . . ., g n : F Ñ C be additive functions such that rngp f q Ă L and rngpgq Ă L K , where f pxq \" p f 1 pxq, . . ., f n pxqq and gpxq \" pg 1 pxq, . . ., g n pxqq px P Fq .\n\n\nIn this case\nn ÿ i\"1 f i pxqg i pxq \" x f pxq, gpxqy \" 0 px P Fq .\nThis shows the necessity of the above assumption.Unfortunately, the sufficiency fails to hold.To see this, let p and q be positive integers and f : F Ñ C be an arbitrary additive function and define the complex-valued functions f 1 , g 1 , f 2 , g 2 on F by\nf 1 pxq \" f pxq g 1 pxq \" f pxq f 2 pxq \" i f pxq g 2 pxq \" i f pxq px P Fq .\nAn immediate computation shows that we have f 1 px p qg 1 px q q `f2 px p qg 2 px q q \" 0 px P Fq .\n\nIn view of the above remarks, from now on, the following assumptions are adopted.\n\nC(i) the positive integers p 1 , . . ., p n are arranged in a strictly increasing order, i.e., p 1 ă ¨¨¨ă p n ; C(ii) for all i \" 1, . . ., n we have p i `qi \" N;\n\nC(iii) for all i, j P t1, . . ., nu, i j we have p i q j .Remark 7. Define the relation \" on F C by f \" g if and only if there exists an automorphism ϕ : C Ñ C such that ϕ ˝f \" g.Obviously \" is an equivalence relation on F C that induces a partition on F C .\n\n\nLemma 5 (Equivalence).\n\nLet n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling the conditions C(i)-C(iii) of Remark 6. Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation (1).Then for an arbitrary automorphism ϕ : C Ñ C the functions ϕ ˝f1 , . . ., ϕ ˝fn , ϕ ˝g1 , . . ., ϕ ˝gn also fulfill equation (1).\n\n\nStructure of solutions\n\nWe can always restrict ourselves to the case when all the involved functions are non-identically zero.Otherwise, the number of the terms appearing in equation ( 1) can be reduced.\n\n\nLemma 6 (Symmetrization).\n\nLet k and n be positive integers, F Ă C be a field and m 1 , . . ., m n :\nF Ñ C be monomials of degree k. If n ÿ i\"1 m i pxq \" 0 holds for all x P F, then n ÿ i\"1 M i px 1 , . . . , x k q \" 0\nis fulfilled for all x 1 , . . ., x k , where for all i \" 1, . . ., n, the mapping M i : F k Ñ C is the uniquely determined symmetric, k-additive function such that M i px, . . ., xq \" m i pxq px P Fq .\n\n\nProof.\n\nLet k and n be positive integers, F Ă C be a field and m 1 , . . ., m n : F Ñ C be monomials of degree k and assume that n ÿ i\"1 m i pxq \" 0 holds for all x P F. Since for all i \" 1, . . ., n, the function m i is a monomial of degree k, there exists a symmetric, k-additive function M i : F k Ñ C such that we have M i px, . . ., xq \" m i pxq px P Fq .\n\n\nObviously the mapping\nn ÿ i\"1\nm i is a monomial of degree k which is, by the assumptions, identically zero.To this monomial there also corresponds a symmetric and k-additive mapping, namely\nF k Q px 1 , . . . , x k q Þ ÝÑ n ÿ i\"1 M i px 1 , . . . x k q.\nObserve that the trace of this symmetric and k-additive mapping is identically zero.At the same time, due to the Polarization formula (Theorem 1), every symmetric and k-additive function is uniquely determined by its trace.Thus\nn ÿ i\"1\nM i px 1 , . . ., x k q \" 0 for all x 1 , . . ., x k P F.\n\n\nLemma 7 (Symmetrization).\n\nLet n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling conditions C(ii), i.e., there is a N P N such p i `qi \" N for all i \" 1, . . ., n. Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation (1) for each x P F. Then\n1 N! ÿ σPS N n ÿ i\"1\nf i px σp1q ¨¨¨x σpp i q q ¨gi px σpp i `1q ¨¨¨x σpNq q \" 0 holds for all x 1 , . . ., x N P F.\n\nProof.Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling conditions C(ii).Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation\nn ÿ i\"1\nf i px p i qg i px q i q \" 0 for each x P F. Due to the additivity of the functions f 1 , . . ., f n and g 1 , . . ., g n for all i \" 1, . . ., n, the mapping\nx Þ ÝÑ f i px p i qg i px q i q\nis a monomial of degree p i `qi \" N. Further, it is the trace of the symmetric and N-additive mapping\nF i px 1 , . . . , x N q \" 1 N! ÿ σPS N f i px σp1q ¨¨¨x σpp i q q ¨gi px σpp i `1q ¨¨¨x σpNq q px 1 , . . . , x N P Fq .\nTherefore, the statement follows from Lemma 6.\n\n\nSolutions of equation (1)\n\nThe main purpose of the subsection is to describe under the conditions C(i)-C(iii), the solution space of equation (1).We first prove that solutions of equation ( 1) are decomposable functions on the multiplicative group F ˆ.In view of Laczkovich [7], this immediately yields that the solutions of equation ( 1) are generalized exponential polynomials of this group.\n\nLemma 8. Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling conditions C(i) and C(ii).Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation (1) for each x P F. Then all the functions f 1 , . . ., f n as well as g 1 , . . ., g n are decomposable functions of the group F ˆ.\n\nProof.Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling conditions C(i) and C(ii).\n\nLet us assume first that condition C(iii) is also satisfied.Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation (1) for each x P F. Let S \" tp 1 , . . ., p n u Y tq 1 , . . ., q n u .\n\nThen max S \" max tp n , q 1 u.By condition C(iii), we have p n q 1 .Without the loss of generality p n ą q 1 can be assumed, otherwise we follow a similar argument.In view of Lemma 7, we have\n1 N! ÿ σPS N n ÿ i\"1\nf i px σp1q ¨¨¨x σpp i q q ¨gi px σpp i `1q ¨¨¨x σpNq q \" 0 for all x 1 , . . ., x N P F, or after some rearrangement,\n1 N! ÿ σPS N f n px σp1q ¨¨¨x σpp n q q ¨gn px σpp n `1q ¨¨¨x σpNq q \" ´1 N! ÿ σPS N n´1 ÿ i\"1\nf i px σp1q ¨¨¨x σpp i q q ¨gi px σpp i `1q ¨¨¨x σpNq q `x1 , . . ., x N P F ˆ˘.\n\n\nLet now\nx p n `1 \" ¨¨¨\" x N \" 1,\nthen the above identity says that g n p1q ¨fn is decomposable.If g n p1q were zero, but g n would not be identically zero, then there would exist a P F ˆsuch that g n paq 0. In this case the above substitutions should be modified to\nx p n `1 \" a, x p n `2 \" ¨¨¨\" x N \" 1,\nto get the same conclusion.If p n \" q j for some j \" t1, . . ., nu (i.e., C(iii) does not hold), then without loss of generality we may assume that j \" 1, otherwise we may change the role of f i and g i , and p i and q i , respectively, and proceed as above.If p n \" q 1 , then we have\n1 N! ÿ σPS n f n px σp1q ¨¨¨x σpp n q q ¨gn px σpp n `1q ¨¨¨x σpNq q `1 N! ÿ σPS n g 1 px σp1q ¨¨¨x σpp n q q ¨f1 px σpp n `1q ¨¨¨x σpNq q \" ´1 N! ÿ σPS n n´1 ÿ i\"2\nf i px σp1q ¨¨¨x σpp i q q ¨gi px σpp i `1q ¨¨¨x σpNq q `x1 , . . ., x N P F ˆ˘.\n\nThis equation with the substitutions x p n `1 \" ¨¨¨\" x N \" 1, yields that a linear combination of f n and g 1 is decomposable.If t f n , g 1 u is linearly dependent, then this obviously means that both f n and g 1 are decomposable functions.If this system is not linearly dependent, then there exist a, b P F ˆ, a b and different complex constants c 1 and c 2 such that\nf n paq \" c 1 g 1 paq f n pbq \" c 2 g 1 pbq.\nWith the substitutions x p n `1 \" a, x p n `2 \" ¨¨¨\" x N \" 1, and\nx p n `1 \" b, x p n `2 \" ¨¨¨\" x N \" 1,\nwe get that the functions px 1 , . . ., x p 1 q Þ ÝÑ g 1 px 1 ¨¨¨x p 1 q f n paq `fn px 1 ¨¨¨x p 1 qg 1 paq and px 1 , . . ., x p 1 q Þ ÝÑ g 1 px 1 ¨¨¨x p 1 q f n pbq `fn px 1 ¨¨¨x p 1 qg 1 pbq are decomposable.Since finite linear combinations of decomposable functions are also decomposable, it follows that f n and g 1 are decomposable, separately.After that, let us consider the set S z tp n u and apply the above argument for this set.With this step-by-step descending argument the statement of the lemma follows.\n\nRemark 8. We emphasize that condition C(iii) in Lemma 8 has not been assumed.Thus, the fact that the additive solutions of (1) are decomposable functions can be deduced only under the conditions C(i) and C(ii).On the other hand, for our purpose to describe the solutions more concretely we have to assume also condition C(iii) to avoid further difficulties.We believe however that most of our methods can work similarly only under the conditions C(i) and C(ii).\n\nRemark 9. Note also that in general we cannot state more than that the involved functions f 1 , . . ., f n and g 1 , . . ., g n are decomposable functions on the commutative group F ˆ.In other words, we can only state that the solutions of the functional equation in question are higher order derivations (see below Corollary 2).In general it is not true that the solutions of this functional equation are differential operators (i.e., exponential polynomials of the multiplicative group F ˆ).To see this, let us consider the functional equation x f 1 px 6 q `x2 f 2 px 5 q `x3 f 3 px 4 q \" 0 px P Fq .\n\nIndeed, using the results of [5], we deduce that\nf 1 , f 2 , f 3 P D 2 pFq.\nDue to a result of Laczkovich [7] and under the assumptions of Lemma 8, there exist a positive integer l, there are generalized polynomials P k,i , Q k,i : F ˆÑ C (k \" 1, . . ., l and i \" 1, . . ., n) and there exist linearly independent exponentials m 1 , . . ., m l : F ˆÑ C such that\nf i pxq \" l ÿ k\"1 P k,i pxqm k pxq and g i pxq \" l ÿ k\"1 Q k,i pxqm k pxq `x P F ˆ˘.\nSince the generalized polynomials on any finitely generated field are polynomials and the exponentials m 1 , . . ., m k are linearly independent, we can apply Theorem 2. This implies that after substituting the above form the functions f 1 , . . ., f n and g 1 , . . ., g n and using that for each κ P N and k \" 1, . . ., n, we have\nm k px κ q \" m k pxq κ `x P F ˆ˘, especially n ÿ i\"1\nP k,i px p i qQ k,i px q i q \" 0 `x P F ˆfollows for all k \" 1, . . ., l.This tells us that it is enough to solve equation ( 1) for generalized polynomials of the group F ˆ.In fact, we can prove the following.\n\nTheorem 5. Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling conditions C(i)-C(iii).Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation (1) for each x P F. Then there exists a positive integer l, there exist exponentials m i : F ˆÑ C and there are generalized polynomials P i , Q i : F ˆÑ C of degree at most l such that f i pxq \" P i pxqm i pxq and g i pxq \" Q i pxqm i pxq `x P F ˆfor each i \" 1, . . ., n.\n\nProof.As we saw above, under the hypothesis of the lemma, if the functions f 1 , . . ., f n and g 1 , . . ., g n solve equation (1), then there exist a positive integer l, there are generalized polynomials P k,i , Q k,i : F ˆÑ C (k \" 1, . . ., l and i \" 1, . . ., n) and there exist linearly independent exponentials m 1 , . . ., m l : F ˆÑ C such that\nf i pxq \" l ÿ k\"1 P k,i pxqm k pxq and g i pxq \" l ÿ k\"1 Q k,i pxqm k pxq `x P F ˆ˘.\nAssume to the contrary that l ě 2. Let S \" tp 1 , . . ., p n u Y tq 1 , . . ., q n u .\n\nThen due to conditions C(i)-C(iii) we have max S \" max tp n , q 1 u.Similarly as in Lemma 8, without the loss of generality p n ą q 1 can be assumed, otherwise we follow a similar argument.By our assumption l ě 2, this yields that there exist different exponential terms in f 1 and g 1 with nonzero polynomial coefficients.For the sake of simplicity, suppose that these different exponentials are m 1 and m 2 .Since max S \" p 1 , the term m p 1 1 m q 1 2 appears only in f 1 px p 1 qg 1 px q 1 q while expanding equation (1).Since generalized polynomials P k,i and exponentials m k satisfies the conditions of Theorem 2 for every finitely generated subfield of F, the coefficient of the above-mentioned term which is P 1,1 px p 1 qQ 1,1 px q 1 q has to vanish on F. From this we can deduce that P 1,1 or Q 1,1 is identically zero, contrary to our assumption.Thus\nf 1 pxq \" P 1 pxqm 1 pxq and g 1 pxq \" Q 1 pxqm 1 pxq px P Fq ,\nwith appropriate generalized polynomials P 1 , Q 1 : F ˆÑ C and exponential m 1 :\nF ˆÑ C.\nSuppose now that there is a positive integer k, less than n such that for all i \" 1, . . ., k we have\nf i pxq \" P i pxqm i pxq and g i pxq \" Q i pxqm i pxq `x P F ˆ˘.\nAssume that in the representation of f k`1 and g k`1 there are different exponentials with nonzero polynomial coefficients, say m j 1 and m j 2 .Observe that while expanding equation ( 1), the term m p k`1 j 1 m q k`1 j 2 appears only at once, namely in the product f k`1 px p k`1 qg k`1 px q k`1 q.Again, due to Theorem 2, we deduce that the appropriate polynomial term, that is, P k`1, j 1 px p k`1 qQ k`1, j 2 px q k`1 q has to vanish.This proves that necessarily\nf k`1 pxq \" P k`1 pxqm k`1 pxq and g k`1 \" Q k`1 pxqm k`1 pxq `x P F ˆhold.\nThis shows that there exist exponentials m 1 , . . ., m n : F ˆÑ C and generalized polynomials P 1 , . . ., P n and Q 1 , . . ., Q n on the group F ˆsuch that for all i \" 1, . . ., n f i pxq \" P i pxqm i pxq and\ng i \" Q i pxqm i pxq `x P F ˆ˘.\nRemark 10.Due to conditions C(i)-C(iii), equation (1) has the form 0 \"\nn ÿ i\"1 f i px p i qg i px q i q \" n ÿ i\"1 P i px p i qm i pxq p i Q i px q i qm i pxq q i \" n ÿ i\"1 P i px p i qQ i px q i qm i pxq N `x P F ˆ˘.\nIf the exponentials appearing on the right hand side would be different, then by Theorem 2, their coefficients would be zero.This implies however that there exists a proper subset J Ă t1, . . ., nu such that ÿ jPJ f j px p j qg j px q j q \" 0 as well as ÿ j J f j px p j qg j px q j q \" 0 `x P F ˆ˘.\n\n(\n)2\nThis leads to the following definition of irreducible solutions.\n\nDefinition 10.A system of solutions t f 1 , . . ., f n , g 1 , . . ., g n u of equation ( 1) is called irreducible if it does not satisfy a sub-term of (1).Otherwise, we say that a system of solutions is reducible.\n\nClearly, a system of solutions t f 1 , . . ., f n , g 1 , . . ., g n u of (1) which fulfills ( 2) is a reducible solution.On the other hand, the argument in Remark 10 shows that every solution of (1) can be given as a sum of irreducible solutions of disjoint sub-terms of (1).Therefore, we restrict ourselves to the irreducible case, since every reducible solution can be deduced as a sum of irreducible solutions.\n\nCorollary 2. Under the conditions of Theorem 5, suppose that system of functions t f 1 , . . ., f n , g 1 , . . ., g n u is an irreducible solution of equation (1).Then there exists an exponential m : F ˆÑ C and there are generalized polynomials P i , Q i : F ˆÑ C such that for each i \" 1, . . ., n f i pxq \" P i pxqmpxq and\ng i pxq \" Q i pxqmpxq `x P F ˆ˘. (3)\nIn other words, for each i \" 1, . . ., n there exists higher order derivations D i , r\nD i : F Ñ C such that f i pxq \" D i pxq and g i pxq \" r D i pxq `x P F ˆ˘,\nwhere \" in the latter two equations is the equivalence relation defined in Remark 7.\n\nProof.By Remark 10, all of the exponentials m i have to be the same in the description of the solutions of Theorem 5. Therefore, equation (3) describes the irreducible solutions of (1).Using Lemma 5, solutions of equation ( 1) are enough to be determined up to the equivalence relation \" defined in Remark 7. Accordingly, we can suppose that the exponential m in the above representation is the identity mapping.Hence, in view of Theorem 4 we get that the functions f 1 , . . ., f n as well as g 1 , . . ., g n are (or more precisely, are equivalent to) higher order derivations, as we stated.\n\n\nThe order of higher order derivation solutions\n\nEvery higher order derivation on F is a differential operator on any finitely generated subfield of F (see Theorem 4 and [6]).Hence on these fields the solutions are differential operators.Moreover, if the solutions on any finitely generated subfield of F are differential operators of order at most n, then every solution on F is a derivation of order n.From now on, instead of finding solutions as higher order derivations we are looking for differential operators as solutions.\n\nFor this purpose, our next aim is to understand the arithmetic of the composition of derivations of the form d 1 ˝¨¨¨˝d r that are building blocks of differential operators.First we show that there is a natural form of composition of derivations that can be taken as a standard basis.\n\nFor this target, the notion of moment function sequences turn out to be useful.Here we follow [4].A composition of a nonnegative integer n is a sequence of nonnegative integers α \" pα k q kPN such that n \"\n8 ÿ k\"1 α k .\nFor a positive integer r, an r-composition of a nonnegative integer n is a composition α \" pα k q kPN with α k \" 0 for k ą r.\n\nGiven a sequence of variables x \" px k q kPN and compositions α \" pα k q kPN and β \" pβ k q kPN we define α! \"\n8 ź k\"1 α k , |α| \" 8 ÿ k\"1 α k , x α \" 8 ź k\"1 x α k k , ˆα β ˙\" 8 ź k\"1 ˆαk β k ˙.\nFurthermore, β ď α means that β k ď α k for all k P N and β ă α stands for β ď α and β α.\n\nDefinition 11.Let G be a commutative group, r a positive integer, and for each multi-index α in N r let f α : G Ñ C be a continuous function.We say that p f α q αPN r is a generalized moment sequence of rank r, if\nf α px `yq \" ÿ βďα ˆα β ˙fβ pxq f α´β pyq(4)\nholds whenever x, y are in G.The function f 0 , where 0 is the zero element in N r , is called the generating function of the sequence.\n\nTheorem 6.Let G be a commutative group, r a positive integer, and for each α in N r let f α : G Ñ C be a function.If the sequence of functions p f α q αPN r forms a generalized moment sequence of rank r, then there exists an exponential m : G Ñ C and a sequence of complex-valued additive functions a \" pa α q αPN r such that for every multi-index α in N r and x in G we have\nf α pxq \" B α papxqqmpxq,\nwhere B α denotes the multivariate Bell polynomial corresponding to the multi-index α.\n\nRemark 11.It is well-known that every polynomial P : C r Ñ C (r P N) can be given as a linear combination of Bell polynomials B α , where α P N r .Hence, momentum generating functions generate the exponential polynomial functions on G.By Lemma 2, polynomials of the form B α ˝a are linearly independent over C.\n\nLemma 9. Let F Ă C be a field, r be a positive integer and d 1 , . . ., d r : F Ñ F be linearly independent derivations.For all multi-index α P N r , α \" pα 1 , . . ., α r q define the function ϕ α : F Ñ C by\nϕ α pxq \" d α pxq \" d α 1 1 ˝¨¨¨˝d α r r pxq \" d 1 ˝¨¨¨˝d 1 looooomooooon α 1 times ˝¨¨¨˝d r ˝¨¨¨˝d r looooomooooon α r times pxq `x P F ˆ˘.\nThen pϕ α q αPN r is a generalized moment sequence of rank r on the commutative group F ˆand the generating function of the sequence is the identity function.\n\nProof.Let F Ă C be a field, r be a positive integer and d 1 , . . ., d r : F Ñ F be linearly independent derivations.For all multi-index α P N r , α \" pα 1 , . . ., α r q define the function ϕ α : F Ñ C as in the lemma.We prove the statement by induction of the length of the multi-index α.Assume that α P N r and |α| \" 1.Then there exists i P t1, . . .ru such that α i \" 1 and α j \" 0 for j i and ϕ α pxyq \" d α i i pxyq \" d i pxyq \" xd i pyq `yd i pxq \" ϕ 0 pxqϕ α pyq `ϕ0 pyqϕ α pxq \" ÿ βďα ˆα β ˙ϕβ pxqϕ α´β pyq `x, y P F ˆ˘.\n\nLet now α P N r and assume that the statement is true for all multi-indices β P N r with β ă α.Then ϕ α pxyq \" d α pxyq \" d α 1 1 ˝¨¨¨˝d α r r pxyq \" d 1 ˝¨¨¨d 1 loooomoooon Corollary 3. By Lemma 9 and Remark 11 implies that the functions d α \" d α 1 1 ˝¨¨¨˝d α r r constitute a basis of the differential operators in F ˆ. Since every d α is additive, by Lemma 2, all elements of the system td α pxqu, where α P Y rPN N r are algebraically independent.\n\nA consequence of the algebraic independence of the elements of d α , where α P Y rPN N r is the following.Let P P Crx 1 , . . ., x n s be a polynomial, d 1 , . . .d r be derivations as in Lemma 9 With the above results, we are now ready to prove an upper bound for the order of derivations appearing in Theorem 5.\n\nTheorem 7.Under the hypotheses of Theorem 5, the solutions f i and g i are derivations D i and r D i for all i \" 1, . . ., n.Let k and l denote the maximal orders of derivations D i and r D i , respectively.Suppose that there exists some i 1 such that the order of D i 1 and r D i 1 is exactly k and l, respectively.Then for all i \" 1, . . ., n the order of D i and r D i is less or equal to n ´1.\n\nProof.Assume contrary that the maximal order k of the above derivations D 1 , . . ., D n is greater than n ´1.\n\nThe argument for the case when the maximal order l of r D 1 , . . ., r D n is greater than n ´1 is analogous.By our assumption there exists an index i 1 such that the orders of the derivations D i 1 and r D i 1 is exactly k and l, respectively.It is important to note that then the sum of the orders of D i and r D i is at most k `l for any i, and it is exactly k `l for some indices if and only if the corresponding D and r D is of order k and of order l, respectively.Furthermore, by the algebraic independence of higher order derivations, there the number of these indices are at least two.\n\nFrom now on we assume that F is finitely generated.Indeed, if we verify the statement for any finitely generated subfield of a field F, then it holds for F itself, as well.On finitely generated fields all of these higher order derivations can be represented as differential operators, that is, on finitely generated fields we have\nD i pxq \" ÿ |α|ďk λ i,α d α i pxq and r D i pxq \" ÿ |β|ďl r λ i,β r d β i pxq px P Fq with appropriate complex constants λ i,α , r λ i,β , (|α| ď k, |β| ď l, i \" 1, .\n. ., n) and higher order derivations\nd α i , r d β i : F Ñ C defined in Lemma 9. Further, we have 0 \" n ÿ i\"1 D i px p i q r D i px q i q \" n ÿ i\"1 ˜ÿ |α|ďk λ i,α d α i px p i q ¸¨˜ÿ |β|ďl r λ i,β r d β i px q i q ¸px P Fq .\nIf we expand the right hand side of the above identity with the aid of Proposition If this identity can be satisfied by different functions, it can also be satisfied by a single one.By Corollary 3, this enables us to substitute the functions d α i , r d β i pi \" 1, . . ., n, |α| ď k, |β| ď lq with suitable compositions of a given derivation d of order 1.In other words, instead of the above identity, we can restrict ourselves to n ÿ i\"1 ˜k ÿ j\"0 λ i, j d j px p i q ¸¨˜l ÿ j\"0 r λ i, j d j px q i q ¸\" 0 px P Fq with appropriate complex constants λ i, j (i \" 1, . . ., n, j \" 0, . . ., k), and r λ i, j (i \" 1, . . ., n, j \" 0, . . ., l) and derivation d : F Ñ C. By our assumptions there are some i 1 such that λ i 1 ,k 0 and r λ i 1 ,l 0. Dividing the above sum to smaller ones, we get\nn ÿ i\"1 ˜k ÿ j\"0 λ i, j d j px p i q ¸¨˜l ÿ j\"0 r λ i, j d j px q i q \" n ÿ i\"1 ˜k´1 ÿ j\"0 λ i, j d j px p i q `λi,k d k px p i q ¸¨˜k ´1 ÿ j\"0 r λ i, j d j px q i q `r λ i,l d l px q i q ņ ÿ i\"1 \" S pp i , k ´1qS pq i , l ´1q `r λ i,l d l px q i qS pp i , k ´1q `λi,k d k px p i qS pq i , l ´1q `λi,k r λ i,k d k px p i qd l px q i q ı \" 0 px P Fq ,\nwhere\nS pp i , k ´1q \" k´1 ÿ j\"0\nλ i, j d j px p i q and S pq i , l ´1q \" l´1 ÿ j\"0 r λ i, j d j px q i q px P Gq .\n\nNote that, by the algebraic independence used in Corollary 3, this sum splits into separate terms of the form d s px p i qd t px q i q, where s `t is a fixed number.By our assumption, when s `t \" k `l, then the only way is that s \" k, t \" l.This implies that using Proposition 1 we get that 0 \"\nn ÿ i\"1 λ i,k r λ i,l d k px p i qd l px q i q \" n ÿ i\"1 λ i,k r λ i,k ÿ j 1 `¨¨¨`j s \"p 1 ăp i j 1 `2 j 2 `¨¨¨`s j s \"k ˆk 1, . . . , 1 loomoon j 1 , . . . , s, . . . , s loomoon j s ˙ˆp i 1, . . . , 1 loomoon p 1 ˙s ź t\"1 pd t pxqq j t p j t !q x p i ´p1 ‹ ‹ ‹ ‚ ÿ j 1 `¨¨¨`j s \"q 1 ăq i j 1 `2 j 2 `¨¨¨`s j s \"l ˆl 1, . . . , 1 loomoon j 1\n, . . ., s, . . ., s loomoon\nj s ˙ˆq i 1, . . . , 1 loomoon q 1 ˙s ź t\"1 pd t pxqq j t p j t !q x q i ´q1 ‹ ‹ ‹ ‚\nwhile the rest in the above sum can be computed similarly.\n\nCase 1.If k ă l, then we compute the coefficients of the terms pdpxqq j d k´j pxqd l pxq p j \" 0, . . ., k ´1q.\n\nFor each j \" 0, . . ., k ´1 this can be taken from the expansion of λ i,k r λ i,l d k px p i qd l px q i q in only one way.Namely, by splitting d k px p i q into j `1 parts and d l px q i q into one.Then the corresponding coefficients are\nn ÿ i\"1 λ i,k r λ i,l ˆk 1, . . . , 1 loomoon j ˙1 j! ˆpi 1, . . . , 1 loomoon j`1 ˙ˆq i 1 ˙\" 0.\nSince each of the terms contains ˆk 1, . . ., 1 loomoon j ˙1 j! \" ˆkj ˙, this can be eliminated from the above equation.The corresponding equations ( j \" 0, . . ., k) can be written in the following matrix equation\n¨`p 1 1 ˘`q 1 1 ˘. . . `pn 1 ˘`q n 1 p1 1,1 ˘`q 1 1 ˘. . . `pn 1,1 ˘`q n 1 .. . . . . . . . `p1 1, . . . , 1 loomoon k ˘`q 1 1 ˘. . . `pn 1, . . . , 1 loomoon k ˘`q n 1 ˘‹ ‹ ‹ ‹ ‹ ‚ ¨¨λ 1,k r λ 1,l λ 2,k r λ 2,l . . . λ n,k r λ n,l ‹ ‹ ‹ ‚ \" ¨0 0 . . . 0 ‹ ‹ ‹ ‚ .\nHere we note that as p i 's are all different positive integers, it follows that p i ě n for some i and hence the first n rows of the matrix are not identically zero, as k ą n ´1.It is straightforward to verify that the first n row of above matrix equation is equivalent to ¨p1 . . .p n p 2 1 . . .p 2 n . . . . . . . . .\np n 1 . . . p n n ‹ ‹ ‹ ‚ ¨¨q 1 λ 1,k r λ 1,l q 2 λ 2,k r λ 2,l\n. . .\nq n λ n,k r λ n,l ‹ ‹ ‹ ‚ \" ¨0 0 . . . 0 ‹ ‹ ‹ ‚ .\nSince this is a Vandermonde type matrix with different p i 's, the only solution of this homogeneous linear system is the zero vector, i.e., q i λ i,k r λ i,l \" 0 for all i \" 1, . . ., n.This contradicts to our assumption that there is some i 1 for which λ i 1 ,k 0 and r λ i 1 ,l 0 (q i 1 0 as q i 0 for all i P t1, . . ., ku).\n\nCase 2. If k \" l, then we compute the coefficients of the terms pdpxqq 2 j pd k´j pxqq 2 p j \" 0, . . ., k ´1q.\n\nIf j ă k 2 , then this term can be taken from the expansion of λ i,k r λ i,l d k px p i qd k px q i q in only one way.Namely, by splitting d k px p i q and d k px q i q into j `1 parts.Then the corresponding coefficients are\nn ÿ i\"1 λ i,k r λ i,l ¨ˆk 1, . . . , 1 loomoon j ˙1 j! ‹ ‹ ‹ ‚ 2 ˆpi 1, . . . , 1 loomoon j`1 ˙ˆq i 1, . . . , 1 loomoon j`1 ˙\" 0.\nSince each of the terms contains ´`k 1, . . ., 1 loomoon j ˘1 j! ¯2 \" `kj ˘2, this can be eliminated from the above equations.\n\nThese equations for j \" 0, . . ., rk{2s ´1 can be written in the following matrix equation\n¨`p 1 1 ˘`q 1 1 ˘. . . `pn 1 ˘`q n 1 p1 1,1 ˘`q 1 1,1 ˘. . . `pn 1,1 ˘`q n 1,1 .. . . . . . . . `p1 1, . . . , 1 loomoon rk{2s´1 ˘`q 1 1, . . . , 1 loomoon rk{2s´1 ˘. . . `pn 1, . . . , 1 loomoon rk{2s´1 ˘`q n 1, . . . , 1 loomoon rk{2s´1 ˘‹ ‹ ‹ ‹ ‹ ‹ ‚ ¨¨λ 1,k r λ 1,k λ 2,k r λ 2,k . . . λ n,k r λ n,k ‹ ‹ ‹ ‚ \" ¨0 0 . . . 0 ‹ ‹ ‹ ‚ .\nHere we note that as p i , q i are all different positive integers, it follow that max i tp i , q i u ě 2n and hence the first n 1 \" minpn, rk{2s ´1q rows of the matrix in not identically zero, as k ą n ´1.It is straightforward to verify that the first n 1 row of above matrix equation is equivalent to ¨p1 q 1 . . .p n q n p 2 1 q 2 1 . . .p 2 n q 2 n . . . . . . . . .\np n 1 1 q n 1 1 . . . p n 1 n q n 1 n ‹ ‹ ‹ ‚ ¨¨λ 1,k r λ 1,k λ 2,k r λ 2,l . . . λ n,k r λ n,k ‹ ‹ ‹ ‚ \" ¨0 0 . . . 0 ‹ ‹ ‹ ‚ .\nNote that if n 1 \" n, then we are done with a Vandermonde matrix argument similar as it is used in Case 1.\n\nSo from now on we assume that n 1 \" rk{2s ´1.This also means that k ă 2n, thus the maximal order of the corresponding derivations is at most 2n ´1.\n\nIf j ě k 2 , then the term pdpxqq 2 j pd k´j pxqq 2 can be taken from the expansion of λ i,k r λ i,l d k px p i qd k px q i q in three ways.One is as above, when we split both d k px p i q and d k px q i q into j `1 parts.Another one is when d k px p i q is split into k parts giving dpxq k , and d k px q i q provides pd k´j pxqq 2 pdpxqq 2 j´k .The third one is given by changing the role of p i and q i .Then the corresponding coefficients are\nn ÿ i\"1 λ i,k r λ i,k ´ˆk 1, . . . , 1 loomoon j ˙1 p j!q ¯2ˆp i 1, . . . , 1 loomoon j ˙ˆq i 1, . . . , 1 loomoon j ˙ǹ ÿ i\"1 λ i,k r λ i,k ˆk 1, . . . , 1 loomoon k ˙1 pk!q ˆk 1, . . . , 1 loomoon 2 j´k , j ˙1 2 1 p2 j ´kq! pi 1, . . . , 1 loomoon k ˙ˆq i 1, . . . , 1 loomoon 2 j´k`2 ˙`ˆp i 1, . . . , 1 loomoon 2 j´k`2 ˙ˆq i 1, . . . , 1 loomoon k ˙¯\" 0.\nFirst we show that the second sum has to vanish for all j \" rk{2s, . . ., k ´1.In such cases, the coefficients ˆk 1, . . ., 1\nloomoon k ˙1 k! ˆk 1, . . . ,\nThis clearly holds, since the term pdpxqq 2 j`1 d 2k´2 j´1 pxq in the expansion of d k px p i qd k px q i q for j \" rk{2s, . . ., k1 can be given in exactly two ways.Either pdpxqq k stems from d k px p i q and pdpxqq 2 j´k`1 ¨d2k´2j´1 pxq stems from d k px q i q, or reversely changing the role of p i and q i .Hence we get ˘`q 1 1 ˘. . .\nn ÿ i\"1 λ i,k r λ i,k ˆk 1, . . . ,`pn 1 ˘`q n 1 p1 1,1 ˘`q 1 1,1 ˘. . . `pn 1,1 ˘`q n 1,1 .. . . . . . . . `p1 1, . . . , 1 loomoon k ˘`q 1 1, . . . , 1 loomoon k ˘. . . `pn 1, . . . , 1 loomoon k ˘`q n 1, . . . , 1 loomoon k ˘‹ ‹ ‹ ‹ ‹ ‚ ¨¨λ 1,k r λ 1,k λ 2,k r λ 2,k . . . λ n,k r λ n,k ‹ ‹ ‹ ‚ \" ¨0 0 . . . 0 ‹ ‹ ‹ ‚ ,\nwhere the first n rows of the matrix is not identically zero as p i and q i are different, hence there is an index i 1 such that p i 1 ě n and q i 1 ě n.Thus this system consisting the first n rows is equivalent to ¨p1 q 1 . . .p n q n p 2 1 q 2 1 . . .p 2 n q 2 n . . . . . . . . .\np n 1 q n 1 . . . p n n q n n ‹ ‹ ‹ ‚ ¨¨λ 1,k r λ 1,k λ 2,k r λ 2,l . . . λ n,k r λ n,k ‹ ‹ ‹ ‚ \" ¨0 0 . . . 0 ‹ ‹ ‹ ‚ .\nBy the usual Vandermonde argument as in Case 1, the only solution of this homogeneous linear system is the zero vector, i.e., λ i,k r λ i,l \" 0 for all i \" 1, . . ., n.This contradicts our assumption that there is some i 1 for which λ i 1 ,k 0 and r λ i 1 ,k 0.\n\nCase 3. If k ą l, then we prove by induction in j that n ÿ i\"1 λ i,k r λ i,l p j`1´s i q s i \" 0 holds for every j \" 0, . . ., n ´1 and every s \" 0, . . ., j.\n\nIn each step we consider how pdpxqq j d k´j pxqd l pxq can be given from the expansion d k px p i qd l px q i q.There are three possible ways, where this term can stem from.\n\n(a) pdpxqq j d k´j pxq stems from d k px p i q and d l pxq stems from d l px q i q.This can happen for every j P t0, . . ., ku.\n\nIn this case the coefficient of pdpxqq j d k´j pxqd l pxq is\nn ÿ i\"1 λ i,k r λ i,l ˆk 1, . . . , 1 loomoon j ˙1 j! ˆpi 1, . . . , 1 loomoon j`1 ˙ˆq i 1 ˙\" n ÿ i\"1 λ i,k r λ i,l ˆkj ˙ˆp i 1, . . . , 1 loomoon j`1\n˙qi .\n\n(b) d k´j pxqd l pxqpdpxqq j´l stems from d k px p i q and pdpxqq l stems from d l px q i q.This can happen if j ě l.In this case the coefficient of pdpxqq j d k´j pxqd l pxq is (c) d l pxqpdpxqq k´l stems from d k px p i q and pdpxqq l´pk´jq d k´j pxq stems from d l px q i q.This can happen if l ě k´j.\nn ÿ i\"1 λ i,k r λ i,l ˆk k ´j,\nIn this case the coefficient of pdpxqq j d k´j pxqd l pxq is ˙.\nn ÿ i\"1 λ i,k r λ i,l ˆk k ´l,1\nFor j \" 0 (and hence s \" 0) only the first term takes into account.This means that\nn ÿ i\"1 λ i,k r λ i,l p i q i \" 0.\nSo the inductive hypothesis holds for j \" 0. Now we assume that n ÿ i\"1 λ i,k r λ i,l p j 1 `1´s i q s i \" 0 holds for every j 1 \" 0, . . ., j ´1 and every s \" 0, . . ., j 1 .We prove that n ÿ i\"1 λ i,k r λ i,l p j`1´s i q s i \" 0 holds for every s \" 0, . . ., j, as well.Generally, some of the previous compositions are possible for a given j but the following argument works in all cases, however we just prove it when all compositions discussed below appear in the expansion.Thus we assume that the coefficient of pdpxqq j d k´j pxqd l pxq is 0 \" By the inductive hypothesis and the fact that j ă n ď min p 1 i , q 1 i ( for some i 1 , this is equivalent to 0 \"\nn ÿ i\"1 λ i,k r λ i,l » - - - -ˆkj ˙ˆp i 1, . . . ,n ÿ i\"1 λ i,k r λ i,l ˆˆk j ˙p j`1 i q i `ˆk k ´j, j ´l, l ˙p j´l`2 i q l i `ˆk k ´l, k ´j, l ´pk ´jq ˙pl`1 i q l´pk´jq`1 i ˙.\nNote that the expressions p j`1´s i q s i for s \" 0, . . ., j can be interchanged in the following sense.By C(ii), p i `qi \" N, thus we have that p j`1´s i q s i \" N p j´s i q s i ´p j´s i q s`1 i .\n\nAs\nn ÿ i\"1\nλ i,k r λ i,l p j´s i q s i \" 0 for every s \" 0, . . ., j ´1 by the inductive hypothesis, the term N p j´s i q s i can be eliminated.After several repetition of this step we get that 0 \"\nn ÿ i\"1 λ i,k r λ i,l ˆˆk j ˙p j`1 i q i ˘ˆk k ´j, j ´l, l ˙p j`1 i q i ˘ˆk k ´l, k ´j, l ´pk ´jq ˙p j`1 i q i ˙.\nWe also note that using this interchange rule and the inductive hypothesis it is clear that n ÿ i\"1 λ i,k r λ i,l p j`1´s i q s i \" 0 for any s \" 0, . . ., j is equivalent to\nn ÿ i\"1 λ i,k r λ i,l p j`1 i q i \" 0.\nTherefore, to finish the proof it is enough to show that ˆkj ˙˘ˆk\nk ´j, j ´l, l ˙˘ˆk k ´l, k ´j, l ´pk ´jq ˙ 0.\nThis is equivalent to verify that 1 j! ˘1 p j ´lq!pl!q ˘1 pk ´lq!pl ´pk ´jq!q 0.\n\nMultiplying by j! this lead to\n1 ˘ˆj l ˙˘ˆj k ´l˙ 0.\nIt is straightforward to show using the growth of `n k ˘in k (if k ď n{2) that one term dominates the others if l k ´l and l j ´pk ´lq.Thus in these cases this (weighted) sum is nonzero.If l \" k ´l or l \" j ´pk ´lq, then either their sign is the same and hence the sum is nonzero, or their sign is different and hence they eliminate each other, hence the sum is 1 which is nonzero.\n\nSummarizing, we get that ˆkj ˙˘ˆk k ´j, j ´l, l ˙˘ˆk k ´l, k ´j, l ´pk ´jq ˙ 0 and hence\nn ÿ i\"1 λ i,k r λ i,l p j`1 i q i \" 0,(8)\nwhich is equivalent to the inductive hypothesis for j as we noted above.\n\nThus equation (8) holds for every j \" 0, . . ., n ´1.In matrix form this means that\n¨1 . . . 1 p 1 . . . p n . . . . . . . . . p n´1 1 . . . p n´1 n ‹ ‹ ‹ ‚ ¨¨p 1 q 1 λ 1,k r λ 1,k p 2 q 2 λ 2,k r λ 2,l . . . p n q n λ n,k r λ n,k ‹ ‹ ‹ ‚ \" ¨0 0 . . . 0 ‹ ‹ ‹ ‚ .\nSince this matrix is a Vandermonde type matrix with different p i 's, as Case 1 and Case 2, this implies that, the only solution of this homogeneous linear system is the zero vector, i.e., p i q i λ i,k r λ i,l \" 0 for all i \" 1, . . ., n.This contradicts to our assumption that there is some i 1 for which λ i 1 ,k 0 and r λ i 1 ,l 0 (and p i 0, q i 0 by C(i)).This also finishes the proof of the theorem, thus the order of all derivations involved in (1) is at most n ´1.\n\nRemark 12.The upper bound appearing in the above theorem is sharp.To see this, let n, N be positive integers, λ 1 , . . ., λ n P C and let f : F Ñ C be an additive function for which n ÿ i\"1 λ i f px i qx N´i \" 0 is fulfilled for all x P F. Then f P D n´1 pRq if and only if λ i \" p´1q i ˆn i ˙for all i \" 1, . . ., n, see [1,3,5].\n\nRemark 13.The proof of Theorem 7 in all the cases is based on the fact that the matrix\n¨`p 1 1 ˘`q 1 1 ˘. . . `pn 1 ˘`q n 1 p1 1,1 ˘`q 1 1 ˘``p 1 1 ˘`q 1,1 1 ˘. . . `pn 1 ˘`q n 1,1 ˘``p n 1,1 ˘`q n 1 .. . . . . . . . `p1 1, . . . , 1 loomoon n´1 ˘`q 1 1, . . . , 1 loomoon n´1 ˘. . . `pn 1, . . . , 1 loomoon n´1 ˘`q n 1, . . . , 1 loomoon n´1 ˘‹ ‹ ‹ ‹ ‹ ‚\nhas rank n, though it is far from being trivial to find the proper sub-matrix, which verifies that.On the other hand, the situation is much more complicated, if the maximal order k of D i , and the maximal order l of r D i is not uniquely determined.Namely, if there are several different pairs pk i , l i q so that k i `li \" K, where K is constant and k i is the maximum order of D i , l i is the maximal order of r D i .Then the equations first can only be determined for subsets of the index set, which satisfy some nontrivial relations.In this case the first task is to show that the problem can be formalized separately for the index sets, which seems a very hard problem in full generality.In this case our method can be applied.Theorem 7 and Remark 13 motivates our conjecture, that we verified for n ď 4.\n\nConjecture 1.Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling conditions C(i)-C(iii).Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy equation (1).Then every solution is a generalized exponential polynomial function of degree at most n ´1 on F ˆ.\n\nIn particular, if f i pxq \" D i pxq and g i pxq \" r D i pxq `x P F ˆ˘ (9) for each i \" 1, . . ., n, then the order of D i , r D i is at most n ´1.\n\nThis conjecture leads to the following more general open question.\n\nOpen Question 1.Is it true that every nonzero additive, irreducible solutions f 1 , . . ., f n of Pp f 1 px p 1 q, . . ., f n px p n qq \" 0, with Pp0, . . ., 0q \" 0 are derivations of order at most n ´1 and the identity function up to a homomorphism, if P : C Ñ C is polynomial and p 1 . . ., p n are distinct positive integers?\n\nFinally we highlight some important special cases when Theorem 7 gives the proper bound of the order of the derivations.Corollary 4. Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling conditions C(i)-C(iii).Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy equation (1) as an irreducible solution.Then f i \" D i and g i \" r D i , where D i and r D i are higher order derivations.Assume further that one of the following holds.(A) All D i have the same order.This is the case, when f i pxq \" c i f pxq px P Fq for some nonzero constants c i P C, i P t1, . . ., nu.\n\n(B) All r D i have the same order.This is the case, when g i pxq \" c i gpxq px P Fq for some nonzero constants c i P C, i P t1, . . ., nu.\n\n(C) f i \" c i ¨gi for all i P t1, . . ., nu with some nonzero constants c i P C, i \" 1, . . ., n.\n\nThen the order D i and r D i is at most n ´1.\n\nProof.Theorem 5 implies that every solution f i (resp.g i ) is an exponential polynomial of the form P i ¨m (resp.Q i ¨m), which means that f i \" D i and g i \" r D i for some derivations D i and r D i .\n\n(A) Let k denote the order of D i , and let l be the maximal order of r D i .Now we are in the position to apply Theorem 7.\n\n(B) Similar to (A), since in case of equation ( 1), the role of the parameters p 1 , . . ., p n and q 1 , . . ., q n is symmetric.\n\n(C) As the maximal degree of f i is the same as the maximal degree of g i and it is taken for the same index we can apply Theorem 7.\n\n\nSpecial cases of equation (1)\n\nIn this subsection we consider special cases of equation (1).All the equations we consider here are of the form f 1 px p 1 qg 1 px q 1 q `f2 px p 2 qg 2 px q 2 q \" 0 px P Fq .\n\nHere f 1 , f 2 , g 1 , g 2 : F Ñ C denote the unknown additive functions and the parameters p 1 , p 2 , q 1 , q 2 fulfill conditions C(i)-C(iii).Due to the results of the previous section, we get that f i pxq \" λ i,0 x `λi,1 d i pxq g i pxq \" µ i,0 x `µi,1 r d i pxq px P F, i \" 1, 2q with appropriate complex constants λ i, j , µ i, j (i \" 1, 2, j \" 0, 1) and derivations d i , di : F Ñ C (i \" 1, 2).\n\nCorollary 5. Let N be a positive integer, F Ă C be a field and p, q be different positive integers (strictly) less than N and assume that q N ´p.If the additive functions f, g : F Ñ C satisfy f px p q f px N´p q \" gpx q qgpx N´q q px P Fq , Corollary 6.Let N be a positive integer, F Ă C be a field and p, q be different positive integers (strictly) less than N and assume that q N ´p.If the additive functions f, g : F Ñ C satisfy f px p qgpx N´p q \" κ f px q qgpx N´q q px P Fq .Both results implies that the nonzero additive solutions of equation f px p q f px N´p q \" κ f px q q f px N´q q px P Fq .\n\n\nThen\n\nare derivations of order 1 (κ \" ppN ´pq qpN ´qq ) or the identity function (κ \" 1) up to a homomorphism.\n\nα 1 times ˝¨¨¨˝d r ˝¨¨¨˝d r looooomooooon α r times pxyq \" d 1 ˝¨¨¨˝d 1 looooomooooon α 1 times ˝¨¨¨˝d r´1 ˝¨¨¨˝d r´1 loooooooomoooooooon α r´1 times pd α r r pxyqq \" d 1 ˝¨¨¨˝d 1 looooomooooon α 1 times ˝¨¨¨˝d r´1 ˝¨¨¨˝d r´1 loooooooomoooooooon α r´1 times 1 1\n1\n˝¨¨¨˝d β r r pxq ¨dα 1 ´β1 1 ˝¨¨¨˝d α r ´βr r pyq \" ÿ βďα ˆα β ˙ϕβ pxqϕ α´β pyq `x, y P F ˆ˘.\n\n\n\n\nthen A) either there exists a homomorphism ϕ : C Ñ C, a derivation d : F Ñ C such that f pxq \" ϕpdpxqq and gpxq \" αϕpdpxqq px P Fq , where α \" ppN ´pq qpN ´qq , B) or there exists a homomorphism ϕ : F Ñ C such that f pxq \" f p1q ¨ϕpxq gpxq \" ˘f p1q ¨ϕpxq px P Fq .\n\n\n\n\nthat is, the notions of first order derivations and derivations coincide.On the other hand for any n P N the set D n pFqzD n´1 pFq is nonempty because d 1 ˝¨¨¨˝d n P D n pFq, but d 1 ˝¨¨¨˝d n D n´1 pRq, where d 1 , . . ., d n P D 1 pFq are non-identically zero derivations.\nRemark 3. Since D 0 pFq \" t0u, the only bi-derivation of order zero is the identically zero function, thusf P D 1 pFq if and only iff pxyq \" x f pyq `f pxqypx, y P Fq ,\n\n\n\noperator of order at most n if D is the linear combination, with coefficients from F, of finitely many maps of the form d 1 ˝¨¨¨˝d k , where d 1 , . . ., d k are derivations on F and k ď n.If k \" 0 then we interpret d 1 ˝¨¨¨˝d k as the identity function.We denote by O n pFq the set of differential operators of order at most n defined on F. We say that the order of a differential operator D is n if D P O n pFqzO n´1 pFq (where O ´1pFq \" H, by definition).\nRemark 4. The term differential operator is justified by the following fact. Let K \" Qpt 1 , . . . , t k q, wheret 1 , . . . , t k are algebraically independent over Q. Then K is the field of all rational functions of t 1 , . . . , t k withrational coefficients. It is clear thatd i \"B Bt i\n\n\n\nand α 1 , . . ., α n P Y rPN N r .Then the following polynomial form Ppd α 1 pxq, . . ., d α ,...,l p ě0 l 1 `¨¨¨`l p \"k ˆk l 1 , . . ., l p ˙¨d l 1 pxq ¨¨¨d l p pxq px 1 , . . ., x p P Fq ,\ndefine the are adopted. Especially, for all positive integer px P Fq . l 1 Reordering the previous expression we can get the following function d k on F by d k pxq \" d ˝¨¨¨˝d loooomoooon k times pxq Then for all positive integer p we have ˆk l 1 , . . . , l p ˙\" k! l 1 ! ¨¨¨l p ! p, we have d k px p q \" ÿ p 1 d where the conventions d 0 \" id and d k px p q \" ÿ ˆk 1, . . . , 1 loomoon , . . . , s, . . . , s loomoon ˙s ź t\"1 1 p p j t !q j 1 `¨¨¨`j s \"p 1 ăp j 1 `2 j 2 `¨¨¨`s j s \"k j 1 j s 1, . . . , 1 loomoon ˙¨pdpxqq j 1 ¨¨¨pd s pxqq j s ¨xp´p 1 , px P Fq\nn pxqq \" 0 px P Gq holds if and only if Pp d |α 1 | pxq, . . ., d |α n | pxqq \" 0 px P Gq , where d is an arbitrary derivation (of order 1).In other words, we can substitute d 1 , . . ., d r by d in d α 1 , . . ., d α n .By Corollary 3, it would be desirable to calculate d k \" d ˝¨¨¨˝d loooomoooon k times , where d is a derivation (of order 1) and k P N. Lemma 9, together with [4, Proposition 1], implies the following statement.Proposition 1.Let F Ă C be a field and d : F Ñ C a derivation.For all positive integer k we k px 1 ¨¨¨x p q \" ÿ l 1 ,...,l p ě0 l 1 `¨¨¨`l p \"k ˆk l 1 , . . ., l p ˙p ź t\"1 d l t px t q px 1 , . . ., x p P Fq , where j 1 , . . ., j s denotes the number of dpxq, . . ., d s pxq in a given composition of d k px p q.\n\n\n\n\nl, 1, . . ., 1\nloomoon ˙1 p j ´lq! ˆl 1, . . . , 1 loomoon ˙1 l!1, . . . , 1 loomoon ˆpi ˙ˆq i 1, . . . , 1 loomoon \"n ÿ i\"1λ i,k r λ i,lj´l ˆk k ´j, j ´l, l ˙ˆp i 1, . . . , 1 loomoon ˙ˆq i l 1, . . . , 1 loomoonj´l`2lj´l`2\nl ˙.\n\n\n\n\nppN ´pq qpN ´qq , then there exists a homomorphism ϕ : C Ñ C and derivations d 1 , d 2 : F Ñ C such that f pxq \" ϕpd 1 pxqq and gpxq \" ϕpd 2 pxqq px P Fq .\nA) if κ\"1,ppN ´pq qpN ´qq*, then f is identically zero,\nB) if κ \" 1, then the only possibility is that f pxq \" f p1q ¨ψpxq and gpxq \" f p1q ¨ψpxq px P Fq , where ψ :F Ñ C is a homomorphism, C) if κ \"\n\nAcknowledgment.The research of E. Gselmann has been supported by project no.K134191 that has been implemented with the support provided by the National Research, Development and Innovation Fund of Hungary, financed under the K 20 funding scheme.The research of G. Kiss has been supported by projects no.K124749 and no.K142993 of the National Research, Development and Innovation Fund of Hungary.The author have supported by Bolyai János Research Fellowship of the Hungarian Academy of Sciences and ÚNKP-22-5 New National Excellence Program of the Ministry for Culture and Innovation.\nCharacterizing ring derivations of all orders via functional equations: results and open problems. Bruce Ebanks, Aequationes Math. 8932015\n\nPolynomially linked additive functions. Bruce Ebanks, Aequationes Math. 9122017\n\nOn the order of a derivation. Bruce Ebanks, Thomas Riedel, Prasanna K Sahoo, Aequationes Math. 9022016\n\n. Żywilla Fechner, Eszter Gselmann, László Székelyhidi, Moment functions on groups. Results Math. 7642021\n\nOn functional equations characterizing derivations: methods and examples. Gergely Eszter Gselmann, Csaba Kiss, Vincze, Paper No. 74. 201873\n\nDerivations and differential operators on rings and fields. Gergely Kiss, Miklós Laczkovich, Ann. Univ. Sci. Budapest. Sect. Comput. 482018\n\nA characterization of generalized exponential polynomials in terms of decomposable functions. Miklós Laczkovich, Acta Math. Hungar. 15822019\n\nDerivationen zweiter Ordnung als Lösungen von Funktionalgleichungen-ein Überblick. Ludwig Reich, Gyula Maurer zum 70. Graz1998337Karl-Franzens-Univ. Graz\n\nDecomposable functions and representations of topological semigroups. Ekaterina Shulman, Aequationes Math. 791-22010\n\nLászló Székelyhidi, Convolution type functional equations on topological abelian groups. Teaneck, NJWorld Scientific Publishing Co., Inc1991\n\nDerivationen höherer Ordnung als Lösungen von Funktionalgleichungen. Josef Unger, Ludwig Reich, Grazer Mathematische Berichte. 336Graz Mathematical Reports\n\n. Karl-Franzens , 1998GrazUniversität Graz\n"},"corpus_id":{"kind":"number","value":253384750,"string":"253,384,750"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2211.03605\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1007/s00025-023-02087-y\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/2211.03605v2.pdf\"\n ],\n \"pdf_sha\": \"64840f61a1a64b4b36a0646a7121a2e7ae8b7088\",\n \"open_access\": {\n \"license\": \"CCBY\",\n \"url\": \"https://link.springer.com/content/pdf/10.1007/s00025-023-02087-y.pdf\",\n \"status\": \"HYBRID\"\n }\n}"},"title":{"kind":"string","value":"Polynomial equations for additive functions I. The inner parameter case"},"authors":{"kind":"list like","value":["Eszter Gselmann gselmann@science.unideb.hu \nUniversity of Debrecen","Gergely Kiss \nUniversity of Debrecen"],"string":"[\n \"Eszter Gselmann gselmann@science.unideb.hu \\nUniversity of Debrecen\",\n \"Gergely Kiss \\nUniversity of Debrecen\"\n]","isConversational":false},"abstract":{"kind":"string","value":"The aim of this sequence of work is to investigate polynomial equations satisfied by additive functions.As a result of this, new characterization theorems for homomorphisms and derivations can be given.More exactly, in this paper the following type of equation is consideredwhere n is a positive integer, F Ă C is a field, f i , g i : F Ñ C are additive functions and p i , q i are positive integers for all i \" 1, . . ., n."},"sections":{"kind":"list like","value":[{"title":"Introduction and preliminaries","content":"Equations satisfied by additive functions play an important role not only in the theory of commutative algebra, but also in the theory of functional equations.It is an important and challenging question how special morphisms (such as homomorphisms and derivations) can be characterized among additive mappings in general.In this paper classes of multivariable algebraic equations are introduced with appropriate solutions as field homomorphisms and derivations.\nConcerning all the cases we consider here, the involved additive functions are defined on a field F Ă C and have values in the complex field, therefore we introduce the preliminaries in this setting.\nWe adopt the standard notations, that is, N and C denote the set of positive integers and the set of complex numbers, respectively.\nHenceforth we assume F Ă C to be a field.\nDefinition 1.We say that a function f : F Ñ C is additive if it fulfills the so-called Cauchy functional equation, that is, f px `yq \" f pxq `f pyq px, y P Fq .\nAn additive function d : F Ñ C is termed to be a derivation (of order 1) if it also fulfills the Leibniz equation, i.e., dpxyq \" dpxqy `xdpyq px, y P Fq .\nAn additive function ϕ : F Ñ C is said to be a homomorphism if it is multiplicative as well, in other words, besides additivity we also have ϕpxyq \" ϕpxqϕpyq px, y P Fq .\nIf F \" C and ϕ is an isomorphism, then ϕ is called a complex automorphism.\nCertain well-known equations are especially important.For instance, the additive solutions of following equation on a ring R with charR 2 f px 2 q \" 2x f pxq px P q are derivations, under some assumptions, where the additive mapping f acts.As an extension of such type of results in [1,2,5] the additive solutions of the equations n ÿ i\"0\nx i f n`1´i px n`1´i q \" 0 px P Rq and n ÿ i\"0 f px p i qx q i \" 0 px P Fq were described, here R denotes a ring with charpRq ě n, while F Ă C is a field.By a polynomial equation of additive functions we mean an equation of the form Pp f r 1 1 px s 1 q, . . ., f r n n px s n qq \" 0, where P : C n Ñ C is a n-variable polynomial, r i , s i are positive integers and f i denote the unknown additive functions.Without further restrictions (e.g., on the polynomial P or on the parameters r i , s i ), the above equation is unfortunately too general for its solutions to be fully determined.Indeed, it is not too hard to specify a polynomial P and parameters r i , s i so that the above equation is satisfied by all additive functions.Therefore, in this series, we will focus on the following classes of equations.\nf i px p i qg i px q i q \" 0, n ÿ i\"1 f i px p i qg i pxq q i \" 0, n ÿ i\"1 f i pxq p i g i pxq q i \" 0,"},{"title":"px P Fq","content":"In this paper the most impressive equation, namely\nf i px p i qg i px q i q \" 0, px P Fq (1) will be studied from this list with a fruitful theoretical description.(This we call 'inner case', as the parameters p i , q i are exponents of the variable x, so they act on the domain of the functions f i , g i , respectively.)Under some natural conditions equation ( 1) is satisfied by compositions of (higher order) derivations and homomorphisms.The purpose of this paper is about the converse by showing proper characterizations for the solutions of (1) in the class of additive functions.\nThe diagonalization or trace of an n-additive function A : G n Ñ S is defined as A ˚pxq \" A px, . . ., xq px P Gq .\nAs a direct consequence of the definition each n-additive function A : G n Ñ S satisfies Apx 1 , . . ., x i´1 , kx i , x i`1 , . . ., x n q \" kApx 1 , . . ., x i´1 , x i , x i`1 , . . ., x n q px 1 , . . ., x n P Gq for all i \" 1, . . ., n, where k P N is arbitrary.The same identity holds for any k P Z provided that G and S are groups, and for k P Q, provided that G and S are linear spaces over the rationals.For the diagonalization of A we have A ˚pk xq \" k n A ˚pxq px P Gq .\nThe above notion can also be extended for the case n \" 0 by letting G 0 \" G and by calling 0-additive any constant function from G to S .\nOne of the most important theoretical results concerning multiadditive functions is the so-called Polarization formula, that briefly expresses that every n-additive symmetric function is uniquely determined by its diagonalization under some conditions on the domain as well as on the range.Suppose that G is a commutative semigroup and S is a commutative group.The action of the difference operator ∆ on a function f : G Ñ S is defined by the formula ∆ y f pxq \" f px `yq ´f pxq px, y P Gq .\nNote that the addition in the argument of the function is the operation of the semigroup G and the subtraction means the inverse of the operation of the group S .\nTheorem 1 (Polarization formula).Suppose that G is a commutative semigroup, S is a commutative group, n P N. If A : G n Ñ S is a symmetric, n-additive function, then for all x, y 1 , . . ., y m P G we have ∆ y 1 ,...,y m A ˚pxq \" \" 0 if m ą n n!Apy 1 , . . ., y m q if m \" n.\nCorollary 1. Suppose that G is a commutative semigroup, S is a commutative group, n P N. If A : G n Ñ S is a symmetric, n-additive function, then for all x, y P G ∆ n y A ˚pxq \" n!A ˚pyq.Lemma 1.Let n P N and suppose that the multiplication by n! is surjective in the commutative semigroup G or injective in the commutative group S .Then for any symmetric, n-additive function A : G n Ñ S , A ˚\" 0 implies that A is identically zero, as well.Definition 3. Let G and S be commutative semigroups, a function p : G Ñ S is called a generalized polynomial from G to S , if it has a representation as the sum of diagonalizations of symmetric multi-additive functions from G to S .In other words, a function p : G Ñ S is a generalized polynomial if and only if, it has a representation\nwhere n is a nonnegative integer and A k : G k Ñ S is a symmetric, k-additive function for each k \" 0, 1, . . ., n.In this case we also say that p is a generalized polynomial of degree at most n.\nLet n be a nonnegative integer, functions p n : G Ñ S of the form\nwhere A n : G n Ñ S is a symmetric and n-additive mapping, are the so-called generalized monomials of degree n.\nIn this subsection pG, ¨q is assumed to be a commutative group (written multiplicatively).\nDefinition 4. Polynomials are elements of the algebra generated by additive functions over G.More exactly, a mapping f : G Ñ C is called a polynomial if there is a positive integer n, there exists a (classical) complex polynomial P : C n Ñ C in n variables and there are additive functions a k : G Ñ C pk \" 1, . . ., nq such that f pxq \" Ppa 1 pxq, . . ., a n pxqq px P Gq .\nRemark 1.We recall that the elements of N n for any positive integer n are called (n-dimensional) multi-indices.Addition, multiplication and inequalities between multi-indices of the same dimension are defined componentwise.Further, we define x α for any n-dimensional multi-index α and for any x \" px 1 , . . ., x n q in C n by\nwhere we always adopt the convention 0 0 \" 0. We also use the notation |α| \" α 1 `¨¨¨`α n .With these notations any polynomial of degree at most N on the commutative semigroup G has the form ppxq \"\nwhere c α P C and a \" pa 1 , . . ., a n q : G Ñ C n is an additive function.Furthermore, the homogeneous term of degree k of p is ÿ\nLemma 2 (Lemma 2.7 of [10]).Let G be a commutative group, n be a positive integer and let a \" pa 1 , . . ., a n q , where a 1 , . . ., a n are linearly independent complex valued additive functions defined on G. Then the monomials ta α u for different multi-indices are linearly independent.\nFurthermore, on a(n) (generalized) exponential polynomial we mean a linear combination of functions of the form p ¨m, where p is a (generalized) polynomial and m is an exponential function.\nThe following lemma shows that generalized exponential polynomial functions are linearly independent.Although it can be stated in a more general way (see [10]), we adopt it to our situation, when the functions are complex valued.Lemma 3 (Lemma 4.3 of [10]).Let G be a commutative group, n a positive integer, m 1 , . . ., m n : G Ñ C pi \" 1, . . ., nq be distinct nonzero exponentials and p 1 , . . ., p n : G Ñ K pi \" 1, . . ., nq be generalized polynomials."},{"title":"If n ÿ","content":"i\"1 p i ¨mi is identically zero, then for all i \" 1, . . ., n the generalized polynomial p i is identically zero.\nAdditionally, we will need the analogous statement for polynomial expressions of generalized exponential polynomials which was proved in [5].\nTheorem 2. Let K be a field of characteristic 0 and k, l, N be positive integers such that k, l ď N. Let m 1 , . . ., m k : K ˆÑ C be distinct exponential functions that are additive on K, let a 1 , . . ., a l : K ˆÑ C be additive functions that are linearly independent over C and for all |s| ď N let P s : C l Ñ C be classical complex polynomials of l variables.If ÿ |s|ďN P s pa 1 , . . ., a l qm s 1 1 ¨¨¨m s k k \" 0 then for all |s| ď N, the polynomials P s vanish identically.\nDefinition 6.Let G be a commutative group and V Ď C G a set of functions.We say that V is translation invariant if for every f P V the function τ g f P V also holds for all g P G, where τ g f phq \" f phgq ph P Gq .\nIn view of Theorem 10.1 of Székelyhidi [10], any finite dimensional translation invariant linear space of complex valued functions on a commutative group consists of exponential polynomials.This implies that if G is a commutative group, then any function f : G Ñ C, satisfying the functional equation\nfor some positive integer n and functions g i , h i : G Ñ C (i \" 1, . . ., n), is an exponential polynomial of degree at most n.This enlightens the connection between generalized polynomials and polynomials.It is easy to see that each polynomial, that is, any function of the form\nwhere n is a positive integer, P : C n Ñ C is a (classical) complex polynomial in n variables and a k : G Ñ C pk \" 1, . . ., nq are additive functions, is a generalized polynomial.The converse however is in general not true.A complex-valued generalized polynomial p defined on a commutative group G is a polynomial if and only if its variety (the linear space spanned by its translates) is of finite dimension.\nHenceforth, not only the notion of (exponential) polynomials, but also that of decomposable functions will be used.The basics of this concept are due to Shulman [9], besides this we heavily rely on the work of Laczkovich [7].Definition 7. Let G be a group and n P N, n ě 2. A function F : G n Ñ C is said to be decomposable if it can be written as a finite sum of products F 1 ¨¨¨F k , where all F i depend on disjoint sets of variables.\nRemark 2. Without loss of generality we can suppose that k \" 2 in the above definition, that is, decomposable functions are those mappings that can be written in the form\nwhere E runs through all non-void proper subsets of t1, . . ., nu and for each E and j the function A E j depends only on variables x i with i P E, while B E j depends only on the variables x i with i E. Theorem 3. Let G be a commutative topological semigroup with unit.A continuous function f : G Ñ C is a generalized exponential polynomial if and only if there is a positive integer n ě 2 such that the mapping\nThe notion of derivations can be extended in several ways.We will employ the concept of higher order derivations according to Reich [8] and Unger-Reich [11].For further results on characterization theorems on higher order derivations consult e.g.[1,2,3] and [5].Definition 8. Let F Ă C be a field.The identically zero map is the only derivation of order zero.For each n P N, an additive mapping f : F Ñ C is termed to be a derivation of order n, if there exists B : F ˆF Ñ C such that B is a bi-derivation of order n ´1 (that is, B is a derivation of order n ´1 in each variable) and f pxyq ´x f pyq ´f pxqy \" Bpx, yq px, y P Fq .\nThe set of derivations of order n of the ring R will be denoted by D n pFq.For our future purposes the notion of differential operators will also be important, see [6].\nDefinition 9. Let F Ă C be a field.We say that the map D : is a derivation on K for every i \" 1, . . ., k.Therefore, every differential operator\nwhere the coefficients c i 1 ,...,i k belong to K, is a differential operator of order at most n, and also conversely, if D is a differential operator of order at most n on the field K \" Qpt 1 , . . ., t k q, then D is of the above form.\nThe main result of [6] is Theorem 1.1 that reads in our settings as follows.\nTheorem 4. Let F Ă C be a field and let n be a positive integer.Then, for every function D : F Ñ C, the following are equivalent.\n(i) D P D n pFq (ii) D P cl pO n pFqq (iii) D is additive on F, Dp1q \" 0, and D{ j, as a map from the group F ˆto C, is a generalized polynomial of degree at most n.Here j stands for the identity map defined on F."},{"title":"Results","content":""},{"title":"Elementary observations: reduction of the problem","content":"This part begins with some elementary, yet fundamental observations.As the following lemmata show, the original problem can be reduced to a more simpler equation.\nLemma 4 (Homogenization).Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers.Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation (1), that is,\nf i px p i qg i px q i q \" 0 for each x P F. If the set tp 1 , . . ., p n u has a partition P 1 , . . ., P k with the property if p α , p β P P j for a certain index j, then p α `qα \" p β `qβ , then the system of equations ÿ p α PP j f α px p α qg α px q α q \" 0 px P F, j \" 1, . . ., kq is satisfied.\nProof.Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers.Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation (1) for each x P F. Assume further that the set tp 1 , . . ., p n u has a partition P 1 , . . ., P k with the property if p α , p β P P j for a certain index j, then p α `qα \" p β `qβ .\nObserve that for all i \" 1, . . ., n, the mapping\nis a generalized monomial of degree p i `qi .Indeed, it is the diagonalization of the symmetric pp i `qi q-additive mapping\nSince F Ă C, we necessarily have Q Ă F. Let now r P Q be arbitrary and substitute rx in place of x in equation (1) to get\nUsing the Q-homogeneity of the additive functions f 1 , . . ., f n and g 1 , . . ., g n , we deduce 0 \"\nNote that the right hand side of this equation is a (classical) polynomial in r which is identically zero.Thus all of its coefficients should be (identically) zero, yielding that the system of equations ÿ p α PP j f α px p α qg α px q α q \" 0 px P F, j \" 1, . . ., kq is fulfilled.\nRemark 5.The above lemma guarantees that ab initio p i `qi \" N pi \" 1, . . ., nq can be assumed.Otherwise, after using the above homogenization, we get a system of functional equations in which this condition is already fulfilled.For instance, due to the above lemma, if the additive functions f 1 , . . ., f 5 : F Ñ C and g 1 , . . ., g 5 : F Ñ C satisfy equation f 1 px 24 qg 1 px 5 q `f2 px 20 qg 2 px 9 q `f3 px 19 qg 3 px 10 q `f4 px 13 qg 4 px 7 q `f5 px 12 qg 4 px 8 q \" 0 px P Fq then the equations f 1 px 24 qg 1 px 5 q `f2 px 20 qg 2 px 9 q `f3 px 19 qg 3 px 10 q \" 0 px P Fq and f 4 px 13 qg 4 px 7 q `f5 px 12 qg 4 px 8 q \" 0 px P Fq are also fulfilled (separately).\nRemark 6.At first glance the assumption that p 1 , . . ., p n are different seems a reasonable and sufficient supposition.Clearly, if the parameters are not necessarily different then we cannot expect anything special for the form of the involved additive functions.Indeed, let L Ă C n be a linear subspace and let f 1 , . . ., f n : F Ñ C and g 1 , . . ., g n : F Ñ C be additive functions such that rngp f q Ă L and rngpgq Ă L K , where f pxq \" p f 1 pxq, . . ., f n pxqq and gpxq \" pg 1 pxq, . . ., g n pxqq px P Fq ."},{"title":"In this case","content":"This shows the necessity of the above assumption.Unfortunately, the sufficiency fails to hold.To see this, let p and q be positive integers and f : F Ñ C be an arbitrary additive function and define the complex-valued functions f 1 , g 1 , f 2 , g 2 on F by\nAn immediate computation shows that we have f 1 px p qg 1 px q q `f2 px p qg 2 px q q \" 0 px P Fq .\nIn view of the above remarks, from now on, the following assumptions are adopted.\nC(i) the positive integers p 1 , . . ., p n are arranged in a strictly increasing order, i.e., p 1 ă ¨¨¨ă p n ; C(ii) for all i \" 1, . . ., n we have p i `qi \" N;\nC(iii) for all i, j P t1, . . ., nu, i j we have p i q j .Remark 7. Define the relation \" on F C by f \" g if and only if there exists an automorphism ϕ : C Ñ C such that ϕ ˝f \" g.Obviously \" is an equivalence relation on F C that induces a partition on F C ."},{"title":"Lemma 5 (Equivalence).","content":"Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling the conditions C(i)-C(iii) of Remark 6. Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation (1).Then for an arbitrary automorphism ϕ : C Ñ C the functions ϕ ˝f1 , . . ., ϕ ˝fn , ϕ ˝g1 , . . ., ϕ ˝gn also fulfill equation (1)."},{"title":"Structure of solutions","content":"We can always restrict ourselves to the case when all the involved functions are non-identically zero.Otherwise, the number of the terms appearing in equation ( 1) can be reduced."},{"title":"Lemma 6 (Symmetrization).","content":"Let k and n be positive integers, F Ă C be a field and m 1 , . . ., m n :\nis fulfilled for all x 1 , . . ., x k , where for all i \" 1, . . ., n, the mapping M i : F k Ñ C is the uniquely determined symmetric, k-additive function such that M i px, . . ., xq \" m i pxq px P Fq ."},{"title":"Proof.","content":"Let k and n be positive integers, F Ă C be a field and m 1 , . . ., m n : F Ñ C be monomials of degree k and assume that n ÿ i\"1 m i pxq \" 0 holds for all x P F. Since for all i \" 1, . . ., n, the function m i is a monomial of degree k, there exists a symmetric, k-additive function M i : F k Ñ C such that we have M i px, . . ., xq \" m i pxq px P Fq ."},{"title":"Obviously the mapping","content":"m i is a monomial of degree k which is, by the assumptions, identically zero.To this monomial there also corresponds a symmetric and k-additive mapping, namely\nObserve that the trace of this symmetric and k-additive mapping is identically zero.At the same time, due to the Polarization formula (Theorem 1), every symmetric and k-additive function is uniquely determined by its trace.Thus\nM i px 1 , . . ., x k q \" 0 for all x 1 , . . ., x k P F."},{"title":"Lemma 7 (Symmetrization).","content":"Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling conditions C(ii), i.e., there is a N P N such p i `qi \" N for all i \" 1, . . ., n. Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation (1) for each x P F. Then\nf i px σp1q ¨¨¨x σpp i q q ¨gi px σpp i `1q ¨¨¨x σpNq q \" 0 holds for all x 1 , . . ., x N P F.\nProof.Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling conditions C(ii).Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation\nf i px p i qg i px q i q \" 0 for each x P F. Due to the additivity of the functions f 1 , . . ., f n and g 1 , . . ., g n for all i \" 1, . . ., n, the mapping\nis a monomial of degree p i `qi \" N. Further, it is the trace of the symmetric and N-additive mapping\nTherefore, the statement follows from Lemma 6."},{"title":"Solutions of equation (1)","content":"The main purpose of the subsection is to describe under the conditions C(i)-C(iii), the solution space of equation (1).We first prove that solutions of equation ( 1) are decomposable functions on the multiplicative group F ˆ.In view of Laczkovich [7], this immediately yields that the solutions of equation ( 1) are generalized exponential polynomials of this group.\nLemma 8. Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling conditions C(i) and C(ii).Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation (1) for each x P F. Then all the functions f 1 , . . ., f n as well as g 1 , . . ., g n are decomposable functions of the group F ˆ.\nProof.Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling conditions C(i) and C(ii).\nLet us assume first that condition C(iii) is also satisfied.Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation (1) for each x P F. Let S \" tp 1 , . . ., p n u Y tq 1 , . . ., q n u .\nThen max S \" max tp n , q 1 u.By condition C(iii), we have p n q 1 .Without the loss of generality p n ą q 1 can be assumed, otherwise we follow a similar argument.In view of Lemma 7, we have\nf i px σp1q ¨¨¨x σpp i q q ¨gi px σpp i `1q ¨¨¨x σpNq q \" 0 for all x 1 , . . ., x N P F, or after some rearrangement,\nf i px σp1q ¨¨¨x σpp i q q ¨gi px σpp i `1q ¨¨¨x σpNq q `x1 , . . ., x N P F ˆ˘."},{"title":"Let now","content":"then the above identity says that g n p1q ¨fn is decomposable.If g n p1q were zero, but g n would not be identically zero, then there would exist a P F ˆsuch that g n paq 0. In this case the above substitutions should be modified to\nto get the same conclusion.If p n \" q j for some j \" t1, . . ., nu (i.e., C(iii) does not hold), then without loss of generality we may assume that j \" 1, otherwise we may change the role of f i and g i , and p i and q i , respectively, and proceed as above.If p n \" q 1 , then we have\nf i px σp1q ¨¨¨x σpp i q q ¨gi px σpp i `1q ¨¨¨x σpNq q `x1 , . . ., x N P F ˆ˘.\nThis equation with the substitutions x p n `1 \" ¨¨¨\" x N \" 1, yields that a linear combination of f n and g 1 is decomposable.If t f n , g 1 u is linearly dependent, then this obviously means that both f n and g 1 are decomposable functions.If this system is not linearly dependent, then there exist a, b P F ˆ, a b and different complex constants c 1 and c 2 such that\nWith the substitutions x p n `1 \" a, x p n `2 \" ¨¨¨\" x N \" 1, and\nwe get that the functions px 1 , . . ., x p 1 q Þ ÝÑ g 1 px 1 ¨¨¨x p 1 q f n paq `fn px 1 ¨¨¨x p 1 qg 1 paq and px 1 , . . ., x p 1 q Þ ÝÑ g 1 px 1 ¨¨¨x p 1 q f n pbq `fn px 1 ¨¨¨x p 1 qg 1 pbq are decomposable.Since finite linear combinations of decomposable functions are also decomposable, it follows that f n and g 1 are decomposable, separately.After that, let us consider the set S z tp n u and apply the above argument for this set.With this step-by-step descending argument the statement of the lemma follows.\nRemark 8. We emphasize that condition C(iii) in Lemma 8 has not been assumed.Thus, the fact that the additive solutions of (1) are decomposable functions can be deduced only under the conditions C(i) and C(ii).On the other hand, for our purpose to describe the solutions more concretely we have to assume also condition C(iii) to avoid further difficulties.We believe however that most of our methods can work similarly only under the conditions C(i) and C(ii).\nRemark 9. Note also that in general we cannot state more than that the involved functions f 1 , . . ., f n and g 1 , . . ., g n are decomposable functions on the commutative group F ˆ.In other words, we can only state that the solutions of the functional equation in question are higher order derivations (see below Corollary 2).In general it is not true that the solutions of this functional equation are differential operators (i.e., exponential polynomials of the multiplicative group F ˆ).To see this, let us consider the functional equation x f 1 px 6 q `x2 f 2 px 5 q `x3 f 3 px 4 q \" 0 px P Fq .\nIndeed, using the results of [5], we deduce that\nDue to a result of Laczkovich [7] and under the assumptions of Lemma 8, there exist a positive integer l, there are generalized polynomials P k,i , Q k,i : F ˆÑ C (k \" 1, . . ., l and i \" 1, . . ., n) and there exist linearly independent exponentials m 1 , . . ., m l : F ˆÑ C such that\nSince the generalized polynomials on any finitely generated field are polynomials and the exponentials m 1 , . . ., m k are linearly independent, we can apply Theorem 2. This implies that after substituting the above form the functions f 1 , . . ., f n and g 1 , . . ., g n and using that for each κ P N and k \" 1, . . ., n, we have\nP k,i px p i qQ k,i px q i q \" 0 `x P F ˆfollows for all k \" 1, . . ., l.This tells us that it is enough to solve equation ( 1) for generalized polynomials of the group F ˆ.In fact, we can prove the following.\nTheorem 5. Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling conditions C(i)-C(iii).Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation (1) for each x P F. Then there exists a positive integer l, there exist exponentials m i : F ˆÑ C and there are generalized polynomials P i , Q i : F ˆÑ C of degree at most l such that f i pxq \" P i pxqm i pxq and g i pxq \" Q i pxqm i pxq `x P F ˆfor each i \" 1, . . ., n.\nProof.As we saw above, under the hypothesis of the lemma, if the functions f 1 , . . ., f n and g 1 , . . ., g n solve equation (1), then there exist a positive integer l, there are generalized polynomials P k,i , Q k,i : F ˆÑ C (k \" 1, . . ., l and i \" 1, . . ., n) and there exist linearly independent exponentials m 1 , . . ., m l : F ˆÑ C such that\nAssume to the contrary that l ě 2. Let S \" tp 1 , . . ., p n u Y tq 1 , . . ., q n u .\nThen due to conditions C(i)-C(iii) we have max S \" max tp n , q 1 u.Similarly as in Lemma 8, without the loss of generality p n ą q 1 can be assumed, otherwise we follow a similar argument.By our assumption l ě 2, this yields that there exist different exponential terms in f 1 and g 1 with nonzero polynomial coefficients.For the sake of simplicity, suppose that these different exponentials are m 1 and m 2 .Since max S \" p 1 , the term m p 1 1 m q 1 2 appears only in f 1 px p 1 qg 1 px q 1 q while expanding equation (1).Since generalized polynomials P k,i and exponentials m k satisfies the conditions of Theorem 2 for every finitely generated subfield of F, the coefficient of the above-mentioned term which is P 1,1 px p 1 qQ 1,1 px q 1 q has to vanish on F. From this we can deduce that P 1,1 or Q 1,1 is identically zero, contrary to our assumption.Thus\nwith appropriate generalized polynomials P 1 , Q 1 : F ˆÑ C and exponential m 1 :\nSuppose now that there is a positive integer k, less than n such that for all i \" 1, . . ., k we have\nAssume that in the representation of f k`1 and g k`1 there are different exponentials with nonzero polynomial coefficients, say m j 1 and m j 2 .Observe that while expanding equation ( 1), the term m p k`1 j 1 m q k`1 j 2 appears only at once, namely in the product f k`1 px p k`1 qg k`1 px q k`1 q.Again, due to Theorem 2, we deduce that the appropriate polynomial term, that is, P k`1, j 1 px p k`1 qQ k`1, j 2 px q k`1 q has to vanish.This proves that necessarily\nThis shows that there exist exponentials m 1 , . . ., m n : F ˆÑ C and generalized polynomials P 1 , . . ., P n and Q 1 , . . ., Q n on the group F ˆsuch that for all i \" 1, . . ., n f i pxq \" P i pxqm i pxq and\nRemark 10.Due to conditions C(i)-C(iii), equation (1) has the form 0 \"\nIf the exponentials appearing on the right hand side would be different, then by Theorem 2, their coefficients would be zero.This implies however that there exists a proper subset J Ă t1, . . ., nu such that ÿ jPJ f j px p j qg j px q j q \" 0 as well as ÿ j J f j px p j qg j px q j q \" 0 `x P F ˆ˘.\n(\nThis leads to the following definition of irreducible solutions.\nDefinition 10.A system of solutions t f 1 , . . ., f n , g 1 , . . ., g n u of equation ( 1) is called irreducible if it does not satisfy a sub-term of (1).Otherwise, we say that a system of solutions is reducible.\nClearly, a system of solutions t f 1 , . . ., f n , g 1 , . . ., g n u of (1) which fulfills ( 2) is a reducible solution.On the other hand, the argument in Remark 10 shows that every solution of (1) can be given as a sum of irreducible solutions of disjoint sub-terms of (1).Therefore, we restrict ourselves to the irreducible case, since every reducible solution can be deduced as a sum of irreducible solutions.\nCorollary 2. Under the conditions of Theorem 5, suppose that system of functions t f 1 , . . ., f n , g 1 , . . ., g n u is an irreducible solution of equation (1).Then there exists an exponential m : F ˆÑ C and there are generalized polynomials P i , Q i : F ˆÑ C such that for each i \" 1, . . ., n f i pxq \" P i pxqmpxq and\nIn other words, for each i \" 1, . . ., n there exists higher order derivations D i , r\nwhere \" in the latter two equations is the equivalence relation defined in Remark 7.\nProof.By Remark 10, all of the exponentials m i have to be the same in the description of the solutions of Theorem 5. Therefore, equation (3) describes the irreducible solutions of (1).Using Lemma 5, solutions of equation ( 1) are enough to be determined up to the equivalence relation \" defined in Remark 7. Accordingly, we can suppose that the exponential m in the above representation is the identity mapping.Hence, in view of Theorem 4 we get that the functions f 1 , . . ., f n as well as g 1 , . . ., g n are (or more precisely, are equivalent to) higher order derivations, as we stated."},{"title":"The order of higher order derivation solutions","content":"Every higher order derivation on F is a differential operator on any finitely generated subfield of F (see Theorem 4 and [6]).Hence on these fields the solutions are differential operators.Moreover, if the solutions on any finitely generated subfield of F are differential operators of order at most n, then every solution on F is a derivation of order n.From now on, instead of finding solutions as higher order derivations we are looking for differential operators as solutions.\nFor this purpose, our next aim is to understand the arithmetic of the composition of derivations of the form d 1 ˝¨¨¨˝d r that are building blocks of differential operators.First we show that there is a natural form of composition of derivations that can be taken as a standard basis.\nFor this target, the notion of moment function sequences turn out to be useful.Here we follow [4].A composition of a nonnegative integer n is a sequence of nonnegative integers α \" pα k q kPN such that n \"\nFor a positive integer r, an r-composition of a nonnegative integer n is a composition α \" pα k q kPN with α k \" 0 for k ą r.\nGiven a sequence of variables x \" px k q kPN and compositions α \" pα k q kPN and β \" pβ k q kPN we define α! \"\nFurthermore, β ď α means that β k ď α k for all k P N and β ă α stands for β ď α and β α.\nDefinition 11.Let G be a commutative group, r a positive integer, and for each multi-index α in N r let f α : G Ñ C be a continuous function.We say that p f α q αPN r is a generalized moment sequence of rank r, if\nholds whenever x, y are in G.The function f 0 , where 0 is the zero element in N r , is called the generating function of the sequence.\nTheorem 6.Let G be a commutative group, r a positive integer, and for each α in N r let f α : G Ñ C be a function.If the sequence of functions p f α q αPN r forms a generalized moment sequence of rank r, then there exists an exponential m : G Ñ C and a sequence of complex-valued additive functions a \" pa α q αPN r such that for every multi-index α in N r and x in G we have\nwhere B α denotes the multivariate Bell polynomial corresponding to the multi-index α.\nRemark 11.It is well-known that every polynomial P : C r Ñ C (r P N) can be given as a linear combination of Bell polynomials B α , where α P N r .Hence, momentum generating functions generate the exponential polynomial functions on G.By Lemma 2, polynomials of the form B α ˝a are linearly independent over C.\nLemma 9. Let F Ă C be a field, r be a positive integer and d 1 , . . ., d r : F Ñ F be linearly independent derivations.For all multi-index α P N r , α \" pα 1 , . . ., α r q define the function ϕ α : F Ñ C by\nThen pϕ α q αPN r is a generalized moment sequence of rank r on the commutative group F ˆand the generating function of the sequence is the identity function.\nProof.Let F Ă C be a field, r be a positive integer and d 1 , . . ., d r : F Ñ F be linearly independent derivations.For all multi-index α P N r , α \" pα 1 , . . ., α r q define the function ϕ α : F Ñ C as in the lemma.We prove the statement by induction of the length of the multi-index α.Assume that α P N r and |α| \" 1.Then there exists i P t1, . . .ru such that α i \" 1 and α j \" 0 for j i and ϕ α pxyq \" d α i i pxyq \" d i pxyq \" xd i pyq `yd i pxq \" ϕ 0 pxqϕ α pyq `ϕ0 pyqϕ α pxq \" ÿ βďα ˆα β ˙ϕβ pxqϕ α´β pyq `x, y P F ˆ˘.\nLet now α P N r and assume that the statement is true for all multi-indices β P N r with β ă α.Then ϕ α pxyq \" d α pxyq \" d α 1 1 ˝¨¨¨˝d α r r pxyq \" d 1 ˝¨¨¨d 1 loooomoooon Corollary 3. By Lemma 9 and Remark 11 implies that the functions d α \" d α 1 1 ˝¨¨¨˝d α r r constitute a basis of the differential operators in F ˆ. Since every d α is additive, by Lemma 2, all elements of the system td α pxqu, where α P Y rPN N r are algebraically independent.\nA consequence of the algebraic independence of the elements of d α , where α P Y rPN N r is the following.Let P P Crx 1 , . . ., x n s be a polynomial, d 1 , . . .d r be derivations as in Lemma 9 With the above results, we are now ready to prove an upper bound for the order of derivations appearing in Theorem 5.\nTheorem 7.Under the hypotheses of Theorem 5, the solutions f i and g i are derivations D i and r D i for all i \" 1, . . ., n.Let k and l denote the maximal orders of derivations D i and r D i , respectively.Suppose that there exists some i 1 such that the order of D i 1 and r D i 1 is exactly k and l, respectively.Then for all i \" 1, . . ., n the order of D i and r D i is less or equal to n ´1.\nProof.Assume contrary that the maximal order k of the above derivations D 1 , . . ., D n is greater than n ´1.\nThe argument for the case when the maximal order l of r D 1 , . . ., r D n is greater than n ´1 is analogous.By our assumption there exists an index i 1 such that the orders of the derivations D i 1 and r D i 1 is exactly k and l, respectively.It is important to note that then the sum of the orders of D i and r D i is at most k `l for any i, and it is exactly k `l for some indices if and only if the corresponding D and r D is of order k and of order l, respectively.Furthermore, by the algebraic independence of higher order derivations, there the number of these indices are at least two.\nFrom now on we assume that F is finitely generated.Indeed, if we verify the statement for any finitely generated subfield of a field F, then it holds for F itself, as well.On finitely generated fields all of these higher order derivations can be represented as differential operators, that is, on finitely generated fields we have\n. ., n) and higher order derivations\nIf we expand the right hand side of the above identity with the aid of Proposition If this identity can be satisfied by different functions, it can also be satisfied by a single one.By Corollary 3, this enables us to substitute the functions d α i , r d β i pi \" 1, . . ., n, |α| ď k, |β| ď lq with suitable compositions of a given derivation d of order 1.In other words, instead of the above identity, we can restrict ourselves to n ÿ i\"1 ˜k ÿ j\"0 λ i, j d j px p i q ¸¨˜l ÿ j\"0 r λ i, j d j px q i q ¸\" 0 px P Fq with appropriate complex constants λ i, j (i \" 1, . . ., n, j \" 0, . . ., k), and r λ i, j (i \" 1, . . ., n, j \" 0, . . ., l) and derivation d : F Ñ C. By our assumptions there are some i 1 such that λ i 1 ,k 0 and r λ i 1 ,l 0. Dividing the above sum to smaller ones, we get\nwhere\nλ i, j d j px p i q and S pq i , l ´1q \" l´1 ÿ j\"0 r λ i, j d j px q i q px P Gq .\nNote that, by the algebraic independence used in Corollary 3, this sum splits into separate terms of the form d s px p i qd t px q i q, where s `t is a fixed number.By our assumption, when s `t \" k `l, then the only way is that s \" k, t \" l.This implies that using Proposition 1 we get that 0 \"\n, . . ., s, . . ., s loomoon\nwhile the rest in the above sum can be computed similarly.\nCase 1.If k ă l, then we compute the coefficients of the terms pdpxqq j d k´j pxqd l pxq p j \" 0, . . ., k ´1q.\nFor each j \" 0, . . ., k ´1 this can be taken from the expansion of λ i,k r λ i,l d k px p i qd l px q i q in only one way.Namely, by splitting d k px p i q into j `1 parts and d l px q i q into one.Then the corresponding coefficients are\nSince each of the terms contains ˆk 1, . . ., 1 loomoon j ˙1 j! \" ˆkj ˙, this can be eliminated from the above equation.The corresponding equations ( j \" 0, . . ., k) can be written in the following matrix equation\nHere we note that as p i 's are all different positive integers, it follows that p i ě n for some i and hence the first n rows of the matrix are not identically zero, as k ą n ´1.It is straightforward to verify that the first n row of above matrix equation is equivalent to ¨p1 . . .p n p 2 1 . . .p 2 n . . . . . . . . .\n. . .\nSince this is a Vandermonde type matrix with different p i 's, the only solution of this homogeneous linear system is the zero vector, i.e., q i λ i,k r λ i,l \" 0 for all i \" 1, . . ., n.This contradicts to our assumption that there is some i 1 for which λ i 1 ,k 0 and r λ i 1 ,l 0 (q i 1 0 as q i 0 for all i P t1, . . ., ku).\nCase 2. If k \" l, then we compute the coefficients of the terms pdpxqq 2 j pd k´j pxqq 2 p j \" 0, . . ., k ´1q.\nIf j ă k 2 , then this term can be taken from the expansion of λ i,k r λ i,l d k px p i qd k px q i q in only one way.Namely, by splitting d k px p i q and d k px q i q into j `1 parts.Then the corresponding coefficients are\nSince each of the terms contains ´`k 1, . . ., 1 loomoon j ˘1 j! ¯2 \" `kj ˘2, this can be eliminated from the above equations.\nThese equations for j \" 0, . . ., rk{2s ´1 can be written in the following matrix equation\nHere we note that as p i , q i are all different positive integers, it follow that max i tp i , q i u ě 2n and hence the first n 1 \" minpn, rk{2s ´1q rows of the matrix in not identically zero, as k ą n ´1.It is straightforward to verify that the first n 1 row of above matrix equation is equivalent to ¨p1 q 1 . . .p n q n p 2 1 q 2 1 . . .p 2 n q 2 n . . . . . . . . .\nNote that if n 1 \" n, then we are done with a Vandermonde matrix argument similar as it is used in Case 1.\nSo from now on we assume that n 1 \" rk{2s ´1.This also means that k ă 2n, thus the maximal order of the corresponding derivations is at most 2n ´1.\nIf j ě k 2 , then the term pdpxqq 2 j pd k´j pxqq 2 can be taken from the expansion of λ i,k r λ i,l d k px p i qd k px q i q in three ways.One is as above, when we split both d k px p i q and d k px q i q into j `1 parts.Another one is when d k px p i q is split into k parts giving dpxq k , and d k px q i q provides pd k´j pxqq 2 pdpxqq 2 j´k .The third one is given by changing the role of p i and q i .Then the corresponding coefficients are\nFirst we show that the second sum has to vanish for all j \" rk{2s, . . ., k ´1.In such cases, the coefficients ˆk 1, . . ., 1\nThis clearly holds, since the term pdpxqq 2 j`1 d 2k´2 j´1 pxq in the expansion of d k px p i qd k px q i q for j \" rk{2s, . . ., k1 can be given in exactly two ways.Either pdpxqq k stems from d k px p i q and pdpxqq 2 j´k`1 ¨d2k´2j´1 pxq stems from d k px q i q, or reversely changing the role of p i and q i .Hence we get ˘`q 1 1 ˘. . .\nwhere the first n rows of the matrix is not identically zero as p i and q i are different, hence there is an index i 1 such that p i 1 ě n and q i 1 ě n.Thus this system consisting the first n rows is equivalent to ¨p1 q 1 . . .p n q n p 2 1 q 2 1 . . .p 2 n q 2 n . . . . . . . . .\nBy the usual Vandermonde argument as in Case 1, the only solution of this homogeneous linear system is the zero vector, i.e., λ i,k r λ i,l \" 0 for all i \" 1, . . ., n.This contradicts our assumption that there is some i 1 for which λ i 1 ,k 0 and r λ i 1 ,k 0.\nCase 3. If k ą l, then we prove by induction in j that n ÿ i\"1 λ i,k r λ i,l p j`1´s i q s i \" 0 holds for every j \" 0, . . ., n ´1 and every s \" 0, . . ., j.\nIn each step we consider how pdpxqq j d k´j pxqd l pxq can be given from the expansion d k px p i qd l px q i q.There are three possible ways, where this term can stem from.\n(a) pdpxqq j d k´j pxq stems from d k px p i q and d l pxq stems from d l px q i q.This can happen for every j P t0, . . ., ku.\nIn this case the coefficient of pdpxqq j d k´j pxqd l pxq is\n˙qi .\n(b) d k´j pxqd l pxqpdpxqq j´l stems from d k px p i q and pdpxqq l stems from d l px q i q.This can happen if j ě l.In this case the coefficient of pdpxqq j d k´j pxqd l pxq is (c) d l pxqpdpxqq k´l stems from d k px p i q and pdpxqq l´pk´jq d k´j pxq stems from d l px q i q.This can happen if l ě k´j.\nIn this case the coefficient of pdpxqq j d k´j pxqd l pxq is ˙.\nFor j \" 0 (and hence s \" 0) only the first term takes into account.This means that\nSo the inductive hypothesis holds for j \" 0. Now we assume that n ÿ i\"1 λ i,k r λ i,l p j 1 `1´s i q s i \" 0 holds for every j 1 \" 0, . . ., j ´1 and every s \" 0, . . ., j 1 .We prove that n ÿ i\"1 λ i,k r λ i,l p j`1´s i q s i \" 0 holds for every s \" 0, . . ., j, as well.Generally, some of the previous compositions are possible for a given j but the following argument works in all cases, however we just prove it when all compositions discussed below appear in the expansion.Thus we assume that the coefficient of pdpxqq j d k´j pxqd l pxq is 0 \" By the inductive hypothesis and the fact that j ă n ď min p 1 i , q 1 i ( for some i 1 , this is equivalent to 0 \"\nNote that the expressions p j`1´s i q s i for s \" 0, . . ., j can be interchanged in the following sense.By C(ii), p i `qi \" N, thus we have that p j`1´s i q s i \" N p j´s i q s i ´p j´s i q s`1 i .\nAs\nλ i,k r λ i,l p j´s i q s i \" 0 for every s \" 0, . . ., j ´1 by the inductive hypothesis, the term N p j´s i q s i can be eliminated.After several repetition of this step we get that 0 \"\nWe also note that using this interchange rule and the inductive hypothesis it is clear that n ÿ i\"1 λ i,k r λ i,l p j`1´s i q s i \" 0 for any s \" 0, . . ., j is equivalent to\nTherefore, to finish the proof it is enough to show that ˆkj ˙˘ˆk\nThis is equivalent to verify that 1 j! ˘1 p j ´lq!pl!q ˘1 pk ´lq!pl ´pk ´jq!q 0.\nMultiplying by j! this lead to\nIt is straightforward to show using the growth of `n k ˘in k (if k ď n{2) that one term dominates the others if l k ´l and l j ´pk ´lq.Thus in these cases this (weighted) sum is nonzero.If l \" k ´l or l \" j ´pk ´lq, then either their sign is the same and hence the sum is nonzero, or their sign is different and hence they eliminate each other, hence the sum is 1 which is nonzero.\nSummarizing, we get that ˆkj ˙˘ˆk k ´j, j ´l, l ˙˘ˆk k ´l, k ´j, l ´pk ´jq ˙ 0 and hence\nwhich is equivalent to the inductive hypothesis for j as we noted above.\nThus equation (8) holds for every j \" 0, . . ., n ´1.In matrix form this means that\nSince this matrix is a Vandermonde type matrix with different p i 's, as Case 1 and Case 2, this implies that, the only solution of this homogeneous linear system is the zero vector, i.e., p i q i λ i,k r λ i,l \" 0 for all i \" 1, . . ., n.This contradicts to our assumption that there is some i 1 for which λ i 1 ,k 0 and r λ i 1 ,l 0 (and p i 0, q i 0 by C(i)).This also finishes the proof of the theorem, thus the order of all derivations involved in (1) is at most n ´1.\nRemark 12.The upper bound appearing in the above theorem is sharp.To see this, let n, N be positive integers, λ 1 , . . ., λ n P C and let f : F Ñ C be an additive function for which n ÿ i\"1 λ i f px i qx N´i \" 0 is fulfilled for all x P F. Then f P D n´1 pRq if and only if λ i \" p´1q i ˆn i ˙for all i \" 1, . . ., n, see [1,3,5].\nRemark 13.The proof of Theorem 7 in all the cases is based on the fact that the matrix\nhas rank n, though it is far from being trivial to find the proper sub-matrix, which verifies that.On the other hand, the situation is much more complicated, if the maximal order k of D i , and the maximal order l of r D i is not uniquely determined.Namely, if there are several different pairs pk i , l i q so that k i `li \" K, where K is constant and k i is the maximum order of D i , l i is the maximal order of r D i .Then the equations first can only be determined for subsets of the index set, which satisfy some nontrivial relations.In this case the first task is to show that the problem can be formalized separately for the index sets, which seems a very hard problem in full generality.In this case our method can be applied.Theorem 7 and Remark 13 motivates our conjecture, that we verified for n ď 4.\nConjecture 1.Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling conditions C(i)-C(iii).Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy equation (1).Then every solution is a generalized exponential polynomial function of degree at most n ´1 on F ˆ.\nIn particular, if f i pxq \" D i pxq and g i pxq \" r D i pxq `x P F ˆ˘ (9) for each i \" 1, . . ., n, then the order of D i , r D i is at most n ´1.\nThis conjecture leads to the following more general open question.\nOpen Question 1.Is it true that every nonzero additive, irreducible solutions f 1 , . . ., f n of Pp f 1 px p 1 q, . . ., f n px p n qq \" 0, with Pp0, . . ., 0q \" 0 are derivations of order at most n ´1 and the identity function up to a homomorphism, if P : C Ñ C is polynomial and p 1 . . ., p n are distinct positive integers?\nFinally we highlight some important special cases when Theorem 7 gives the proper bound of the order of the derivations.Corollary 4. Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling conditions C(i)-C(iii).Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy equation (1) as an irreducible solution.Then f i \" D i and g i \" r D i , where D i and r D i are higher order derivations.Assume further that one of the following holds.(A) All D i have the same order.This is the case, when f i pxq \" c i f pxq px P Fq for some nonzero constants c i P C, i P t1, . . ., nu.\n(B) All r D i have the same order.This is the case, when g i pxq \" c i gpxq px P Fq for some nonzero constants c i P C, i P t1, . . ., nu.\n(C) f i \" c i ¨gi for all i P t1, . . ., nu with some nonzero constants c i P C, i \" 1, . . ., n.\nThen the order D i and r D i is at most n ´1.\nProof.Theorem 5 implies that every solution f i (resp.g i ) is an exponential polynomial of the form P i ¨m (resp.Q i ¨m), which means that f i \" D i and g i \" r D i for some derivations D i and r D i .\n(A) Let k denote the order of D i , and let l be the maximal order of r D i .Now we are in the position to apply Theorem 7.\n(B) Similar to (A), since in case of equation ( 1), the role of the parameters p 1 , . . ., p n and q 1 , . . ., q n is symmetric.\n(C) As the maximal degree of f i is the same as the maximal degree of g i and it is taken for the same index we can apply Theorem 7."},{"title":"Special cases of equation (1)","content":"In this subsection we consider special cases of equation (1).All the equations we consider here are of the form f 1 px p 1 qg 1 px q 1 q `f2 px p 2 qg 2 px q 2 q \" 0 px P Fq .\nHere f 1 , f 2 , g 1 , g 2 : F Ñ C denote the unknown additive functions and the parameters p 1 , p 2 , q 1 , q 2 fulfill conditions C(i)-C(iii).Due to the results of the previous section, we get that f i pxq \" λ i,0 x `λi,1 d i pxq g i pxq \" µ i,0 x `µi,1 r d i pxq px P F, i \" 1, 2q with appropriate complex constants λ i, j , µ i, j (i \" 1, 2, j \" 0, 1) and derivations d i , di : F Ñ C (i \" 1, 2).\nCorollary 5. Let N be a positive integer, F Ă C be a field and p, q be different positive integers (strictly) less than N and assume that q N ´p.If the additive functions f, g : F Ñ C satisfy f px p q f px N´p q \" gpx q qgpx N´q q px P Fq , Corollary 6.Let N be a positive integer, F Ă C be a field and p, q be different positive integers (strictly) less than N and assume that q N ´p.If the additive functions f, g : F Ñ C satisfy f px p qgpx N´p q \" κ f px q qgpx N´q q px P Fq .Both results implies that the nonzero additive solutions of equation f px p q f px N´p q \" κ f px q q f px N´q q px P Fq ."},{"title":"Then","content":"are derivations of order 1 (κ \" ppN ´pq qpN ´qq ) or the identity function (κ \" 1) up to a homomorphism."},{"title":"α 1 times ˝¨¨¨˝d r ˝¨¨¨˝d r looooomooooon α r times pxyq \" d 1 ˝¨¨¨˝d 1 looooomooooon α 1 times ˝¨¨¨˝d r´1 ˝¨¨¨˝d r´1 loooooooomoooooooon α r´1 times pd α r r pxyqq \" d 1 ˝¨¨¨˝d 1 looooomooooon α 1 times ˝¨¨¨˝d r´1 ˝¨¨¨˝d r´1 loooooooomoooooooon α r´1 times 1 1","content":"˝¨¨¨˝d β r r pxq ¨dα 1 ´β1 1 ˝¨¨¨˝d α r ´βr r pyq \" ÿ βďα ˆα β ˙ϕβ pxqϕ α´β pyq `x, y P F ˆ˘.\nthen A) either there exists a homomorphism ϕ : C Ñ C, a derivation d : F Ñ C such that f pxq \" ϕpdpxqq and gpxq \" αϕpdpxqq px P Fq , where α \" ppN ´pq qpN ´qq , B) or there exists a homomorphism ϕ : F Ñ C such that f pxq \" f p1q ¨ϕpxq gpxq \" ˘f p1q ¨ϕpxq px P Fq .\nthat is, the notions of first order derivations and derivations coincide.On the other hand for any n P N the set D n pFqzD n´1 pFq is nonempty because d 1 ˝¨¨¨˝d n P D n pFq, but d 1 ˝¨¨¨˝d n D n´1 pRq, where d 1 , . . ., d n P D 1 pFq are non-identically zero derivations.\noperator of order at most n if D is the linear combination, with coefficients from F, of finitely many maps of the form d 1 ˝¨¨¨˝d k , where d 1 , . . ., d k are derivations on F and k ď n.If k \" 0 then we interpret d 1 ˝¨¨¨˝d k as the identity function.We denote by O n pFq the set of differential operators of order at most n defined on F. We say that the order of a differential operator D is n if D P O n pFqzO n´1 pFq (where O ´1pFq \" H, by definition).\nand α 1 , . . ., α n P Y rPN N r .Then the following polynomial form Ppd α 1 pxq, . . ., d α ,...,l p ě0 l 1 `¨¨¨`l p \"k ˆk l 1 , . . ., l p ˙¨d l 1 pxq ¨¨¨d l p pxq px 1 , . . ., x p P Fq ,\nn pxqq \" 0 px P Gq holds if and only if Pp d |α 1 | pxq, . . ., d |α n | pxqq \" 0 px P Gq , where d is an arbitrary derivation (of order 1).In other words, we can substitute d 1 , . . ., d r by d in d α 1 , . . ., d α n .By Corollary 3, it would be desirable to calculate d k \" d ˝¨¨¨˝d loooomoooon k times , where d is a derivation (of order 1) and k P N. Lemma 9, together with [4, Proposition 1], implies the following statement.Proposition 1.Let F Ă C be a field and d : F Ñ C a derivation.For all positive integer k we k px 1 ¨¨¨x p q \" ÿ l 1 ,...,l p ě0 l 1 `¨¨¨`l p \"k ˆk l 1 , . . ., l p ˙p ź t\"1 d l t px t q px 1 , . . ., x p P Fq , where j 1 , . . ., j s denotes the number of dpxq, . . ., d s pxq in a given composition of d k px p q.\nl, 1, . . ., 1\nl ˙.\nppN ´pq qpN ´qq , then there exists a homomorphism ϕ : C Ñ C and derivations d 1 , d 2 : F Ñ C such that f pxq \" ϕpd 1 pxqq and gpxq \" ϕpd 2 pxqq px P Fq .\nB) if κ \" 1, then the only possibility is that f pxq \" f p1q ¨ψpxq and gpxq \" f p1q ¨ψpxq px P Fq , where ψ :F Ñ C is a homomorphism, C) if κ \""}],"string":"[\n {\n \"title\": \"Introduction and preliminaries\",\n \"content\": \"Equations satisfied by additive functions play an important role not only in the theory of commutative algebra, but also in the theory of functional equations.It is an important and challenging question how special morphisms (such as homomorphisms and derivations) can be characterized among additive mappings in general.In this paper classes of multivariable algebraic equations are introduced with appropriate solutions as field homomorphisms and derivations.\\nConcerning all the cases we consider here, the involved additive functions are defined on a field F Ă C and have values in the complex field, therefore we introduce the preliminaries in this setting.\\nWe adopt the standard notations, that is, N and C denote the set of positive integers and the set of complex numbers, respectively.\\nHenceforth we assume F Ă C to be a field.\\nDefinition 1.We say that a function f : F Ñ C is additive if it fulfills the so-called Cauchy functional equation, that is, f px `yq \\\" f pxq `f pyq px, y P Fq .\\nAn additive function d : F Ñ C is termed to be a derivation (of order 1) if it also fulfills the Leibniz equation, i.e., dpxyq \\\" dpxqy `xdpyq px, y P Fq .\\nAn additive function ϕ : F Ñ C is said to be a homomorphism if it is multiplicative as well, in other words, besides additivity we also have ϕpxyq \\\" ϕpxqϕpyq px, y P Fq .\\nIf F \\\" C and ϕ is an isomorphism, then ϕ is called a complex automorphism.\\nCertain well-known equations are especially important.For instance, the additive solutions of following equation on a ring R with charR 2 f px 2 q \\\" 2x f pxq px P q are derivations, under some assumptions, where the additive mapping f acts.As an extension of such type of results in [1,2,5] the additive solutions of the equations n ÿ i\\\"0\\nx i f n`1´i px n`1´i q \\\" 0 px P Rq and n ÿ i\\\"0 f px p i qx q i \\\" 0 px P Fq were described, here R denotes a ring with charpRq ě n, while F Ă C is a field.By a polynomial equation of additive functions we mean an equation of the form Pp f r 1 1 px s 1 q, . . ., f r n n px s n qq \\\" 0, where P : C n Ñ C is a n-variable polynomial, r i , s i are positive integers and f i denote the unknown additive functions.Without further restrictions (e.g., on the polynomial P or on the parameters r i , s i ), the above equation is unfortunately too general for its solutions to be fully determined.Indeed, it is not too hard to specify a polynomial P and parameters r i , s i so that the above equation is satisfied by all additive functions.Therefore, in this series, we will focus on the following classes of equations.\\nf i px p i qg i px q i q \\\" 0, n ÿ i\\\"1 f i px p i qg i pxq q i \\\" 0, n ÿ i\\\"1 f i pxq p i g i pxq q i \\\" 0,\"\n },\n {\n \"title\": \"px P Fq\",\n \"content\": \"In this paper the most impressive equation, namely\\nf i px p i qg i px q i q \\\" 0, px P Fq (1) will be studied from this list with a fruitful theoretical description.(This we call 'inner case', as the parameters p i , q i are exponents of the variable x, so they act on the domain of the functions f i , g i , respectively.)Under some natural conditions equation ( 1) is satisfied by compositions of (higher order) derivations and homomorphisms.The purpose of this paper is about the converse by showing proper characterizations for the solutions of (1) in the class of additive functions.\\nThe diagonalization or trace of an n-additive function A : G n Ñ S is defined as A ˚pxq \\\" A px, . . ., xq px P Gq .\\nAs a direct consequence of the definition each n-additive function A : G n Ñ S satisfies Apx 1 , . . ., x i´1 , kx i , x i`1 , . . ., x n q \\\" kApx 1 , . . ., x i´1 , x i , x i`1 , . . ., x n q px 1 , . . ., x n P Gq for all i \\\" 1, . . ., n, where k P N is arbitrary.The same identity holds for any k P Z provided that G and S are groups, and for k P Q, provided that G and S are linear spaces over the rationals.For the diagonalization of A we have A ˚pk xq \\\" k n A ˚pxq px P Gq .\\nThe above notion can also be extended for the case n \\\" 0 by letting G 0 \\\" G and by calling 0-additive any constant function from G to S .\\nOne of the most important theoretical results concerning multiadditive functions is the so-called Polarization formula, that briefly expresses that every n-additive symmetric function is uniquely determined by its diagonalization under some conditions on the domain as well as on the range.Suppose that G is a commutative semigroup and S is a commutative group.The action of the difference operator ∆ on a function f : G Ñ S is defined by the formula ∆ y f pxq \\\" f px `yq ´f pxq px, y P Gq .\\nNote that the addition in the argument of the function is the operation of the semigroup G and the subtraction means the inverse of the operation of the group S .\\nTheorem 1 (Polarization formula).Suppose that G is a commutative semigroup, S is a commutative group, n P N. If A : G n Ñ S is a symmetric, n-additive function, then for all x, y 1 , . . ., y m P G we have ∆ y 1 ,...,y m A ˚pxq \\\" \\\" 0 if m ą n n!Apy 1 , . . ., y m q if m \\\" n.\\nCorollary 1. Suppose that G is a commutative semigroup, S is a commutative group, n P N. If A : G n Ñ S is a symmetric, n-additive function, then for all x, y P G ∆ n y A ˚pxq \\\" n!A ˚pyq.Lemma 1.Let n P N and suppose that the multiplication by n! is surjective in the commutative semigroup G or injective in the commutative group S .Then for any symmetric, n-additive function A : G n Ñ S , A ˚\\\" 0 implies that A is identically zero, as well.Definition 3. Let G and S be commutative semigroups, a function p : G Ñ S is called a generalized polynomial from G to S , if it has a representation as the sum of diagonalizations of symmetric multi-additive functions from G to S .In other words, a function p : G Ñ S is a generalized polynomial if and only if, it has a representation\\nwhere n is a nonnegative integer and A k : G k Ñ S is a symmetric, k-additive function for each k \\\" 0, 1, . . ., n.In this case we also say that p is a generalized polynomial of degree at most n.\\nLet n be a nonnegative integer, functions p n : G Ñ S of the form\\nwhere A n : G n Ñ S is a symmetric and n-additive mapping, are the so-called generalized monomials of degree n.\\nIn this subsection pG, ¨q is assumed to be a commutative group (written multiplicatively).\\nDefinition 4. Polynomials are elements of the algebra generated by additive functions over G.More exactly, a mapping f : G Ñ C is called a polynomial if there is a positive integer n, there exists a (classical) complex polynomial P : C n Ñ C in n variables and there are additive functions a k : G Ñ C pk \\\" 1, . . ., nq such that f pxq \\\" Ppa 1 pxq, . . ., a n pxqq px P Gq .\\nRemark 1.We recall that the elements of N n for any positive integer n are called (n-dimensional) multi-indices.Addition, multiplication and inequalities between multi-indices of the same dimension are defined componentwise.Further, we define x α for any n-dimensional multi-index α and for any x \\\" px 1 , . . ., x n q in C n by\\nwhere we always adopt the convention 0 0 \\\" 0. We also use the notation |α| \\\" α 1 `¨¨¨`α n .With these notations any polynomial of degree at most N on the commutative semigroup G has the form ppxq \\\"\\nwhere c α P C and a \\\" pa 1 , . . ., a n q : G Ñ C n is an additive function.Furthermore, the homogeneous term of degree k of p is ÿ\\nLemma 2 (Lemma 2.7 of [10]).Let G be a commutative group, n be a positive integer and let a \\\" pa 1 , . . ., a n q , where a 1 , . . ., a n are linearly independent complex valued additive functions defined on G. Then the monomials ta α u for different multi-indices are linearly independent.\\nFurthermore, on a(n) (generalized) exponential polynomial we mean a linear combination of functions of the form p ¨m, where p is a (generalized) polynomial and m is an exponential function.\\nThe following lemma shows that generalized exponential polynomial functions are linearly independent.Although it can be stated in a more general way (see [10]), we adopt it to our situation, when the functions are complex valued.Lemma 3 (Lemma 4.3 of [10]).Let G be a commutative group, n a positive integer, m 1 , . . ., m n : G Ñ C pi \\\" 1, . . ., nq be distinct nonzero exponentials and p 1 , . . ., p n : G Ñ K pi \\\" 1, . . ., nq be generalized polynomials.\"\n },\n {\n \"title\": \"If n ÿ\",\n \"content\": \"i\\\"1 p i ¨mi is identically zero, then for all i \\\" 1, . . ., n the generalized polynomial p i is identically zero.\\nAdditionally, we will need the analogous statement for polynomial expressions of generalized exponential polynomials which was proved in [5].\\nTheorem 2. Let K be a field of characteristic 0 and k, l, N be positive integers such that k, l ď N. Let m 1 , . . ., m k : K ˆÑ C be distinct exponential functions that are additive on K, let a 1 , . . ., a l : K ˆÑ C be additive functions that are linearly independent over C and for all |s| ď N let P s : C l Ñ C be classical complex polynomials of l variables.If ÿ |s|ďN P s pa 1 , . . ., a l qm s 1 1 ¨¨¨m s k k \\\" 0 then for all |s| ď N, the polynomials P s vanish identically.\\nDefinition 6.Let G be a commutative group and V Ď C G a set of functions.We say that V is translation invariant if for every f P V the function τ g f P V also holds for all g P G, where τ g f phq \\\" f phgq ph P Gq .\\nIn view of Theorem 10.1 of Székelyhidi [10], any finite dimensional translation invariant linear space of complex valued functions on a commutative group consists of exponential polynomials.This implies that if G is a commutative group, then any function f : G Ñ C, satisfying the functional equation\\nfor some positive integer n and functions g i , h i : G Ñ C (i \\\" 1, . . ., n), is an exponential polynomial of degree at most n.This enlightens the connection between generalized polynomials and polynomials.It is easy to see that each polynomial, that is, any function of the form\\nwhere n is a positive integer, P : C n Ñ C is a (classical) complex polynomial in n variables and a k : G Ñ C pk \\\" 1, . . ., nq are additive functions, is a generalized polynomial.The converse however is in general not true.A complex-valued generalized polynomial p defined on a commutative group G is a polynomial if and only if its variety (the linear space spanned by its translates) is of finite dimension.\\nHenceforth, not only the notion of (exponential) polynomials, but also that of decomposable functions will be used.The basics of this concept are due to Shulman [9], besides this we heavily rely on the work of Laczkovich [7].Definition 7. Let G be a group and n P N, n ě 2. A function F : G n Ñ C is said to be decomposable if it can be written as a finite sum of products F 1 ¨¨¨F k , where all F i depend on disjoint sets of variables.\\nRemark 2. Without loss of generality we can suppose that k \\\" 2 in the above definition, that is, decomposable functions are those mappings that can be written in the form\\nwhere E runs through all non-void proper subsets of t1, . . ., nu and for each E and j the function A E j depends only on variables x i with i P E, while B E j depends only on the variables x i with i E. Theorem 3. Let G be a commutative topological semigroup with unit.A continuous function f : G Ñ C is a generalized exponential polynomial if and only if there is a positive integer n ě 2 such that the mapping\\nThe notion of derivations can be extended in several ways.We will employ the concept of higher order derivations according to Reich [8] and Unger-Reich [11].For further results on characterization theorems on higher order derivations consult e.g.[1,2,3] and [5].Definition 8. Let F Ă C be a field.The identically zero map is the only derivation of order zero.For each n P N, an additive mapping f : F Ñ C is termed to be a derivation of order n, if there exists B : F ˆF Ñ C such that B is a bi-derivation of order n ´1 (that is, B is a derivation of order n ´1 in each variable) and f pxyq ´x f pyq ´f pxqy \\\" Bpx, yq px, y P Fq .\\nThe set of derivations of order n of the ring R will be denoted by D n pFq.For our future purposes the notion of differential operators will also be important, see [6].\\nDefinition 9. Let F Ă C be a field.We say that the map D : is a derivation on K for every i \\\" 1, . . ., k.Therefore, every differential operator\\nwhere the coefficients c i 1 ,...,i k belong to K, is a differential operator of order at most n, and also conversely, if D is a differential operator of order at most n on the field K \\\" Qpt 1 , . . ., t k q, then D is of the above form.\\nThe main result of [6] is Theorem 1.1 that reads in our settings as follows.\\nTheorem 4. Let F Ă C be a field and let n be a positive integer.Then, for every function D : F Ñ C, the following are equivalent.\\n(i) D P D n pFq (ii) D P cl pO n pFqq (iii) D is additive on F, Dp1q \\\" 0, and D{ j, as a map from the group F ˆto C, is a generalized polynomial of degree at most n.Here j stands for the identity map defined on F.\"\n },\n {\n \"title\": \"Results\",\n \"content\": \"\"\n },\n {\n \"title\": \"Elementary observations: reduction of the problem\",\n \"content\": \"This part begins with some elementary, yet fundamental observations.As the following lemmata show, the original problem can be reduced to a more simpler equation.\\nLemma 4 (Homogenization).Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers.Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation (1), that is,\\nf i px p i qg i px q i q \\\" 0 for each x P F. If the set tp 1 , . . ., p n u has a partition P 1 , . . ., P k with the property if p α , p β P P j for a certain index j, then p α `qα \\\" p β `qβ , then the system of equations ÿ p α PP j f α px p α qg α px q α q \\\" 0 px P F, j \\\" 1, . . ., kq is satisfied.\\nProof.Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers.Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation (1) for each x P F. Assume further that the set tp 1 , . . ., p n u has a partition P 1 , . . ., P k with the property if p α , p β P P j for a certain index j, then p α `qα \\\" p β `qβ .\\nObserve that for all i \\\" 1, . . ., n, the mapping\\nis a generalized monomial of degree p i `qi .Indeed, it is the diagonalization of the symmetric pp i `qi q-additive mapping\\nSince F Ă C, we necessarily have Q Ă F. Let now r P Q be arbitrary and substitute rx in place of x in equation (1) to get\\nUsing the Q-homogeneity of the additive functions f 1 , . . ., f n and g 1 , . . ., g n , we deduce 0 \\\"\\nNote that the right hand side of this equation is a (classical) polynomial in r which is identically zero.Thus all of its coefficients should be (identically) zero, yielding that the system of equations ÿ p α PP j f α px p α qg α px q α q \\\" 0 px P F, j \\\" 1, . . ., kq is fulfilled.\\nRemark 5.The above lemma guarantees that ab initio p i `qi \\\" N pi \\\" 1, . . ., nq can be assumed.Otherwise, after using the above homogenization, we get a system of functional equations in which this condition is already fulfilled.For instance, due to the above lemma, if the additive functions f 1 , . . ., f 5 : F Ñ C and g 1 , . . ., g 5 : F Ñ C satisfy equation f 1 px 24 qg 1 px 5 q `f2 px 20 qg 2 px 9 q `f3 px 19 qg 3 px 10 q `f4 px 13 qg 4 px 7 q `f5 px 12 qg 4 px 8 q \\\" 0 px P Fq then the equations f 1 px 24 qg 1 px 5 q `f2 px 20 qg 2 px 9 q `f3 px 19 qg 3 px 10 q \\\" 0 px P Fq and f 4 px 13 qg 4 px 7 q `f5 px 12 qg 4 px 8 q \\\" 0 px P Fq are also fulfilled (separately).\\nRemark 6.At first glance the assumption that p 1 , . . ., p n are different seems a reasonable and sufficient supposition.Clearly, if the parameters are not necessarily different then we cannot expect anything special for the form of the involved additive functions.Indeed, let L Ă C n be a linear subspace and let f 1 , . . ., f n : F Ñ C and g 1 , . . ., g n : F Ñ C be additive functions such that rngp f q Ă L and rngpgq Ă L K , where f pxq \\\" p f 1 pxq, . . ., f n pxqq and gpxq \\\" pg 1 pxq, . . ., g n pxqq px P Fq .\"\n },\n {\n \"title\": \"In this case\",\n \"content\": \"This shows the necessity of the above assumption.Unfortunately, the sufficiency fails to hold.To see this, let p and q be positive integers and f : F Ñ C be an arbitrary additive function and define the complex-valued functions f 1 , g 1 , f 2 , g 2 on F by\\nAn immediate computation shows that we have f 1 px p qg 1 px q q `f2 px p qg 2 px q q \\\" 0 px P Fq .\\nIn view of the above remarks, from now on, the following assumptions are adopted.\\nC(i) the positive integers p 1 , . . ., p n are arranged in a strictly increasing order, i.e., p 1 ă ¨¨¨ă p n ; C(ii) for all i \\\" 1, . . ., n we have p i `qi \\\" N;\\nC(iii) for all i, j P t1, . . ., nu, i j we have p i q j .Remark 7. Define the relation \\\" on F C by f \\\" g if and only if there exists an automorphism ϕ : C Ñ C such that ϕ ˝f \\\" g.Obviously \\\" is an equivalence relation on F C that induces a partition on F C .\"\n },\n {\n \"title\": \"Lemma 5 (Equivalence).\",\n \"content\": \"Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling the conditions C(i)-C(iii) of Remark 6. Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation (1).Then for an arbitrary automorphism ϕ : C Ñ C the functions ϕ ˝f1 , . . ., ϕ ˝fn , ϕ ˝g1 , . . ., ϕ ˝gn also fulfill equation (1).\"\n },\n {\n \"title\": \"Structure of solutions\",\n \"content\": \"We can always restrict ourselves to the case when all the involved functions are non-identically zero.Otherwise, the number of the terms appearing in equation ( 1) can be reduced.\"\n },\n {\n \"title\": \"Lemma 6 (Symmetrization).\",\n \"content\": \"Let k and n be positive integers, F Ă C be a field and m 1 , . . ., m n :\\nis fulfilled for all x 1 , . . ., x k , where for all i \\\" 1, . . ., n, the mapping M i : F k Ñ C is the uniquely determined symmetric, k-additive function such that M i px, . . ., xq \\\" m i pxq px P Fq .\"\n },\n {\n \"title\": \"Proof.\",\n \"content\": \"Let k and n be positive integers, F Ă C be a field and m 1 , . . ., m n : F Ñ C be monomials of degree k and assume that n ÿ i\\\"1 m i pxq \\\" 0 holds for all x P F. Since for all i \\\" 1, . . ., n, the function m i is a monomial of degree k, there exists a symmetric, k-additive function M i : F k Ñ C such that we have M i px, . . ., xq \\\" m i pxq px P Fq .\"\n },\n {\n \"title\": \"Obviously the mapping\",\n \"content\": \"m i is a monomial of degree k which is, by the assumptions, identically zero.To this monomial there also corresponds a symmetric and k-additive mapping, namely\\nObserve that the trace of this symmetric and k-additive mapping is identically zero.At the same time, due to the Polarization formula (Theorem 1), every symmetric and k-additive function is uniquely determined by its trace.Thus\\nM i px 1 , . . ., x k q \\\" 0 for all x 1 , . . ., x k P F.\"\n },\n {\n \"title\": \"Lemma 7 (Symmetrization).\",\n \"content\": \"Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling conditions C(ii), i.e., there is a N P N such p i `qi \\\" N for all i \\\" 1, . . ., n. Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation (1) for each x P F. Then\\nf i px σp1q ¨¨¨x σpp i q q ¨gi px σpp i `1q ¨¨¨x σpNq q \\\" 0 holds for all x 1 , . . ., x N P F.\\nProof.Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling conditions C(ii).Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation\\nf i px p i qg i px q i q \\\" 0 for each x P F. Due to the additivity of the functions f 1 , . . ., f n and g 1 , . . ., g n for all i \\\" 1, . . ., n, the mapping\\nis a monomial of degree p i `qi \\\" N. Further, it is the trace of the symmetric and N-additive mapping\\nTherefore, the statement follows from Lemma 6.\"\n },\n {\n \"title\": \"Solutions of equation (1)\",\n \"content\": \"The main purpose of the subsection is to describe under the conditions C(i)-C(iii), the solution space of equation (1).We first prove that solutions of equation ( 1) are decomposable functions on the multiplicative group F ˆ.In view of Laczkovich [7], this immediately yields that the solutions of equation ( 1) are generalized exponential polynomials of this group.\\nLemma 8. Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling conditions C(i) and C(ii).Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation (1) for each x P F. Then all the functions f 1 , . . ., f n as well as g 1 , . . ., g n are decomposable functions of the group F ˆ.\\nProof.Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling conditions C(i) and C(ii).\\nLet us assume first that condition C(iii) is also satisfied.Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation (1) for each x P F. Let S \\\" tp 1 , . . ., p n u Y tq 1 , . . ., q n u .\\nThen max S \\\" max tp n , q 1 u.By condition C(iii), we have p n q 1 .Without the loss of generality p n ą q 1 can be assumed, otherwise we follow a similar argument.In view of Lemma 7, we have\\nf i px σp1q ¨¨¨x σpp i q q ¨gi px σpp i `1q ¨¨¨x σpNq q \\\" 0 for all x 1 , . . ., x N P F, or after some rearrangement,\\nf i px σp1q ¨¨¨x σpp i q q ¨gi px σpp i `1q ¨¨¨x σpNq q `x1 , . . ., x N P F ˆ˘.\"\n },\n {\n \"title\": \"Let now\",\n \"content\": \"then the above identity says that g n p1q ¨fn is decomposable.If g n p1q were zero, but g n would not be identically zero, then there would exist a P F ˆsuch that g n paq 0. In this case the above substitutions should be modified to\\nto get the same conclusion.If p n \\\" q j for some j \\\" t1, . . ., nu (i.e., C(iii) does not hold), then without loss of generality we may assume that j \\\" 1, otherwise we may change the role of f i and g i , and p i and q i , respectively, and proceed as above.If p n \\\" q 1 , then we have\\nf i px σp1q ¨¨¨x σpp i q q ¨gi px σpp i `1q ¨¨¨x σpNq q `x1 , . . ., x N P F ˆ˘.\\nThis equation with the substitutions x p n `1 \\\" ¨¨¨\\\" x N \\\" 1, yields that a linear combination of f n and g 1 is decomposable.If t f n , g 1 u is linearly dependent, then this obviously means that both f n and g 1 are decomposable functions.If this system is not linearly dependent, then there exist a, b P F ˆ, a b and different complex constants c 1 and c 2 such that\\nWith the substitutions x p n `1 \\\" a, x p n `2 \\\" ¨¨¨\\\" x N \\\" 1, and\\nwe get that the functions px 1 , . . ., x p 1 q Þ ÝÑ g 1 px 1 ¨¨¨x p 1 q f n paq `fn px 1 ¨¨¨x p 1 qg 1 paq and px 1 , . . ., x p 1 q Þ ÝÑ g 1 px 1 ¨¨¨x p 1 q f n pbq `fn px 1 ¨¨¨x p 1 qg 1 pbq are decomposable.Since finite linear combinations of decomposable functions are also decomposable, it follows that f n and g 1 are decomposable, separately.After that, let us consider the set S z tp n u and apply the above argument for this set.With this step-by-step descending argument the statement of the lemma follows.\\nRemark 8. We emphasize that condition C(iii) in Lemma 8 has not been assumed.Thus, the fact that the additive solutions of (1) are decomposable functions can be deduced only under the conditions C(i) and C(ii).On the other hand, for our purpose to describe the solutions more concretely we have to assume also condition C(iii) to avoid further difficulties.We believe however that most of our methods can work similarly only under the conditions C(i) and C(ii).\\nRemark 9. Note also that in general we cannot state more than that the involved functions f 1 , . . ., f n and g 1 , . . ., g n are decomposable functions on the commutative group F ˆ.In other words, we can only state that the solutions of the functional equation in question are higher order derivations (see below Corollary 2).In general it is not true that the solutions of this functional equation are differential operators (i.e., exponential polynomials of the multiplicative group F ˆ).To see this, let us consider the functional equation x f 1 px 6 q `x2 f 2 px 5 q `x3 f 3 px 4 q \\\" 0 px P Fq .\\nIndeed, using the results of [5], we deduce that\\nDue to a result of Laczkovich [7] and under the assumptions of Lemma 8, there exist a positive integer l, there are generalized polynomials P k,i , Q k,i : F ˆÑ C (k \\\" 1, . . ., l and i \\\" 1, . . ., n) and there exist linearly independent exponentials m 1 , . . ., m l : F ˆÑ C such that\\nSince the generalized polynomials on any finitely generated field are polynomials and the exponentials m 1 , . . ., m k are linearly independent, we can apply Theorem 2. This implies that after substituting the above form the functions f 1 , . . ., f n and g 1 , . . ., g n and using that for each κ P N and k \\\" 1, . . ., n, we have\\nP k,i px p i qQ k,i px q i q \\\" 0 `x P F ˆfollows for all k \\\" 1, . . ., l.This tells us that it is enough to solve equation ( 1) for generalized polynomials of the group F ˆ.In fact, we can prove the following.\\nTheorem 5. Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling conditions C(i)-C(iii).Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy functional equation (1) for each x P F. Then there exists a positive integer l, there exist exponentials m i : F ˆÑ C and there are generalized polynomials P i , Q i : F ˆÑ C of degree at most l such that f i pxq \\\" P i pxqm i pxq and g i pxq \\\" Q i pxqm i pxq `x P F ˆfor each i \\\" 1, . . ., n.\\nProof.As we saw above, under the hypothesis of the lemma, if the functions f 1 , . . ., f n and g 1 , . . ., g n solve equation (1), then there exist a positive integer l, there are generalized polynomials P k,i , Q k,i : F ˆÑ C (k \\\" 1, . . ., l and i \\\" 1, . . ., n) and there exist linearly independent exponentials m 1 , . . ., m l : F ˆÑ C such that\\nAssume to the contrary that l ě 2. Let S \\\" tp 1 , . . ., p n u Y tq 1 , . . ., q n u .\\nThen due to conditions C(i)-C(iii) we have max S \\\" max tp n , q 1 u.Similarly as in Lemma 8, without the loss of generality p n ą q 1 can be assumed, otherwise we follow a similar argument.By our assumption l ě 2, this yields that there exist different exponential terms in f 1 and g 1 with nonzero polynomial coefficients.For the sake of simplicity, suppose that these different exponentials are m 1 and m 2 .Since max S \\\" p 1 , the term m p 1 1 m q 1 2 appears only in f 1 px p 1 qg 1 px q 1 q while expanding equation (1).Since generalized polynomials P k,i and exponentials m k satisfies the conditions of Theorem 2 for every finitely generated subfield of F, the coefficient of the above-mentioned term which is P 1,1 px p 1 qQ 1,1 px q 1 q has to vanish on F. From this we can deduce that P 1,1 or Q 1,1 is identically zero, contrary to our assumption.Thus\\nwith appropriate generalized polynomials P 1 , Q 1 : F ˆÑ C and exponential m 1 :\\nSuppose now that there is a positive integer k, less than n such that for all i \\\" 1, . . ., k we have\\nAssume that in the representation of f k`1 and g k`1 there are different exponentials with nonzero polynomial coefficients, say m j 1 and m j 2 .Observe that while expanding equation ( 1), the term m p k`1 j 1 m q k`1 j 2 appears only at once, namely in the product f k`1 px p k`1 qg k`1 px q k`1 q.Again, due to Theorem 2, we deduce that the appropriate polynomial term, that is, P k`1, j 1 px p k`1 qQ k`1, j 2 px q k`1 q has to vanish.This proves that necessarily\\nThis shows that there exist exponentials m 1 , . . ., m n : F ˆÑ C and generalized polynomials P 1 , . . ., P n and Q 1 , . . ., Q n on the group F ˆsuch that for all i \\\" 1, . . ., n f i pxq \\\" P i pxqm i pxq and\\nRemark 10.Due to conditions C(i)-C(iii), equation (1) has the form 0 \\\"\\nIf the exponentials appearing on the right hand side would be different, then by Theorem 2, their coefficients would be zero.This implies however that there exists a proper subset J Ă t1, . . ., nu such that ÿ jPJ f j px p j qg j px q j q \\\" 0 as well as ÿ j J f j px p j qg j px q j q \\\" 0 `x P F ˆ˘.\\n(\\nThis leads to the following definition of irreducible solutions.\\nDefinition 10.A system of solutions t f 1 , . . ., f n , g 1 , . . ., g n u of equation ( 1) is called irreducible if it does not satisfy a sub-term of (1).Otherwise, we say that a system of solutions is reducible.\\nClearly, a system of solutions t f 1 , . . ., f n , g 1 , . . ., g n u of (1) which fulfills ( 2) is a reducible solution.On the other hand, the argument in Remark 10 shows that every solution of (1) can be given as a sum of irreducible solutions of disjoint sub-terms of (1).Therefore, we restrict ourselves to the irreducible case, since every reducible solution can be deduced as a sum of irreducible solutions.\\nCorollary 2. Under the conditions of Theorem 5, suppose that system of functions t f 1 , . . ., f n , g 1 , . . ., g n u is an irreducible solution of equation (1).Then there exists an exponential m : F ˆÑ C and there are generalized polynomials P i , Q i : F ˆÑ C such that for each i \\\" 1, . . ., n f i pxq \\\" P i pxqmpxq and\\nIn other words, for each i \\\" 1, . . ., n there exists higher order derivations D i , r\\nwhere \\\" in the latter two equations is the equivalence relation defined in Remark 7.\\nProof.By Remark 10, all of the exponentials m i have to be the same in the description of the solutions of Theorem 5. Therefore, equation (3) describes the irreducible solutions of (1).Using Lemma 5, solutions of equation ( 1) are enough to be determined up to the equivalence relation \\\" defined in Remark 7. Accordingly, we can suppose that the exponential m in the above representation is the identity mapping.Hence, in view of Theorem 4 we get that the functions f 1 , . . ., f n as well as g 1 , . . ., g n are (or more precisely, are equivalent to) higher order derivations, as we stated.\"\n },\n {\n \"title\": \"The order of higher order derivation solutions\",\n \"content\": \"Every higher order derivation on F is a differential operator on any finitely generated subfield of F (see Theorem 4 and [6]).Hence on these fields the solutions are differential operators.Moreover, if the solutions on any finitely generated subfield of F are differential operators of order at most n, then every solution on F is a derivation of order n.From now on, instead of finding solutions as higher order derivations we are looking for differential operators as solutions.\\nFor this purpose, our next aim is to understand the arithmetic of the composition of derivations of the form d 1 ˝¨¨¨˝d r that are building blocks of differential operators.First we show that there is a natural form of composition of derivations that can be taken as a standard basis.\\nFor this target, the notion of moment function sequences turn out to be useful.Here we follow [4].A composition of a nonnegative integer n is a sequence of nonnegative integers α \\\" pα k q kPN such that n \\\"\\nFor a positive integer r, an r-composition of a nonnegative integer n is a composition α \\\" pα k q kPN with α k \\\" 0 for k ą r.\\nGiven a sequence of variables x \\\" px k q kPN and compositions α \\\" pα k q kPN and β \\\" pβ k q kPN we define α! \\\"\\nFurthermore, β ď α means that β k ď α k for all k P N and β ă α stands for β ď α and β α.\\nDefinition 11.Let G be a commutative group, r a positive integer, and for each multi-index α in N r let f α : G Ñ C be a continuous function.We say that p f α q αPN r is a generalized moment sequence of rank r, if\\nholds whenever x, y are in G.The function f 0 , where 0 is the zero element in N r , is called the generating function of the sequence.\\nTheorem 6.Let G be a commutative group, r a positive integer, and for each α in N r let f α : G Ñ C be a function.If the sequence of functions p f α q αPN r forms a generalized moment sequence of rank r, then there exists an exponential m : G Ñ C and a sequence of complex-valued additive functions a \\\" pa α q αPN r such that for every multi-index α in N r and x in G we have\\nwhere B α denotes the multivariate Bell polynomial corresponding to the multi-index α.\\nRemark 11.It is well-known that every polynomial P : C r Ñ C (r P N) can be given as a linear combination of Bell polynomials B α , where α P N r .Hence, momentum generating functions generate the exponential polynomial functions on G.By Lemma 2, polynomials of the form B α ˝a are linearly independent over C.\\nLemma 9. Let F Ă C be a field, r be a positive integer and d 1 , . . ., d r : F Ñ F be linearly independent derivations.For all multi-index α P N r , α \\\" pα 1 , . . ., α r q define the function ϕ α : F Ñ C by\\nThen pϕ α q αPN r is a generalized moment sequence of rank r on the commutative group F ˆand the generating function of the sequence is the identity function.\\nProof.Let F Ă C be a field, r be a positive integer and d 1 , . . ., d r : F Ñ F be linearly independent derivations.For all multi-index α P N r , α \\\" pα 1 , . . ., α r q define the function ϕ α : F Ñ C as in the lemma.We prove the statement by induction of the length of the multi-index α.Assume that α P N r and |α| \\\" 1.Then there exists i P t1, . . .ru such that α i \\\" 1 and α j \\\" 0 for j i and ϕ α pxyq \\\" d α i i pxyq \\\" d i pxyq \\\" xd i pyq `yd i pxq \\\" ϕ 0 pxqϕ α pyq `ϕ0 pyqϕ α pxq \\\" ÿ βďα ˆα β ˙ϕβ pxqϕ α´β pyq `x, y P F ˆ˘.\\nLet now α P N r and assume that the statement is true for all multi-indices β P N r with β ă α.Then ϕ α pxyq \\\" d α pxyq \\\" d α 1 1 ˝¨¨¨˝d α r r pxyq \\\" d 1 ˝¨¨¨d 1 loooomoooon Corollary 3. By Lemma 9 and Remark 11 implies that the functions d α \\\" d α 1 1 ˝¨¨¨˝d α r r constitute a basis of the differential operators in F ˆ. Since every d α is additive, by Lemma 2, all elements of the system td α pxqu, where α P Y rPN N r are algebraically independent.\\nA consequence of the algebraic independence of the elements of d α , where α P Y rPN N r is the following.Let P P Crx 1 , . . ., x n s be a polynomial, d 1 , . . .d r be derivations as in Lemma 9 With the above results, we are now ready to prove an upper bound for the order of derivations appearing in Theorem 5.\\nTheorem 7.Under the hypotheses of Theorem 5, the solutions f i and g i are derivations D i and r D i for all i \\\" 1, . . ., n.Let k and l denote the maximal orders of derivations D i and r D i , respectively.Suppose that there exists some i 1 such that the order of D i 1 and r D i 1 is exactly k and l, respectively.Then for all i \\\" 1, . . ., n the order of D i and r D i is less or equal to n ´1.\\nProof.Assume contrary that the maximal order k of the above derivations D 1 , . . ., D n is greater than n ´1.\\nThe argument for the case when the maximal order l of r D 1 , . . ., r D n is greater than n ´1 is analogous.By our assumption there exists an index i 1 such that the orders of the derivations D i 1 and r D i 1 is exactly k and l, respectively.It is important to note that then the sum of the orders of D i and r D i is at most k `l for any i, and it is exactly k `l for some indices if and only if the corresponding D and r D is of order k and of order l, respectively.Furthermore, by the algebraic independence of higher order derivations, there the number of these indices are at least two.\\nFrom now on we assume that F is finitely generated.Indeed, if we verify the statement for any finitely generated subfield of a field F, then it holds for F itself, as well.On finitely generated fields all of these higher order derivations can be represented as differential operators, that is, on finitely generated fields we have\\n. ., n) and higher order derivations\\nIf we expand the right hand side of the above identity with the aid of Proposition If this identity can be satisfied by different functions, it can also be satisfied by a single one.By Corollary 3, this enables us to substitute the functions d α i , r d β i pi \\\" 1, . . ., n, |α| ď k, |β| ď lq with suitable compositions of a given derivation d of order 1.In other words, instead of the above identity, we can restrict ourselves to n ÿ i\\\"1 ˜k ÿ j\\\"0 λ i, j d j px p i q ¸¨˜l ÿ j\\\"0 r λ i, j d j px q i q ¸\\\" 0 px P Fq with appropriate complex constants λ i, j (i \\\" 1, . . ., n, j \\\" 0, . . ., k), and r λ i, j (i \\\" 1, . . ., n, j \\\" 0, . . ., l) and derivation d : F Ñ C. By our assumptions there are some i 1 such that λ i 1 ,k 0 and r λ i 1 ,l 0. Dividing the above sum to smaller ones, we get\\nwhere\\nλ i, j d j px p i q and S pq i , l ´1q \\\" l´1 ÿ j\\\"0 r λ i, j d j px q i q px P Gq .\\nNote that, by the algebraic independence used in Corollary 3, this sum splits into separate terms of the form d s px p i qd t px q i q, where s `t is a fixed number.By our assumption, when s `t \\\" k `l, then the only way is that s \\\" k, t \\\" l.This implies that using Proposition 1 we get that 0 \\\"\\n, . . ., s, . . ., s loomoon\\nwhile the rest in the above sum can be computed similarly.\\nCase 1.If k ă l, then we compute the coefficients of the terms pdpxqq j d k´j pxqd l pxq p j \\\" 0, . . ., k ´1q.\\nFor each j \\\" 0, . . ., k ´1 this can be taken from the expansion of λ i,k r λ i,l d k px p i qd l px q i q in only one way.Namely, by splitting d k px p i q into j `1 parts and d l px q i q into one.Then the corresponding coefficients are\\nSince each of the terms contains ˆk 1, . . ., 1 loomoon j ˙1 j! \\\" ˆkj ˙, this can be eliminated from the above equation.The corresponding equations ( j \\\" 0, . . ., k) can be written in the following matrix equation\\nHere we note that as p i 's are all different positive integers, it follows that p i ě n for some i and hence the first n rows of the matrix are not identically zero, as k ą n ´1.It is straightforward to verify that the first n row of above matrix equation is equivalent to ¨p1 . . .p n p 2 1 . . .p 2 n . . . . . . . . .\\n. . .\\nSince this is a Vandermonde type matrix with different p i 's, the only solution of this homogeneous linear system is the zero vector, i.e., q i λ i,k r λ i,l \\\" 0 for all i \\\" 1, . . ., n.This contradicts to our assumption that there is some i 1 for which λ i 1 ,k 0 and r λ i 1 ,l 0 (q i 1 0 as q i 0 for all i P t1, . . ., ku).\\nCase 2. If k \\\" l, then we compute the coefficients of the terms pdpxqq 2 j pd k´j pxqq 2 p j \\\" 0, . . ., k ´1q.\\nIf j ă k 2 , then this term can be taken from the expansion of λ i,k r λ i,l d k px p i qd k px q i q in only one way.Namely, by splitting d k px p i q and d k px q i q into j `1 parts.Then the corresponding coefficients are\\nSince each of the terms contains ´`k 1, . . ., 1 loomoon j ˘1 j! ¯2 \\\" `kj ˘2, this can be eliminated from the above equations.\\nThese equations for j \\\" 0, . . ., rk{2s ´1 can be written in the following matrix equation\\nHere we note that as p i , q i are all different positive integers, it follow that max i tp i , q i u ě 2n and hence the first n 1 \\\" minpn, rk{2s ´1q rows of the matrix in not identically zero, as k ą n ´1.It is straightforward to verify that the first n 1 row of above matrix equation is equivalent to ¨p1 q 1 . . .p n q n p 2 1 q 2 1 . . .p 2 n q 2 n . . . . . . . . .\\nNote that if n 1 \\\" n, then we are done with a Vandermonde matrix argument similar as it is used in Case 1.\\nSo from now on we assume that n 1 \\\" rk{2s ´1.This also means that k ă 2n, thus the maximal order of the corresponding derivations is at most 2n ´1.\\nIf j ě k 2 , then the term pdpxqq 2 j pd k´j pxqq 2 can be taken from the expansion of λ i,k r λ i,l d k px p i qd k px q i q in three ways.One is as above, when we split both d k px p i q and d k px q i q into j `1 parts.Another one is when d k px p i q is split into k parts giving dpxq k , and d k px q i q provides pd k´j pxqq 2 pdpxqq 2 j´k .The third one is given by changing the role of p i and q i .Then the corresponding coefficients are\\nFirst we show that the second sum has to vanish for all j \\\" rk{2s, . . ., k ´1.In such cases, the coefficients ˆk 1, . . ., 1\\nThis clearly holds, since the term pdpxqq 2 j`1 d 2k´2 j´1 pxq in the expansion of d k px p i qd k px q i q for j \\\" rk{2s, . . ., k1 can be given in exactly two ways.Either pdpxqq k stems from d k px p i q and pdpxqq 2 j´k`1 ¨d2k´2j´1 pxq stems from d k px q i q, or reversely changing the role of p i and q i .Hence we get ˘`q 1 1 ˘. . .\\nwhere the first n rows of the matrix is not identically zero as p i and q i are different, hence there is an index i 1 such that p i 1 ě n and q i 1 ě n.Thus this system consisting the first n rows is equivalent to ¨p1 q 1 . . .p n q n p 2 1 q 2 1 . . .p 2 n q 2 n . . . . . . . . .\\nBy the usual Vandermonde argument as in Case 1, the only solution of this homogeneous linear system is the zero vector, i.e., λ i,k r λ i,l \\\" 0 for all i \\\" 1, . . ., n.This contradicts our assumption that there is some i 1 for which λ i 1 ,k 0 and r λ i 1 ,k 0.\\nCase 3. If k ą l, then we prove by induction in j that n ÿ i\\\"1 λ i,k r λ i,l p j`1´s i q s i \\\" 0 holds for every j \\\" 0, . . ., n ´1 and every s \\\" 0, . . ., j.\\nIn each step we consider how pdpxqq j d k´j pxqd l pxq can be given from the expansion d k px p i qd l px q i q.There are three possible ways, where this term can stem from.\\n(a) pdpxqq j d k´j pxq stems from d k px p i q and d l pxq stems from d l px q i q.This can happen for every j P t0, . . ., ku.\\nIn this case the coefficient of pdpxqq j d k´j pxqd l pxq is\\n˙qi .\\n(b) d k´j pxqd l pxqpdpxqq j´l stems from d k px p i q and pdpxqq l stems from d l px q i q.This can happen if j ě l.In this case the coefficient of pdpxqq j d k´j pxqd l pxq is (c) d l pxqpdpxqq k´l stems from d k px p i q and pdpxqq l´pk´jq d k´j pxq stems from d l px q i q.This can happen if l ě k´j.\\nIn this case the coefficient of pdpxqq j d k´j pxqd l pxq is ˙.\\nFor j \\\" 0 (and hence s \\\" 0) only the first term takes into account.This means that\\nSo the inductive hypothesis holds for j \\\" 0. Now we assume that n ÿ i\\\"1 λ i,k r λ i,l p j 1 `1´s i q s i \\\" 0 holds for every j 1 \\\" 0, . . ., j ´1 and every s \\\" 0, . . ., j 1 .We prove that n ÿ i\\\"1 λ i,k r λ i,l p j`1´s i q s i \\\" 0 holds for every s \\\" 0, . . ., j, as well.Generally, some of the previous compositions are possible for a given j but the following argument works in all cases, however we just prove it when all compositions discussed below appear in the expansion.Thus we assume that the coefficient of pdpxqq j d k´j pxqd l pxq is 0 \\\" By the inductive hypothesis and the fact that j ă n ď min p 1 i , q 1 i ( for some i 1 , this is equivalent to 0 \\\"\\nNote that the expressions p j`1´s i q s i for s \\\" 0, . . ., j can be interchanged in the following sense.By C(ii), p i `qi \\\" N, thus we have that p j`1´s i q s i \\\" N p j´s i q s i ´p j´s i q s`1 i .\\nAs\\nλ i,k r λ i,l p j´s i q s i \\\" 0 for every s \\\" 0, . . ., j ´1 by the inductive hypothesis, the term N p j´s i q s i can be eliminated.After several repetition of this step we get that 0 \\\"\\nWe also note that using this interchange rule and the inductive hypothesis it is clear that n ÿ i\\\"1 λ i,k r λ i,l p j`1´s i q s i \\\" 0 for any s \\\" 0, . . ., j is equivalent to\\nTherefore, to finish the proof it is enough to show that ˆkj ˙˘ˆk\\nThis is equivalent to verify that 1 j! ˘1 p j ´lq!pl!q ˘1 pk ´lq!pl ´pk ´jq!q 0.\\nMultiplying by j! this lead to\\nIt is straightforward to show using the growth of `n k ˘in k (if k ď n{2) that one term dominates the others if l k ´l and l j ´pk ´lq.Thus in these cases this (weighted) sum is nonzero.If l \\\" k ´l or l \\\" j ´pk ´lq, then either their sign is the same and hence the sum is nonzero, or their sign is different and hence they eliminate each other, hence the sum is 1 which is nonzero.\\nSummarizing, we get that ˆkj ˙˘ˆk k ´j, j ´l, l ˙˘ˆk k ´l, k ´j, l ´pk ´jq ˙ 0 and hence\\nwhich is equivalent to the inductive hypothesis for j as we noted above.\\nThus equation (8) holds for every j \\\" 0, . . ., n ´1.In matrix form this means that\\nSince this matrix is a Vandermonde type matrix with different p i 's, as Case 1 and Case 2, this implies that, the only solution of this homogeneous linear system is the zero vector, i.e., p i q i λ i,k r λ i,l \\\" 0 for all i \\\" 1, . . ., n.This contradicts to our assumption that there is some i 1 for which λ i 1 ,k 0 and r λ i 1 ,l 0 (and p i 0, q i 0 by C(i)).This also finishes the proof of the theorem, thus the order of all derivations involved in (1) is at most n ´1.\\nRemark 12.The upper bound appearing in the above theorem is sharp.To see this, let n, N be positive integers, λ 1 , . . ., λ n P C and let f : F Ñ C be an additive function for which n ÿ i\\\"1 λ i f px i qx N´i \\\" 0 is fulfilled for all x P F. Then f P D n´1 pRq if and only if λ i \\\" p´1q i ˆn i ˙for all i \\\" 1, . . ., n, see [1,3,5].\\nRemark 13.The proof of Theorem 7 in all the cases is based on the fact that the matrix\\nhas rank n, though it is far from being trivial to find the proper sub-matrix, which verifies that.On the other hand, the situation is much more complicated, if the maximal order k of D i , and the maximal order l of r D i is not uniquely determined.Namely, if there are several different pairs pk i , l i q so that k i `li \\\" K, where K is constant and k i is the maximum order of D i , l i is the maximal order of r D i .Then the equations first can only be determined for subsets of the index set, which satisfy some nontrivial relations.In this case the first task is to show that the problem can be formalized separately for the index sets, which seems a very hard problem in full generality.In this case our method can be applied.Theorem 7 and Remark 13 motivates our conjecture, that we verified for n ď 4.\\nConjecture 1.Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling conditions C(i)-C(iii).Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy equation (1).Then every solution is a generalized exponential polynomial function of degree at most n ´1 on F ˆ.\\nIn particular, if f i pxq \\\" D i pxq and g i pxq \\\" r D i pxq `x P F ˆ˘ (9) for each i \\\" 1, . . ., n, then the order of D i , r D i is at most n ´1.\\nThis conjecture leads to the following more general open question.\\nOpen Question 1.Is it true that every nonzero additive, irreducible solutions f 1 , . . ., f n of Pp f 1 px p 1 q, . . ., f n px p n qq \\\" 0, with Pp0, . . ., 0q \\\" 0 are derivations of order at most n ´1 and the identity function up to a homomorphism, if P : C Ñ C is polynomial and p 1 . . ., p n are distinct positive integers?\\nFinally we highlight some important special cases when Theorem 7 gives the proper bound of the order of the derivations.Corollary 4. Let n be a positive integer, F Ă C be a field and p 1 , . . ., p n , q 1 , . . ., q n be fixed positive integers fulfilling conditions C(i)-C(iii).Assume that the additive functions f 1 , . . ., f n , g 1 , . . ., g n : F Ñ C satisfy equation (1) as an irreducible solution.Then f i \\\" D i and g i \\\" r D i , where D i and r D i are higher order derivations.Assume further that one of the following holds.(A) All D i have the same order.This is the case, when f i pxq \\\" c i f pxq px P Fq for some nonzero constants c i P C, i P t1, . . ., nu.\\n(B) All r D i have the same order.This is the case, when g i pxq \\\" c i gpxq px P Fq for some nonzero constants c i P C, i P t1, . . ., nu.\\n(C) f i \\\" c i ¨gi for all i P t1, . . ., nu with some nonzero constants c i P C, i \\\" 1, . . ., n.\\nThen the order D i and r D i is at most n ´1.\\nProof.Theorem 5 implies that every solution f i (resp.g i ) is an exponential polynomial of the form P i ¨m (resp.Q i ¨m), which means that f i \\\" D i and g i \\\" r D i for some derivations D i and r D i .\\n(A) Let k denote the order of D i , and let l be the maximal order of r D i .Now we are in the position to apply Theorem 7.\\n(B) Similar to (A), since in case of equation ( 1), the role of the parameters p 1 , . . ., p n and q 1 , . . ., q n is symmetric.\\n(C) As the maximal degree of f i is the same as the maximal degree of g i and it is taken for the same index we can apply Theorem 7.\"\n },\n {\n \"title\": \"Special cases of equation (1)\",\n \"content\": \"In this subsection we consider special cases of equation (1).All the equations we consider here are of the form f 1 px p 1 qg 1 px q 1 q `f2 px p 2 qg 2 px q 2 q \\\" 0 px P Fq .\\nHere f 1 , f 2 , g 1 , g 2 : F Ñ C denote the unknown additive functions and the parameters p 1 , p 2 , q 1 , q 2 fulfill conditions C(i)-C(iii).Due to the results of the previous section, we get that f i pxq \\\" λ i,0 x `λi,1 d i pxq g i pxq \\\" µ i,0 x `µi,1 r d i pxq px P F, i \\\" 1, 2q with appropriate complex constants λ i, j , µ i, j (i \\\" 1, 2, j \\\" 0, 1) and derivations d i , di : F Ñ C (i \\\" 1, 2).\\nCorollary 5. Let N be a positive integer, F Ă C be a field and p, q be different positive integers (strictly) less than N and assume that q N ´p.If the additive functions f, g : F Ñ C satisfy f px p q f px N´p q \\\" gpx q qgpx N´q q px P Fq , Corollary 6.Let N be a positive integer, F Ă C be a field and p, q be different positive integers (strictly) less than N and assume that q N ´p.If the additive functions f, g : F Ñ C satisfy f px p qgpx N´p q \\\" κ f px q qgpx N´q q px P Fq .Both results implies that the nonzero additive solutions of equation f px p q f px N´p q \\\" κ f px q q f px N´q q px P Fq .\"\n },\n {\n \"title\": \"Then\",\n \"content\": \"are derivations of order 1 (κ \\\" ppN ´pq qpN ´qq ) or the identity function (κ \\\" 1) up to a homomorphism.\"\n },\n {\n \"title\": \"α 1 times ˝¨¨¨˝d r ˝¨¨¨˝d r looooomooooon α r times pxyq \\\" d 1 ˝¨¨¨˝d 1 looooomooooon α 1 times ˝¨¨¨˝d r´1 ˝¨¨¨˝d r´1 loooooooomoooooooon α r´1 times pd α r r pxyqq \\\" d 1 ˝¨¨¨˝d 1 looooomooooon α 1 times ˝¨¨¨˝d r´1 ˝¨¨¨˝d r´1 loooooooomoooooooon α r´1 times 1 1\",\n \"content\": \"˝¨¨¨˝d β r r pxq ¨dα 1 ´β1 1 ˝¨¨¨˝d α r ´βr r pyq \\\" ÿ βďα ˆα β ˙ϕβ pxqϕ α´β pyq `x, y P F ˆ˘.\\nthen A) either there exists a homomorphism ϕ : C Ñ C, a derivation d : F Ñ C such that f pxq \\\" ϕpdpxqq and gpxq \\\" αϕpdpxqq px P Fq , where α \\\" ppN ´pq qpN ´qq , B) or there exists a homomorphism ϕ : F Ñ C such that f pxq \\\" f p1q ¨ϕpxq gpxq \\\" ˘f p1q ¨ϕpxq px P Fq .\\nthat is, the notions of first order derivations and derivations coincide.On the other hand for any n P N the set D n pFqzD n´1 pFq is nonempty because d 1 ˝¨¨¨˝d n P D n pFq, but d 1 ˝¨¨¨˝d n D n´1 pRq, where d 1 , . . ., d n P D 1 pFq are non-identically zero derivations.\\noperator of order at most n if D is the linear combination, with coefficients from F, of finitely many maps of the form d 1 ˝¨¨¨˝d k , where d 1 , . . ., d k are derivations on F and k ď n.If k \\\" 0 then we interpret d 1 ˝¨¨¨˝d k as the identity function.We denote by O n pFq the set of differential operators of order at most n defined on F. We say that the order of a differential operator D is n if D P O n pFqzO n´1 pFq (where O ´1pFq \\\" H, by definition).\\nand α 1 , . . ., α n P Y rPN N r .Then the following polynomial form Ppd α 1 pxq, . . ., d α ,...,l p ě0 l 1 `¨¨¨`l p \\\"k ˆk l 1 , . . ., l p ˙¨d l 1 pxq ¨¨¨d l p pxq px 1 , . . ., x p P Fq ,\\nn pxqq \\\" 0 px P Gq holds if and only if Pp d |α 1 | pxq, . . ., d |α n | pxqq \\\" 0 px P Gq , where d is an arbitrary derivation (of order 1).In other words, we can substitute d 1 , . . ., d r by d in d α 1 , . . ., d α n .By Corollary 3, it would be desirable to calculate d k \\\" d ˝¨¨¨˝d loooomoooon k times , where d is a derivation (of order 1) and k P N. Lemma 9, together with [4, Proposition 1], implies the following statement.Proposition 1.Let F Ă C be a field and d : F Ñ C a derivation.For all positive integer k we k px 1 ¨¨¨x p q \\\" ÿ l 1 ,...,l p ě0 l 1 `¨¨¨`l p \\\"k ˆk l 1 , . . ., l p ˙p ź t\\\"1 d l t px t q px 1 , . . ., x p P Fq , where j 1 , . . ., j s denotes the number of dpxq, . . ., d s pxq in a given composition of d k px p q.\\nl, 1, . . ., 1\\nl ˙.\\nppN ´pq qpN ´qq , then there exists a homomorphism ϕ : C Ñ C and derivations d 1 , d 2 : F Ñ C such that f pxq \\\" ϕpd 1 pxqq and gpxq \\\" ϕpd 2 pxqq px P Fq .\\nB) if κ \\\" 1, then the only possibility is that f pxq \\\" f p1q ¨ψpxq and gpxq \\\" f p1q ¨ψpxq px P Fq , where ψ :F Ñ C is a homomorphism, C) if κ \\\"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"˝¨¨¨˝d β r r pxq ¨dα 1 ´β1 1 ˝¨¨¨˝d α r ´βr r pyq \" ÿ βďα ˆα β ˙ϕβ pxqϕ α´β pyq `x, y P F ˆ˘.","content":"α 1 times ˝¨¨¨˝d r ˝¨¨¨˝d r looooomooooon α r times pxyq \" d 1 ˝¨¨¨˝d 1 looooomooooon α 1 times ˝¨¨¨˝d r´1 ˝¨¨¨˝d r´1 loooooooomoooooooon α r´1 times pd α r r pxyqq \" d 1 ˝¨¨¨˝d 1 looooomooooon α 1 times ˝¨¨¨˝d r´1 ˝¨¨¨˝d r´1 loooooooomoooooooon α r´1 times 1 1\n1\n˝¨¨¨˝d β r r pxq ¨dα 1 ´β1 1 ˝¨¨¨˝d α r ´βr r pyq \" ÿ βďα ˆα β ˙ϕβ pxqϕ α´β pyq `x, y P F ˆ˘."},{"id":"","caption":"then A) either there exists a homomorphism ϕ : C Ñ C, a derivation d : F Ñ C such that f pxq \" ϕpdpxqq and gpxq \" αϕpdpxqq px P Fq , where α \" ppN ´pq qpN ´qq , B) or there exists a homomorphism ϕ : F Ñ C such that f pxq \" f p1q ¨ϕpxq gpxq \" ˘f p1q ¨ϕpxq px P Fq .","content":"then A) either there exists a homomorphism ϕ : C Ñ C, a derivation d : F Ñ C such that f pxq \" ϕpdpxqq and gpxq \" αϕpdpxqq px P Fq , where α \" ppN ´pq qpN ´qq , B) or there exists a homomorphism ϕ : F Ñ C such that f pxq \" f p1q ¨ϕpxq gpxq \" ˘f p1q ¨ϕpxq px P Fq ."},{"id":"","caption":"that is, the notions of first order derivations and derivations coincide.On the other hand for any n P N the set D n pFqzD n´1 pFq is nonempty because d 1 ˝¨¨¨˝d n P D n pFq, but d 1 ˝¨¨¨˝d n D n´1 pRq, where d 1 , . . ., d n P D 1 pFq are non-identically zero derivations.","content":"that is, the notions of first order derivations and derivations coincide.On the other hand for any n P N the set D n pFqzD n´1 pFq is nonempty because d 1 ˝¨¨¨˝d n P D n pFq, but d 1 ˝¨¨¨˝d n D n´1 pRq, where d 1 , . . ., d n P D 1 pFq are non-identically zero derivations.\nRemark 3. Since D 0 pFq \" t0u, the only bi-derivation of order zero is the identically zero function, thusf P D 1 pFq if and only iff pxyq \" x f pyq `f pxqypx, y P Fq ,"},{"id":"","caption":"operator of order at most n if D is the linear combination, with coefficients from F, of finitely many maps of the form d 1 ˝¨¨¨˝d k , where d 1 , . . ., d k are derivations on F and k ď n.If k \" 0 then we interpret d 1 ˝¨¨¨˝d k as the identity function.We denote by O n pFq the set of differential operators of order at most n defined on F. We say that the order of a differential operator D is n if D P O n pFqzO n´1 pFq (where O ´1pFq \" H, by definition).","content":"operator of order at most n if D is the linear combination, with coefficients from F, of finitely many maps of the form d 1 ˝¨¨¨˝d k , where d 1 , . . ., d k are derivations on F and k ď n.If k \" 0 then we interpret d 1 ˝¨¨¨˝d k as the identity function.We denote by O n pFq the set of differential operators of order at most n defined on F. We say that the order of a differential operator D is n if D P O n pFqzO n´1 pFq (where O ´1pFq \" H, by definition).\nRemark 4. The term differential operator is justified by the following fact. Let K \" Qpt 1 , . . . , t k q, wheret 1 , . . . , t k are algebraically independent over Q. Then K is the field of all rational functions of t 1 , . . . , t k withrational coefficients. It is clear thatd i \"B Bt i"},{"id":"","caption":"and α 1 , . . ., α n P Y rPN N r .Then the following polynomial form Ppd α 1 pxq, . . ., d α ,...,l p ě0 l 1 `¨¨¨`l p \"k ˆk l 1 , . . ., l p ˙¨d l 1 pxq ¨¨¨d l p pxq px 1 , . . ., x p P Fq ,","content":"and α 1 , . . ., α n P Y rPN N r .Then the following polynomial form Ppd α 1 pxq, . . ., d α ,...,l p ě0 l 1 `¨¨¨`l p \"k ˆk l 1 , . . ., l p ˙¨d l 1 pxq ¨¨¨d l p pxq px 1 , . . ., x p P Fq ,\ndefine the are adopted. Especially, for all positive integer px P Fq . l 1 Reordering the previous expression we can get the following function d k on F by d k pxq \" d ˝¨¨¨˝d loooomoooon k times pxq Then for all positive integer p we have ˆk l 1 , . . . , l p ˙\" k! l 1 ! ¨¨¨l p ! p, we have d k px p q \" ÿ p 1 d where the conventions d 0 \" id and d k px p q \" ÿ ˆk 1, . . . , 1 loomoon , . . . , s, . . . , s loomoon ˙s ź t\"1 1 p p j t !q j 1 `¨¨¨`j s \"p 1 ăp j 1 `2 j 2 `¨¨¨`s j s \"k j 1 j s 1, . . . , 1 loomoon ˙¨pdpxqq j 1 ¨¨¨pd s pxqq j s ¨xp´p 1 , px P Fq\nn pxqq \" 0 px P Gq holds if and only if Pp d |α 1 | pxq, . . ., d |α n | pxqq \" 0 px P Gq , where d is an arbitrary derivation (of order 1).In other words, we can substitute d 1 , . . ., d r by d in d α 1 , . . ., d α n .By Corollary 3, it would be desirable to calculate d k \" d ˝¨¨¨˝d loooomoooon k times , where d is a derivation (of order 1) and k P N. Lemma 9, together with [4, Proposition 1], implies the following statement.Proposition 1.Let F Ă C be a field and d : F Ñ C a derivation.For all positive integer k we k px 1 ¨¨¨x p q \" ÿ l 1 ,...,l p ě0 l 1 `¨¨¨`l p \"k ˆk l 1 , . . ., l p ˙p ź t\"1 d l t px t q px 1 , . . ., x p P Fq , where j 1 , . . ., j s denotes the number of dpxq, . . ., d s pxq in a given composition of d k px p q."},{"id":"","caption":"l, 1, . . ., 1","content":"l, 1, . . ., 1\nloomoon ˙1 p j ´lq! ˆl 1, . . . , 1 loomoon ˙1 l!1, . . . , 1 loomoon ˆpi ˙ˆq i 1, . . . , 1 loomoon \"n ÿ i\"1λ i,k r λ i,lj´l ˆk k ´j, j ´l, l ˙ˆp i 1, . . . , 1 loomoon ˙ˆq i l 1, . . . , 1 loomoonj´l`2lj´l`2\nl ˙."},{"id":"","caption":"ppN ´pq qpN ´qq , then there exists a homomorphism ϕ : C Ñ C and derivations d 1 , d 2 : F Ñ C such that f pxq \" ϕpd 1 pxqq and gpxq \" ϕpd 2 pxqq px P Fq .","content":"ppN ´pq qpN ´qq , then there exists a homomorphism ϕ : C Ñ C and derivations d 1 , d 2 : F Ñ C such that f pxq \" ϕpd 1 pxqq and gpxq \" ϕpd 2 pxqq px P Fq .\nA) if κ\"1,ppN ´pq qpN ´qq*, then f is identically zero,\nB) if κ \" 1, then the only possibility is that f pxq \" f p1q ¨ψpxq and gpxq \" f p1q ¨ψpxq px P Fq , where ψ :F Ñ C is a homomorphism, C) if κ \""}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"˝¨¨¨˝d β r r pxq ¨dα 1 ´β1 1 ˝¨¨¨˝d α r ´βr r pyq \\\" ÿ βďα ˆα β ˙ϕβ pxqϕ α´β pyq `x, y P F ˆ˘.\",\n \"content\": \"α 1 times ˝¨¨¨˝d r ˝¨¨¨˝d r looooomooooon α r times pxyq \\\" d 1 ˝¨¨¨˝d 1 looooomooooon α 1 times ˝¨¨¨˝d r´1 ˝¨¨¨˝d r´1 loooooooomoooooooon α r´1 times pd α r r pxyqq \\\" d 1 ˝¨¨¨˝d 1 looooomooooon α 1 times ˝¨¨¨˝d r´1 ˝¨¨¨˝d r´1 loooooooomoooooooon α r´1 times 1 1\\n1\\n˝¨¨¨˝d β r r pxq ¨dα 1 ´β1 1 ˝¨¨¨˝d α r ´βr r pyq \\\" ÿ βďα ˆα β ˙ϕβ pxqϕ α´β pyq `x, y P F ˆ˘.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"then A) either there exists a homomorphism ϕ : C Ñ C, a derivation d : F Ñ C such that f pxq \\\" ϕpdpxqq and gpxq \\\" αϕpdpxqq px P Fq , where α \\\" ppN ´pq qpN ´qq , B) or there exists a homomorphism ϕ : F Ñ C such that f pxq \\\" f p1q ¨ϕpxq gpxq \\\" ˘f p1q ¨ϕpxq px P Fq .\",\n \"content\": \"then A) either there exists a homomorphism ϕ : C Ñ C, a derivation d : F Ñ C such that f pxq \\\" ϕpdpxqq and gpxq \\\" αϕpdpxqq px P Fq , where α \\\" ppN ´pq qpN ´qq , B) or there exists a homomorphism ϕ : F Ñ C such that f pxq \\\" f p1q ¨ϕpxq gpxq \\\" ˘f p1q ¨ϕpxq px P Fq .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"that is, the notions of first order derivations and derivations coincide.On the other hand for any n P N the set D n pFqzD n´1 pFq is nonempty because d 1 ˝¨¨¨˝d n P D n pFq, but d 1 ˝¨¨¨˝d n D n´1 pRq, where d 1 , . . ., d n P D 1 pFq are non-identically zero derivations.\",\n \"content\": \"that is, the notions of first order derivations and derivations coincide.On the other hand for any n P N the set D n pFqzD n´1 pFq is nonempty because d 1 ˝¨¨¨˝d n P D n pFq, but d 1 ˝¨¨¨˝d n D n´1 pRq, where d 1 , . . ., d n P D 1 pFq are non-identically zero derivations.\\nRemark 3. Since D 0 pFq \\\" t0u, the only bi-derivation of order zero is the identically zero function, thusf P D 1 pFq if and only iff pxyq \\\" x f pyq `f pxqypx, y P Fq ,\"\n },\n {\n \"id\": \"\",\n \"caption\": \"operator of order at most n if D is the linear combination, with coefficients from F, of finitely many maps of the form d 1 ˝¨¨¨˝d k , where d 1 , . . ., d k are derivations on F and k ď n.If k \\\" 0 then we interpret d 1 ˝¨¨¨˝d k as the identity function.We denote by O n pFq the set of differential operators of order at most n defined on F. We say that the order of a differential operator D is n if D P O n pFqzO n´1 pFq (where O ´1pFq \\\" H, by definition).\",\n \"content\": \"operator of order at most n if D is the linear combination, with coefficients from F, of finitely many maps of the form d 1 ˝¨¨¨˝d k , where d 1 , . . ., d k are derivations on F and k ď n.If k \\\" 0 then we interpret d 1 ˝¨¨¨˝d k as the identity function.We denote by O n pFq the set of differential operators of order at most n defined on F. We say that the order of a differential operator D is n if D P O n pFqzO n´1 pFq (where O ´1pFq \\\" H, by definition).\\nRemark 4. The term differential operator is justified by the following fact. Let K \\\" Qpt 1 , . . . , t k q, wheret 1 , . . . , t k are algebraically independent over Q. Then K is the field of all rational functions of t 1 , . . . , t k withrational coefficients. It is clear thatd i \\\"B Bt i\"\n },\n {\n \"id\": \"\",\n \"caption\": \"and α 1 , . . ., α n P Y rPN N r .Then the following polynomial form Ppd α 1 pxq, . . ., d α ,...,l p ě0 l 1 `¨¨¨`l p \\\"k ˆk l 1 , . . ., l p ˙¨d l 1 pxq ¨¨¨d l p pxq px 1 , . . ., x p P Fq ,\",\n \"content\": \"and α 1 , . . ., α n P Y rPN N r .Then the following polynomial form Ppd α 1 pxq, . . ., d α ,...,l p ě0 l 1 `¨¨¨`l p \\\"k ˆk l 1 , . . ., l p ˙¨d l 1 pxq ¨¨¨d l p pxq px 1 , . . ., x p P Fq ,\\ndefine the are adopted. Especially, for all positive integer px P Fq . l 1 Reordering the previous expression we can get the following function d k on F by d k pxq \\\" d ˝¨¨¨˝d loooomoooon k times pxq Then for all positive integer p we have ˆk l 1 , . . . , l p ˙\\\" k! l 1 ! ¨¨¨l p ! p, we have d k px p q \\\" ÿ p 1 d where the conventions d 0 \\\" id and d k px p q \\\" ÿ ˆk 1, . . . , 1 loomoon , . . . , s, . . . , s loomoon ˙s ź t\\\"1 1 p p j t !q j 1 `¨¨¨`j s \\\"p 1 ăp j 1 `2 j 2 `¨¨¨`s j s \\\"k j 1 j s 1, . . . , 1 loomoon ˙¨pdpxqq j 1 ¨¨¨pd s pxqq j s ¨xp´p 1 , px P Fq\\nn pxqq \\\" 0 px P Gq holds if and only if Pp d |α 1 | pxq, . . ., d |α n | pxqq \\\" 0 px P Gq , where d is an arbitrary derivation (of order 1).In other words, we can substitute d 1 , . . ., d r by d in d α 1 , . . ., d α n .By Corollary 3, it would be desirable to calculate d k \\\" d ˝¨¨¨˝d loooomoooon k times , where d is a derivation (of order 1) and k P N. Lemma 9, together with [4, Proposition 1], implies the following statement.Proposition 1.Let F Ă C be a field and d : F Ñ C a derivation.For all positive integer k we k px 1 ¨¨¨x p q \\\" ÿ l 1 ,...,l p ě0 l 1 `¨¨¨`l p \\\"k ˆk l 1 , . . ., l p ˙p ź t\\\"1 d l t px t q px 1 , . . ., x p P Fq , where j 1 , . . ., j s denotes the number of dpxq, . . ., d s pxq in a given composition of d k px p q.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"l, 1, . . ., 1\",\n \"content\": \"l, 1, . . ., 1\\nloomoon ˙1 p j ´lq! ˆl 1, . . . , 1 loomoon ˙1 l!1, . . . , 1 loomoon ˆpi ˙ˆq i 1, . . . , 1 loomoon \\\"n ÿ i\\\"1λ i,k r λ i,lj´l ˆk k ´j, j ´l, l ˙ˆp i 1, . . . , 1 loomoon ˙ˆq i l 1, . . . , 1 loomoonj´l`2lj´l`2\\nl ˙.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"ppN ´pq qpN ´qq , then there exists a homomorphism ϕ : C Ñ C and derivations d 1 , d 2 : F Ñ C such that f pxq \\\" ϕpd 1 pxqq and gpxq \\\" ϕpd 2 pxqq px P Fq .\",\n \"content\": \"ppN ´pq qpN ´qq , then there exists a homomorphism ϕ : C Ñ C and derivations d 1 , d 2 : F Ñ C such that f pxq \\\" ϕpd 1 pxqq and gpxq \\\" ϕpd 2 pxqq px P Fq .\\nA) if κ\\\"1,ppN ´pq qpN ´qq*, then f is identically zero,\\nB) if κ \\\" 1, then the only possibility is that f pxq \\\" f p1q ¨ψpxq and gpxq \\\" f p1q ¨ψpxq px P Fq , where ψ :F Ñ C is a homomorphism, C) if κ \\\"\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Characterizing ring derivations of all orders via functional equations: results and open problems. Bruce Ebanks, Aequationes Math. 8932015","doi":""},{"text":"Polynomially linked additive functions. Bruce Ebanks, Aequationes Math. 9122017","doi":""},{"text":"On the order of a derivation. Bruce Ebanks, Thomas Riedel, Prasanna K Sahoo, Aequationes Math. 9022016","doi":""},{"text":". Żywilla Fechner, Eszter Gselmann, László Székelyhidi, Moment functions on groups. Results Math. 7642021","doi":""},{"text":"On functional equations characterizing derivations: methods and examples. Gergely Eszter Gselmann, Csaba Kiss, Vincze, Paper No. 74. 201873","doi":""},{"text":"Derivations and differential operators on rings and fields. Gergely Kiss, Miklós Laczkovich, Ann. Univ. Sci. Budapest. Sect. Comput. 482018","doi":""},{"text":"A characterization of generalized exponential polynomials in terms of decomposable functions. Miklós Laczkovich, Acta Math. Hungar. 15822019","doi":""},{"text":"Derivationen zweiter Ordnung als Lösungen von Funktionalgleichungen-ein Überblick. Ludwig Reich, Gyula Maurer zum 70. Graz1998337Karl-Franzens-Univ. Graz","doi":""},{"text":"Decomposable functions and representations of topological semigroups. Ekaterina Shulman, Aequationes Math. 791-22010","doi":""},{"text":"László Székelyhidi, Convolution type functional equations on topological abelian groups. Teaneck, NJWorld Scientific Publishing Co., Inc1991","doi":""},{"text":"Derivationen höherer Ordnung als Lösungen von Funktionalgleichungen. Josef Unger, Ludwig Reich, Grazer Mathematische Berichte. 336Graz Mathematical Reports","doi":""},{"text":". Karl-Franzens , 1998GrazUniversität Graz","doi":""}],"string":"[\n {\n \"text\": \"Characterizing ring derivations of all orders via functional equations: results and open problems. Bruce Ebanks, Aequationes Math. 8932015\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polynomially linked additive functions. Bruce Ebanks, Aequationes Math. 9122017\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the order of a derivation. Bruce Ebanks, Thomas Riedel, Prasanna K Sahoo, Aequationes Math. 9022016\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Żywilla Fechner, Eszter Gselmann, László Székelyhidi, Moment functions on groups. Results Math. 7642021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On functional equations characterizing derivations: methods and examples. Gergely Eszter Gselmann, Csaba Kiss, Vincze, Paper No. 74. 201873\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Derivations and differential operators on rings and fields. Gergely Kiss, Miklós Laczkovich, Ann. Univ. Sci. Budapest. Sect. Comput. 482018\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A characterization of generalized exponential polynomials in terms of decomposable functions. Miklós Laczkovich, Acta Math. Hungar. 15822019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Derivationen zweiter Ordnung als Lösungen von Funktionalgleichungen-ein Überblick. Ludwig Reich, Gyula Maurer zum 70. Graz1998337Karl-Franzens-Univ. Graz\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Decomposable functions and representations of topological semigroups. Ekaterina Shulman, Aequationes Math. 791-22010\",\n \"doi\": \"\"\n },\n {\n \"text\": \"László Székelyhidi, Convolution type functional equations on topological abelian groups. Teaneck, NJWorld Scientific Publishing Co., Inc1991\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Derivationen höherer Ordnung als Lösungen von Funktionalgleichungen. Josef Unger, Ludwig Reich, Grazer Mathematische Berichte. 336Graz Mathematical Reports\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Karl-Franzens , 1998GrazUniversität Graz\",\n \"doi\": \"\"\n }\n]","isConversational":false}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":4,"numItemsPerPage":100,"numTotalItems":2579762,"offset":400,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sInJlcG9JZHMiOlsiNjc3NDI4YmI1YThjMWM5YzgyNWVmYTYwIl0sImlhdCI6MTc5NDUwNTAyNiwianRpIjoiNGJlNTgyMmItOWZiNi00MzlkLWE0NmYtMzBjZTdkNmZhNzZkIiwic3ViIjoiL2RhdGFzZXRzL0FsZ29yaXRobWljUmVzZWFyY2hHcm91cC9zMm9yY19hcnhpdiIsImV4cCI6MTc5NDUwODYyNiwiaXNzIjoiaHR0cHM6Ly9odWdnaW5nZmFjZS5jbyJ9.BPLGG9cf8xA36yPmXBUSqG9wDfINPdk-NT1HfKHnmenWUyg5RQ0FR3IAnABA-EgyL9I14rVoR65G547M9r0OBQ","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":2579762,"numBytesParquet":266094171612}]},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
text stringlengths 4 760k | corpus_id int64 230 275M | external_ids dict | source dict | title stringlengths 0 3.69k | authors listlengths 0 4.6k | abstract stringlengths 0 34.2k | sections listlengths 0 734 | figures listlengths 0 570 | references listlengths 0 1.74k |
|---|---|---|---|---|---|---|---|---|---|
Parametrized Quantum Circuits and their approximation capacities in the context of quantum machine learning
Alberto Manzano
Department of Mathematics and CITIC
Universidade da Coruña
Campus de Elviña s/nA CoruñaSpain
David Dechant
⟨aQa L ⟩ Applied Quantum Algorithms Leiden
The Netherlands
Instituut-Lorentz
Unive... | 260,203,221 | {
"arxiv": "2307.14792",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2307.14792v1.pdf"
],
"pdf_sha": "e97c860d446443518d89209b4037b5cda5cd4166",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Parametrized Quantum Circuits and their approximation capacities in the context of quantum machine learning | [
"Alberto Manzano \nDepartment of Mathematics and CITIC\nUniversidade da Coruña\nCampus de Elviña s/nA CoruñaSpain",
"David Dechant \n⟨aQa L ⟩ Applied Quantum Algorithms Leiden\nThe Netherlands\n\nInstituut-Lorentz\nUniversiteit Leiden\nP.O. Box 95062300 RALeidenThe Netherlands",
"Jordi Tura \n⟨aQa L ⟩ Applied Q... | Parametrized quantum circuits (PQC) are quantum circuits which consist of both fixed and parametrized gates. In recent approaches to quantum machine learning (QML), PQCs are essentially ubiquitous and play the role analogous to classical neural networks. They are used to learn various types of data, with an underlying ... | [
{
"title": "Introduction",
"content": "Machine learning has gained significant attention in recent years for its practical applications and Alberto Manzano: alberto.manzano.herrero@udc.es U (x, θ) represents a quantum circuit that takes x as input and with variational parameters θ, f θ (x) is the expected v... | [
{
"id": "",
"caption": "Sketch of a hybrid variational algorithm.",
"content": "Figure 1 :\n1Sketch of a hybrid variational algorithm."
},
{
"id": "",
"caption": "Let {H m } be a universal Hamiltonian family, and {f m } the associated quantum model family, defined via",
"content": "Theor... | [
{
"text": "Sobolev spaces. A Robert, John Jf Adams, Fournier, ElsevierRobert A Adams and John JF Fournier. Sobolev spaces. Elsevier, 2003.",
"doi": ""
},
{
"text": "On Maz'ya's work in functional analysis, partial differential equations and applications. Victor Burenkov, 10.1007/978-3-0348-8675-8_13... |
BACKGROUND COHOMOLOGY AND SYMPLECTIC REDUCTION
4 Nov 2024
Maxim Braverman
BACKGROUND COHOMOLOGY AND SYMPLECTIC REDUCTION
4 Nov 2024A254DA9570DC876CCA39821C2E79070AarXiv:2410.05532v2[math.SG]2020 Mathematics Subject Classification. 53D2032L1032Q20 Geometric quantizationSymplectic reductionQuantization commutes with r... | 273,229,333 | {
"arxiv": "2410.05532",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2410.05532v2.pdf"
],
"pdf_sha": "f9f5f9fc9ae505329b1ec3bd8fbe6eb6c67eab5e",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | BACKGROUND COHOMOLOGY AND SYMPLECTIC REDUCTION | [
"Maxim Braverman"
] | We consider a Hamiltonian action of a compact Lie group G on a complete Kähler manifold M with a proper moment map.In a previous paper, we defined a regularized version of the Dolbeault cohomology of a G-equivariant holomorphic vector bundle, called the background cohomology.In this paper, we show that the background c... | [
{
"title": "Introduction",
"content": "If E is a holomorphic vector bundle over a compact Kähler manifold M , the Dolbeault cohomology H 0, * (M, E) is finite-dimensional and has a lot of nice properties.In particular, given a Hamiltonian action of a compact group G on M and a prequantum G-equivariant line ... | [
{
"id": "",
"caption": "A Hamiltonian group action.Suppose a compact Lie group G acts holomorphically on a complete Kähler manifold M .The moment map µ : M → g * is defined by the formula d µ, u = ι(u) ω, where u ∈ g, ω ∈ Ω 2 (M ) is the Kähler form on M and u(x) := d dt t=0 exp(tu) x ∈ T x M is the vector ... | [
{
"text": "M Braverman, Quantization of Singular Symplectic Quotients. Progress in Mathematics. N P Landsman, M Pflaum, M Schlichenmaier, Birkhäuser198Cohomology of the Mumford quotient",
"doi": ""
},
{
"text": "M Braverman, Index theorem for equivariant Dirac operators on noncompact manifolds, K-Th... |
MULE: Multimodal Universal Language Embedding
Donghyun Kim
Boston University
Kuniaki Saito keisaito@bu.edu
Boston University
Kate Saenko saenko@bu.edu
Boston University
Stan Sclaroff sclaroff@bu.edu
Boston University
Bryan A Plummer
Boston University
MULE: Multimodal Universal Language Embedding
B14BC... | 202,537,512 | {
"arxiv": "1909.03493",
"mag": 2997786945,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/aaai/KimSSSP20",
"doi": "10.1609/aaai.v34i07.6785"
} | {
"pdf_urls": [
"https://ojs.aaai.org/index.php/AAAI/article/download/6785/6639"
],
"pdf_sha": "f45eee9da54ee844f5b010bd9a5ede4cd8637977",
"open_access": {
"license": null,
"url": "https://ojs.aaai.org/index.php/AAAI/article/download/6785/6639",
"status": "GOLD"
}
} | MULE: Multimodal Universal Language Embedding | [
"Donghyun Kim \nBoston University",
"Kuniaki Saito keisaito@bu.edu \nBoston University",
"Kate Saenko saenko@bu.edu \nBoston University",
"Stan Sclaroff sclaroff@bu.edu \nBoston University",
"Bryan A Plummer \nBoston University"
] | Existing vision-language methods typically support two languages at a time at most.In this paper, we present a modular approach which can easily be incorporated into existing vision-language methods in order to support many languages.We accomplish this by learning a single shared Multimodal Universal Language Embedding... | [
{
"title": "Introduction",
"content": "Vision-language understanding has been an active area of research addressing many tasks such as image captioning (Fang et al. 2015;Gu et al. 2018), visual question answering (Antol et al. 2015;Goyal et al. 2017), imagesentence retrieval (Wang et al. 2019;Nam, Ha, and K... | [
{
"id": "",
"caption": "Machine Translations N 72.0 60.3 54.8 46.3 76.8 71.4 73.2 PARALLEL-EmbN Human Generated + Machine Translations Y 69.0 62.6 60.6 54.1 72.5 72.3 73.3 MULE EmbN -Full En → Others, Machine Translations Only Y 69.3 62.1 61.5 55.5 70.1 71.6 73.7 MULE EmbN -Full Human Generated + Machine Tr... | [
{
"text": "VQA: Visual Question Answering. S Antol, A Agrawal, J Lu, M Mitchell, D Batra, C L Zitnick, D Parikh, D Cho, K Bengio, Y , arXiv:1409.0473Neural machine translation by jointly learning to align and translate. 2015. 2014ICCV",
"doi": ""
},
{
"text": "Findings of the third shared task on mu... |
String Memories ... openly retold
8 Oct 2020
A Aldi
M Bianchi
M Firrotta mfirrotta@roma2.infn.it
Dipartimento di Fisica
Università di Roma
Tor Vergata Via della Ricerca Scientifica 100133RomaItaly
INFN sezione di Roma
Tor Vergata Via della Ricerca Scientifica 100133RomaItaly
String Memories ... openly retold
8... | 222,208,962 | {
"arxiv": "2010.04082",
"mag": 3111325126,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1016/j.physletb.2020.136037"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2010.04082v1.pdf"
],
"pdf_sha": "f22ff6862561b1a427bbf4eb94c04e36c44733f2",
"open_access": {
"license": "CCBY",
"url": "https://doi.org/10.1016/j.physletb.2020.136037",
"status": "GOLD"
}
} | String Memories ... openly retold | [
"A Aldi",
"M Bianchi",
"M Firrotta mfirrotta@roma2.infn.it",
"Dipartimento di Fisica\nUniversità di Roma\nTor Vergata Via della Ricerca Scientifica 100133RomaItaly",
"INFN sezione di Roma\nTor Vergata Via della Ricerca Scientifica 100133RomaItaly"
] | We identify string corrections to the EM memory effect. Though largely negligible in the low-energy limit, the effect become relevant in high-energy collisions and in extreme events. We illustrate our findings in a simple unoriented bosonic string model. Thanks to the coherent effect of the infinite tower of open strin... | [
{
"title": "Introduction",
"content": "Thanks to the universal behaviour of soft photons and gravitons [1], a memory effect is expected to take place both in electro-dynamics [2] and in geometro-dynamics [3].\nIn a series of papers [4] a triangle of equivalences Soft Behaviour [1,5] -BMS Symmetry [6][7][8] ... | [
{
"id": "",
"caption": "Representative picture of the Ω3/D3 setup.",
"content": "Figure 1 .\n1Representative picture of the Ω3/D3 setup."
},
{
"id": "",
"caption": "Amplitude with one U (1) photon and 3 'tachyons'.",
"content": "Figure 2 .\n2Amplitude with one U (1) photon and 3 'tachyon... | [
{
"text": ". F E Low, Phys. Rev. 110974F. E. Low, Phys. Rev. 110 (1958) 974.",
"doi": ""
},
{
"text": ". S Weinberg, Phys. Rev. 140516S. Weinberg, Phys. Rev. 140 (1965) B516.",
"doi": ""
},
{
"text": ". D J Gross, R Jackiw, Phys. Rev. 1661287D. J. Gross and R. Jackiw, Phys. Rev. 166 (196... |
An Effective Automatic Image Annotation Model Via Attention Model and Data Equilibrium
Amir Vatani
Milad Taleby Ahvanooey
Mostafa Rahimi
Department of Computer Engineering
Islamic Azad University, Science & Research Branch of Tehran
P.O. Box 14515/775Iran
School of Computer Science and Engineering
Nanjing Univ... | 4,657,643 | {
"arxiv": "2001.10590",
"mag": 2794856123,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-2001-10590",
"doi": "10.14569/ijacsa.2018.090338"
} | {
"pdf_urls": [
"http://thesai.org/Downloads/Volume9No3/Paper_38-An_Effective_Automatic_Image_Annotation_Model.pdf"
],
"pdf_sha": "afca5e921965b78920b264545ba8edfeab699b83",
"open_access": {
"license": "CCBY",
"url": "http://thesai.org/Downloads/Volume9No3/Paper_38-An_Effective_Automatic_Image_Annot... | An Effective Automatic Image Annotation Model Via Attention Model and Data Equilibrium | [
"Amir Vatani",
"Milad Taleby Ahvanooey",
"Mostafa Rahimi",
"Department of Computer Engineering\nIslamic Azad University, Science & Research Branch of Tehran\nP.O. Box 14515/775Iran",
"School of Computer Science and Engineering\nNanjing University of Science and Technology\nP.O. Box 210094NanjingP.R. China",... | Nowadays, a huge number of images are available.However, retrieving a required image for an ordinary user is a challenging task in computer vision systems.During the past two decades, many types of research have been introduced to improve the performance of the automatic annotation of images, which are traditionally fo... | [
{
"title": "INTRODUCTION",
"content": "Automatic image annotation (AIA) is one of the image retrieval techniques in that the images can be retrieved in the same way as text documents.In the AIA, the main idea is to automatically learn the semantic concept models from a huge number of image samples and utili... | [
{
"id": "",
"caption": "Fig. 1.The overall structure of proposed automatic image annotation model.",
"content": "Fig. 1 .\n1\nFig. 1.The overall structure of proposed automatic image annotation model."
},
{
"id": "",
"caption": "Fig.2.Basic LSTM unit[48].",
"content": "Fig. 2 .\n2\nFig.2... | [
{
"text": "A review on automatic image annotation techniques. D Zhang, Md M Islam, G Lu, Pattern Recognition. 452012",
"doi": ""
},
{
"text": "Automatic Image Annotation via Label Transfer in the Semantic Space. L B Tiberio Uricchio, Lorenzo Seidenari, Alberto Del Bimbo, , , P R , 10.1016/j.patcog.2... |
Inhomogeneous spacetimes in Weyl integrable geometry with matter source
2 Jun 2020
Andronikos Paliathanasis
Institute of Systems Science
Durban University of Technology
4000DurbanSouth Africa
Instituto de Ciencias Físicas y Matemáticas
Universidad Austral de Chile
5090000ValdiviaChile
Genly Leon
Departamento de M... | 219,179,774 | {
"arxiv": "2006.01793",
"mag": 3100721468,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1140/epjc/s10052-020-8277-z"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2006.01793v1.pdf"
],
"pdf_sha": "dcb3b1e1999f150d9ad45a8126921563c7a93387",
"open_access": {
"license": "CCBY",
"url": "https://link.springer.com/content/pdf/10.1140/epjc/s10052-020-8277-z.pdf",
"status": "GOLD"
}
} | Inhomogeneous spacetimes in Weyl integrable geometry with matter source | [
"Andronikos Paliathanasis \nInstitute of Systems Science\nDurban University of Technology\n4000DurbanSouth Africa\n\nInstituto de Ciencias Físicas y Matemáticas\nUniversidad Austral de Chile\n5090000ValdiviaChile",
"Genly Leon \nDepartamento de Matemáticas\nUniversidad Católica del Norte\nAvda. Angamos 06101280Ca... | We investigate the existence of inhomogeneous exact solutions in Weyl Integrable theory with a matter source. In particular we consider the existence of a dust fluid source while for the underlying geometry we assume a line element which belongs to the family of silent universes. We solve explicitly the field equations... | [
{
"title": "INTRODUCTION",
"content": "Analytical and exact solutions play a significant role in the study of gravitational physics. The existence of exact spacetimes is essential in order to understand the physical properties and the nature of the physical space. Inhomogeneous and anisotropic exact spaceti... | [
{
"id": "",
"caption": "The phase portrait of the dynamical system (32), (33) for λ = 2 and λ = −1. Note that for negative values of λ, we apply the transformation Θ → iΘ.",
"content": "For λ 1 FIG. 1 :\n11The phase portrait of the dynamical system (32), (33) for λ = 2 and λ = −1. Note that for negative... | [
{
"text": ". P Szekeres, Commun. Math. Phys. 4155P. Szekeres, Commun. Math. Phys. 41, 55 (1975)",
"doi": ""
},
{
"text": "Inhomogeneous Cosmological Models. A Krasiński, Cambridge U.P., CambridgeA. Krasiński, Inhomogeneous Cosmological Models, Cambridge U.P., Cambridge, (1997)",
"doi": ""
},
... |
Ab-Initio Study on the Hard Magnetic Properties of MnBi
Peter Toson peter.toson@tuwien.ac.at
Institute of Solid State Physics
Vienna University of Technology
ViennaAustria
Ahmad Asali
Institute of Solid State Physics
Vienna University of Technology
ViennaAustria
Gregor A Zickler
Institute of Solid State Physics... | 119,292,672 | {
"arxiv": "1508.03208",
"mag": 2963882090,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1016/j.phpro.2015.12.159"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1508.03208v1.pdf"
],
"pdf_sha": "146f9a8a315a7c570a20700993c8d686611a2887",
"open_access": {
"license": null,
"url": "https://doi.org/10.1016/j.phpro.2015.12.159",
"status": "GOLD"
}
} | Ab-Initio Study on the Hard Magnetic Properties of MnBi | [
"Peter Toson peter.toson@tuwien.ac.at \nInstitute of Solid State Physics\nVienna University of Technology\nViennaAustria",
"Ahmad Asali \nInstitute of Solid State Physics\nVienna University of Technology\nViennaAustria",
"Gregor A Zickler \nInstitute of Solid State Physics\nVienna University of Technology\nVien... | We have studied the hard magnetic properties of the low-temperature phase of MnBi with first principle calculations based on the density functional theory. The calculations have been carried out on two distinct unit cell configurations MnBi and BiMn with the element in the unit cell origin named first. Our results show... | [
{
"title": "Introduction",
"content": "The low-temperature phase (LTP) of MnBi has interesting properties suitable for permanent magnetic applications [1]. At low temperatures its spin configuration is in-plane but at T = 80K a spin reorientation takes place and the magneto-crystalline anisotropy energy (MA... | [
{
"id": "",
"caption": "Atomic environments in the MnBi unit cell. (a) Trigonal prism around 2c cite. (b) Distorted square anti-prism surrounding 2a site. Figures created with VESTA [7].",
"content": "Figure 1 :\n1Atomic environments in the MnBi unit cell. (a) Trigonal prism around 2c cite. (b) Distorte... | [
{
"text": "Anisotropic fully dense MnBi permanent magnet with high energy product and high coercivity at elevated temperatures. N V Rama Rao, A M Gabay, G C Hadjipanayis, Journal of Physics D: Applied Physics. 46662001N. V. Rama Rao, A. M. Gabay, and G. C. Hadjipanayis. Anisotropic fully dense MnBi permanent ma... |
18 Mar 2024
Mahfuzur Rahaman mahfuzurrahaman01@gmail.com
Department of Physics
Darjeeling Government College
Darjeeling-734101India
Md Hasanujjaman
Department of Physics
Darjeeling Government College
Darjeeling-734101India
Golam Sarwar golamsarwar1990@gmail.com
Department of Physics
University of Calcutta
92, A... | 259,138,869 | {
"arxiv": "2306.06905",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1140/epjc/s10052-024-12642-8"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2306.06905v2.pdf"
],
"pdf_sha": "9c0915847dad1124a4af355af100b3b1fa1ff0de",
"open_access": {
"license": "CCBY",
"url": "https://link.springer.com/content/pdf/10.1140/epjc/s10052-024-12642-8.pdf",
"status": "GOLD"
}
} | [
"Mahfuzur Rahaman mahfuzurrahaman01@gmail.com \nDepartment of Physics\nDarjeeling Government College\nDarjeeling-734101India",
"Md Hasanujjaman \nDepartment of Physics\nDarjeeling Government College\nDarjeeling-734101India",
"Golam Sarwar golamsarwar1990@gmail.com \nDepartment of Physics\nUniversity of Calcutta... | The dynamical correlation of density fluctuation in quark gluon plasma with a critical end point has been investigated within the scope of the Müller-Israel-Stewart theory in the presence of static ultra-high external magnetic field.The dynamic structure factor of the density fluctuation exhibits three Lorentzian peaks... | [
{
"title": "I. INTRODUCTION",
"content": "One of the primary objectives of heavy-ion collision (HIC) experiments at Relativistic Heavy Ion Collider (RHIC) and at Large Hadron Collider (LHC) is to create and characterize a deconfined state of thermal quarks and gluons, called quark gluon plasma (QGP).It is e... | [
{
"id": "",
"caption": "and D µ ≡ H µν u ν .In the local rest frame of the fluid, the non-trivial components of these tensors are(F 10 , F 20 , F 30 ) = E, (F 32 , F 13 , F 21 ) = B, (M 10 , M 20 , M 30 ) = −P, (M 32 , M 13 , M 21 ) = M, (H 10 , H 20 , H 30 ) = D,and (H 32 , H 13 , H 21 ) = H, where P and M... | [
{
"text": "idealistic scenario of non-expanding QGP in a static magnetic field. VI. ACKNOWLEDGEMENT MR, MH would like to thank Department of Higher Education. Govt. of West Bengal. India for the support",
"doi": ""
},
{
"text": "A K Chaudhuri, 10.1088/978-0-750-31060-4arXiv:1207.7028A short course o... | |
SciPost Physics Proceedings Submission Probing Lepton Universality with (Semi)-Leptonic B decays
September 2018
G Banelli
Science Park 105NL-1098 XGNikhef, AmsterdamNetherlands
R Fleischer
Science Park 105NL-1098 XGNikhef, AmsterdamNetherlands
Faculty of Science
Vrije Universiteit Amsterdam
NL-1081 HVAmsterdamNet... | 119,478,570 | {
"arxiv": "1812.05200",
"mag": 2962925789,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.21468/scipostphysproc.1.013"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1812.05200v1.pdf"
],
"pdf_sha": "cbacf3985b4c3cf77cfd35566d34be80d1202d7d",
"open_access": {
"license": "CCBY",
"url": "https://scipost.org/10.21468/SciPostPhysProc.1.013/pdf",
"status": "GOLD"
}
} | SciPost Physics Proceedings Submission Probing Lepton Universality with (Semi)-Leptonic B decays | [
"G Banelli \nScience Park 105NL-1098 XGNikhef, AmsterdamNetherlands",
"R Fleischer \nScience Park 105NL-1098 XGNikhef, AmsterdamNetherlands\n\nFaculty of Science\nVrije Universiteit Amsterdam\nNL-1081 HVAmsterdamNetherlands",
"R Jaarsma \nScience Park 105NL-1098 XGNikhef, AmsterdamNetherlands",
"G Tetlalmatzi... | The most recent measurements of the observables R D ( * ) are in tension with the Standard Model offering hints of New Physics in b → c ν transitions. Motivated by these results, in this work we present an analysis on their b → u ν counterparts (for = e, µ, τ ). Our study has three main objectives. Firstly, using ratio... | [
{
"title": "Introduction",
"content": "Recent measurements of the observables R D ( * ) ≡ B(B → D ( * ) τν τ )/B(B → D ( * ) ν τ ), with = e, µ, have caused a lot of excitement in the high-energy physics community. As a matter of fact, the combined measurements of the BaBar, LHCb and Belle collaborations sh... | [
{
"id": "",
"caption": "Figure 1: Allowed regions in the C µ P -C τ P plane following from the leptonic ratio R τ µ .",
"content": "Figure 1\n1Figure 1: Allowed regions in the C µ P -C τ P plane following from the leptonic ratio R τ µ ."
},
{
"id": "",
"caption": "R e,µ ;ρ [q 2 min ≤q 2 ≤q 2... | [
{
"text": "Averages of b-hadron, c-hadron, and τ -lepton properties as of summer 2016. Y Amhis, 10.1140/epjc/s10052-017-5058-4Eur. Phys. J. C77. 12895Y. Amhis et al., Averages of b-hadron, c-hadron, and τ -lepton properties as of summer 2016, Eur. Phys. J. C77(12), 895 (2017), doi:10.1140/epjc/s10052-017-5058-4... |
Speaker Diarization using Two-pass Leave-One-Out Gaussian PLDA Clustering of DNN Embeddings
Kiran Karra kiran.karra@jhuapl.edu
Human Language Technology Center of Excellence
The Johns Hopkins University
BaltimoreMDUSA
Alan Mccree alan.mccree@jhu.edu
Human Language Technology Center of Excellence
The Johns Hopkins... | 233,033,813 | {
"arxiv": "2104.02469",
"mag": 3147362063,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/interspeech/KarraM21",
"doi": "10.21437/interspeech.2021-1807"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2104.02469v3.pdf"
],
"pdf_sha": "e91e7eb03a17f42b318a38031bb2cf12ea66edb1",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/2104.02469",
"status": "GREEN"
}
} | Speaker Diarization using Two-pass Leave-One-Out Gaussian PLDA Clustering of DNN Embeddings | [
"Kiran Karra kiran.karra@jhuapl.edu \nHuman Language Technology Center of Excellence\nThe Johns Hopkins University\nBaltimoreMDUSA",
"Alan Mccree alan.mccree@jhu.edu \nHuman Language Technology Center of Excellence\nThe Johns Hopkins University\nBaltimoreMDUSA"
] | Many modern systems for speaker diarization, such as the recently-developed VBx approach, rely on clustering of DNN speaker embeddings followed by resegmentation. Two problems with this approach are that the DNN is not directly optimized for this task, and the parameters need significant retuning for different applicat... | [
{
"title": "Introduction",
"content": "As highlighted in the recent DIHARD challenges (2018, 2019), there are many techniques for speaker diarization [1,2,3,4,5,6,7,8]. Traditional approaches involve clustering of speaker embeddings (xvectors) followed by resegmentation, but we have concerns that the DNN is... | [
{
"id": "",
"caption": "Learning rate schedule used for training xvector DNN.",
"content": "Figure 1 :\n1Learning rate schedule used for training xvector DNN."
},
{
"id": "",
"caption": "Comparison of functions for approximating effective number of samples.",
"content": "Figure 2 :\n2Com... | [
{
"text": "Diarization is hard: Some experiences and lessons learned for the jhu team in the inaugural DIHARD challenge. G Sell, D Snyder, A Mccree, D Garcia-Romero, J Villalba, M Maciejewski, V Manohar, N Dehak, D Povey, S Watanabe, Proc. Interspeech. InterspeechG. Sell, D. Snyder, A. McCree, D. Garcia-Romero,... |
What is the Mass of a Gap-Opening Planet?
24 Jul 2017
Ruobing Dong rdong@email.arizona.edu
Steward Observatory
University of Arizona
Jeffrey Fung
Department of Astronomy
University of California
Berkeley
NASA Sagan Fellow
What is the Mass of a Gap-Opening Planet?
24 Jul 2017Subject headings: protoplanetary dis... | 118,989,219 | {
"arxiv": "1612.04821",
"mag": 3098814278,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.3847/1538-4357/835/2/146"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1612.04821v2.pdf"
],
"pdf_sha": "69df23ae70d837d62a13682c5b3c0e4b1ddb220b",
"open_access": {
"license": null,
"url": "https://repository.arizona.edu/bitstream/10150/624383/1/Dong__2017_ApJ_835_146.pdf",
"status": "GREEN"
}
} | What is the Mass of a Gap-Opening Planet? | [
"Ruobing Dong rdong@email.arizona.edu \nSteward Observatory\nUniversity of Arizona",
"Jeffrey Fung \nDepartment of Astronomy\nUniversity of California\nBerkeley\n\nNASA Sagan Fellow"
] | High contrast imaging instruments such as GPI and SPHERE are discovering gap structures in protoplanetary disks at an ever faster pace. Some of these gaps may be opened by planets forming in the disks. In order to constrain planet formation models using disk observations, it is crucial to find a robust way to quantitat... | [
{
"title": "Introduction",
"content": "In the past few years, high angular resolution observations of protoplanetary disks have revolutionized our understanding of planet formation. Rich structures have been identified in disk images in near infrared (NIR) and in mm/cm dust continuum and gas emission. Among... | [
{
"id": "",
"caption": "23 compose a complete set of equations to link [∆ I , δ I ] with [M p , h/r, α]. Once a narrow gap (∆ I (η, i) 0.5) is identified in scattered light of a disk at a modest inclination (i 45 • ) with both the inner and outer gap edges clearly revealed, one can follow the following step... | [
{
"text": ". E Akiyama, T Muto, N Kusakabe, ApJ. 80217Akiyama, E., Muto, T., Kusakabe, N., et al. 2015, ApJ, 802, L17",
"doi": ""
},
{
"text": "F Allard, D Homeier, B Freytag, 16th Cambridge Workshop on Cool Stars, Stellar Systems, and the Sun. C. Johns-Krull, M. K. Browning, & A. A. West44891Allard... |
A CONSTRUCTION OF THE MEASURABLE POISSON BOUNDARY: FROM DISCRETE TO CONTINUOUS GROUPS
11 Mar 2015 March 12, 2015
Sara Brofferio
A CONSTRUCTION OF THE MEASURABLE POISSON BOUNDARY: FROM DISCRETE TO CONTINUOUS GROUPS
11 Mar 2015 March 12, 2015arXiv:1503.03333v1 [math.PR]
Let Γ be a dense countable subgroup of a locally... | 119,608,559 | {
"arxiv": "1503.03333",
"mag": 1956993254,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1017/9781316576571.006"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1503.03333v1.pdf"
],
"pdf_sha": "33efca2d5ce3066ebd9b5589f3e66b7d16dd1579",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1503.03333",
"status": "GREEN"
}
} | A CONSTRUCTION OF THE MEASURABLE POISSON BOUNDARY: FROM DISCRETE TO CONTINUOUS GROUPS | [
"Sara Brofferio"
] | Let Γ be a dense countable subgroup of a locally compact continuous group G. Take a probability measure µ on Γ. There are two natural spaces of harmonic functions: the space of µ-harmonic functions on the countable group Γ and the space of µ-harmonic functions seen as functions on G defined a.s. with respect to its Haa... | [
{
"title": "G-harmonic functions and G-Poisson boundary",
"content": "We present in this section a brief introduction to measurable Poisson boundary following the notation of Babillot [1] and Kaimanovich [13] G-harmonic functions. Let G be locally compact second countable (thus metrizable and complete) grou... | [
{
"id": "",
"caption": "[4]), while there still a lot of open questions concerning the Poisson boundaries of the corresponding real groups (see section 1 for more detailed examples).",
"content": "[4]), while there still a lot of open questions concerning the Poisson boundaries of the corresponding real... | [
{
"text": "Martine Babillot, An introduction to Poisson boundaries of Lie groups. Probability measures on groups: recent directions and trends. MumbaiBabillot, Martine: An introduction to Poisson boundaries of Lie groups. Probability mea- sures on groups: recent directions and trends, 1-90, Tata Inst. Fund. Res... |
Centrality properties of directed module members in social networks
3 Sep 2008
Péter Pollner
Statistical and Biological Physics Research Group of Hungarian Academy of Sciences
Pázmány P. stny. 11117Budapest
Gergely Palla A Dánielábel
Statistical and Biological Physics Research Group of Hungarian Academy of Science... | 15,239,261 | {
"arxiv": "0809.0669",
"mag": 2157551912,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1016/j.physa.2008.04.025"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/0809.0669v1.pdf"
],
"pdf_sha": "f2666eb099bf8e47bc0761189160d7b41ff87445",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/0809.0669",
"status": "GREEN"
}
} | Centrality properties of directed module members in social networks | [
"Péter Pollner \nStatistical and Biological Physics Research Group of Hungarian Academy of Sciences\nPázmány P. stny. 11117Budapest",
"Gergely Palla A Dánielábel \nStatistical and Biological Physics Research Group of Hungarian Academy of Sciences\nPázmány P. stny. 11117Budapest\n\nDepartment of Biological Physics... | Several recent studies of complex networks have suggested algorithms for locating network communities, also called modules or clusters, which are mostly defined as groups of nodes with dense internal connections. Along with the rapid development of these clustering techniques, the ability of revealing overlaps between ... | [
{
"title": "Introduction",
"content": "A widespread approach to the analysis of complex social and economic phenomena is to assemble the participating individuals or objects and their interactions into a network (nodes and links) and to infer functional characteristics of the entire system from this static ... | [
{
"id": "",
"caption": "Two example for directed graphs, that are k-cliques of size k = 4 if neglecting the directionality of the links. However, the graph a) forms a directed k-clique, but the graph b) does not. The numbers near the nodes of graph a) show the ordering among the nodes for defining the overa... | [
{
"text": ". A L Barabási, R Albert, Rev. Mod. Phys. 7447Barabási, A. L. and Albert, R., Rev. Mod. Phys. 74, 47 (2002).",
"doi": ""
},
{
"text": "Evolution of Networks: From Biological Nets to the Internet and WWW. S N Dorogovtsev, J F F Mendes, Oxford University PressDorogovtsev, S. N. and Mendes, ... |
Singular sectors of the 1-layer Benney and dToda systems and their interrelations
May 2011 January 26, 2013
B Konopelchenko
Dipartimento di Fisica
Universitá del Salento
INFN 7 3100Sezione, LecceItaly
L Martínez Alonso
Departamento de Física Teórica II
Universidad Complutense E28040 Madrid
Spain
E Medina
Departa... | 52,992,603 | {
"arxiv": "1105.5931",
"mag": 2149295173,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1007/s11232-011-0078-y"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1105.5931v1.pdf"
],
"pdf_sha": "e98ad5dc6b8a1c2974a95a4626a004e56ab00fde",
"open_access": {
"license": null,
"url": "https://eprints.ucm.es/34173/1/martinezalonso07preprint.pdf",
"status": "GREEN"
}
} | Singular sectors of the 1-layer Benney and dToda systems and their interrelations | [
"B Konopelchenko \nDipartimento di Fisica\nUniversitá del Salento\nINFN 7 3100Sezione, LecceItaly",
"L Martínez Alonso \nDepartamento de Física Teórica II\nUniversidad Complutense E28040 Madrid\nSpain",
"E Medina \nDepartamento de Matemáticas\nUniversidad de Cádiz E11510 Puerto Real\nCádizSpain",
"B Konopelch... | Complete description of the singular sectors of the 1-layer Benney system ( classical long wave equation) and dToda system is presented. Associated Euler-Poisson-Darboux equations E(1/2,1/2) and E(-1/2,-1/2) are the main tool in the analysis. A complete list of solutions of the 1-layer Benney system depending on two pa... | [
{
"title": "Introduction",
"content": ""
},
{
"title": "Introduction",
"content": "The 1-layer Benney system (classical long wave equation)\nThe 1-layer Benney system (classical long wave equation)\nand dToda equation v xx = (logv) tt or equivalently the system\nand dToda equation v xx = (logv) ... | [
{
"id": "",
"caption": "singular sectors M reg and M sing are defined as the sets (t, β, β) of solutions of equation (25) such that the hermitian form",
"content": "singular sectors M reg and M sing are defined as the sets (t, β, β) of solutions of equation (25) such that the hermitian form"
},
{
... | [
{
"text": ". B A Dubrovin, S P Novikov, Russian Math. Surveys. 4435B. A. Dubrovin and S. P. Novikov, Russian Math. Surveys 44, 35 (1989)",
"doi": ""
},
{
"text": ". D Y Benney, Stud. Appl. Math. 52D. Y. Benney, Stud. Appl. Math. 52 45-50 (1973).",
"doi": ""
},
{
"text": ". V E Zakharov, ... |
Kinetics of local "magnetic" moment and non-stationary spin-polarized current in the single impurity Anderson-model
N S Maslova
Department of Physics
Moscow State University
119991Moscow
V N Mantsevich
Department of Physics
Moscow State University
119991Moscow
P I Arseyev
Russia P.N.Lebedev
Physical Institute o... | 118,612,958 | {
"arxiv": "1606.08975",
"mag": 3097996080,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1134/s002136401704004x"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1606.08975v1.pdf"
],
"pdf_sha": "baa93a5a7e25081df8f97309ab19173c413b0c52",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1606.08975",
"status": "GREEN"
}
} | Kinetics of local "magnetic" moment and non-stationary spin-polarized current in the single impurity Anderson-model | [
"N S Maslova \nDepartment of Physics\nMoscow State University\n119991Moscow",
"V N Mantsevich \nDepartment of Physics\nMoscow State University\n119991Moscow",
"P I Arseyev \nRussia P.N.Lebedev\nPhysical Institute of RAS\n119991Moscow3 Russia\n\nNational Research University Higher School of Economics\nMoscowRuss... | We perform theoretical investigation of the localized state dynamics in the presence of interaction with the reservoir and Coulomb correlations. We analyze kinetic equations for electron occupation numbers with different spins taking into account high order correlation functions for the localized electrons. We reveal t... | [
{
"title": "I. INTRODUCTION",
"content": "The creation, diagnostics and controlled manipulation of charge and spin states of the impurity atoms or quantum dots (QDs) is one of the most important problems in nano-electronics now a days 1 , 2 , 3 , 4 , 5 . Modern ultra small size electronic devices design wit... | [
{
"id": "",
"caption": "online) Normalized relaxation rates |λ1,2(ε)|/2Γ as a functions of the localized state energy level position ε/2Γ for U/2Γ = 7.5 and Γ = 1.",
"content": "online) Normalized relaxation rates |λ1,2(ε)|/2Γ as a functions of the localized state energy level position ε/2Γ for U/2Γ = 7... | [
{
"text": ". * vmantsev@gmail.com. * vmantsev@gmail.com;",
"doi": ""
},
{
"text": "L Jacak, P Hawrylak, A Wojs, Quantum Dots. BerlinSpringerL. Jacak, P. Hawrylak, A. Wojs, Quantum Dots, (Springer, Berlin, 1998).",
"doi": ""
},
{
"text": ". W G Van Der Wiel, S De Franceschi, J M Elzerman,... |
Ergodic capacity analysis of reconfigurable intelligent surface assisted wireless systems
Alexandros-Apostolos A Boulogeorgos s:al.boulogeorgos@ieee.org
Department of Digital Systems
University of Piraeus
PiraeusGreece
Angeliki Alexiou alexiou@unipi.gr
Department of Digital Systems
University of Piraeus
PiraeusGr... | 220,969,094 | {
"arxiv": "2008.01931",
"mag": 3094078794,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/5gwf/BoulogeorgosA20",
"doi": "10.1109/5gwf49715.2020.9221372"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2008.01931v1.pdf"
],
"pdf_sha": "f92f64e905834f112e163a2d35d9132484131f9a",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/2008.01931",
"status": "GREEN"
}
} | Ergodic capacity analysis of reconfigurable intelligent surface assisted wireless systems | [
"Alexandros-Apostolos A Boulogeorgos s:al.boulogeorgos@ieee.org \nDepartment of Digital Systems\nUniversity of Piraeus\nPiraeusGreece",
"Angeliki Alexiou alexiou@unipi.gr \nDepartment of Digital Systems\nUniversity of Piraeus\nPiraeusGreece"
] | This paper presents the analytic framework for evaluating the ergodic capacity (EC) of the reconfigurable intelligent surface (RIS) assisted systems. Moreover, high-signal-to-noise-ratio and high-number of reflection units (RUs) approximations for the EC are provided. Finally, the special case in which the RIS is equip... | [
{
"title": "I. INTRODUCTION",
"content": "While the wireless world moves towards the sixth generation (6G) era, the data-rate and network traffic demands have been exponential increased [1]- [4]. Technological advances, such as massive multiple-input multiple-output, full-duplexing, and high-frequency commu... | [
{
"id": "",
"caption": "(6.4.1)], the elliptic integral function [24, eq. (17.1.1)], Fig. 1: System model of the RIS-assisted wireless system. and the complete elliptic integral function of the first kind [24, eq. (17.3.1)]. Furthermore, 2 F 1 (·, ·; ·; ·) stands for the Gauss hypergeometric function [24, e... | [
{
"text": "A vision of 6G wireless systems: Applications, trends, technologies, and open research problems. W Saad, M Bennis, M Chen, IEEE Netw. W. Saad, M. Bennis, and M. Chen, \"A vision of 6G wireless systems: Applications, trends, technologies, and open research problems,\" IEEE Netw., pp. 1-9, Oct. 2019.",... |
Noise Fit, Estimation Error and a Sharpe Information Criterion: Linear Case
September 8, 2017 8 Sep 2017
Dirk Paulsen dirk.h.paulsen@googlemail.com
John Street Capital
Jakob Söhl
Faculty of Electrical Engineering, Mathematics and Computer Science
John Street Capital LLP
Delft, London, DelftTUUnited Kingdom, The N... | 157,704,930 | {
"arxiv": "1602.06186",
"mag": 3005640052,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.2139/ssrn.2928607"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1602.06186v2.pdf"
],
"pdf_sha": "9e35be6c1ed81cc488a1d4402ac009580548e49f",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1602.06186",
"status": "GREEN"
}
} | Noise Fit, Estimation Error and a Sharpe Information Criterion: Linear Case | [
"Dirk Paulsen dirk.h.paulsen@googlemail.com \nJohn Street Capital",
"Jakob Söhl \nFaculty of Electrical Engineering, Mathematics and Computer Science\nJohn Street Capital LLP\nDelft, London, DelftTUUnited Kingdom, The Netherlands"
] | We derive (1) an unbiased estimator for the out-of-sample Sharpe ratio when the in-sample Sharpe ratio is obtained by optimizing over a k-dimensional parameter space. The estimator corrects the in-sample Sharpe ratio for both: noise fit and estimation error. We then show (2) how to use the corrected Sharpe ratio as mod... | [
{
"title": "Introduction",
"content": "A convenient measure for return predictability is the Sharpe ratio, sometimes called information ratio. The Sharpe ratio is the ratio of annualized mean (excess) returns over return volatility. It is a useful statistic as it summarizes the first two moments of the retu... | [
{
"id": "",
"caption": "Noise-Fit and Estimation Error for Mean-Variance). Under the assumptions of this section holds E [N M V ] = k + 1 γT and E [N M V + E M V + U M V unbiased estimator for the out-of-sample utility u(θ).",
"content": "Theorem 3 . 5 (\n35Noise-Fit and Estimation Error for Mean-Varian... | [
{
"text": "A new look at the statistical model identification. Automatic Control. H Akaike, IEEE Transactions on. 196Akaike, H. (1974). A new look at the statistical model identification. Automatic Control, IEEE Transactions on, 19(6):716-723.",
"doi": ""
},
{
"text": "A bayesian analysis of the min... |
SOUNDLOCD: AN EFFICIENT CONDITIONAL DISCRETE CONTRASTIVE LATENT DIFFUSION MODEL FOR TEXT-TO-SOUND GENERATION
24 May 2024
Xinlei Niu
Australian National University
CanberraAustralia
Jing Zhang
Australian National University
CanberraAustralia
Christian Walder
Google DeepMind
MontrealCanada
Charles Patrick Martin ... | 268,568,238 | {
"arxiv": "2405.15338",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/icassp/Niu0WM24",
"doi": "10.1109/icassp48485.2024.10446349"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2405.15338v1.pdf"
],
"pdf_sha": "3cbc75c37e8f9abd03aac0e60d0053dd546a83b9",
"open_access": {
"license": null,
"url": "https://arxiv.org/pdf/2405.15338",
"status": "GREEN"
}
} | SOUNDLOCD: AN EFFICIENT CONDITIONAL DISCRETE CONTRASTIVE LATENT DIFFUSION MODEL FOR TEXT-TO-SOUND GENERATION | [
"Xinlei Niu \nAustralian National University\nCanberraAustralia",
"Jing Zhang \nAustralian National University\nCanberraAustralia",
"Christian Walder \nGoogle DeepMind\nMontrealCanada",
"Charles Patrick Martin \nAustralian National University\nCanberraAustralia"
] | We present SoundLoCD, a novel text-to-sound generation framework, which incorporates a LoRA-based conditional discrete contrastive latent diffusion model.Unlike recent large-scale sound generation models, our model can be efficiently trained under limited computational resources.The integration of a contrastive learnin... | [
{
"title": "INTRODUCTION",
"content": "Sound synthesis spans a diverse array of applications, including game development, film post-production, virtual reality, and musical performance.In these domains, automated sound effects generation could address tasks such as replicating environmental ambience and sim... | [
{
"id": "",
"caption": "Fig. 1.Overall pipeline of SoundLoCD, which is performed based on a pre-trained spectrogram VQ-VAE.SoundLoCD involves N + 1 parallel discrete diffusion processes on original data and N randomly shuffled negative data.",
"content": "Fig. 1 .\n1\nFig. 1.Overall pipeline of SoundLoC... | [
{
"text": "Diffsound: Discrete diffusion model for text-to-sound generation. Dongchao Yang, Jianwei Yu, Helin Wang, Wen Wang, Chao Weng, Yuexian Zou, Dong Yu, Speech, and Language Processing. 2023",
"doi": ""
},
{
"text": "Vector quantized diffusion model for text-toimage synthesis. Shuyang Gu, Dong... |
TORIC DEGENERATIONS OF TORIC VARIETIES AND TROPICAL CURVES
6 Mar 2006
Takeo Nishinou
Bernd Siebert
TORIC DEGENERATIONS OF TORIC VARIETIES AND TROPICAL CURVES
6 Mar 2006arXiv:math/0409060v3 [math.AG]
We show that the counting of rational curves on a complete toric variety that are in general position to the toric pr... | 14,934,551 | {
"arxiv": "math/0409060",
"mag": 2952606334,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1215/s0012-7094-06-13511-1"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/math/0409060v3.pdf"
],
"pdf_sha": "8cb4533299e9be269504306db7678ef3b22607f1",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/math/0409060",
"status": "GREEN"
}
} | TORIC DEGENERATIONS OF TORIC VARIETIES AND TROPICAL CURVES | [
"Takeo Nishinou",
"Bernd Siebert"
] | We show that the counting of rational curves on a complete toric variety that are in general position to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraic-geometric and relies on degeneration techniques and log deformation theory. | [
{
"title": "Introduction.",
"content": "The advent of mirror symmetry has rekindled the interest in the classical enumerative question of counting algebraic curves on a smooth projective variety. In fact, the unexpected relation of this problem with period computations for the mirror family [COGP] really ca... | [
{
"id": "",
"caption": "For d = (d 1 , . . . , d l ) ∈ N l an affine constraint of codimension d is an l-tupleA = (A 1 , . . . , A l ) of affine subspaces A i ⊂ N Q with dim A i = n − d i − 1. An l-marked tropical curve (Γ, E, h) matches the affine constraint A if h(E i ) ∩ A i = ∅, i = 1, . . . , l.",
... | [
{
"text": "Manin: Stacks of stable maps and Gromov-Witten invariants. K Behrend, Y , Duke. Math. Journ. 85K. Behrend, Y. Manin: Stacks of stable maps and Gromov-Witten invariants, Duke. Math. Journ. 85 (1996) 1-60.",
"doi": ""
},
{
"text": "A pair of Calabi-Yau manifolds as an exactly soluble superc... |
Low temperature expansion for the 3-d Ising Model
24 Jun 1992 June 1992
Gyan Bhanot
School of Natural Sciences Institute for Advanced Study Princeton
School of Natural Sciences Institute for Advanced Study Princeton
Physics Department Brookhaven National Laboratory Upton
Thinking Machines Corporation
245 First Stree... | 13,967,448 | {
"arxiv": "hep-lat/9206020",
"mag": 2058408058,
"acl": null,
"pubmed": "10046329",
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevlett.69.1841"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/hep-lat/9206020v1.pdf"
],
"pdf_sha": "16fd3c73a4a0670db4fb64e9626a5487c15ff8fc",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/hep-lat/9206020",
"status": "GREEN"
}
} | Low temperature expansion for the 3-d Ising Model | [
"Gyan Bhanot \nSchool of Natural Sciences Institute for Advanced Study Princeton\nSchool of Natural Sciences Institute for Advanced Study Princeton\nPhysics Department Brookhaven National Laboratory Upton\nThinking Machines Corporation\n245 First Street08540, 02142, 11973, 08540Cambridge, NYNJ, MA, NJ",
"Michael ... | We compute the weak coupling expansion for the energy of the three dimensional Ising model through 48 excited bonds. We also compute the magnetization through 40 excited bonds. This was achieved via a recursive enumeration of states of fixed energy on a set of finite lattices. We use a linear combination of lattices wi... | [
{
"title": "Figure Captions",
"content": ""
},
{
"title": "z",
"content": ""
},
{
"title": "Fig. 1 .",
"content": ""
},
{
"title": "Fig. 2 .",
"content": ""
},
{
"title": "Table I .",
"content": ""
}
] | [
{
"id": "",
"caption": "where n i represents the number of steps in the ith direction. The length of such a loop is n = |n x |+|n y |+|n z |. On an infinite cubic lattice the only solution to these equations is the trivial case n i = 0. On a finite lattice, any other solution represents a finite size correc... | [
{
"text": ". K Binder, Physica. 62508K. Binder, Physica 62 (1972) 508.",
"doi": ""
},
{
"text": ". G Bhanot, J. Stat. Phys. 6055G. Bhanot, J. Stat. Phys. 60 (1990) 55;",
"doi": ""
},
{
"text": ". G Bhanot, S Sastry, J. Stat. Phys. 60333G. Bhanot and S. Sastry, J. Stat. Phys. 60 (1990) 33... |
Ion detection in the photoionization of a Rb Bose-Einstein condensate
1 Feb 2010
M Viteau
J Radogostowicz
A Chotia
Dipartimento di Fisica E. Fermi
CNR-INO
Università di Pisa
Largo Pontecorvo 3I-56127PisaItaly
M G Bason +
Dipartimento di Fisica E. Fermi
CNR-INO
Università di Pisa
Largo Pontecorvo 3I-56127PisaItal... | 119,303,334 | {
"arxiv": "1002.0212",
"mag": 2057710262,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1088/0953-4075/43/15/155301"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1002.0212v1.pdf"
],
"pdf_sha": "eaab0cc1dda3e8ea5b3d2f5e17b4aad4e5bc1e19",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1002.0212",
"status": "GREEN"
}
} | Ion detection in the photoionization of a Rb Bose-Einstein condensate | [
"M Viteau",
"J Radogostowicz",
"A Chotia \nDipartimento di Fisica E. Fermi\nCNR-INO\nUniversità di Pisa\nLargo Pontecorvo 3I-56127PisaItaly",
"M G Bason + \nDipartimento di Fisica E. Fermi\nCNR-INO\nUniversità di Pisa\nLargo Pontecorvo 3I-56127PisaItaly",
"N Malossi",
"F Fuso",
"D Ciampini",
"O Morsch... | Two-photon ionization of Rubidium atoms in a magneto-optical trap and a Bose-Einstein condensate (BEC) is experimentally investigated. Using 100 ns laser pulses, we detect single ions photoionized from the condenstate with a 35(10)% efficiency. The measurements are performed using a quartz cell with external electrodes... | [
{
"title": "Introduction",
"content": "The development of laser cooling and trapping techniques has enabled the creation of dense and ultracold atomic samples, thus opening the way for laser spectroscopic investigations in regimes not accessible with conventional techniques. A wide range of applications are... | [
{
"id": "",
"caption": "In (a) and (b) the atomic levels involved in the two photoionization schemes are presented. A schematic of the ion collection system produced by the MOT/BEC ionization within the vacuum quartz cell is shown in (c). The laser beams are superimposed so as to excite atoms within the sam... | [
{
"text": ". I Bloch, Nature. 4531016I. Bloch, Nature 453 1016 (2008).",
"doi": ""
},
{
"text": ". T P Dinneen, C D Wallace, K N Tan, P L Gould, Opt. Lett. 171706T. P. Dinneen, C. D. Wallace, K. N. Tan, and P. L. Gould, Opt. Lett. 17, 1706 (1992).",
"doi": ""
},
{
"text": ". M Anderlini,... |
Noncommutative gravity coupled to fermions: second order expansion via Seiberg-Witten map
29 Apr 2012
Paolo Aschieri aschieri@to.infn.it
Leonardo Castellani leonardo.castellani@mfn.unipmn.it
Dipartimento di Scienze e Tecnologie Avanzate
INFN Gruppo collegato di Alessandria
Università del Piemonte Orientale
Viale T... | 119,288,213 | {
"arxiv": "1111.4822",
"mag": 2005516134,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1007/jhep07(2012)184"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1111.4822v2.pdf"
],
"pdf_sha": "d1180120f50d9270648523f4079a9dd48d626271",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1111.4822",
"status": "GREEN"
}
} | Noncommutative gravity coupled to fermions: second order expansion via Seiberg-Witten map | [
"Paolo Aschieri aschieri@to.infn.it",
"Leonardo Castellani leonardo.castellani@mfn.unipmn.it",
"Dipartimento di Scienze e Tecnologie Avanzate\nINFN Gruppo collegato di Alessandria\nUniversità del Piemonte Orientale\nViale T. Michel 1115121AlessandriaItaly",
"Introduction"
] | We use the Seiberg-Witten map (SW map) to expand noncommutative gravity coupled to fermions in terms of ordinary commuting fields. The action is invariant under general coordinate transformations and local Lorentz rotations, and has the same degrees of freedom as the commutative gravity action. The expansion is given u... | [
{
"title": "Introduction",
"content": "Field theories defined on noncommutative spaces can be systematically constructed by use of an associative and noncommutative ‹-product. This product between fields generates infinitely many derivatives and introduces a dimensionful noncommutativity parameter θ. The pr... | [
{
"id": "",
"caption": "X a \" B Bx a . In this instance we have the Moyal twist, cf. (A.",
"content": "X a \" B Bx a . In this instance we have the Moyal twist, cf. (A."
},
{
"id": "",
"caption": "p ǫ : \"´γ 0 p ǫγ 0 , p ǫ T \" C q ǫ C ; (B.33) p V : \" γ 0 p V γ 0 , p V T \" C | V C ; (B.3... | [
{
"text": "Quantum mechanics as a statistical theory. J E , Proc. Camb. Phil. Soc. 4599J. E. Moyal, Quantum mechanics as a statistical theory, Proc. Camb. Phil. Soc. 45 (1949) 99 ;",
"doi": ""
},
{
"text": ". H Groenewold, Physica. 12405H. Groenewold, Physica 12 (1946) 405.",
"doi": ""
},
{
... |
A CES ÀRO-LIKE OPERATOR FROM BESOV SPACES TO SOME SPACES OF ANALYTIC FUNCTIONS
4 May 2023
FANGQIN YE
SHANTOU UNIVERSITY
SHANTOU 515063CHINA
LIUCHANG ZHOU
SHANTOU UNIVERSITY
SHANTOU 515063CHINA
A CES ÀRO-LIKE OPERATOR FROM BESOV SPACES TO SOME SPACES OF ANALYTIC FUNCTIONS
4 May 20235C6A36E3E5E5CA0E7234CE3EB90B1F89... | 258,479,778 | {
"arxiv": "2305.02717",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1007/s40315-024-00542-7"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2305.02717v1.pdf"
],
"pdf_sha": "0dd07a374a10eac9e316b6a1698f8ae7d43a1d31",
"open_access": {
"license": null,
"url": "https://arxiv.org/pdf/2305.02717",
"status": "GREEN"
}
} | A CES ÀRO-LIKE OPERATOR FROM BESOV SPACES TO SOME SPACES OF ANALYTIC FUNCTIONS | [
"FANGQIN YE\nSHANTOU UNIVERSITY\nSHANTOU 515063CHINA",
"LIUCHANG ZHOU\nSHANTOU UNIVERSITY\nSHANTOU 515063CHINA"
] | In this paper, for p > 1 and s > 1, we give a complete description of the boundedness and compactness of a Cesàro-like operator from the Besov space Bp into a Banach space X between the mean Lipschitz space Λ s 1/s and the Bloch space. | [
{
"title": "INTRODUCTION",
"content": "Let µ be a finite positive Borel measure on [0, 1).For a nonnegative integer n, denote by µ n the moment of order n of µ.More precisely,\nt n dµ(t).\nLet f (z) = ∞ n=0 a n z n belong to H(D), the space of functions analytic in the open unit disk D of the complex plane ... | [
{
"id": "",
"caption": "and a finite positive Borel measure ν on D is called a vanishing α-logarithmic s-Carleson measure if lim |I|→0 ν(S(I))(log e |I| ) α |I| s = 0.",
"content": "and a finite positive Borel measure ν on D is called a vanishing α-logarithmic s-Carleson measure if lim |I|→0 ν(S(I))(log... | [
{
"text": "Hankel matrices acting on Dirichlet spaces. G Bao, H Wulan, J. Math. Anal. Appl. 4092014",
"doi": ""
},
{
"text": "Carleson measures and the range of a Cesàro-like operator acting on H ∞. G Bao, F Sun, H Wulan, Anal. Math. Phys. 12142022",
"doi": ""
},
{
"text": "Cesàro operat... |
Correlation entropy of synaptic input-output dynamics
10 Oct 2006
Ingo C Kleppe
Department of Physiology
University of Cambridge
Downing StreetCB2 3EGCambridgeU.K
Hugh P C Robinson
Department of Physiology
University of Cambridge
Downing StreetCB2 3EGCambridgeU.K
Correlation entropy of synaptic input-output dynam... | 13,248,373 | {
"arxiv": "q-bio/0610019",
"mag": 1989899277,
"acl": null,
"pubmed": "17155098",
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physreve.74.041909"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/q-bio/0610019v1.pdf"
],
"pdf_sha": "985e2099126d07a98ceaadf5aff3b6e90b06cea9",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/q-bio/0610019",
"status": "GREEN"
}
} | Correlation entropy of synaptic input-output dynamics | [
"Ingo C Kleppe \nDepartment of Physiology\nUniversity of Cambridge\nDowning StreetCB2 3EGCambridgeU.K",
"Hugh P C Robinson \nDepartment of Physiology\nUniversity of Cambridge\nDowning StreetCB2 3EGCambridgeU.K"
] | The responses of synapses in the neocortex show highly stochastic and nonlinear behavior. The microscopic dynamics underlying this behavior, and its computational consequences during natural patterns of synaptic input, are not explained by conventional macroscopic models of deterministic ensemble mean dynamics. Here, w... | [
{
"title": "I. INTRODUCTION",
"content": "The excitatory cortical synapse is an example of a nonlinear biological system with a high level of intrinsic noise. This leads to a complex relationship between the input -the times of presynaptic action potentials (APs) -and the output -the variable amplitudes of ... | [
{
"id": "",
"caption": "Input-output recurrence plot of synaptic transmission data. Stimulation parameters as in Fig. 3B, D-E. A: Excerpt of a recurrence plot for neighborhood dimensions ǫ = 0.35 mV, δ = 290ms. Matrix elements [i, j] are colored black when Θ(ǫ − Ai − Aj )Θ(δ − ∆ti − ∆tj ) = 1. m = n = 2. B:... | [
{
"text": "M V Tsodyks, H Markram, Proc. Natl. Acad. Sci. USA. Natl. Acad. Sci. USA94719M. V. Tsodyks and H. Markram, Proc. Natl. Acad. Sci. USA 94, 719 (1997).",
"doi": ""
},
{
"text": ". L F Abbott, J A Varela, K Sen, S B Nelson, Science. 275220L. F. Abbott, J. A. Varela, K. Sen, and S. B. Nelson,... |
Heterogeneous Multilayer Generalized Operational Perceptron
28 Aug 2018
Dat Thanh Tran
Laboratory of Signal Processing
Tampere University of Technology
TampereFinland
Serkan Kiranyaz mkiranyaz@qu.edu.qa
Department of Electrical Engineering
Qatar University
Qatar
Moncef Gabbouj moncef.gabbouj@tut.fi
Laboratory of... | 4,890,277 | {
"arxiv": "1804.05093",
"mag": 2950544156,
"acl": null,
"pubmed": "31170081v1",
"pubmedcentral": null,
"dblp": "journals/tnn/TranKGI20",
"doi": "10.1109/tnnls.2019.2914082"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1804.05093v2.pdf"
],
"pdf_sha": "e483f644eb20d79402bab1f5d96025598e101f82",
"open_access": {
"license": null,
"url": "https://arxiv.org/pdf/1804.05093",
"status": "GREEN"
}
} | Heterogeneous Multilayer Generalized Operational Perceptron | [
"Dat Thanh Tran \nLaboratory of Signal Processing\nTampere University of Technology\nTampereFinland",
"Serkan Kiranyaz mkiranyaz@qu.edu.qa \nDepartment of Electrical Engineering\nQatar University\nQatar",
"Moncef Gabbouj moncef.gabbouj@tut.fi \nLaboratory of Signal Processing\nTampere University of Technology\n... | The traditional Multilayer Perceptron (MLP) using McCulloch-Pitts neuron model is inherently limited to a set of neuronal activities, i.e., linear weighted sum followed by nonlinear thresholding step. Previously, Generalized Operational Perceptron (GOP) was proposed to extend conventional perceptron model by defining a... | [
{
"title": "I. INTRODUCTION",
"content": "In recent years, learning systems based on neural networks have gained tremendous popularity in a variety of application domains such as machine vision, natural language processing, biomedical analysis or financial data analysis [1], [2], [3], [4], [5], [6], [7], [8... | [
{
"id": "",
"caption": "=H * c , Φ * = Φ * + {Φ * l } 24: and W * = W * + {W * l }. 25: Finetune W * , B * c . 26: Outputs: Φ * , W * , B * c r l+1 = L l − L l+1 L l",
"content": "=H * c , Φ * = Φ * + {Φ * l } 24: and W * = W * + {W * l }. 25: Finetune W * , B * c . 26: Outputs: Φ * , W * , B * c r l+1 ... | [
{
"text": "You only look once: Unified, real-time object detection. J Redmon, S Divvala, R Girshick, A Farhadi, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionJ. Redmon, S. Divvala, R. Girshick, and A. Farhadi, \"You o... |
Static, Quasi-static and Dynamic Analysis for Scaled Perona-Malik Functionals
Andrea Braides braides@mat.uniroma2.it
Università di Roma "Tor Vergata"
Università di Roma "Tor Vergata"
Valerio Vallocchia vallocch@mat.uniroma2.it
Università di Roma "Tor Vergata"
Università di Roma "Tor Vergata"
Static, Quasi-stat... | 119,146,301 | {
"arxiv": "1610.08295",
"mag": 2963262620,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1007/s10440-018-0155-4"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1610.08295v1.pdf"
],
"pdf_sha": "08eb1b4f267aa728c60e304ea01c0d9448a2e75a",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1610.08295",
"status": "GREEN"
}
} | Static, Quasi-static and Dynamic Analysis for Scaled Perona-Malik Functionals | [
"Andrea Braides braides@mat.uniroma2.it \nUniversità di Roma \"Tor Vergata\"\nUniversità di Roma \"Tor Vergata\"",
"Valerio Vallocchia vallocch@mat.uniroma2.it \nUniversità di Roma \"Tor Vergata\"\nUniversità di Roma \"Tor Vergata\""
] | We present an asymptotic description of local minimization problems, and of quasi-static and dynamic evolutions of scaled Perona-Malik functionals. The scaling is chosen such that these energies Γ-converge to the Mumford-Shah functional by a result by Morini and Negri [20]. | [
{
"title": "Introduction",
"content": "The Perona-Malik anisotropic diffusion technique in Image Processing [22] is formally based on a gradient-flow dynamics related to the non-convex energy\nwhere u represents the output signal or picture defined on Ω and K a tuning parameter. In the convexity domain of t... | [
{
"id": "",
"caption": "J(z) = log(1 + z 2 )",
"content": "Figure 1 :\n1J(z) = log(1 + z 2 )"
},
{
"id": "",
"caption": "Perona-Malik and Mumford-Shah local minima",
"content": "Figure 3 :\n3Perona-Malik and Mumford-Shah local minima"
},
{
"id": "",
"caption": "Perona-Malik a... | [
{
"text": "Energies in SBV and variational models in fracture mechanics. L Ambrosio, A Braides, Homogenization and Applications to Material Sciences. D. Cioranescu, A. Damlamian, P. Donato eds.), GAKUTOGakkōtosho, Tokio, JapanAmbrosio L. and Braides A. Energies in SBV and variational models in fracture mechanic... |
epl draft Quasi-Compact Q-balls
D Bazeia
Departamento de Física
Universidade Federal da Paraíba
58051-970João PessoaPBBrazil
M A Marques
Departamento de Física
Universidade Federal da Paraíba
58051-970João PessoaPBBrazil
R Menezes
Departamento de Física
Universidade Federal da Paraíba
58051-970João PessoaPBBraz... | 202,133,522 | {
"arxiv": "1909.01163",
"mag": 3106534652,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1209/0295-5075/127/21001"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1909.01163v1.pdf"
],
"pdf_sha": "6685b94de466183bd5277719881032fbdfd9ee1a",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1909.01163",
"status": "GREEN"
}
} | epl draft Quasi-Compact Q-balls | [
"D Bazeia \nDepartamento de Física\nUniversidade Federal da Paraíba\n58051-970João PessoaPBBrazil",
"M A Marques \nDepartamento de Física\nUniversidade Federal da Paraíba\n58051-970João PessoaPBBrazil",
"R Menezes \nDepartamento de Física\nUniversidade Federal da Paraíba\n58051-970João PessoaPBBrazil\n\nDeparta... | This work deals with charged nontopological solutions that appear in relativistic models described by a single complex scalar field in two-dimensional spacetime. We study a model which supports novel analytical configurations of the Q-ball type, that engender double exponential tails, in this sense being different from... | [
{
"title": "2.A. Standard Q-ball.",
"content": "Our first example is with the model, driven by the |ϕ| 4 potential\nwith a being a real and positive parameter, a ≥ 0. This model was studied in Refs. [2,3,28]. As one knows, due to the presence of time in the ansatz (3), this potential is influenced by the an... | [
{
"id": "",
"caption": "in which the solution lives, and b is a parameter that depends on a and ω as b = arccoth",
"content": "in which the solution lives, and b is a parameter that depends on a and ω as b = arccoth"
},
{
"id": "",
"caption": "The effective potential in Eq. (32) depicted for... | [
{
"text": ". S Coleman, Nucl. Phys. B. 262263S. Coleman, Nucl. Phys. B 262 (1985) 263;",
"doi": ""
},
{
"text": ". S Coleman, Nucl. Phys. B. 269744Erratum)S. Coleman, Nucl. Phys. B 269 (1986) 744 (Erratum).",
"doi": ""
},
{
"text": ". J M Cerveró, P G Estévez, Phys. Lett. B. 176139J.M. C... |
LARGE TIME ASYMPTOTIC BEHAVIOR OF GRAIN BOUNDARIES MOTION WITH DYNAMIC LATTICE MISORIENTATIONS AND WITH TRIPLE JUNCTIONS DRAG
17 Oct 2019
Yekaterina Epshteyn
ANDChun Liu
Masashi Mizuno
LARGE TIME ASYMPTOTIC BEHAVIOR OF GRAIN BOUNDARIES MOTION WITH DYNAMIC LATTICE MISORIENTATIONS AND WITH TRIPLE JUNCTIONS DRAG
17 O... | 204,744,015 | {
"arxiv": "1910.08022",
"mag": 2981135086,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.4310/cms.2021.v19.n5.a10"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1910.08022v1.pdf"
],
"pdf_sha": "c8530ad4d88e676c89a33bde73210eebac823a3c",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1910.08022",
"status": "GREEN"
}
} | LARGE TIME ASYMPTOTIC BEHAVIOR OF GRAIN BOUNDARIES MOTION WITH DYNAMIC LATTICE MISORIENTATIONS AND WITH TRIPLE JUNCTIONS DRAG | [
"Yekaterina Epshteyn",
"ANDChun Liu",
"Masashi Mizuno"
] | A. Most technologically useful materials are polycrystalline microstructures composed of a myriad of small monocrystalline grains separated by grain boundaries. Dynamics of grain boundaries play an essential role in defining the materials properties across multiple scales. In this work, we study the large time asymptot... | [
{
"title": "I",
"content": "Most technologically useful materials are polycrystalline microstructures composed of a myriad of small monocrystalline grains separated by grain boundaries. Dynamics of grain boundaries play an essential role in defining the materials properties across multiple scales. Experimen... | [
{
"id": "",
"caption": "The figure illustrates the model of the equilibrium state (2.5). There are no misorientation effects on the grain boundaries. A solution of (2.5) is unique under the assumption (2.4), and it minimizes the total length of Γ ( j) ∞ .",
"content": "The figure illustrates the model o... | [
{
"text": "Geometric growth and character development in large metastable networks. K Barmak, E Eggeling, M Emelianenko, Y Epshteyn, D Kinderlehrer, S Ta'asan, Rend. Mat. Appl. 297K. Barmak, E. Eggeling, M. Emelianenko, Y. Epshteyn, D. Kinderlehrer, and S. Ta'asan. Geometric growth and character developmen... |
THE BMM SYMMETRISING TRACE CONJECTURE FOR THE EXCEPTIONAL 2-REFLECTION GROUPS OF RANK 2
16 Jan 2020
Christina Boura
Michel Broué
ANDEirini Chavli
Michel Broué
Maria Chlouveraki
Michel Broué
THE BMM SYMMETRISING TRACE CONJECTURE FOR THE EXCEPTIONAL 2-REFLECTION GROUPS OF RANK 2
16 Jan 2020
We prove the symmet... | 62,791,325 | {
"arxiv": "1810.13370",
"mag": 2966069513,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1016/j.jalgebra.2019.07.010"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1810.13370v2.pdf"
],
"pdf_sha": "25ff042a72ea2068c6fbf51006ddf400368aa7f2",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1810.13370",
"status": "GREEN"
}
} | THE BMM SYMMETRISING TRACE CONJECTURE FOR THE EXCEPTIONAL 2-REFLECTION GROUPS OF RANK 2 | [
"Christina Boura \nMichel Broué",
"ANDEirini Chavli \nMichel Broué",
"Maria Chlouveraki \nMichel Broué"
] | We prove the symmetrising trace conjecture of Broué, Malle and Michel for the generic Hecke algebra associated to the exceptional irreducible complex reflection group G13. Our result completes the proof of the conjecture for the exceptional 2-reflection groups of rank 2. | [
{
"title": "Introduction",
"content": "Real reflection groups are finite groups of real matrices generated by reflections. They are also known as finite Coxeter groups. Complex reflection groups are finite groups of complex matrices generated by pseudo-reflections, that is, non-trivial elements that fix a h... | [
{
"id": "",
"caption": "es ) with f k denoting the k-th elementary symmetric polynomial, for k = 1, . . . , e s . Therefore, in the presentation of H(W ), we have two kinds of relations: the braid relations, coming from the Artin-like presentation of B(W ), and the positive Hecke relations:",
"content":... | [
{
"text": "Representation theory of a Hecke algebra of G(r, p, n). S Ariki, J. Algebra. 177S. Ariki, Representation theory of a Hecke algebra of G(r, p, n), J. Algebra 177 (1995), 164-185.",
"doi": ""
},
{
"text": "A Hecke algebra of (Z/rZ) ≀ Sn and construction of its irreducible representations. S... |
Fivebrane Instanton Corrections to the Universal Hypermultiplet
2 May 2006
Marijn Davidse
Institute for Theoretical Physics
Spinoza Institute Utrecht University
3508 TDUtrechtThe Netherlands
Ulrich Theis theis@hep.itp.tuwien.ac.at
Institute for Theoretical Physics
Vienna University of Technology
Wiedner Hauptstras... | 15,527,905 | {
"arxiv": "hep-th/0404147",
"mag": 2145315226,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1016/j.nuclphysb.2004.07.021"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/hep-th/0404147v2.pdf"
],
"pdf_sha": "72828d4a29842ab924a71970245da5db4c03cb4b",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/hep-th/0404147",
"status": "GREEN"
}
} | Fivebrane Instanton Corrections to the Universal Hypermultiplet | [
"Marijn Davidse \nInstitute for Theoretical Physics\nSpinoza Institute Utrecht University\n3508 TDUtrechtThe Netherlands",
"Ulrich Theis theis@hep.itp.tuwien.ac.at \nInstitute for Theoretical Physics\nVienna University of Technology\nWiedner Hauptstrasse 8-10A-1040ViennaAustria",
"Stefan Vandoren \nInstitute fo... | We analyze the Neveu-Schwarz fivebrane instanton in type IIA string theory compactifications on rigid Calabi-Yau threefolds, in the low-energy supergravity approximation. It there appears as a finite action solution to the Euclidean equations of motion of a double-tensor multiplet (dual to the universal hypermultiplet)... | [
{
"title": "Introduction",
"content": "One particular object that plays an important role in nonperturbative string theory is the Neveu-Schwarz fivebrane (NS5-brane). It is the magnetic dual of the fundamental string, and appears as a soliton-like solution preserving half of the supersymmetry, for instance ... | [
{
"id": "",
"caption": "/ k is defined in (7.3). We have shown in sections 5 and 7 that (8.18) indeed produces the correct field equations. We can construct hermitean operators M k = D / −kD / k andM k =D / −k D / k . The cases of interest are when k = ±3/4. The spectrum of nonzero modes of M −3/4 andM 3/4 ... | [
{
"text": "Heterotic solitons. A Strominger, Nucl. Phys. 343Erratum: B353 (1991) 565A. Strominger, Heterotic solitons. Nucl. Phys. B343 (1990) 167. [Erratum: B353 (1991) 565];",
"doi": ""
},
{
"text": "World sheet approach to heterotic instantons and solitons. C CallanJr, J Harvey, A Strominger, Nuc... |
Classical and quantum harmonic oscillators subject to a time dependent force
20 Apr 2022
Henryk Gzyl henryk.gzyl@iesa.edu
Centro de Finanzas
IESA
Caracas
Classical and quantum harmonic oscillators subject to a time dependent force
20 Apr 2022arXiv:2204.03460v2 [quant-ph]Forced harmonic oscillatorcanonical transform... | 248,006,337 | {
"arxiv": "2204.03460",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.2139/ssrn.4070993"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2204.03460v2.pdf"
],
"pdf_sha": "82876045eb6a1dc94e0068dbc45a55277d267b73",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/2204.03460",
"status": "GREEN"
}
} | Classical and quantum harmonic oscillators subject to a time dependent force | [
"Henryk Gzyl henryk.gzyl@iesa.edu \nCentro de Finanzas\nIESA\nCaracas"
] | In this work we address the problem of the quantization of a simple harmonic oscillator that is perturbed by a time dependent force. The approach consists of removing the perturbation by a canonical change of coordinates. Since the quantization procedure uses the classical Hamiltonian formalism as staring point, the ch... | [
{
"title": "Introduction and Preliminaries",
"content": "When confronted with a classical or quantum system subject to a perturbation, one possible approach to describe it consists in removing the perturbation by means of a change of coordinates. So the first problem to consider is to determine the change o... | [
{
"id": "",
"caption": "where we prove that the unitary transformation maps the position, momentum and Hamiltonian operators as in the classical case. On the one hand this means that the quantization rules behave consistently under canonical transformations, and on the other, that we can transform the solut... | [
{
"text": "R P Feynman, R B Leighton, M Sands, The Feynman Lectures on Physics. ReadingAddison Wesley Pub. Co. IncIFeynman R.P., Leighton, R.B. and Sands, M. (1963).The Feynman Lectures on Physics, Vol. I, Addison Wesley Pub. Co. Inc., Reading.",
"doi": ""
},
{
"text": ". V I Arnold, Mathematical Me... |
Generalized quark number susceptibilities from fugacity expansion at finite chemical potential for N f = 2 Wilson fermions
Christof Gattringer
Institute of Physics
University of Graz
8010GrazAustria
Hans-Peter Schadler
Institute of Physics
University of Graz
8010GrazAustria
Generalized quark number susceptibilit... | 118,498,909 | {
"arxiv": "1411.5133",
"mag": 1996654111,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevd.91.074511"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1411.5133v2.pdf"
],
"pdf_sha": "d2098ccfb7552162b0fdfe905a0cee2d562904da",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1411.5133",
"status": "GREEN"
}
} | Generalized quark number susceptibilities from fugacity expansion at finite chemical potential for N f = 2 Wilson fermions | [
"Christof Gattringer \nInstitute of Physics\nUniversity of Graz\n8010GrazAustria",
"Hans-Peter Schadler \nInstitute of Physics\nUniversity of Graz\n8010GrazAustria"
] | Generalized susceptibilities of the net quark number have been proposed to be good probes for the transitions in the QCD phase diagram and for the search of a possible critical end point. In this article we explore a new strategy for computing quark number susceptibilities from lattice QCD via an expansion in the fugac... | [
{
"title": "I. INTRODUCTION",
"content": "In recent years heavy ion collision experiments started to directly probe the phase diagram of QCD. In particular results from RHIC and LHC shed some light on properties of the quark gluon plasma and the position of the crossover line. To fully explore the phase dia... | [
{
"id": "",
"caption": "Distribution of the absolute values of the canonical determinants normalized by |D (0) | and weighted with the fugacity factors as a function of the net quark number q. The results are from our 12 3 × 6, κ = 0.162, β = 5.40 ensemble.",
"content": "Figure 1 :\n1Distribution of the... | [
{
"text": ". † − E −µβd, 4) H − γ 5 ] det[γ 5† − e −µβD(4) H − γ 5 ] det[γ 5 ] .",
"doi": ""
},
{
"text": ". D Sexty, arXiv:1410.8813PoS. 2014D. Sexty, PoS LATTICE 2014 (2014) [arXiv:1410.8813];",
"doi": ""
},
{
"text": ". C Gattringer, arXiv:1401.7788PoS. 20132C. Gattringer, PoS LATTICE... |
Dipolar broadening of nuclear spin resonance under dynamical pumping
10 May 2012
O Tsyplyatyev
Department of Physics and Astronomy
University of Sheffield
S3 7RHSheffieldUK
D M Whittaker
Department of Physics and Astronomy
University of Sheffield
S3 7RHSheffieldUK
Dipolar broadening of nuclear spin resonance unde... | 119,214,281 | {
"arxiv": "1205.2189",
"mag": 2031516781,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevb.85.125123"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/1205.2189"
],
"pdf_sha": "78bc69a21c939be1dec4cb15d2f56f26b1fdd612",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1205.2189",
"status": "GREEN"
}
} | Dipolar broadening of nuclear spin resonance under dynamical pumping | [
"O Tsyplyatyev \nDepartment of Physics and Astronomy\nUniversity of Sheffield\nS3 7RHSheffieldUK",
"D M Whittaker \nDepartment of Physics and Astronomy\nUniversity of Sheffield\nS3 7RHSheffieldUK"
] | We study the polarisation dependence of the homogeneously broadened nuclear spin resonance in a crystal. We employ a combinatorial method to restrict the nuclear states to a fixed polarisation and show that the centre of the resonance is shifted linearly with the nuclear polarisation by up to the zero polarisation line... | [
{
"title": "I. INTRODUCTION",
"content": "Recent interest in studying nuclear spins was fuelled [1][2][3][4][5][6][7][8] by the dephasing effect in GaAs quantum dots which is induced by fluctuations of the nuclear (Overhauser) field and blocks prospective application of electron spins for spintronics [9] an... | [
{
"id": "",
"caption": "Substituting the product of the two commutators [H, I − ] [H, I + ] into Eq. (6), and using the above expression, we obtain the second cumulant ν 2 = m 2 − m 2 that corresponds to the line width,",
"content": "Substituting the product of the two commutators [H, I − ] [H, I + ] in... | [
{
"text": ". J R Petta, A C Johnson, J M Taylor, E A Laird, A Yacoby, M D Lukin, C M Marcus, M P Hanson, A C Gossard, Science. 3092180J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Science 309, 2180 (2005).",
"doi": ""
},
{
... |
Multilayer Block Models for Exploratory Analysis of Computer Event Logs
21 Jun 2022
Corentin Larroche corentin.larroche@ssi.gouv.fr
French National Cybersecurity Agency (ANSSI)
ParisFrance
Multilayer Block Models for Exploratory Analysis of Computer Event Logs
21 Jun 2022575B38DC0C44C8A8FD57BE419CD388A2arXiv:2206.1... | 249,890,153 | {
"arxiv": "2206.10413",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/complexnetworks/Larroche22",
"doi": "10.48550/arxiv.2206.10413"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2206.10413v1.pdf"
],
"pdf_sha": "2dfa64c86f03aca01f0392544ba8c92609edbc0a",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/2206.10413",
"status": "GREEN"
}
} | Multilayer Block Models for Exploratory Analysis of Computer Event Logs | [
"Corentin Larroche corentin.larroche@ssi.gouv.fr \nFrench National Cybersecurity Agency (ANSSI)\nParisFrance"
] | We investigate a graph-based approach to exploratory data analysis in the context of network security monitoring.Given a possibly large batch of event logs describing ongoing activity, we first represent these events as a bipartite multiplex graph.We then apply a modelbased biclustering algorithm to extract relevant cl... | [
{
"title": "Introduction",
"content": "The ever-increasing prevalence of cyber threats compels network administrators and information security specialists to constantly monitor ongoing activity inside sensitive networks.Such monitoring often boils down to collecting and analyzing event logs, which can be ge... | [
{
"id": "",
"caption": "all i ∈ [I] and j ∈ [J].It then maximizes the fuzzy criterion",
"content": "all i ∈ [I] and j ∈ [J].It then maximizes the fuzzy criterion"
},
{
"id": "",
"caption": "TCP/20 (FTP-Data), TCP/21 (FTP), TCP/22 (SSH), TCP/23 (Telnet), TCP/25 (SMTP), TCP/53 (DNS), TCP/80 (H... | [
{
"text": "Home-centric visualization of network traffic for security administration. R Ball, G A Fink, C North, VizSec/DMSec2004",
"doi": ""
},
{
"text": "Stochastic block models for multiplex networks: an application to a multilevel network of researchers. P Barbillon, S Donnet, E Lazega, A Bar-He... |
Cosmography beyond Standard Candles and Rulers
25 Jan 2012
Jun-Qing Xia
Scuola Internazionale Superiore di Studi Avanzati
Via Bonomea 26534136TriesteItaly
Vincenzo Vitagliano
Scuola Internazionale Superiore di Studi Avanzati
Via Bonomea 26534136TriesteItaly
INFN sez. Trieste
Via Valerio 234127TriesteItaly
Stefan... | 119,191,028 | {
"arxiv": "1103.0378",
"mag": 2057950870,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevd.85.043520"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1103.0378v3.pdf"
],
"pdf_sha": "ede3c4390e42f3a3b59abe4ee2f76e022f9c42b7",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1103.0378",
"status": "GREEN"
}
} | Cosmography beyond Standard Candles and Rulers | [
"Jun-Qing Xia \nScuola Internazionale Superiore di Studi Avanzati\nVia Bonomea 26534136TriesteItaly",
"Vincenzo Vitagliano \nScuola Internazionale Superiore di Studi Avanzati\nVia Bonomea 26534136TriesteItaly\n\nINFN sez. Trieste\nVia Valerio 234127TriesteItaly",
"Stefano Liberati \nScuola Internazionale Superi... | We perform a cosmographic analysis using several cosmological observables such as the luminosity distance moduli, the volume distance, the angular diameter distance and the Hubble parameter. These quantities are determined using different data sets: Supernovae type Ia and Gamma Ray Bursts, the Baryonic Acoustic Oscilla... | [
{
"title": "I. INTRODUCTION",
"content": "Cosmology is going through a golden age as we can nowadays start reconstructing the expansion history of the universe with unprecedented precision. The huge range of data sets spans a wide realm of observations with heterogeneous nature, providing us with a much mor... | [
{
"id": "",
"caption": "One-dimensional likelihood distributions for the parameters q0, j0, s0 and c0 for the data combinations SNIa+GRB+BAO+CMB+Hub.",
"content": "FIG. 1 :\n1One-dimensional likelihood distributions for the parameters q0, j0, s0 and c0 for the data combinations SNIa+GRB+BAO+CMB+Hub."
... | [
{
"text": "INFN/PD-51, ASI/AAE grant and the European ERC-StG \"cosmoIGM\" grant. MV is supported by PRIN-MIUR. MV is supported by PRIN-MIUR, PRIN-INAF 2009, INFN/PD-51, ASI/AAE grant and the European ERC-StG \"cosmoIGM\" grant.",
"doi": ""
},
{
"text": ". M Kunz, A R Liddle, D Parkinson, C Gao, Phy... |
Majorana Zero-modes and Topological Phases of Multi-flavored Jackiw-Rebbi model
23 Jul 2012
Shih-Hao Ho
Center for Theoretical Physics
Massachusetts Institute of Technology
02139CambridgeMAUSA
Feng-Li Lin blinfengli@phy.ntnu.edu.tw
Department of Physics
Massachusetts Institute of Technology
02139CambridgeMassachus... | 16,745,316 | {
"arxiv": "1207.1620",
"mag": 1622352708,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1007/jhep12(2012)074"
} | {
"pdf_urls": [
"http://export.arxiv.org/pdf/1207.1620"
],
"pdf_sha": "a54a39f9315e7f20aef6c206e859a208da68c5e6",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1207.1620",
"status": "GREEN"
}
} | Majorana Zero-modes and Topological Phases of Multi-flavored Jackiw-Rebbi model | [
"Shih-Hao Ho \nCenter for Theoretical Physics\nMassachusetts Institute of Technology\n02139CambridgeMAUSA",
"Feng-Li Lin blinfengli@phy.ntnu.edu.tw \nDepartment of Physics\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUSA\n\nDepartment of Physics\nNational Taiwan Normal University\n116TaipeiT... | Motivated by the recent Kitaev's K-theory analysis of topological insulators and superconductors, we adopt the same framework to study the topological phase structure of Jackiw-Rebbi model in 3+1 dimensions. According to the K-theory analysis based on the properties of the charge conjugation and time reversal symmetrie... | [
{
"title": "I. INTRODUCTION",
"content": "In the recent years, Majorana zero-modes in condensed matter systems have attracted much attention due to its topological and/or symmetry protected degeneracy (similar to the topological ground state degeneracy of topological ordered states). The degeneracy leads to... | [
{
"id": "",
"caption": "The above mentioned 3+1 dimensional Majorana zero modes are one of the examples in this general classification schemes.",
"content": "The above mentioned 3+1 dimensional Majorana zero modes are one of the examples in this general classification schemes."
},
{
"id": "",
... | [
{
"text": "However, these two choices lead us to nowhere since all λ's have to be zero in order to satisfy the condition. 3However, these two choices lead us to nowhere since all λ's have to be zero in order to satisfy the condition (A3).",
"doi": ""
},
{
"text": "Majorana Fermions and Non-Abelian S... |
Testing the Origin of Cosmological Magnetic Fields through the Large-Scale Structure Consistency Relations
5 Feb 2014
P Berger
Department of Theoretical Physics and Center for Astroparticle Physics (CAP)
24 quai E. AnsermetCH-1211Geneva 4Switzerland
A Kehagias
Department of Theoretical Physics and Center for Astro... | 119,239,830 | {
"arxiv": "1402.1044",
"mag": 2000187614,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1088/1475-7516/2014/05/025"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1402.1044v1.pdf"
],
"pdf_sha": "54e9a33322f4bd20b5849928341e7ab8c53f916f",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1402.1044",
"status": "GREEN"
}
} | Testing the Origin of Cosmological Magnetic Fields through the Large-Scale Structure Consistency Relations | [
"P Berger \nDepartment of Theoretical Physics and Center for Astroparticle Physics (CAP)\n24 quai E. AnsermetCH-1211Geneva 4Switzerland",
"A Kehagias \nDepartment of Theoretical Physics and Center for Astroparticle Physics (CAP)\n24 quai E. AnsermetCH-1211Geneva 4Switzerland\n\nPhysics Division\nNational Technica... | We study the symmetries of the post-recombination cosmological magnetohydrodynamical equations which describe the evolution of dark matter, baryons and magnetic fields in a self-consistent way. This is done both at the level of fluid equations and of Vlasov-Poisson-Maxwell equations in phase space. We discuss some cons... | [
{
"title": "Introduction",
"content": "Magnetic fields appear to be ubiquitous in astrophysical and cosmological environments [1]. The largest observable magnetic fields are found inside the atmospheres of galaxy clusters with a strength of the order of the µG in the core regions. Magnetic fields are found ... | [
{
"id": "",
"caption": "the Alfvén speed. Strictly speaking, Alfvén waves are oscillatory perturbations superimposed on a homogeneous magnetic component and the Alfvén speed depends on the amplitude of the homogeneous component. In the cosmological context where the magnetic field is purely stochastic, the ... | [
{
"text": "For a review and references therein, see R. Durrer and A. Neronov. arXiv:1303.7121Astron. Astrophys. Rev. 21astro-ph.COFor a review and references therein, see R. Durrer and A. Neronov, Astron. Astrophys. Rev. 21, 62 (2013) [arXiv:1303.7121 [astro-ph.CO]].",
"doi": "arXiv:1303.7121"
},
{
... |
On the S-Euclidean minimum of an ideal class
11 Aug 2013
Kevin J Mcgown
On the S-Euclidean minimum of an ideal class
11 Aug 2013FD3E37E23F821FEB6BFD68AD22768E4BarXiv:1304.1815v2[math.NT]
We show that the S-Euclidean minimum of an ideal class is a rational number, generalizing a result of Cerri.We also give some coro... | 116,933,265 | {
"arxiv": "1304.1815",
"mag": 2964349186,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.4064/aa171-2-2"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1304.1815v2.pdf"
],
"pdf_sha": "6f0cfb0a70441a192bb345d2c3754b4fbcdb5abb",
"open_access": {
"license": null,
"url": "https://www.impan.pl/shop/publication/transaction/download/product/91018?download.pdf",
"status": "BRONZE"
}
} | On the S-Euclidean minimum of an ideal class | [
"Kevin J Mcgown"
] | We show that the S-Euclidean minimum of an ideal class is a rational number, generalizing a result of Cerri.We also give some corollaries which explain the relationship of our results with Lenstra's notion of a norm-Euclidean ideal class and the conjecture of Barnes and Swinnerton-Dyer on quadratic forms.The proof is s... | [
{
"title": "Introduction",
"content": "Let K be a number field of degree n = r 1 + 2r 2 .Let S be a finite set of primes containing the infinite primes S ∞ .Let O S denote the S-integers of K, let U S denote the S-units of K, and let N S denote the S-norm map. 1ecall that the S-norm of a number ξ ∈ K is def... | [
{
"id": "",
"caption": "2 follows immediately from: Let a, b ∈ R + .Suppose a and b are multiplicatively independent.Then {a n b m | n, m ∈ Z} is dense in R + .Proof.Taking the logarithm to the base a of a n b m gives m + nα where α = log b/ log a.Thus it suffices to show that {m + nα | m, n ∈ Z} is dense i... | [
{
"text": "The inhomogeneous minima of binary quadratic forms. E S Barnes, H P F Swinnerton-Dyer, II. Acta Math. 881952",
"doi": ""
},
{
"text": "Multi-invariant sets on tori. Daniel Berend, Trans. Amer. Math. Soc. 28021983",
"doi": ""
},
{
"text": "Minimal sets on tori. Daniel Berend, E... |
R-PROCESS NUCLEOSYNTHESIS IN COMMON ENVELOPE JETS SUPERNOVAE
Draft version October 10, 2018 Draft version October 10, 2018
Aldana Grichener
Noam Soker
R-PROCESS NUCLEOSYNTHESIS IN COMMON ENVELOPE JETS SUPERNOVAE
Draft version October 10, 2018 Draft version October 10, 2018Preprint typeset using L A T E X style emul... | 119,396,863 | {
"arxiv": "1810.03889",
"mag": 2895959435,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.3847/1538-4357/ab1d5d"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1810.03889v1.pdf"
],
"pdf_sha": "1c80ba2bf62739e52a06bc147f3516da5c5f295e",
"open_access": {
"license": null,
"url": "https://iopscience.iop.org/article/10.3847/1538-4357/ab1d5d/pdf",
"status": "BRONZE"
}
} | R-PROCESS NUCLEOSYNTHESIS IN COMMON ENVELOPE JETS SUPERNOVAE | [
"Aldana Grichener",
"Noam Soker"
] | We study r-process feasibility inside jets launched by a cold neutron star (NS) spiralling-in inside the core of a giant star, and find that such common envelope jets supernova (CEJSN) events might be a significant source of heavy r-process elements in the early Universe. We run the stellar evolution code MESA to follo... | [
{
"title": "INTRODUCTION",
"content": "One of the topics of intensive research in astrophysics concerns the sites where r-process nucleosynthesis takes place, and in particular the sites of \"strong r-process\" nucleosynthesis where elements of atomic weight A 130, are formed. Over the years, two of the mai... | [
{
"id": "",
"caption": "-The mass (black) and density (blue) profiles of our M ZAMS = 15M stellar model at the onset of the CEE assuming that the NS is swallowed by the giant star when it reaches a radius of R = 150R (upper panel),R = 300R (middle panel) or R = 600R (lower panel). The red dashed-dotted line... | [
{
"text": ". D Argast, M Samland, F.-K Thielemann, Y.-Z Qian, A&A. 416997Argast, D., Samland, M., Thielemann, F.-K., & Qian, Y.-Z. 2004, A&A, 416, 997",
"doi": ""
},
{
"text": ". P J Armitage, M Livio, ApJ. 532540Armitage, P. J., & Livio, M. 2000, ApJ, 532, 540",
"doi": ""
},
{
"text": "... |
THE EULER TOP AND CANONICAL LIFTS
8 Aug 2017
Alexandru Buium
Emma Previato
THE EULER TOP AND CANONICAL LIFTS
8 Aug 2017
In this note, we prove a finiteness result for fibers that are canonical lifts in a given elliptic fibration. The question was motivated by the authors' construction of an arithmetic Euler top, an... | 119,123,769 | {
"arxiv": "1708.02558",
"mag": 2963818740,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1016/j.jnt.2018.02.008"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1708.02558v1.pdf"
],
"pdf_sha": "a9c9be986759024f41bbcd45005cdada1922474c",
"open_access": {
"license": null,
"url": "https://www.sciencedirect.com/science/article/am/pii/S0022314X1830057X",
"status": "BRONZE"
}
} | THE EULER TOP AND CANONICAL LIFTS | [
"Alexandru Buium",
"Emma Previato",
"Alexandru Buium",
"Emma Previato"
] | In this note, we prove a finiteness result for fibers that are canonical lifts in a given elliptic fibration. The question was motivated by the authors' construction of an arithmetic Euler top, and it highlights an interesting discrepancy between the arithmetic and the classical case: in the former, it is impossible to... | [
{
"title": "Introduction",
"content": ""
},
{
"title": "Introduction",
"content": "In this note, we present an analogue, for the multiplicative group G m , of one of the main finiteness results for elliptic curves proved in [4]. This is then applied to a question left open in [3,Remark 6.6], nam... | [
{
"id": "",
"caption": "0 the curves E c given by 2.5 are tangent to the vector field so the derivation δ induces derivations δ c on the rings O(E c ). The remarkable fact about the situation is that δ c on O(E c ) are linearized in the sense that for all c ∈ A 2 with N (c) = 0, (2.9)δ c extends to a vector... | [
{
"text": "S Bosch, W Lütkebohmert, M Raynaud, Néron Models. SpringerS. Bosch, W. Lütkebohmert, M. Raynaud, Néron Models, Springer, 1980.",
"doi": ""
},
{
"text": "Differential characters of abelian varieties over p-adic fields. A Buium, Invent. Math. 122A. Buium, Differential characters of abelian ... |
CONJECTURES FOR THE INTEGRAL MOMENTS AND RATIOS OF L-FUNCTIONS OVER FUNCTION FIELDS
11 Jul 2014
J C Andrade
J P Keating
CONJECTURES FOR THE INTEGRAL MOMENTS AND RATIOS OF L-FUNCTIONS OVER FUNCTION FIELDS
11 Jul 2014
We extend to the function field setting the heuristic previously developed, by Conrey, Farmer, Keati... | 119,309,636 | {
"arxiv": "1407.3099",
"mag": 2090082281,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1016/j.jnt.2014.02.019"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1407.3099v1.pdf"
],
"pdf_sha": "94e782e78c8fcaf6bb9b1fc443f2797673f51daa",
"open_access": {
"license": "elsevier-specific: oa user license",
"url": "https://doi.org/10.1016/j.jnt.2014.02.019",
"status": "BRONZE"
}
} | CONJECTURES FOR THE INTEGRAL MOMENTS AND RATIOS OF L-FUNCTIONS OVER FUNCTION FIELDS | [
"J C Andrade",
"J P Keating"
] | We extend to the function field setting the heuristic previously developed, by Conrey, Farmer, Keating, Rubinstein and Snaith, for the integral moments and ratios of L-functions defined over number fields. Specifically, we give a heuristic for the moments and ratios of a family of L-functions associated with hyperellip... | [
{
"title": "Some Basic facts about L-functions in function fields",
"content": "We begin by fixing a finite field F q of odd cardinality and letting A = F q [x] be the polynomial ring over F q in the variable x. We will denote by C any smooth, projective, geometrically connected curve of genus g ≥ 1 defined... | [
{
"id": "",
"caption": "Basic facts about L-functions in function fields 7 2.1. Background on F q [x] 7 2.2. Quadratic Characters and the Corresponding L-functions 8 2.3. The Hyperelliptic Ensemble H 2g+1,q 9 3. Statement of the Main Results 10 4. Integral Moments of L-functions in the Hyperelliptic Ensembl... | [
{
"text": "Conjectures and experiments concerning the moments of L(1/2, χ d ). M Alderson, M O Rubinstein, Experimental Mathematics. 213M. Alderson and M.O. Rubinstein, Conjectures and experiments concerning the moments of L(1/2, χ d ), Experimental Mathematics, Volume 21, Issue 3 (2012), 307-328.",
"doi": ... |
On parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems
10 Mar 2014 March, 7 2014
A Lastra
S Malek
Departamento de Física y Matemáticas
University of Alcalá
Ap. de Correos 20E-28871Alcalá de Henares (Madrid)Spain
University of Lille 1
Laboratoire Paul Painlevé
59655Villeneuve d'Ascq ced... | 8,078,007 | {
"arxiv": "1403.2350",
"mag": 2963525691,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1016/j.jde.2015.06.020"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1403.2350v1.pdf"
],
"pdf_sha": "adbe678076266a908fdd8fafa39483473bc175db",
"open_access": {
"license": "elsevier-specific: oa user license",
"url": "https://doi.org/10.1016/j.jde.2015.06.020",
"status": "BRONZE"
}
} | On parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems | [
"A Lastra",
"S Malek",
"Departamento de Física y Matemáticas\nUniversity of Alcalá\nAp. de Correos 20E-28871Alcalá de Henares (Madrid)Spain",
"University of Lille 1\nLaboratoire Paul Painlevé\n59655Villeneuve d'Ascq cedexFrance"
] | We study a nonlinear initial value Cauchy problem depending upon a complex perturbation parameter ǫ with vanishing initial data at complex time t = 0 and whose coefficients depend analytically on (ǫ, t) near the origin in C 2 and are bounded holomorphic on some horizontal strip in C w.r.t the space variable. This probl... | [
{
"title": "Introduction",
"content": "In this paper, we consider a family of parameter depending nonlinear initial value Cauchy problems of the form (1) Q(∂ z )(∂ t u(t, z, ǫ)) = (Q 1 (∂ z )u(t, z, ǫ))(Q 2 (∂ z )u(t, z, ǫ)) + D l=1 ǫ ∆ l t d l ∂ δ l t R l (∂ z )u(t, z, ǫ) + c 0 (t, z, ǫ)R 0 (∂ z )u(t, z, ǫ... | [
{
"id": "",
"caption": "Let k ≥ 1 be an integer. Let m k (n) be the sequence defined bym k (n) n T n ∈ T E[[T ]]with coefficients in a Banach space (E, ||.|| E ) is said to be m k −summable with respect to t in the direction d ∈ [0, 2π) if i) there exists ρ ∈ R + such that the following formal series, calle... | [
{
"text": "From divergent power series to analytic functions. Theory and application of multisummable power series. W Balser, Lecture Notes in Mathematics. 1582Springer-Verlagx+108 ppW. Balser, From divergent power series to analytic functions. Theory and application of multisummable power series. Lecture Notes... |
PHYSICAL STRUCTURE OF PLANETARY NEBULAE. II. NGC 7662
Martí Ín
A Guerrero
Elizabeth G Jaxon
You-Hua Chu
PHYSICAL STRUCTURE OF PLANETARY NEBULAE. II. NGC 7662
Receiv ved 2004 May 25; accepted 2004 June 29planetary nebulae: general -planetary nebulae: individual (NGC 7662)
We present a study of the structure and k... | 7,142,085 | {
"arxiv": "astro-ph/0407029",
"mag": 2022868277,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1086/423913"
} | {
"pdf_urls": [],
"pdf_sha": "a000fca900522fd906a22c4aac1d16f393159676",
"open_access": {
"license": null,
"url": null,
"status": "CLOSED"
}
} | PHYSICAL STRUCTURE OF PLANETARY NEBULAE. II. NGC 7662 | [
"Martí Ín",
"A Guerrero",
"Elizabeth G Jaxon",
"You-Hua Chu"
] | We present a study of the structure and kinematics of the triple-shell planetary nebula NGC 7662 based on long-slit echelle spectroscopic observations and Hubble Space Telescope archival narrowband images. The structure of the main nebula consists of a central cavity surrounded by two concentric shells. The bright inne... | [
{
"title": "INTRODUCTION",
"content": "Planetary nebulae (PNs) exhibit a wide variety of morphologies, including round, elliptical, bipolar, and pointsymmetric (Balick 1987;Schwarz et al. 1992;Manchado et al. 1996). These main morphological classes can be accounted for by the interacting stellar winds model... | [
{
"id": "",
"caption": "-Negative gray-scale representation of the HST WFPC2 narrowband images of NGC 7662. The contribution of the H line in the [ N ii] k6584 filter has been subtracted. The slit positions of the echelle observations are marked on the [O iii] image. The images are displayed with a square r... | [
{
"text": ". J Alexander, B Balick, AJ. 114713Alexander, J., & Balick, B. 1997, AJ, 114, 713",
"doi": ""
},
{
"text": ". B Balick, AJ. 94671Balick, B. 1987, AJ, 94, 671",
"doi": ""
},
{
"text": ". B Balick, J Alexander, A R Hajian, Y Terzian, M Perinotto, P Patriarchi, AJ. 116360Balick, ... |
Learning Human-Robot Handovers Through π-STAM: Policy Improvement With Spatio-Temporal Affordance Maps
Francesco Riccio
Roberto Capobianco
Daniele Nardi
Learning Human-Robot Handovers Through π-STAM: Policy Improvement With Spatio-Temporal Affordance Maps
Human-robot handovers are characterized by high uncertain... | 14,045,519 | {
"arxiv": "1610.02609",
"mag": 2949577940,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/humanoids/RiccioCN16",
"doi": "10.1109/humanoids.2016.7803373"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1610.02609v2.pdf"
],
"pdf_sha": "743f3ee37b9d61720dd17a499c1efb243d78feae",
"open_access": {
"license": null,
"url": null,
"status": "CLOSED"
}
} | Learning Human-Robot Handovers Through π-STAM: Policy Improvement With Spatio-Temporal Affordance Maps | [
"Francesco Riccio",
"Roberto Capobianco",
"Daniele Nardi"
] | Human-robot handovers are characterized by high uncertainty and poor structure of the problem that make them difficult tasks. While machine learning methods have shown promising results, their application to problems with large state dimensionality, such as in the case of humanoid robots, is still limited. Additionally... | [
{
"title": "I. INTRODUCTION",
"content": "Human-robot handovers are fundamental but difficult tasks, due to the unpredictability of human behaviors and the need to appropriately recover from robot failures. They are, in fact, characterized by high uncertainty and poor structure of the problem that make it d... | [
{
"id": "",
"caption": "Learned affordance function for the action left_arm_up both in a simulated environment and real scenario.",
"content": "Fig. 1 .\n1Learned affordance function for the action left_arm_up both in a simulated environment and real scenario."
},
{
"id": "",
"caption": "Dec... | [
{
"text": "A bayesian approach for policy learning from trajectory preference queries. A Wilson, A Fern, P Tadepalli, Advances in Neural Information Processing Systems. P. Bartlett, F. Pereira, C. Burges, L. Bottou, and K. Weinberger25A. Wilson, A. Fern, and P. Tadepalli, \"A bayesian approach for policy learni... |
THE EFFECT OF STAR FORMATION HISTORY ON THE INFERRED STELLAR INITIAL MASS FUNCTION
Bruce G Elmegreen
Department of Astronomy
IBM Research Division
T. J. Watson Research Center
P.O. Box 21810598Yorktown HeightsNY
John Scalo
University of Texas
78712AustinTX
THE EFFECT OF STAR FORMATION HISTORY ON THE INFERRED STE... | 119,338,607 | {
"arxiv": "astro-ph/0509282",
"mag": 1968468356,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1086/497889"
} | {
"pdf_urls": [
"https://web.archive.org/web/20181030080025/http:/iopscience.iop.org/article/10.1086/497889/pdf"
],
"pdf_sha": "112e86294c7e2f2931c639b71af24b100a12fda3",
"open_access": {
"license": null,
"url": null,
"status": "CLOSED"
}
} | THE EFFECT OF STAR FORMATION HISTORY ON THE INFERRED STELLAR INITIAL MASS FUNCTION | [
"Bruce G Elmegreen \nDepartment of Astronomy\nIBM Research Division\nT. J. Watson Research Center\nP.O. Box 21810598Yorktown HeightsNY",
"John Scalo \nUniversity of Texas\n78712AustinTX"
] | Peaks and lulls in the star formation rate (SFR) over the history of the Galaxy produce plateaus and declines in the present-day mass function (PDMF) where the main-sequence lifetime overlaps the age and duration of the SFR variation. These PDMF features can be misinterpreted as the form of the intrinsic stellar initia... | [
{
"title": "INTRODUCTION",
"content": "The star formation rate (SFR) in galaxies varies over a wide range of timescales. The ratio of recent to past-average SFR (denoted b; see Scalo 1986) highlights the most recent changes. This ratio has been estimated using the equivalent width of H (Kennicutt et al. 199... | [
{
"id": "",
"caption": "-Three sample star formation histories (left), each of which consists of a constant rate with a burst at various times. Present time is to the left. Successive plots are",
"content": "Fig. 1 .\n1-Three sample star formation histories (left), each of which consists of a constant r... | [
{
"text": ". D C Barry, ApJ. 334436Barry, D. C. 1988, ApJ, 334, 436",
"doi": ""
},
{
"text": ". S Basu, N C Rana, ApJ. 393373Basu, S., & Rana, N. C. 1992, ApJ, 393, 373",
"doi": ""
},
{
"text": ". A E Bauer, N Drory, G J Hill, G Feulner, ApJ. 62189Bauer, A. E., Drory, N., Hill, G. J., & ... |
FRACTIONAL HARDY-SOBOLEV TYPE INEQUALITIES FOR HALF SPACES AND JOHN DOMAINS
11 Sep 2017
Bart Lomiej Dyda
ANDJuha Lehrbäck
Antti V Vähäkangas
FRACTIONAL HARDY-SOBOLEV TYPE INEQUALITIES FOR HALF SPACES AND JOHN DOMAINS
11 Sep 2017
As our main result we prove a variant of the fractional Hardy-Sobolev-Maz'ya inequalit... | 119,173,495 | {
"arxiv": "1709.03296",
"mag": 2963672719,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1090/proc/14051"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1709.03296v1.pdf"
],
"pdf_sha": "65d6baa3a5050bde618000340ab9d7ba9d33cf40",
"open_access": {
"license": "publisher-specific, author manuscript",
"url": "https://doi.org/10.1090/proc/14051",
"status": "HYBRID"
}
} | FRACTIONAL HARDY-SOBOLEV TYPE INEQUALITIES FOR HALF SPACES AND JOHN DOMAINS | [
"Bart Lomiej Dyda",
"ANDJuha Lehrbäck",
"Antti V Vähäkangas"
] | As our main result we prove a variant of the fractional Hardy-Sobolev-Maz'ya inequality for half spaces. This result contains a complete answer to a recent open question by Musina and Nazarov. In the proof we apply a new version of the fractional Hardy-Sobolev inequality that we establish also for more general unbounde... | [
{
"title": "Introduction",
"content": "The main result in this note is the following fractional Hardy-Sobolev-Maz'ya inequality for functions u ∈ C ∞ 0 (R n + ), where R n + is the upper half space of R n with n ≥ 2:\nHere 2 ≤ p, q < ∞ and 0 < s < 1 are such that sp < n and p < q ≤ np/(n − sp), and b = n(1/... | [
{
"id": "",
"caption": "The point x 0 appearing in Lemma 2.2 is called a John center of D. The following engulfing property of John domains can be found in [17, Theorem 4.6]. Lemma 2.3. A c-John domain D R n can be written as the union of c 1 -John domains D 1 , D 2 , . . ., where c 1 = c 1 (c, n) and D i i... | [
{
"text": "The best constant in a fractional Hardy inequality. K Bogdan, B Dyda, Math. Nachr. 2845-6K. Bogdan and B. Dyda. The best constant in a fractional Hardy inequality. Math. Nachr., 284(5-6):629- 638, 2011.",
"doi": ""
},
{
"text": "Fractional Hardy-Sobolev-Maz'ya inequality for domains. B Dy... |
Two-Stage TMLE to reduce bias and improve efficiency in cluster randomized trials Two-Stage TMLE for CRTs
2023
Laura B Balzer lbalzer@umass.edu
Mark Van Der Laan
James Ayieko
Moses Kamya
Gabriel Chamie
Joshua Schwab
Diane V Havlir
Maya L Petersen
Department of Biostatistics & Epidemiology
Division of Biostat... | 235,683,416 | {
"arxiv": "2106.15737",
"mag": null,
"acl": null,
"pubmed": "34939083",
"pubmedcentral": 10102904,
"dblp": null,
"doi": "10.1093/biostatistics/kxab043"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2106.15737v2.pdf"
],
"pdf_sha": "b5271e44f20e849b434455916009d3c30c3a325b",
"open_access": {
"license": "CCBY",
"url": "https://academic.oup.com/biostatistics/advance-article-pdf/doi/10.1093/biostatistics/kxab043/41901265/kxab043.pdf",
"status": "HYBRID"
... | Two-Stage TMLE to reduce bias and improve efficiency in cluster randomized trials Two-Stage TMLE for CRTs | [
"Laura B Balzer lbalzer@umass.edu",
"Mark Van Der Laan",
"James Ayieko",
"Moses Kamya",
"Gabriel Chamie",
"Joshua Schwab",
"Diane V Havlir",
"Maya L Petersen",
"Department of Biostatistics & Epidemiology\nDivision of Biostatistics\nUniversity of Massachusetts Amherst\n715 North Pleasant StAmherstMAU... | Cluster randomized trials (CRTs) randomly assign an intervention to groups of individuals (e.g., clinics or communities) and measure outcomes on individuals in those groups. While offering many advantages, this experimental design introduces challenges that are only partially addressed by existing analytic approaches. ... | [
{
"title": "INTRODUCTION",
"content": "In many trials, treatments are randomly allocated to groups of individuals, such as hospitals, schools, or communities, and outcomes are measured on individuals in those groups. These studies are known as group or cluster randomized trials (CRTs). They are implemented ... | [
{
"id": "",
"caption": "Simplified causal graph to illustrate the challenges of a single stage approach to identifying effects defined by interventions on both the cluster-level treatment and individual-level measurement mechanism.",
"content": "Fig. 1 .\n1Simplified causal graph to illustrate the chall... | [
{
"text": "Far from MCAR: obtaining population-level estimates of HIV viral suppression. L B Balzer, J Ayieko, D Kwarisiima, G Chamie, E D Charlebois, J Schwab, M J Van Der Laan, M R Kamya, D V Havlir, M L Petersen, Epidemiology. 31BALZER, L. B., AYIEKO, J., KWARISIIMA, D., CHAMIE, G., CHARLEBOIS, E. D., SCHWAB... |
On Partially Massless Theory in 3 Dimensions
30 Oct 2014
Sergei Alexandrov
Laboratoire Charles Coulomb
UMR 5221
Université Montpellier 2
F-34095MontpellierFrance
Laboratoire Charles Coulomb
UMR 5221
CNRS
F-34095MontpellierFrance
Cédric Deffayet deffayet@iap.fr
UMR7095
UPMC-CNRS
Institut d'Astrophysique de Paris
9... | 119,270,541 | {
"arxiv": "1410.2897",
"mag": 2950081928,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1088/1475-7516/2015/03/043"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1410.2897v3.pdf"
],
"pdf_sha": "a50cbf77f41713625f18fa3f1248c096b52ab81e",
"open_access": {
"license": "CCBY",
"url": "https://doi.org/10.1088/1475-7516/2015/03/043",
"status": "HYBRID"
}
} | On Partially Massless Theory in 3 Dimensions | [
"Sergei Alexandrov \nLaboratoire Charles Coulomb\nUMR 5221\nUniversité Montpellier 2\nF-34095MontpellierFrance\n\nLaboratoire Charles Coulomb\nUMR 5221\nCNRS\nF-34095MontpellierFrance",
"Cédric Deffayet deffayet@iap.fr \nUMR7095\nUPMC-CNRS\nInstitut d'Astrophysique de Paris\n98bis boulevard AragoF-75014GReCO, Par... | We analyze the first-order formulation of the ghost-free bigravity model in threedimensions known as zwei-dreibein gravity. For a special choice of parameters, it was argued to have an additional gauge symmetry and give rise to a partially massless theory. We provide a thorough canonical analysis and identify that whet... | [
{
"title": "Introduction",
"content": "Massive gravity (see [1,2,3] for reviews) has recently attracted a lot of attention with motivations ranging from cosmology [4,5,6,7,8] to more abstract ones [9]. Since the seminal work of Fierz and Pauli [10,11], it is well known that a sound massive graviton on a fou... | [
{
"id": "",
"caption": "[e ± , ω ± ] = S HP [e + , ω + ] + S HP [e − , ω − ] + S int [e + , e − ], (2.1) curvature terms.",
"content": "S\n[e ± , ω ± ] = S HP [e + , ω + ] + S HP [e − , ω − ] + S int [e + , e − ], (2.1) curvature terms."
},
{
"id": "",
"caption": "e I +,a − e I −,a ∼ ǫ ≪ 1. ... | [
{
"text": "Infrared-modified gravities and massive gravitons. V Rubakov, P Tinyakov, 0802.4379Phys.Usp. 51V. Rubakov and P. Tinyakov, \"Infrared-modified gravities and massive gravitons,\" Phys.Usp. 51 (2008) 759-792, 0802.4379.",
"doi": ""
},
{
"text": "Theoretical Aspects of Massive Gravity. K Hin... |
A MODIFIED ALGORITHM FOR OPTIMAL PICKER ROUTING IN A SINGLE BLOCK WAREHOUSE
20 Sep 2024
George Dunn
School of Information and Physical Sciences
University of Newcastle
NSW
Australia
Hadi Charkhgard
Department of Industrial and Management Systems Engineering
University of South Florida
FLUSA
Ali Eshragh
Carey Bus... | 272,770,375 | {
"arxiv": "2409.13219",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2409.13219v1.pdf"
],
"pdf_sha": "35d318bf4d0adf1e15dd8f4d304dc9936d91f91b",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | A MODIFIED ALGORITHM FOR OPTIMAL PICKER ROUTING IN A SINGLE BLOCK WAREHOUSE | [
"George Dunn \nSchool of Information and Physical Sciences\nUniversity of Newcastle\nNSW\nAustralia",
"Hadi Charkhgard \nDepartment of Industrial and Management Systems Engineering\nUniversity of South Florida\nFLUSA",
"Ali Eshragh \nCarey Business School\nJohns Hopkins University\nMDUSA\n\nInternational Comput... | The order picker routing problem involves finding the optimal tour of a warehouse that collects all the required items on a given pick list.Ratliff and Rosenthal introduced a dynamic programming algorithm for solving this problem in polynomial time by sequentially adding edges inside and between each aisle to construct... | [
{
"title": "Introduction",
"content": "Order picking is the process of collecting goods in a warehouse to meet the requirements of the customer.Given a pick list detailing the specific items needed, these must be retrieved from their respective locations within the warehouse and transported to a depot for p... | [
{
"id": "",
"caption": "Figure 1: Standard Parallel Aisle Warehouse",
"content": "Figure 1: Standard Parallel Aisle Warehouse"
},
{
"id": "",
"caption": "Figure 2: Possible edge configurations in a optimal tour graph.",
"content": "Figure 2 :\n2\nFigure 2: Possible edge configurations in... | [
{
"text": "Order picking under random and turnover-based storage policies in fishbone aisle warehouses. M Çelk, H Süral, IIE transactions. 4632014",
"doi": ""
},
{
"text": "Design and control of warehouse order picking: A literature review. De Koster, R Le-Duc, T Roodbergen, K J , European journal o... |
DEEPTalk: Dynamic Emotion Embedding for Probabilistic Speech-Driven 3D Face Animation
12 Aug 2024
Jisoo Kim
Yonsei University
Jungbin Cho
Yonsei University
Joonho Park joonho.park@giantstepcorp.com
GIANTSTEP Inc
Soonmin Hwang
Yonsei University
Da Eun Kim
GIANTSTEP Inc
Geon Kim geon.kim@giantstepcorp.co... | 271,854,861 | {
"arxiv": "2408.06010",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-2408-06010",
"doi": "10.48550/arxiv.2408.06010"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2408.06010v1.pdf"
],
"pdf_sha": "5d4e2349ccbde193d5cc91c7638ba5ee1ad10850",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | DEEPTalk: Dynamic Emotion Embedding for Probabilistic Speech-Driven 3D Face Animation | [
"Jisoo Kim \nYonsei University",
"Jungbin Cho \nYonsei University",
"Joonho Park joonho.park@giantstepcorp.com \nGIANTSTEP Inc",
"Soonmin Hwang \nYonsei University",
"Da Eun Kim \nGIANTSTEP Inc",
"Geon Kim geon.kim@giantstepcorp.com \nGIANTSTEP Inc",
"Youngjae Yu \nYonsei University"
] | Speech-driven 3D facial animation has garnered lots of attention thanks to its broad range of applications.Despite recent advancements in achieving realistic lip motion, current methods fail to capture the nuanced emotional undertones conveyed through speech and produce monotonous facial motion.These limitations result... | [
{
"title": "Introduction",
"content": "Speech-driven 3D facial motion generation has a wide range of applications, encompassing avatar animation for gaming or cinematic productions, virtual chatbots, and immersive virtual meetings within virtual reality environments.Despite substantial advancements in accur... | [
{
"id": "",
"caption": "Figure 2: Overall Architecture of Our Method.(a) E audio and E exp are trained to predict mean and variance for a joint audio-facial emotion embedding space, DEE.(b) We train TH-VQVAE with separate codebooks, Z b and Z t , for low and highfrequency motions, respectively.(c) DEEPTalk ... | [
{
"text": "Emotion Recognition in Speech using Cross-Modal Transfer in the Wild. S Albanie, A Nagrani, A Vedaldi, A Zisserman, CoRR, abs/1808.055612018",
"doi": ""
},
{
"text": "Fac-eTalk: Audio-Driven Motion Diffusion for Neural Parametric Head Models. S Aneja, J Thies, A Dai, M Nießner, Proc. IEEE... |
OPERATORS ON ANTI-DUAL PAIRS: LEBESGUE DECOMPOSITION VIA ARLINSKII'S ITERATION
21 Sep 2024
Ábel G Öde
Zsigmond Tarcsay
OPERATORS ON ANTI-DUAL PAIRS: LEBESGUE DECOMPOSITION VIA ARLINSKII'S ITERATION
21 Sep 2024C5ABCB872FF856FF379D2D1E59C0C5B0arXiv:2409.14068v1[math.FA]2020 Mathematics Subject Classification. Primary... | 272,828,086 | {
"arxiv": "2409.14068",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2409.14068v1.pdf"
],
"pdf_sha": "2d036ef787d7c87393f63e8ac1f82e0c57b9bebd",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | OPERATORS ON ANTI-DUAL PAIRS: LEBESGUE DECOMPOSITION VIA ARLINSKII'S ITERATION | [
"Ábel G Öde",
"Zsigmond Tarcsay"
] | The aim of this paper is to prove a general Lebesgue decomposition theorem for positive operators on so-called anti-dual pairs, following the iterative approach introduced by Arlinskii.This procedure and the resulting theorem encompass several special cases, including positive operators on Hilbert spaces, non-negative ... | [
{
"title": "Introduction",
"content": "The theory of non-commutative Lebesgue decomposition dates back at least to the pioneering paper by T. Ando [3].According to his results, any positive operator on a Hilbert space can be decomposed into the sum of absolutely continuous and singular parts with respect to... | [
{
"id": "",
"caption": "A and B are called mutually singular (notated by A ⊥ B), if for any positive operator C ∈ L + (E, F ), the relations C ≤ A and C ≤ B imply C = 0.It is important to note that B ≪ A if and only if there exist an increasing sequence of positive operators (B n ) n∈N strongly converging t... | [
{
"text": "Shorted operators II. W N AndersonJr, G E Trapp, SIAM Journal on Applied Mathematics. 2811975",
"doi": ""
},
{
"text": "Series and parallel addition of matrices. William , Anderson Jr, Richard James Duffin, Journal of Mathematical Analysis and Applications. 2631969",
"doi": ""
},
... |
VD3D: Taming Large Video Diffusion Transformers for 3D Camera Control
20 Jul 2024
Sherwin Bahmani
University of Toronto
Ivan Skorokhodov
University of Toronto
Aliaksandr Siarohin
University of Toronto
Willi Menapace
University of Toronto
Guocheng Qian
University of Toronto
Michael Vasilkovsky
Universi... | 271,244,756 | {
"arxiv": "2407.12781",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-2407-12781",
"doi": "10.48550/arxiv.2407.12781"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2407.12781v2.pdf"
],
"pdf_sha": "cfabe8e31cd348d8c15933197721cd6bb4cc4477",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | VD3D: Taming Large Video Diffusion Transformers for 3D Camera Control | [
"Sherwin Bahmani \nUniversity of Toronto",
"Ivan Skorokhodov \nUniversity of Toronto",
"Aliaksandr Siarohin \nUniversity of Toronto",
"Willi Menapace \nUniversity of Toronto",
"Guocheng Qian \nUniversity of Toronto",
"Michael Vasilkovsky \nUniversity of Toronto",
"Hsin-Ying Lee \nUniversity of Toronto",... | 2 Vector Institute 3 Snap Inc. 4 SFU https://snap-research.github.io/vd3d"A dog wearing vr goggles on a boat" "Otters with chef hats, skillfully preparing a miniature sushi feast on a lily pad" CameraFigure 1: 3D camera control for text-to-video generation.We introduce a method that can control camera poses for text-to... | [
{
"title": "Introduction",
"content": "Text-to-video foundation models achieve unprecedented visual quality [8,90].They are trained on massive collections of images and videos and learn to synthesize remarkably consistent and physically plausible visualizations of the world.Yet, they lack built-in mechanism... | [
{
"id": "",
"caption": "shuttle launching, camera zoom\" \"A space shuttle launching\"",
"content": "shuttle launching, camera zoom\" \"A space shuttle launching\""
},
{
"id": "",
"caption": "Figure 2: Comparing text-to-video, text-to-4D, and camera-conditioned text-to-video generation.Text-... | [
{
"text": "Tc4d: Trajectory-conditioned text-to-4d generation. Sherwin Bahmani, Xian Liu, Wang Yifan, Ivan Skorokhodov, Victor Rong, Ziwei Liu, Xihui Liu, Jeong Joon Park, Sergey Tulyakov, Gordon Wetzstein, Andrea Tagliasacchi, David B Lindell, Proc. ECCV. ECCV2024",
"doi": ""
},
{
"text": "Luc Van ... |
PRESCRIBED MEAN CURVATURE MIN-MAX THEORY IN SOME NON-COMPACT MANIFOLDS
15 Apr 2022
Liam Mazurowski
PRESCRIBED MEAN CURVATURE MIN-MAX THEORY IN SOME NON-COMPACT MANIFOLDS
15 Apr 2022arXiv:2204.07493v1 [math.DG]
This paper develops a technique for applying one-parameter prescribed mean curvature min-max theory in cert... | 248,218,657 | {
"arxiv": "2204.07493",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2204.07493v1.pdf"
],
"pdf_sha": "9692f12531eafced4df60a9ed6014609179571b8",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | PRESCRIBED MEAN CURVATURE MIN-MAX THEORY IN SOME NON-COMPACT MANIFOLDS | [
"Liam Mazurowski"
] | This paper develops a technique for applying one-parameter prescribed mean curvature min-max theory in certain non-compact manifolds. We give two main applications. First, fix a dimension 3 ≤ n + 1 ≤ 7 and consider a smooth function h : R n+1 → R which is asymptotic to a positive constant near infinity. We show that, u... | [
{
"title": "CMC Surfaces in Asymptotically Flat",
"content": "Manifolds. The proof of Theorem 1 uses a cutoff trick to reduce min-max on R n+1 to min-max on a sequence of larger and larger balls. We are hopeful that this cutoff trick may be adapted to perform min-max in other non-compact manifolds. As an ex... | [
{
"id": "",
"caption": "Fix a constant c > 0. Let (M 3 , g) be an asymptotically flat manifold which satisfies hypothesis (H) for the constant c. Then there exists a closed, almost-embedded c-CMC in M.",
"content": "Theorem 2 .\n2Fix a constant c > 0. Let (M 3 , g) be an asymptotically flat manifold whi... | [
{
"text": "Uniqueness theorems for surfaces in the large. A , Amer. Math. Soc. Transl. 21A. Aleksandrov. Uniqueness theorems for surfaces in the large. Amer. Math. Soc. Transl., 21:412-416, 1962.",
"doi": ""
},
{
"text": "F Almgren, The theory of varifolds. Mimeographed notes. F. Almgren. The theory... |
ON THE TOP-DIMENSIONAL COHOMOLOGY OF ARITHMETIC CHEVALLEY GROUPS
13 May 2024
Benjamin Brück benjamin.brueck@uni-muenster.de
ANDYuri Santos Rego
Robin J Sroka robinjsroka@uni-muenster.de
Universität Münster
Institut für mathematische Logik und Grundlagen-forschung
Einsteinstr. 6248149MünsterGermany
School of Mat... | 253,098,705 | {
"arxiv": "2210.12784",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1090/proc/16948"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2210.12784v2.pdf"
],
"pdf_sha": "d5d41b98d7788020a05189263c546cc544e462b6",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | ON THE TOP-DIMENSIONAL COHOMOLOGY OF ARITHMETIC CHEVALLEY GROUPS | [
"Benjamin Brück benjamin.brueck@uni-muenster.de",
"ANDYuri Santos Rego",
"Robin J Sroka robinjsroka@uni-muenster.de",
"Universität Münster\nInstitut für mathematische Logik und Grundlagen-forschung\nEinsteinstr. 6248149MünsterGermany",
"School of Mathematics and Physics\nUniversity of Lincoln\nCharlotte Sco... | Let K be a number field with ring of integers O and let G be a Chevalley group scheme not of type E 8 , F 4 or G 2 .We use the theory of Tits buildings and a result of Tóth on Steinberg modules to prove that | [
{
"title": "Introduction",
"content": "In this article, we obtain the following result about the cohomology of arithmetic groups:\nTheorem 1.1.Let K be a number field, O the ring of integers in K and G a Chevalley-Demazure group scheme of type A n , B n , C n , D n , E 6 or E 7 .If O is Euclidean, then the ... | [
{
"id": "",
"caption": "BB and RJS were partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -Project-ID 427320536 -SFB 1442, as well as by Germany's Excellence Strategy EXC 2044 -390685587, Mathematics Münster: Dynamics-Geometry-Structure.YSR was partially supported ... | [
{
"text": ". Peter Abramenko, S Kenneth, Brown, Buildings, Graduate Texts in Mathematics. 24832008Springer",
"doi": ""
},
{
"text": "The modular symbol and continued fractions in higher dimensions. Avner Ash, Lee Rudolph, Invent. Math. 55361979",
"doi": ""
},
{
"text": "Top-degree ration... |
arXiv:physics/0205030v2 [physics.bio-ph] Thinking about the brain
Springer-VerlagCopyright Springer-Verlag2002
Physics Of Biomolecules
H Cells
F Flyvbjerg
P Jülicher
arXiv:physics/0205030v2 [physics.bio-ph] Thinking about the brain
Les Ulis; BerlinSpringer-Verlag2002Based on lectures at Les Houches Session LXXV, ... | 5,095,582 | {
"arxiv": "physics/0205030",
"mag": 1508923040,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1007/3-540-45701-1_12"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/physics/0205030v2.pdf"
],
"pdf_sha": "22a251f0b15b70b62cbfeb6ee79ac4623e88be1f",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | arXiv:physics/0205030v2 [physics.bio-ph] Thinking about the brain | [
"Physics Of Biomolecules",
"H Cells",
"F Flyvbjerg",
"P Jülicher"
] | We all are fascinated by the phenomena of intelligent behavior, as generated both by our own brains and by the brains of other animals. As physicists we would like to understand if there are some general principles that govern the structure and dynamics of the neural circuits that underlie these phenomena. At the molec... | [
{
"title": "Introduction",
"content": "Here in Les Houches we are surrounded by many beautiful and dramatic phenomena of nature. In the last century we have come to understand the powerful physical forces that shaped the landscape, creating the peaks that reach thousands of meters into the sky. As we stand ... | [
{
"id": "",
"caption": "; this is a functional Pcorrect[f (x)].",
"content": "( 5 )\n5; this is a functional Pcorrect[f (x)]."
},
{
"id": "",
"caption": "Positivity of information. Prove that the mutual information I(D → W ), defined in Eq. (44), is positive.",
"content": "Problem 7 :\n7... | [
{
"text": "Intelligence, guesswork, language. H B Barlow, Nature. 304Barlow, H. B. (1983). Intelligence, guesswork, language, Nature 304, 207-209.",
"doi": ""
},
{
"text": "Critical limiting factors in the design of the eye and visual cortex. H B Barlow, Proc. R. Soc. Lond. Ser. B. 212Barlow, H. B. ... |
Eliciting the Double-edged Impact of Digitalisation: a Case Study in Rural Areas
Alessio Ferrari alessio.ferrari@isti.cnr.it
Fabio Lepore fabio.lepore@phd.unipi.it
Livia Ortolani livia.ortolani@agr.unipi.it
Gianluca Brunori gianluca.brunori@unipi.it
CNR-ISTI
PisaItaly
Department of Agriculture, Food and Envir... | 259,108,673 | {
"arxiv": "2306.05078",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/re/FerrariLOB23",
"doi": "10.1109/re57278.2023.00024"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2306.05078v1.pdf"
],
"pdf_sha": "fb13692bc4059b395612fad09f958f4d1ccc803c",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Eliciting the Double-edged Impact of Digitalisation: a Case Study in Rural Areas | [
"Alessio Ferrari alessio.ferrari@isti.cnr.it",
"Fabio Lepore fabio.lepore@phd.unipi.it",
"Livia Ortolani livia.ortolani@agr.unipi.it",
"Gianluca Brunori gianluca.brunori@unipi.it",
"CNR-ISTI\nPisaItaly",
"Department of Agriculture, Food and Environment\nUniversity of Pisa\nPisaItaly"
] | Designing systems that account for sustainability concerns demands for a better understanding of the impact that digital technology interventions can have on a certain sociotechnical context. However, limited studies are available about the elicitation of impact-related information from stakeholders, and strategies are... | [
{
"title": "I. INTRODUCTION",
"content": "Sustainable innovation asks software engineers to reflect on the potential short-and long-term impacts of their technical solutions in relation to the environment in which they are deployed [1]. Requirements engineering (RE) is recognised as the key area to address ... | [
{
"id": "",
"caption": "I spend two hours in the car in the morning and two hours in the evening, if I didn't use voice messages I don't think my performance in giving directions to colleagues would be any better. (PA) Different positive and negative impacts, considering past and present, as well as future ... | [
{
"text": "Sustainability Design and Software: the Karlskrona Manifesto. C Becker, R Chitchyan, L Duboc, S Easterbrook, B Penzenstadler, N Seyff, C C Venters, 2015 IEEE/ACM 37th IEEE International Conference on Software Engineering. IEEE2C. Becker, R. Chitchyan, L. Duboc, S. Easterbrook, B. Penzenstadler, N. Se... |
Prepared for submission to JHEP On searches for anomalous WWW interaction at the LHC
30 Apr 2019
B A Arbuzov arbuzov@theory.sinp.msu.ru
M.V. Lomonosov Moscow State University
119991MoscowRussia
I V Zaitsev zaitsev@theory.sinp.msu.ru
M.V. Lomonosov Moscow State University
119991MoscowRussia
Prepared for submission... | 140,232,997 | {
"arxiv": "1904.13193",
"mag": 2943291581,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1904.13193v1.pdf"
],
"pdf_sha": "020a4abc5d002d98614665226ff4d778095aa854",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Prepared for submission to JHEP On searches for anomalous WWW interaction at the LHC | [
"B A Arbuzov arbuzov@theory.sinp.msu.ru \nM.V. Lomonosov Moscow State University\n119991MoscowRussia",
"I V Zaitsev zaitsev@theory.sinp.msu.ru \nM.V. Lomonosov Moscow State University\n119991MoscowRussia"
] | The process p + p → W ± W + W − with lepton decays of W -bosons is shown to be promising for improving conditions for studies of parameter λ, which is the anomalous triple W -bosons interaction coupling constant. | [
{
"title": "Introduction",
"content": "The totality of data acquired by LHC confirms predictions of the Standard Model. Meanwhile the numerous attempts for obtaining even indications for deviations from the SM were not successful. Mostly these efforts were aimed at a quest for effects of the so-called New P... | [
{
"id": "",
"caption": "we have in processes (2.1,2.2) no jets, three charged leptons in the final state and a missing energy, which is carried by neutrinos. Decays (2.3,2.4) contain leptons of the same flavour bearing same electric charges, and different flavour leptons with opposite charges. This signatur... | [
{
"text": "Probing of the weak boson sector in e + e − → W + W −. K Hagiwara, R D Peccei, D Zeppenfeld, K Hikasa, Nucl. Phys. 282253K. Hagiwara, R. D. Peccei, D. Zeppenfeld and K. Hikasa, Probing of the weak boson sector in e + e − → W + W − , Nucl. Phys. B282 (1987) p. 253.",
"doi": ""
},
{
"text":... |
Collisional charging of dust particles by suprathermal particles. I -Standard anisotropic Kappa distributions
5 Nov 2024
L F Ziebell
Instituto de Física
Universidade Federal do Rio Grande do Sul
Caixa Postal
CEP
15051, 91501-970Porto AlegreRSBrasil
R Gaelzer
Instituto de Física
Universidade Federal do Rio Grande d... | 273,849,922 | {
"arxiv": "2411.03536",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2411.03536v1.pdf"
],
"pdf_sha": "5c3d75b5371ff61ef2697c6111791ff7a77288c7",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Collisional charging of dust particles by suprathermal particles. I -Standard anisotropic Kappa distributions | [
"L F Ziebell \nInstituto de Física\nUniversidade Federal do Rio Grande do Sul\nCaixa Postal\nCEP\n15051, 91501-970Porto AlegreRSBrasil",
"R Gaelzer \nInstituto de Física\nUniversidade Federal do Rio Grande do Sul\nCaixa Postal\nCEP\n15051, 91501-970Porto AlegreRSBrasil"
] | We study the effect of the velocity distributions of the plasma particles on the equilibrium charge of dust particles which suffer collisional charging, considering different forms of both isotropic and anisotropic Kappa distributions for ions and electrons.This paper is the first of a series of two papers on this subj... | [
{
"title": "I. INTRODUCTION",
"content": "Plasmas are generally defined as being ionized gases which exhibit collective behavior.As such, they are basically composed by ions and electrons, sometimes containing also a population of neutral atoms or molecules.In space environments, plasmas are composed mainly... | [
{
"id": "",
"caption": "Max, T e⊥ /T e = 0.2, i Max 8,774 e bi-Max, T e⊥ /T e = 5.0, i Max 44,382 e Max, i bi-Max, T i⊥ /T i = 0.2 14,182 e Max, i bi-Max, T i⊥ /T i = 5.0 15,749 TABLE I. Charge number on spherical dust particles collisionally charged, for different cases of isotropic and anisotropic Maxwell... | [
{
"text": ". E Marsch, Living Rev. Solar Phys. 312006",
"doi": ""
},
{
"text": ". 10.12942/lrsp-2006-1Sep. 2006",
"doi": ""
},
{
"text": ". R Schwenn, 10.1007/s11214-006-9099-5Space Sci. Rev. 1242006",
"doi": ""
},
{
"text": ". H Kruger, N Altobelli, B Anweiler, S F Dermott, ... |
NEW NOTES ON PLANAR SEMIMODULAR LATTICES. I. TWO REMARKS ON SLIM RECTANGULAR LATTICES
George Grätzer
NEW NOTES ON PLANAR SEMIMODULAR LATTICES. I. TWO REMARKS ON SLIM RECTANGULAR LATTICES
Let L be a slim, planar, semimodular lattice (slim means that it does not contain an M 3 -sublattice). We call the intervalThe f... | 248,986,526 | {
"arxiv": "2205.10966",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2205.10966v1.pdf"
],
"pdf_sha": "ac732694716412d2e0c568a3e8b47e8265387599",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | NEW NOTES ON PLANAR SEMIMODULAR LATTICES. I. TWO REMARKS ON SLIM RECTANGULAR LATTICES | [
"George Grätzer"
] | Let L be a slim, planar, semimodular lattice (slim means that it does not contain an M 3 -sublattice). We call the intervalThe first remark : a rectangular interval of a rectangular lattice is a rectangular lattice. As an application, we get a recent result of G. Czédli.In a 2017 paper, G. Czédli introduced a very powe... | [
{
"title": "Introduction",
"content": "In 2006, we started studying planar, semimodular lattices in my papers with E. Knapp [9]- [13]. More than four dozen publications have been devoted to this topic since; see G. Czédli's list http://www.math.u-szeged.hu/~czedli/m/listak/publ-psml.pdf\nAn SPS lattice L is... | [
{
"id": "",
"caption": "Deleting a fork minimal, that is, there is no S a covering S 7 with top element t satisfying that t < t.Lemma 1. Let L be an SPS lattice and let S = {o, m, b l , b r , t} be a minimal covering S 7 in L. Then L has a sublattice L − with 4",
"content": "C"
},
{
"id": "",
... | [
{
"text": "Diagrams and rectangular extensions of planar semimodular lattices. G Czédli, Algebra Universalis. 77G. Czédli, Diagrams and rectangular extensions of planar semimodular lattices. Algebra Universalis 77 (2017), 443-498.",
"doi": ""
},
{
"text": "Lamps in slim rectangular planar semimodula... |
Remote Time Manipulation
25 Mar 2019
David Trillo
Institute for Quantum Optics and Quantum Information (IQOQI)
Austrian Academy of Sciences
Boltzmanngasse 31090Vienna
Benjamin Dive
Institute for Quantum Optics and Quantum Information (IQOQI)
Austrian Academy of Sciences
Boltzmanngasse 31090Vienna
Miguel Navascués... | 85,517,015 | {
"arxiv": "1903.10568",
"mag": 2924792615,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1903.10568v1.pdf"
],
"pdf_sha": "c5c6009f491bfc5ebc6ea435d0aa6912aeeb1cee",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Remote Time Manipulation | [
"David Trillo \nInstitute for Quantum Optics and Quantum Information (IQOQI)\nAustrian Academy of Sciences\nBoltzmanngasse 31090Vienna",
"Benjamin Dive \nInstitute for Quantum Optics and Quantum Information (IQOQI)\nAustrian Academy of Sciences\nBoltzmanngasse 31090Vienna",
"Miguel Navascués \nInstitute for Qua... | Harnessing the flow of proper time of arbitrary external systems over which we exert little or no control has been a recurring theme in both science and science-fiction. Unfortunately, all relativistic schemes to achieve this effect beyond mere time dilation are utterly unrealistic. In this work, we find that there exi... | [
{
"title": "The model",
"content": "We consider a scenario where the experimental setup consists in two parts: a controlled lab, where we can prepare any quantum state and conduct any quantum operation; and a scattering region. The latter contains n identical physical target systems of Hilbert space dimensi... | [
{
"id": "",
"caption": "The setup.",
"content": "FIG. 1 :\n1The setup."
},
{
"id": "",
"caption": ", T allowed by Eq.(3) just by applying the previous resetting protocol for time αT , with 0 ≤ α ≤ 1 and then letting the system evolve naturally for time(1 − α)T .Let us now move to the case wh... | [
{
"text": "D Ivey, J Hume, no. v. 1 in Physics. Ronald PressD. Ivey and J. Hume, Physics, no. v. 1 in Physics (Ronald Press, 1974).",
"doi": ""
},
{
"text": ". C Simak, Time is the Simplest Thing, Gateway Essentials. ISBN 9780575122383C. Simak, Time is the Simplest Thing, Gateway Essen- tials (Orion... |
Characterizing Social Movement Narratives in Online Communities: The 2021 Cuban Protests on Reddit
Brian Keith Norambuena
Chris North north@vt.edu
Virginia Tech Blacksburg
VirginiaUSA
Tanushree Mitra
University of Washington Seattle
WashingtonUSA
Virginia Tech Blacksburg
VirginiaUSA
Characterizing Social Mov... | 249,980,189 | {
"arxiv": "2208.04465",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-2208-04465",
"doi": "10.48550/arxiv.2208.04465"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2208.04465v1.pdf"
],
"pdf_sha": "1cbf3c0d16ac5aa64c2c73537f2a13242b3e5516",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Characterizing Social Movement Narratives in Online Communities: The 2021 Cuban Protests on Reddit | [
"Brian Keith Norambuena",
"Chris North north@vt.edu",
"Virginia Tech Blacksburg\nVirginiaUSA",
"Tanushree Mitra\nUniversity of Washington Seattle\nWashingtonUSA",
"Virginia Tech Blacksburg\nVirginiaUSA"
] | Social movements are dominated by storytelling, as narratives play a key role in how communities involved in these movements shape their identities. Thus, recognizing the accepted narratives of different communities is central to understanding social movements. In this context, journalists face the challenge of making ... | [
{
"title": "INTRODUCTION",
"content": "Social movements are dominated by storytelling [12]. In fact, narratives play a key role in social change and social movements [7], as people use them to learn and exercise agency, shape identity, and motivate actions [13,17]. Social movements are made possible by the ... | [
{
"id": "",
"caption": "Linear program formulation. Parts highlighted in red are our contributions to the model.",
"content": "Figure 1 :\n1Linear program formulation. Parts highlighted in red are our contributions to the model."
},
{
"id": "",
"caption": "Narrative map for r/cuba. The numbe... | [
{
"text": "The Cambridge introduction to narrative. H Porter, Abbott , Cambridge University PressNY, USAH Porter Abbott. 2008. The Cambridge introduction to narrative. Cambridge University Press, NY, USA.",
"doi": ""
},
{
"text": "The Impact of the US Economic Sanctions on Health in Cuba. Mohsen Nar... |
MSKdeX: Musculoskeletal (MSK) decomposition from an X-ray image for fine-grained estimation of lean muscle mass and muscle volume
21 Jul 2023
Yi Gu
Division of Information Science
Graduate School of Science and Technology
Nara Institute of Science and Technology
Japan
CentraleSupélec
Université Paris-Saclay
France
... | 258,987,892 | {
"arxiv": "2305.19920",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/miccai/GuOUTSHTOSS23",
"doi": "10.48550/arxiv.2305.19920"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2305.19920v2.pdf"
],
"pdf_sha": "67f05258f162691a4fbf17f82d22a9249255e9fd",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | MSKdeX: Musculoskeletal (MSK) decomposition from an X-ray image for fine-grained estimation of lean muscle mass and muscle volume | [
"Yi Gu \nDivision of Information Science\nGraduate School of Science and Technology\nNara Institute of Science and Technology\nJapan\n\nCentraleSupélec\nUniversité Paris-Saclay\nFrance",
"Yoshito Otake otake@is.naist.jp \nDivision of Information Science\nGraduate School of Science and Technology\nNara Institute o... | Musculoskeletal diseases such as sarcopenia and osteoporosis are major obstacles to health during aging. Although dual-energy X-ray absorptiometry (DXA) and computed tomography (CT) can be used to evaluate musculoskeletal conditions, frequent monitoring is difficult due to the cost and accessibility (as well as high ra... | [
{
"title": "Introduction",
"content": "Sarcopenia is a prevalent musculoskeletal disease characterized by the inevitable loss of skeletal muscle, causing increased risks of all-cause mortality and disability that result in heavy healthcare costs [1,2,3,4,6,5]. Measuring body composition, such as lean muscle... | [
{
"id": "",
"caption": "Lean muscle mass (left) and muscle volume (right) estimation results by the conventional and proposed methods for the gluteus medius (top) and iliacus (bottom).",
"content": "Fig. 3 .\n3Lean muscle mass (left) and muscle volume (right) estimation results by the conventional and p... | [
{
"text": "Sarcopenia: prevalence, associated factors, and the risk of mortality and disability in Japanese older adults. A Kitamura, S Seino, A Takumi, 10.1002/jcsm.12651J. Cachexia Sarcopenia Muscle. 121Kitamura, A., Seino, S., Takumi, A., et al.: Sarcopenia: prevalence, associated fac- tors, and the risk of ... |
Deep Learning for CSI Feedback: One-Sided Model and Joint Multi-Module Learning Perspectives
9 May 2024
Yiran Guo
Feifei Sun
Jiaming Cheng
Fellow, IEEEMichail Matthaiou
Fellow, IEEEBo Ai
Deep Learning for CSI Feedback: One-Sided Model and Joint Multi-Module Learning Perspectives
9 May 20244407F8348C905FC824EFFA7... | 269,635,314 | {
"arxiv": "2405.05522",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2405.05522v1.pdf"
],
"pdf_sha": "cc726097c82596be4401c73aca8a5a68b9ca029d",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Deep Learning for CSI Feedback: One-Sided Model and Joint Multi-Module Learning Perspectives | [
"Yiran Guo",
"Feifei Sun",
"Jiaming Cheng",
"Fellow, IEEEMichail Matthaiou",
"Fellow, IEEEBo Ai"
] | The use of deep learning (DL) for channel state information (CSI) feedback has garnered widespread attention across academia and industry.The mainstream DL architectures, e.g., CsiNet, deploy DL models on the base station (BS) side and the user equipment (UE) side, which are highly coupled and need to be trained jointl... | [
{
"title": "I. INTRODUCTION",
"content": "Given the global success of fifth-generation (5G) technologies, anticipation is growing for the enhanced performance that sixth-generation (6G) would offer in a wide range of applications.Massive multiple-input multiple-output (MIMO) technology has the capability to... | [
{
"id": "",
"caption": "Fig. 1.A communication framework with CSI feedback.",
"content": "Fig. 1 .\n1\nFig. 1.A communication framework with CSI feedback."
},
{
"id": "",
"caption": "Fig. 2. The construction of AI-based CSI feedback: (a) one-sided AI-based CSI feedback; (b) two-sided AI-base... | [
{
"text": "5G-advanced toward 6G: Past, present, and future. W Chen, X Lin, J Lee, A Toskala, S Sun, C F Chiasserini, L Liu, IEEE J. Sel. Areas Commun. 416Jun. 2023",
"doi": ""
},
{
"text": "AI for CSI feedback enhancement in 5G-advanced. J Guo, C.-K Wen, S Jin, X Li, 110.1109/MWC.010.2200304IEEE Wi... |
Longitudinal and transverse Greens functions in ϕ 4 model below and near the critical point
29 Jun 2004 October 28, 2018
J Kaupužs
Institute of Mathematics and Computer Science
University of Latvia
29 Rainja BoulevardLV-1459RigaLatvia
Longitudinal and transverse Greens functions in ϕ 4 model below and near the crit... | 118,852,156 | {
"arxiv": "cond-mat/0202416",
"mag": 1629577645,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/cond-mat/0202416v4.pdf"
],
"pdf_sha": "b067420a9ca7f5aa069efeb159bfd363977a1945",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Longitudinal and transverse Greens functions in ϕ 4 model below and near the critical point | [
"J Kaupužs \nInstitute of Mathematics and Computer Science\nUniversity of Latvia\n29 Rainja BoulevardLV-1459RigaLatvia"
] | We have extended our method of grouping of Feynman diagrams (GFD theory) to study the transverse and longitudinal Greens functions G ⊥ (k) and G (k) in ϕ 4 model below the critical point (T < T c ) in presence of an infinitesimal external field. Our method allows a qualitative analysis not cutting the perturbation seri... | [
{
"title": "Introduction",
"content": "Phase transitions and critical phenomena is a widely investigated field in physics and natural sciences [1,2,3,4]. The current paper is devoted to further development of our original diagrammatic method introduced in [5], to study the ϕ 4 phase transition model below t... | [
{
"id": "",
"caption": "verify the theoretical predictions for the exponent ρ = (d/λ ⊥ ) − 1 in (84), we have made Monte Carlo (MC) simulations of 3D XY model on simple cubic lattice with the Hamiltonian",
"content": "verify the theoretical predictions for the exponent ρ = (d/λ ⊥ ) − 1 in (84), we have ... | [
{
"text": "Since there is no reason to assume that one of us has made a wrong simulation, the small but remarkable (5 their standard deviations at K = 0.5) discrepancies, obviously, are due to the different extrapolation procedures used. Our values are more precise and reliable, since our method allows to estim... |
A Research Note on the Implementation of Star Formation and Stellar Feedback in Semi-Analytic Models
17 Jan 2013 December 11, 2013
Fabio Fontanot fabio.fontanot@h-its.org
HITS-Heidelberger Institut für Theoretische Studien
Schloss-Wolfsbrunnenweg 3569118HeidelbergGermany
Institut für Theoretische Physik
Universität... | 118,459,030 | {
"arxiv": "1301.4220",
"mag": 1581079906,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1301.4220v1.pdf"
],
"pdf_sha": "0bd0aa1061d5a03fb2cad35f662877edbbdc114f",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | A Research Note on the Implementation of Star Formation and Stellar Feedback in Semi-Analytic Models | [
"Fabio Fontanot fabio.fontanot@h-its.org \nHITS-Heidelberger Institut für Theoretische Studien\nSchloss-Wolfsbrunnenweg 3569118HeidelbergGermany\n\nInstitut für Theoretische Physik\nUniversität Heidelberg\nPhilosophenweg, 1669120HeidelbergGermany",
"Gabriella De Lucia \nINAF-Osservatorio Astronomico di Trieste\nV... | We study the impact of star formation and stellar feedback prescriptions on galaxy properties predicted by means of "stripped-down" versions of independently developed semi-analytic models (SAMs). These include cooling, star formation, feedback from supernovae (SNe) and simplified prescriptions for galaxy merging, but ... | [
{
"title": "Introduction",
"content": "In order to understand the complex process of galaxy formation over the entire cosmic history of the Universe, many different physical processes need to be taken into account. These processes act on different scales and their interplay appears critical for an appropria... | [
{
"id": "",
"caption": "Redshift evolution of the baryonic content for a MW-like representative DMH (with quiet mass accretion history; fig.1 right panels in De Lucia et al. 2010). Upper row: cold gas fractions associated with the central galaxy;",
"content": "Figure 1 :\n1Redshift evolution of the bary... | [
{
"text": ". C M Baugh, Reports of Progress in Physics. 693101Baugh C. M., 2006, Reports of Progress in Physics, 69, 3101",
"doi": ""
},
{
"text": ". A J Benson, 17175NewABenson A. J., 2012, NewA, 17, 175",
"doi": ""
},
{
"text": ". A J Benson, R G Bower, C S Frenk, C G Lacey, C M Baugh,... |
Advancing Gasoline Consumption Forecasting: A Novel Hybrid Model Integrating Transformers, LSTM, and CNN
Mahmoud Ranjbar mahmoodranjbar2010@gmail.com
Department of Environmental Engineering
University of Tehran
TehranIran
Mohammad Rahimzadeh
AI Research Lab
NovaVision AI Co
TallinEstonia
Advancing Gasoline Consu... | 273,507,550 | {
"arxiv": "2410.16336",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-2410-16336",
"doi": "10.48550/arxiv.2410.16336"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2410.16336v2.pdf"
],
"pdf_sha": "f5fe8f9b1ca05bb3288fb7ef7d9ef37377dad0e1",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Advancing Gasoline Consumption Forecasting: A Novel Hybrid Model Integrating Transformers, LSTM, and CNN | [
"Mahmoud Ranjbar mahmoodranjbar2010@gmail.com \nDepartment of Environmental Engineering\nUniversity of Tehran\nTehranIran",
"Mohammad Rahimzadeh \nAI Research Lab\nNovaVision AI Co\nTallinEstonia"
] | Iran, endowed with abundant hydrocarbon resources, plays a crucial role in the global energy landscape.Gasoline, as a critical fuel, significantly supports the nation's transportation sector.Accurate forecasting of gasoline consumption is essential for strategic resource management and environmental planning.This resea... | [
{
"title": "Introduction",
"content": "Energy is one of the most critical needs of modern humanity.Economic development and information technology are closely linked to energy consumption.Given the rising global population growth rate, alongside rapid industrial expansion and technological advancements in t... | [
{
"id": "",
"caption": "Figure 1: Gasoline Consumption: Actual vs Predicted (ANN, Regression, Hybrid Model)",
"content": "Figure 1 :\n1\nFigure 1: Gasoline Consumption: Actual vs Predicted (ANN, Regression, Hybrid Model)"
},
{
"id": "",
"caption": "Figure 2: RMSE Comparison of Hybrid Model, ... | [
{
"text": "Economic instability and aggregate investment. R S Pindyck, A Solimano, NBER macroeconomics annual. 81993",
"doi": ""
},
{
"text": "A survey on renewable energy sources and its impact on sustainable development. A A Khan, J A Hussain, Renewable and Sustainable Energy Reviews. 892018",
... |
The Role of Type Ia Supernovae in Constraining the Hubble Constant
28 Nov 2023
Dan Scolnic daniel.scolnic@duke.edu
Maria Vincenzi maria.vincenzi@duke.edu
Department of Physics
Duke University
27708DurhamNCUSA
Department of Physics
Duke University
27708DurhamNCUSA
The Role of Type Ia Supernovae in Constraining t... | 265,466,777 | {
"arxiv": "2311.16830",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2311.16830v1.pdf"
],
"pdf_sha": "34988c6054e7a4ba845c55d9ea7772cded9084af",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | The Role of Type Ia Supernovae in Constraining the Hubble Constant | [
"Dan Scolnic daniel.scolnic@duke.edu",
"Maria Vincenzi maria.vincenzi@duke.edu",
"Department of Physics\nDuke University\n27708DurhamNCUSA",
"Department of Physics\nDuke University\n27708DurhamNCUSA"
] | In the conventional / most studied local distance ladder measurements, Type Ia supernovae (SNe Ia) are used in two of the three rungs.In the second rung, their luminosities are calibrated by standard candles like Cepheids or Tip of the Red Giant Branch (TRGB).In the third rung, the high luminosities and standardizabili... | [
{
"title": "Introduction",
"content": "When Edwin Hubble first measured the expansion of the universe, no supernovae (SNe) were used in this measurement [1].Instead, Hubble measured distances to galaxies by observing the Cepheids in the galaxy and Cepheids' properties discovered by Henrietta Leavitt [2].Tod... | [
{
"id": "",
"caption": "Fig. 1.1 Left panel: The 'distance ladder' from [3].Right panel: The Pantheon+ \"Hubble diagram\" from [11] showing the distance modulus µ versus redshift z.The SNe from the second rung make up the majority of the z < 0.01 SNe in the Pantheon+ Hubble diagram, and the third rung makes... | [
{
"text": "E Hubble, 10.1073/pnas.15.3.168Proceedings of the National Academy of Science. the National Academy of Science192915168",
"doi": ""
},
{
"text": "Harvard College Observatory Circular. Henrietta S Leavitt, Edward C Pickering, 1912173",
"doi": ""
},
{
"text": ". A G Riess, W Yua... |
Computation of Composition Functions and Invariant Vector Fields in Terms of Structure Constants of Associated Lie Algebras
2015
Integrability Symmetry
Geometry
Computation of Composition Functions and Invariant Vector Fields in Terms of Structure Constants of Associated Lie Algebras
Methods and Applications SIGMA... | 15,549,400 | {
"arxiv": "1312.0362",
"mag": 1875470045,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.3842/sigma.2015.066"
} | {
"pdf_urls": [
"http://emis.maths.tcd.ie/EMIS/journals/SIGMA/2015/066/sigma15-066.pdf",
"http://www.emis.de/journals/SIGMA/2015/066/sigma15-066.pdf",
"http://www.maths.tcd.ie/EMIS/journals/SIGMA/2015/066/sigma15-066.pdf"
],
"pdf_sha": "72b7217ccf8e7d049d7ef612644c46b1af3763ff",
"open_access": {
... | Computation of Composition Functions and Invariant Vector Fields in Terms of Structure Constants of Associated Lie Algebras | [
"Integrability Symmetry",
"Geometry"
] | Methods of construction of the composition function, left-and right-invariant vector fields and differential 1-forms of a Lie group from the structure constants of the associated Lie algebra are proposed. It is shown that in the second canonical coordinates these problems are reduced to the matrix inversions and matrix... | [
{
"title": "Introduction",
"content": "Researchers in the field of theoretical and mathematical physics who use methods of Lie theory face the problem on realizations of a finite-dimensional Lie algebra by means of vector fields on a certain domain of a finite-dimensional real space. This problem is vitally... | [
{
"id": "",
"caption": "This implies that each a-th component of the composition function Φ(x, z) = Φ((q, y), z) does not depend on the coordinates in H and coincides with the respective component of the composition function Ψ(q, z) Ψ a (q, z) = Φ a ((q, y), z), a = 1, . . . , m. (2.16) The equalities (2.16... | [
{
"text": "Algebraic approach to the scattering matrix. Y Alhassid, J Engel, J Wu, 10.1103/PhysRevLett.53.17Phys. Rev. Lett. 53Alhassid Y., Engel J., Wu J., Algebraic approach to the scattering matrix, Phys. Rev. Lett. 53 (1984), 17-20.",
"doi": "10.1103/PhysRevLett.53.17"
},
{
"text": "Extensions o... |
Doping dependence and electron-boson coupling in the ultrafast relaxation of hot electron populations in Ba(Fe 1−x Co x ) 2 As 2
17 May 2016
I Avigo
Fakultät für Physik
Universität Duisburg-Essen
Lotharstr. 147057DuisburgGermany
S Thirupathaiah
Fakultät für Physik
Universität Duisburg-Essen
Lotharstr. 147057Duisbu... | 119,284,387 | {
"arxiv": "1605.05177",
"mag": 3105845006,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1088/1367-2630/18/9/093028"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1605.05177v1.pdf"
],
"pdf_sha": "e0e9978db51a32ac5a9543d431856490b79ab914",
"open_access": {
"license": "CCBY",
"url": "https://iopscience.iop.org/article/10.1088/1367-2630/18/9/093028/pdf",
"status": "GOLD"
}
} | Doping dependence and electron-boson coupling in the ultrafast relaxation of hot electron populations in Ba(Fe 1−x Co x ) 2 As 2 | [
"I Avigo \nFakultät für Physik\nUniversität Duisburg-Essen\nLotharstr. 147057DuisburgGermany",
"S Thirupathaiah \nFakultät für Physik\nUniversität Duisburg-Essen\nLotharstr. 147057DuisburgGermany\n\ncurrent address: Solid State and Structural Chemistry Unit\nIndian Institute of Science\nKar-nataka560012BangaloreI... | Using femtosecond time-and angle-resolved photoemission spectroscopy we investigate the effect of electron doping on the electron dynamics in Ba(Fe 1−x Co x ) 2 As 2 in a range of 0 ≤ x < 0.15 at temperatures slightly above the Néel temperature. By analyzing the time-dependent photoemission intensity of the pump laser ... | [
{
"title": "I. INTRODUCTION",
"content": "The discovery of Fe-based high-T c superconductivity is one of the breakthroughs in the last two decades of condensed matter research [1] because in addition to the cuprates scientists identified a second class of compounds displaying superconductivity with high T c... | [
{
"id": "",
"caption": "Ti:Sapphire amplifier (Coherent RegA 9040) generates infrared laser pulses with 800 nm central wavelength, an energy of 1.5 eV with a repetition rate of 250 kHz and 40 fs pulse duration. The ∼ 6 µJ pulses are split and 50 percent is used as pump, while the other half of the beam is f... | [
{
"text": "Iron-based layered superconductor. Y Kamihara, T Watanabe, M Hirano, H Hosono, Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono. Iron-based layered superconductor",
"doi": ""
},
{
"text": ". J. Am. Chem. Soc. 1303296J. Am. Chem. Soc., 130:3296, 2008.",
"doi": ""
},
{
"text":... |
Easy bootstrap for the 3D Ising model: a hybrid approach of the lightcone bootstrap and error minimization methods
8 Jul 2024
Wenliang Li liwliang3@mail.sysu.edu.cn
School of Physics
Sun Yat-Sen University
510275GuangzhouChina
Easy bootstrap for the 3D Ising model: a hybrid approach of the lightcone bootstrap and e... | 266,191,513 | {
"arxiv": "2312.07866",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1007/jhep07(2024)047"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2312.07866v2.pdf"
],
"pdf_sha": "156f66a13d5b43567ff18d6321ce649c5c464af3",
"open_access": {
"license": "CCBY",
"url": "https://doi.org/10.1007/jhep07(2024)047",
"status": "GOLD"
}
} | Easy bootstrap for the 3D Ising model: a hybrid approach of the lightcone bootstrap and error minimization methods | [
"Wenliang Li liwliang3@mail.sysu.edu.cn \nSchool of Physics\nSun Yat-Sen University\n510275GuangzhouChina"
] | As a simple lattice model that exhibits a phase transition, the Ising model plays a fundamental role in statistical and condensed matter physics.The Ising transition is realized by physical systems, such as the liquid-vapor transition.Its continuum limit also furnishes a basic example of interacting quantum field theor... | [
{
"title": "Introduction",
"content": "Onsager's exact solution for the partition function of the 2D square-lattice Ising model is a landmark in theoretical physics [1].However, an exact solution for the 3D cubic-lattice Ising partition function has remained elusive.Since the 2D Ising model is exactly solva... | [
{
"id": "",
"caption": "Figure 1:The approximate solution for ∆ ϵ at various ∆ σ .The red dots represent the free theory and the Ising solutions in three dimensions.The blue dots are obtained from solving (3.5) with GFF approximations for the high-lying contributions.The orange dots are derived from solving... | [
{
"text": "Crystal statistics. 1. A Two-dimensional model with an order disorder transition. L Onsager, 10.1103/PhysRev.65.117Phys. Rev. 651944",
"doi": ""
},
{
"text": "Combinatorial and topological approach to the 3-D Ising model. T Regge, R Zecchina, 10.1088/0305-4470/33/4/308arXiv:cond-mat/99091... |
An Extended Kalman Filter for Data-enabled Predictive Control
Daniele Alpago
Florian Dörfler
John Lygeros
An Extended Kalman Filter for Data-enabled Predictive Control
The literature dealing with data-driven analysis and control problems has significantly grown in the recent years. Most of the recent literature ... | 212,747,784 | {
"arxiv": "2003.08269",
"mag": 3029097646,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/csysl/AlpagoDL20",
"doi": "10.1109/lcsys.2020.2998296"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2003.08269v2.pdf"
],
"pdf_sha": "d1a421a5e3a9388a27dd4c7edfbbed27c4a51c90",
"open_access": {
"license": null,
"url": "https://www.research-collection.ethz.ch/bitstream/20.500.11850/425063/3/EKF_DeePC_AAM.pdf",
"status": "GREEN"
}
} | An Extended Kalman Filter for Data-enabled Predictive Control | [
"Daniele Alpago",
"Florian Dörfler",
"John Lygeros"
] | The literature dealing with data-driven analysis and control problems has significantly grown in the recent years. Most of the recent literature deals with linear timeinvariant systems in which the uncertainty (if any) is assumed to be deterministic and bounded; relatively little attention has been devoted to stochasti... | [
{
"title": "I. INTRODUCTION",
"content": "With the increasing complexity of applications in science and engineering, the use of model-based control techniques is becoming more and more challenging as they usually require accurate descriptions of model and uncertainties, often difficult to obtain. The need o... | [
{
"id": "",
"caption": "Research supported by the ERC under the project OCAL, the SEMP and the Gini Foundation. D. Alpago is with the Department of Information Engineering at University of Padova, Italy, dalpago@dei.unipd.it; F. Dörfler and J. Lygeros are with the Department of Information Technology and El... | [
{
"text": "System Identification: Theory for the User. Prentice Hall information and system sciences series. L Ljung, Prentice Hall PTRL. Ljung, System Identification: Theory for the User. Prentice Hall information and system sciences series, Prentice Hall PTR, 1999.",
"doi": ""
},
{
"text": "Adapti... |
Multiple Object Tracking with Correlation Learning
Qiang Wang
Machine Intelligence Technology Lab, Alibaba Group
Yun Zheng zhengyun.zy@alibaba-inc.com
Machine Intelligence Technology Lab, Alibaba Group
Pan Pan
Machine Intelligence Technology Lab, Alibaba Group
Yinghui Xu
Machine Intelligence Technology Lab... | 233,182,057 | {
"arxiv": "2104.03541",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/cvpr/WangZPX21",
"doi": "10.1109/cvpr46437.2021.00387"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2104.03541v1.pdf"
],
"pdf_sha": "cecfb3f6f900a01b8495f5075e3d62390f0a0bdd",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/2104.03541",
"status": "GREEN"
}
} | Multiple Object Tracking with Correlation Learning | [
"Qiang Wang \nMachine Intelligence Technology Lab, Alibaba Group",
"Yun Zheng zhengyun.zy@alibaba-inc.com \nMachine Intelligence Technology Lab, Alibaba Group",
"Pan Pan \nMachine Intelligence Technology Lab, Alibaba Group",
"Yinghui Xu \nMachine Intelligence Technology Lab, Alibaba Group"
] | Recent works have shown that convolutional networks have substantially improved the performance of multiple object tracking by simultaneously learning detection and appearance features. However, due to the local perception of the convolutional network structure itself, the long-range dependencies in both the spatial an... | [
{
"title": "Introduction",
"content": "Multi-Object Tracking (MOT) is an essential component for computer vision with many applications, such as video surveillance [31] and modern autonomous driving [19,41]. It aims to continuously locate trajectories of multiple targets in video frames. Decades of research... | [
{
"id": "",
"caption": "Visualization of the matching confidences (a)-(b) computed between the indicated target (white cross) in the reference image and all locations of the query image. The appearance feature based tracker[58] (a) generates undistinctive and inaccurate confidences due to the existing of si... | [
{
"text": "Tracking without bells and whistles. Philipp Bergmann, Tim Meinhardt, Laura Leal-Taixe, Proceedings of the IEEE international conference on computer vision. the IEEE international conference on computer visionPhilipp Bergmann, Tim Meinhardt, and Laura Leal-Taixe. Tracking without bells and whistles. ... |
Computers Can't Give Credit: How Automatic Attribution Falls Short in an Online Remixing Community
Andrés Monroy-Hernández
Massachusetts Institute of Technology
02139CambridgeMA
Microsoft Research
02142CambridgeMA
Benjamin Mako Hill
Massachusetts Institute of Technology
02139CambridgeMA
Jazmin Gonzalez-Rivero
... | 7,494,330 | {
"arxiv": "1507.01285",
"mag": 3123727583,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/Monroy-Hernandez15a",
"doi": "10.1145/1978942.1979452"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1507.01285v1.pdf"
],
"pdf_sha": "d825d5aeb21acfc950efee6211bb130f7c701083",
"open_access": {
"license": null,
"url": "https://arxiv.org/pdf/1507.01285",
"status": "GREEN"
}
} | Computers Can't Give Credit: How Automatic Attribution Falls Short in an Online Remixing Community | [
"Andrés Monroy-Hernández \nMassachusetts Institute of Technology\n02139CambridgeMA\n\nMicrosoft Research\n02142CambridgeMA",
"Benjamin Mako Hill \nMassachusetts Institute of Technology\n02139CambridgeMA",
"Jazmin Gonzalez-Rivero \nMicrosoft Research\n02142CambridgeMA",
"Danah Boyd \nMicrosoft Research\n02142C... | In this paper, we explore the role that attribution plays in shaping user reactions to content reuse, or remixing, in a large user-generated content community. We present two studies using data from the Scratch online communitya social media platform where hundreds of thousands of young people share and remix animation... | [
{
"title": "INTRODUCTION",
"content": "Networked information technologies have changed the way people use and reuse creative -and frequently copyrightedmaterials. This change has generated excitement, and heated Best Paper Honorable Mention at CHI 2011. debate, among content-creators, technologists, legal a... | [
{
"id": "",
"caption": "An example Scratch program written using programming \"blocks\".",
"content": "Figure 1 .\n1An example Scratch program written using programming \"blocks\"."
},
{
"id": "",
"caption": "Screenshot of a Scratch remix project highlighting automatic (the area inside the b... | [
{
"text": "The Wealth of Networks: How Social Production Transforms Markets and Freedom. Y Benkler, Yale University PressBenkler, Y. The Wealth of Networks: How Social Production Transforms Markets and Freedom. Yale University Press, Oct. 2007.",
"doi": ""
},
{
"text": "Announcing (and explaining) o... |
A quantum topological phase transition at the microscopic level
1 Mar 2008
Claudio Castelnovo
Rudolf Peierls Centre for Theoretical Physics
University of Oxford
OX1 3NPOxfordUK
Claudio Chamon
Physics Department
Boston University
02215BostonMAUSA
A quantum topological phase transition at the microscopic level
1 Ma... | 16,676,596 | {
"arxiv": "0707.2084",
"mag": 2108036131,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevb.77.054433"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/0707.2084v3.pdf"
],
"pdf_sha": "5b78ee6cbb5d6deb5ac5c4b40cd7e8e336e5e181",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/0707.2084",
"status": "GREEN"
}
} | A quantum topological phase transition at the microscopic level | [
"Claudio Castelnovo \nRudolf Peierls Centre for Theoretical Physics\nUniversity of Oxford\nOX1 3NPOxfordUK",
"Claudio Chamon \nPhysics Department\nBoston University\n02215BostonMAUSA"
] | We study a quantum phase transition between a phase which is topologically ordered and one which is not. We focus on a spin model, an extension of the toric code, for which we obtain the exact ground state for all values of the coupling constant that takes the system across the phase transition. We compute the entangle... | [
{
"title": "I. INTRODUCTION",
"content": "Some strongly correlated quantum many body systems display a type of order which cannot be characterized by any local order parameter. Instead, such order is topological in nature 1 , with the fractional quantum Hall systems being the primary example so far: they ar... | [
{
"id": "",
"caption": "(Color online) -Examples of star and plaquette operators, centered at a lattice site s (blue open circle) and at a dual lattice site p (red open square), respectively. The solid black dots represent the spin-1/2 degrees of freedom living on the bonds of the lattice, and the dashed li... | [
{
"text": "DMR-0403997 (CC and CC), and by EPSRC Grant No. DMR-0403997 (CC and CC), and by EPSRC Grant No.",
"doi": ""
},
{
"text": "Castelnovo). C. Castelnovo would like to acknowledge the I2CAM NSF Grant DMR No. 0645461 for travel support, during which part of this work was carried out. GR/R83712/... |
JOINT LEARNING OF OBJECT GRAPH AND RELATION GRAPH FOR VISUAL QUESTION ANSWERING
9 May 2022
Hao Li
School of Electronic and Computer Engineering
Peking University
China
Xu Li
Cognitive Computing Lab
Baidu ResearchChina
Belhal Karimi belhalkarimi@baidu.com
Cognitive Computing Lab
Baidu ResearchChina
Jie Chen chen... | 248,571,502 | {
"arxiv": "2205.04188",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/icmcs/LiLKCS22",
"doi": "10.1109/icme52920.2022.9859766"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2205.04188v1.pdf"
],
"pdf_sha": "744ec954802334820e8cd2faa6e53c5e90d28920",
"open_access": {
"license": null,
"url": "https://arxiv.org/pdf/2205.04188",
"status": "GREEN"
}
} | JOINT LEARNING OF OBJECT GRAPH AND RELATION GRAPH FOR VISUAL QUESTION ANSWERING | [
"Hao Li \nSchool of Electronic and Computer Engineering\nPeking University\nChina",
"Xu Li \nCognitive Computing Lab\nBaidu ResearchChina",
"Belhal Karimi belhalkarimi@baidu.com \nCognitive Computing Lab\nBaidu ResearchChina",
"Jie Chen chenj@pcl.ac.cn \nSchool of Electronic and Computer Engineering\nPeking U... | Modeling visual question answering (VQA) through scene graphs can significantly improve the reasoning accuracy and interpretability.However, existing models answer poorly for complex reasoning questions with attributes or relations, which causes false attribute selection or missing relation in Figure1(a).It is because ... | [
{
"title": "INTRODUCTION",
"content": "VQA tasks require a model to answer a free-form natural language question using visual information from an image.Scene graph (SG) reasoning is an essential instance of VQA tasks [1].The model extracts objects' names, attributes, and relations from the input images and ... | [
{
"id": "",
"caption": "Fig. 1.(a) Two key issues of traditional scene graph based models that we address: false attribute selection (top: select attribute \"black\" instead of attribute \"off\") and missing relation (bottom: missing \"Behind\") (b) Overview of our DM-GNN model.F N and F E are the object fe... | [
{
"text": "Vg- Dataset, GroundTruth Motif-VG Question type What Where Who Why Overall What Where Who Why Overall Percentage (54%) (17%) (5%) (3%) (100%) (54%) (17%).",
"doi": ""
},
{
"text": "Scene graph reasoning for visual question answering. M Hildebrandt, H Li, R Koner, V Tresp, S Günnemann, 202... |
Experimental designs for binary data in switching measurements on superconducting Josephson junctions
2007
Juha Karvanen
National Public Health Institute
HelsinkiFinland
Juha J Vartiainen
Helsinki University of Technology
EspooFinland
Andrey Timofeev
Helsinki University of Technology
EspooFinland
Jukka Pekola
... | 16,899,178 | {
"arxiv": "cond-mat/0610507",
"mag": 1975915405,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1111/j.1467-9876.2007.00572.x"
} | {
"pdf_urls": [],
"pdf_sha": "69ac5b6bb2dd1fb450fc0f5f2eba30ef41e7a0fe",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/cond-mat/0610507v1.pdf",
"status": "GREEN"
}
} | Experimental designs for binary data in switching measurements on superconducting Josephson junctions | [
"Juha Karvanen \nNational Public Health Institute\nHelsinkiFinland",
"Juha J Vartiainen \nHelsinki University of Technology\nEspooFinland",
"Andrey Timofeev \nHelsinki University of Technology\nEspooFinland",
"Jukka Pekola \nHelsinki University of Technology\nEspooFinland"
] | We study the optimal design of switching measurements of small Josephson junction circuits which operate in the macroscopic quantum tunnelling regime. In the experiment, sequences of current pulses are applied to the Josephson junction sample, while the voltage over the structure is monitored. The appearance of a volta... | [
{
"title": "Introduction",
"content": "Randomness plays a central role in quantum physics. On the fundamental level, randomness is not caused only by unknown covariates or measurement error but is understood to be an intrinsic property of nature itself (Ballentine, 1998). From the statistical point of view,... | [
{
"id": "",
"caption": "Fig. 2. Switching rate γ as a function of the relative bias current I=I C ( , switching rate γ MQT in the macroscopic quantum tunnelling; -------, switching rate γ TA under the thermal activation model at temperature 1 K; ---, γ TA at temperature 0.1 K); the JJ parameters are set to ... | [
{
"text": "Experimental design for binary data. K M Abdelbasit, R L Plackett, J. Am. Statist. Ass. 78Abdelbasit, K. M. and Plackett, R. L. (1983) Experimental design for binary data. J. Am. Statist. Ass., 78, 90-98.",
"doi": ""
},
{
"text": "On the existence of maximum likelihood estimates in logist... |
BoxSup: Exploiting Bounding Boxes to Supervise Convolutional Networks for Semantic Segmentation
Jifeng Dai jifdai@microsoft.com
Microsoft Research
Kaiming He kahe@microsoft.com
Microsoft Research
Jian Sun jiansun@microsoft.com
Microsoft Research
BoxSup: Exploiting Bounding Boxes to Supervise Convolutional N... | 1,613,420 | {
"arxiv": "1503.01640",
"mag": 2949086864,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/iccv/DaiHS15",
"doi": "10.1109/iccv.2015.191"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1503.01640v2.pdf"
],
"pdf_sha": "fc882ab8af22e25eacd5909256ed480417aa61bc",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1503.01640",
"status": "GREEN"
}
} | BoxSup: Exploiting Bounding Boxes to Supervise Convolutional Networks for Semantic Segmentation | [
"Jifeng Dai jifdai@microsoft.com \nMicrosoft Research",
"Kaiming He kahe@microsoft.com \nMicrosoft Research",
"Jian Sun jiansun@microsoft.com \nMicrosoft Research"
] | Recent leading approaches to semantic segmentation rely on deep convolutional networks trained with humanannotated, pixel-level segmentation masks. Such pixelaccurate supervision demands expensive labeling effort and limits the performance of deep networks that usually benefit from more training data. In this paper, we... | [
{
"title": "Introduction",
"content": "In the past few months, tremendous progress has been made in the field of semantic segmentation [12,22,13,6,5,23]. Deep convolutional neural networks (CNNs) [19,18] that play as rich hierarchical feature extractors are a key to these methods. These networks are trained... | [
{
"id": "",
"caption": "Overview of our training approach supervised by bounding boxes.",
"content": "Figure 1 :\n1Overview of our training approach supervised by bounding boxes."
},
{
"id": "",
"caption": "Segmentation masks used as supervision. (a) A training image. (b) Ground-truth. (c) E... | [
{
"text": "Analyzing the performance of multilayer neural networks for object recognition. P Agrawal, R Girshick, J Malik, ECCV. P. Agrawal, R. Girshick, and J. Malik. Analyzing the perfor- mance of multilayer neural networks for object recognition. In ECCV, 2014.",
"doi": ""
},
{
"text": "Multiscal... |
Effect of phase relaxation on quantum superpositions in complex collisions
28 Mar 2006
L Benet
Centro de Ciencias Físicas
National University of Mexico (UNAM)
Mor62210CuernavacaMexico
S Yu Kun
Centro de Ciencias Físicas
National University of Mexico (UNAM)
Mor62210CuernavacaMexico
Nonlinear Physics Center
The Aus... | 39,198,288 | {
"arxiv": "quant-ph/0503046",
"mag": 2060521203,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevc.73.064602"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/quant-ph/0503046v2.pdf"
],
"pdf_sha": "021f5666d73fa04c76b4468f838cb14615b2e5e3",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/quant-ph/0503046",
"status": "GREEN"
}
} | Effect of phase relaxation on quantum superpositions in complex collisions | [
"L Benet \nCentro de Ciencias Físicas\nNational University of Mexico (UNAM)\nMor62210CuernavacaMexico",
"S Yu Kun \nCentro de Ciencias Físicas\nNational University of Mexico (UNAM)\nMor62210CuernavacaMexico\n\nNonlinear Physics Center\nThe Australian National University\n0200RSPhysSE, CanberraACTAustralia\n\nDepa... | We study the effect of phase relaxation on coherent superpositions of rotating clockwise and anticlockwise wave packets in the regime of strongly overlapping resonances of the intermediate complex. Such highly excited deformed complexes may be created in binary collisions of heavy ions, molecules and atomic clusters. I... | [
{
"title": "I. INTRODUCTION",
"content": "For highly excited strongly interacting many-body systems the independent particle picture has very limited validity [1]. For high excitations, the interaction results in a quick decay of single-particle [2,3,4] as well as collective modes [2,4], which are not eigen... | [
{
"id": "",
"caption": "(Color online) Angular dependence of the decay intensity of the highly excited intermediate complex at different moments of time for different combinations of β and d. Panel (a) corresponds to time t = 3T /8; (b) t = 7T /16; (c) t = T /2; (d) t = 5T /8. Here, T is the period of one c... | [
{
"text": "E Wigner, Statistical Properties of Nuclei. J.B. GardNew York-LondonPlenum Press11E. Wigner, in Statistical Properties of Nuclei, editor J.B. Gard (Plenum Press, New York-London, 1972), p. 11.",
"doi": ""
},
{
"text": ". O Bohigas, H A Weidenmüller, Annu. Rev. Nucl. Part. Sci. 38421O. Boh... |
SIGNATURES OF PHOTON-AXION CONVERSION IN THE THERMAL SPECTRA AND POLARIZATION OF NEUTRON STARS
25 Jan 2012 December 21, 2013
Rosalba Perna
Wynn C G Ho
Licia Verde
Matthew Van Adelsberg
Raul Jimenez
SIGNATURES OF PHOTON-AXION CONVERSION IN THE THERMAL SPECTRA AND POLARIZATION OF NEUTRON STARS
25 Jan 2012 December... | 119,242,279 | {
"arxiv": "1201.5390",
"mag": 2097726156,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1088/0004-637x/748/2/116"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1201.5390v1.pdf"
],
"pdf_sha": "bb78eb5c48e8cdaf07f3a4e5f55767da50778065",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1201.5390",
"status": "GREEN"
}
} | SIGNATURES OF PHOTON-AXION CONVERSION IN THE THERMAL SPECTRA AND POLARIZATION OF NEUTRON STARS | [
"Rosalba Perna",
"Wynn C G Ho",
"Licia Verde",
"Matthew Van Adelsberg",
"Raul Jimenez"
] | Conversion of photons into axions under the presence of a strong magnetic field can dim the radiation from magnetized astrophysical objects. Here we perform a detailed calculation aimed at quantifying the signatures of photon-axion conversion in the spectra, light curves, and polarization of neutron stars (NSs). We tak... | [
{
"title": "INTRODUCTION",
"content": "Axions with low mass (below 1 meV) are a direct prediction of the solution for the strong CP violation problem. Peccei & Quinn (1977) proposed axions as Pseudo-Goldstone bosons arising from the spontaneous breakdown of the U(1) symmetry. The Peccei-Quinn axion occupies... | [
{
"id": "",
"caption": "-Fractional contributions to the flux from the emergent O ( thick/green lines) and X (thin/red lines) modes for four angles δ with respect to the surface normal and three magnetic field strengths. The effective temperature is Ts = 5 × 10 6 K in all cases.",
"content": "FIG. 1 .\n... | [
{
"text": ". S L Adler, Ann. Phys. 67599Adler, S. L. 1971, Ann. Phys., 67, 599",
"doi": ""
},
{
"text": ". C Alcock, A Illarionov, ApJ. 235534Alcock, C. & Illarionov, A. 1980, ApJ, 235, 534",
"doi": ""
},
{
"text": ". S Andriamonje, J. Cosm. Astropart. Phys. 410Andriamonje, S. et al. 200... |
21 Oct 2017
21 Oct 2017EQUIVALENCE OF VISCOSITY AND WEAK SOLUTIONS FOR THE NORMALIZED p(x)-LAPLACIAN JARKKO SILTAKOSKI
We show that viscosity solutions to the normalized p(x)-Laplace equation coincide with distributional weak solutions to the strong p(x)-Laplace equation when p is Lipschitz and inf p > 1. This yields... | 119,566,555 | {
"arxiv": "1710.07760",
"mag": 2963997759,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1007/s00526-018-1375-1"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1710.07760v1.pdf"
],
"pdf_sha": "30a77bdf240d7376b838c3d10ea98da9698126e1",
"open_access": {
"license": "CC0",
"url": "https://jyx.jyu.fi/bitstream/123456789/62921/1/siltakoskiequivalence%20revised.pdf",
"status": "GREEN"
}
} | [] | We show that viscosity solutions to the normalized p(x)-Laplace equation coincide with distributional weak solutions to the strong p(x)-Laplace equation when p is Lipschitz and inf p > 1. This yields C 1,α regularity for the viscosity solutions of the normalized p(x)-Laplace equation. As an additional application, we p... | [
{
"title": "Introduction",
"content": "In this paper, we study viscosity solutions to the normalized p(x)-Laplace equation which is defined by There has been recent interest in normalized equations, see for example [JS17,IJS,BG15]. We are partly motivated by the connection to stochastic tug-of-war games [PS... | [
{
"id": "",
"caption": "∆ N p(x) u := −∆u − p(x) − 2 |Du| 2 ∆ ∞ u := −∆u − p(x) − 2 |Du| 2 D 2 uDu, Du = 0. (1.1)",
"content": "−\n∆ N p(x) u := −∆u − p(x) − 2 |Du| 2 ∆ ∞ u := −∆u − p(x) − 2 |Du| 2 D 2 uDu, Du = 0. (1.1)"
},
{
"id": "",
"caption": "The strong and normalized p(x)-Laplace equa... | [
{
"text": "Mappings of finite distortion and PDE with nonstandard growth. T Adamowicz, P Hästö, Int. Math. Res. Not. IMRN. T. Adamowicz and P. Hästö. Mappings of finite distortion and PDE with nonstandard growth. Int. Math. Res. Not. IMRN, pages 1940-1965, 2010.",
"doi": ""
},
{
"text": "Harnack's i... | |
PDD Graph: Bridging Electronic Medical Records and Biomedical Knowledge Graphs via Entity Linking
Meng Wang wangmengsd@stu.xjtu.edu.cn
MOEKLINNS lab
Xi'an Jiaotong University
Xi'anChina
Jiaheng Zhang
MOEKLINNS lab
Xi'an Jiaotong University
Xi'anChina
Jun Liu
MOEKLINNS lab
Xi'an Jiaotong University
Xi'anChina
W... | 237,451 | {
"arxiv": "1707.05340",
"mag": 2964109414,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/WangZLHWLL17",
"doi": "10.1007/978-3-319-68204-4_23"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1707.05340v2.pdf"
],
"pdf_sha": "5b8c7905fe427aa7c26a41d11894e849b487e72a",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1707.05340",
"status": "GREEN"
}
} | PDD Graph: Bridging Electronic Medical Records and Biomedical Knowledge Graphs via Entity Linking | [
"Meng Wang wangmengsd@stu.xjtu.edu.cn \nMOEKLINNS lab\nXi'an Jiaotong University\nXi'anChina",
"Jiaheng Zhang \nMOEKLINNS lab\nXi'an Jiaotong University\nXi'anChina",
"Jun Liu \nMOEKLINNS lab\nXi'an Jiaotong University\nXi'anChina",
"Wei Hu \nState Key Laboratory for Novel Software Technology\nNanjing Univers... | Electronic medical records contain multi-format electronic medical data that consist of an abundance of medical knowledge. Facing with patients symptoms, experienced caregivers make right medical decisions based on their professional knowledge that accurately grasps relationships between symptoms, diagnosis and corresp... | [
{
"title": "Introduction",
"content": "Big data vendors collect and store large number of electronic medical records (EMRs) in hospital, with the goal of instantly accessing to comprehensive medical patient histories for caregivers at a lower cost. Public availability of EMRs collections has attracted much ... | [
{
"id": "",
"caption": "Left part is the Linked Data Cloud 1 , which contains interlinked biomedical knowledge graphs. Right part is the MIMIC-III database.",
"content": "Fig. 1 .\n1Left part is the Linked Data Cloud 1 , which contains interlinked biomedical knowledge graphs. Right part is the MIMIC-III... | [
{
"text": "Linked data-the story so far. Semantic services, interoperability and web applications: emerging concepts pp. C Bizer, T Heath, T Berners-Lee, Bizer, C., Heath, T., Berners-Lee, T.: Linked data-the story so far. Semantic ser- vices, interoperability and web applications: emerging concepts pp. 205-227... |
First-principles study of the low-temperature charge density wave phase in the quasi-one-dimensional Weyl chiral compound (TaSe 4 ) 2 I
Yang Zhang
Department of Physics and Astronomy
University of Tennessee
37996KnoxvilleTNUSA
School of Physics
Southeast University
211189NanjingChina
Ling-Fang Lin
Department of ... | 211,204,724 | {
"arxiv": "2002.08826",
"mag": 3024063761,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevb.101.174106"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2002.08826v2.pdf"
],
"pdf_sha": "403523603f500908370ccf002e97635883606d01",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/2002.08826",
"status": "GREEN"
}
} | First-principles study of the low-temperature charge density wave phase in the quasi-one-dimensional Weyl chiral compound (TaSe 4 ) 2 I | [
"Yang Zhang \nDepartment of Physics and Astronomy\nUniversity of Tennessee\n37996KnoxvilleTNUSA\n\nSchool of Physics\nSoutheast University\n211189NanjingChina",
"Ling-Fang Lin \nDepartment of Physics and Astronomy\nUniversity of Tennessee\n37996KnoxvilleTNUSA\n\nSchool of Physics\nSoutheast University\n211189Nanj... | Using ab initio density functional theory, we study the lattice phase transition of quasi-onedimensional (TaSe4)2I. In the undistorted state, the strongly anisotropic semimetal band structure presents two non-equivalent Weyl points. In previous efforts, two possible Ta-tetramerization patterns were proposed to be assoc... | [
{
"title": "I. INTRODUCTION",
"content": "One-dimensional (1D) systems continue attracting considerable attention due to their rich physical properties and reduced dimensional phase space. In particular, in low dimension systems the electron-electron, phononphonon, electron-phonon, and spin-phonon couplings... | [
{
"id": "",
"caption": "Schematic crystal structure of the conventional cell of (TaSe4)2I with the convention: Blue = Ta; green = Se; purple = I. (a-b) The crystal structure is shown from the perspectives of the (a) bc plane [(100) direction] and (b) top [(001) direction]. (c-d) The two different Ta-tetrame... | [
{
"text": "Density Waves in Solids (Perseus. G Grüner, Cambridge, MAG. Grüner, Density Waves in Solids (Perseus, Cam- bridge, MA, 2000).",
"doi": ""
},
{
"text": ". P Monceau, 10.1080/00018732.2012.719674Adv. Phys. 61325P. Monceau, Adv. Phys. 61, 325 (2012).",
"doi": "10.1080/00018732.2012.71967... |
Negative modes of Schwarzschild black holes in Einstein-Gauss-Bonnet theory
2008
Takayuki Hirayama
Physics Division
National Center for Theoretical Sciences
Department of Physics
National Taiwan Normal University
300, 116Hsinchu, TaipeiTaiwan, Taiwan
Negative modes of Schwarzschild black holes in Einstein-Gauss-Bon... | 18,991,817 | {
"arxiv": "0804.3694",
"mag": 2049028784,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1088/0264-9381/25/24/245006"
} | {
"pdf_urls": [],
"pdf_sha": "21a714fe1824430c075bb6d30810eb128ff659db",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/0804.3694",
"status": "GREEN"
}
} | Negative modes of Schwarzschild black holes in Einstein-Gauss-Bonnet theory | [
"Takayuki Hirayama \nPhysics Division\nNational Center for Theoretical Sciences\nDepartment of Physics\nNational Taiwan Normal University\n300, 116Hsinchu, TaipeiTaiwan, Taiwan"
] | We study non-conformal negative modes of black holes in the five-dimensional Einstein-Gauss-Bonnet gravity theory with a zero or negative cosmological constant (which become Schwarzschild black holes in the Einstein limit), in the context of the semiclassical quantum gravity formulated by the path integral. We find a u... | [
{
"title": "Introduction",
"content": "Black holes are very important objects in classical and quantum gravity. One of the most important and profound properties in black hole physics is its thermodynamics. The thermodynamic quantities can be derived from the path integral formulation of quantum gravity, an... | [
{
"id": "",
"caption": "the region where the black hole has a negative heat capacity disappears and the black hole is always stable as a thermodynamic system. Seefigure 4. C V > 0 for all r h .",
"content": "the region where the black hole has a negative heat capacity disappears and the black hole is al... | [
{
"text": ". D J Gross, M J Perry, Yaffe L G , 10.1103/PhysRevD.25.330Phys. Rev. D. 25330Gross D J, Perry M J and Yaffe L G 1982 Phys. Rev. D 25 330",
"doi": ""
},
{
"text": ". T Prestidge, 10.1103/PhysRevD.61.084002arXiv:hep-th/9907163Phys. Rev. D. 6184002Prestidge T 2000 Phys. Rev. D 61 084002 (ar... |
Approximate Task Tree Retrieval in a Knowledge Network for Robotic Cooking
8 Jul 2022
Md Sadman Sakib
David Paulius
Yu Sun
Approximate Task Tree Retrieval in a Knowledge Network for Robotic Cooking
8 Jul 2022B2A168BBE8E57E0B15A27CAF981EDE90arXiv:2207.03693v1[cs.RO]Service RoboticsTask PlanningPlanning under Uncert... | 250,407,773 | {
"arxiv": "2207.03693",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/ral/SakibPS22",
"doi": "10.1109/lra.2022.3191068"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2207.03693v1.pdf"
],
"pdf_sha": "badc46d8915e9adce6859456b78036f6b1edec5c",
"open_access": {
"license": null,
"url": "https://arxiv.org/pdf/2207.03693",
"status": "GREEN"
}
} | Approximate Task Tree Retrieval in a Knowledge Network for Robotic Cooking | [
"Md Sadman Sakib",
"David Paulius",
"Yu Sun"
] | Flexible task planning continues to pose a difficult challenge for robots, where a robot is unable to creatively adapt their task plans to new or unseen problems, which is mainly due to the limited knowledge it has about its actions and world.Motivated by a human's ability to adapt, we explore how task plans from a kno... | [
{
"title": "I. INTRODUCTION",
"content": "M AJOR efforts in robotics research have been devoted to the development of intelligent agents with the ability to understand human intentions and to perform actions for problems in human-centered domains.Such domains include, but are not limited to, the application... | [
{
"id": "",
"caption": "Fig. 2. Illustration of a universal FOON made of 140 recipe subgraphs, and three (3) connected functional units, where object nodes are denoted by green circles and motion nodes are denoted by red squares.",
"content": "Fig. 2 .\n2\nFig. 2. Illustration of a universal FOON made o... | [
{
"text": "Joint Object and State Recognition using Language Knowledge. A B Jelodar, Y Sun, 2019 IEEE International Conference on Image Processing (ICIP). IEEE2019",
"doi": ""
},
{
"text": "Functional Object-Oriented Network for Manipulation Learning. D Paulius, Y Huang, R Milton, W D Buchanan, J Sa... |
Entanglement Entropies in Conformal Systems with Boundaries
(Dated: September 2, 2013)
L Taddia
Dipartimento di Fisica e Astronomia dell'Università di Bologna and INFN
via Irnerio 4640126BolognaItaly
Instituto de Física Teórica UAM/CSIC
Universidad Autónoma de Madrid
28049Cantoblanco, MadridSpain
J C Xavier
Insti... | 53,462,122 | {
"arxiv": "1302.6222",
"mag": 2327779959,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevb.88.075112"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1302.6222v2.pdf"
],
"pdf_sha": "dfe1c34eb6db35475e00401506c4418b1ba62c73",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1302.6222",
"status": "GREEN"
}
} | Entanglement Entropies in Conformal Systems with Boundaries | [
"L Taddia \nDipartimento di Fisica e Astronomia dell'Università di Bologna and INFN\nvia Irnerio 4640126BolognaItaly\n\nInstituto de Física Teórica UAM/CSIC\nUniversidad Autónoma de Madrid\n28049Cantoblanco, MadridSpain",
"J C Xavier \nInstituto de Física\nUniversidade Federal de Uberlândia\nCaixa Postal 59338400... | We study the entanglement entropies in one-dimensional open critical systems, whose effective description is given by a conformal field theory with boundaries. We show that for pure-state systems formed by the ground state or by the excited states associated to primary fields, the entanglement entropies have a finite-s... | [
{
"title": "FIG. 1 :",
"content": ""
},
{
"title": "FIG. 2 :",
"content": ""
},
{
"title": "FIG. 3 :",
"content": ""
}
] | [
{
"id": "",
"caption": "PACS numbers: 03.67.-a, 03.67.Mn, 64.60.an, 11.25.Hf",
"content": "PACS numbers: 03.67.-a, 03.67.Mn, 64.60.an, 11.25.Hf"
},
{
"id": "",
"caption": "(2), and finite-size scaling (FSS) techniques, we obtain the asymptotic values c = 0.499 and c η=2 1 = 0.241. The value ... | [
{
"text": ". P Calabrese, J Cardy, B Doyon, J. Phys. A.: Math. Theor. 42500301P. Calabrese, J. Cardy, B. Doyon, J. Phys. A.: Math. Theor. 42, 500301 (2009).",
"doi": ""
},
{
"text": ". J Eisert, M Cramer, M B Plenio, Rev. Mod. Phys. 82277J. Eisert, M. Cramer, M. B. Plenio, Rev. Mod. Phys. 82, 277 (2... |
Visual Attention-based Self-supervised Absolute Depth Estimation using Geometric Priors in Autonomous Driving
Jie Xiang
Yun Wang
Lifeng An
Haiyang Liu
Zijun Wang
Jian Liu
Visual Attention-based Self-supervised Absolute Depth Estimation using Geometric Priors in Autonomous Driving
10.1109/LRA.2022.3210298
Altho... | 248,863,189 | {
"arxiv": "2205.08780",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/ral/XiangWALWL22",
"doi": "10.1109/lra.2022.3210298"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2205.08780v3.pdf"
],
"pdf_sha": "2869090a3a6dd3f51c33bc0aaa6563c6cd5c96fa",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/2205.08780",
"status": "GREEN"
}
} | Visual Attention-based Self-supervised Absolute Depth Estimation using Geometric Priors in Autonomous Driving | [
"Jie Xiang",
"Yun Wang",
"Lifeng An",
"Haiyang Liu",
"Zijun Wang",
"Jian Liu"
] | Although existing monocular depth estimation methods have made great progress, predicting an accurate absolute depth map from a single image is still challenging due to the limited modeling capacity of networks and the scale ambiguity issue. In this paper, we introduce a fully Visual Attentionbased Depth (VADepth) netw... | [
{
"title": "I. INTRODUCTION",
"content": "M ONOCULAR Depth Estimation (MDE) is a fundamental problem in the field of computer vision, which refers to predicting the corresponding depth map from a single image. Depth Estimation has broad application prospects in autonomous driving, such as 3D object detectio... | [
{
"id": "",
"caption": "Network structure of VADepth. The encoder extracts feature maps X e i at different resolutions. The Dual Attention Module (DAM) takes the last level feature map X e 4 as input and emphasizes the important features to produce the initial input of the decoder. The decoder has five succ... | [
{
"text": "Is pseudolidar needed for monocular 3d object detection. D Park, R Ambrus, V Guizilini, J Li, A Gaidon, Proc. IEEE/CVF Int. Conf. Comput. Vis. IEEE/CVF Int. Conf. Comput. VisD. Park, R. Ambrus, V. Guizilini, J. Li, and A. Gaidon, \"Is pseudo- lidar needed for monocular 3d object detection?\" in Proc.... |
Enhancing synchronizability of weighted dynamical networks using betweenness centrality
Mahdi Jalili
School of Computer and Communication Sciences
Laboratory of Nonlinear Systems
Ecole Polytechnique Fédérale de Lausanne
1015LausanneCHSwitzerland
Ali Ajdari Rad
School of Computer and Communication Sciences
Laborat... | 19,797,201 | {
"arxiv": "0812.4720",
"mag": 2079897686,
"acl": null,
"pubmed": "18764018",
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physreve.78.016105"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/0812.4720v1.pdf"
],
"pdf_sha": "6a6e20664c2fa7838851898b5dd455167668802b",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/0812.4720",
"status": "GREEN"
}
} | Enhancing synchronizability of weighted dynamical networks using betweenness centrality | [
"Mahdi Jalili \nSchool of Computer and Communication Sciences\nLaboratory of Nonlinear Systems\nEcole Polytechnique Fédérale de Lausanne\n1015LausanneCHSwitzerland",
"Ali Ajdari Rad \nSchool of Computer and Communication Sciences\nLaboratory of Nonlinear Systems\nEcole Polytechnique Fédérale de Lausanne\n1015Laus... | By considering the eigenratio of the Laplacian of the connection graph as synchronizability measure, we propose a procedure for weighting dynamical networks to enhance their synchronizability. The method is based on node and edge betweenness centrality measures and is tested on artificially constructed scale-free, Watt... | [
{
"title": "II. WEIGHTING ALGORITHM",
"content": "Let us start with a dynamical network of N linearly coupled identical systems with the following equations of the motion The variational equations of manifold synchronized solution ( )\nλ are the eigenvalues of G , ordered as 0 = λ 1 ≤ λ 2 ≤ … ≤ λ N and λ 1 ... | [
{
"id": "",
"caption": "introduced in[7]. Namely, starting with a network of m + 1 all-to-all connected nodes, at each step a new node is added with m links that are connected to node i with probability k i is the degree of the node and B a tunable real parameter controlling the heterogeneity of the network... | [
{
"text": ". S H Strogatz, Nature. 410268S. H. Strogatz, Nature 410 (2001) 268.",
"doi": ""
},
{
"text": "M Newman, A.-L Barabasi, D J Watts, The structure and dynamics of networks. Princeton University PressM. Newman, A.-L. Barabasi, and D. J. Watts, The structure and dynamics of networks, Princeto... |
Signed Distance Laplacian Matrices for Signed Graphs
8 Oct 2020
Roshni T Roy roshnitroy@gmail.com
Department of Mathematics
Central University of Kerala
Kasaragod -671316, Ker-alaIndia
Department of Mathematics
Central University of Kerala
Kasaragod -671316, Ker-alaIndia
Shahul Hameed
Department of Mathematics
K ... | 222,272,048 | {
"arxiv": "2010.04204",
"mag": 3092419419,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1080/03081087.2022.2158165"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2010.04204v1.pdf"
],
"pdf_sha": "4f42f693b90293d8b352f226594aced3e52683f1",
"open_access": {
"license": null,
"url": "https://arxiv.org/pdf/2010.04204",
"status": "GREEN"
}
} | Signed Distance Laplacian Matrices for Signed Graphs | [
"Roshni T Roy roshnitroy@gmail.com \nDepartment of Mathematics\nCentral University of Kerala\nKasaragod -671316, Ker-alaIndia\n\nDepartment of Mathematics\nCentral University of Kerala\nKasaragod -671316, Ker-alaIndia",
"Shahul Hameed \nDepartment of Mathematics\nK M M Government Women's College\nKannur -670004Ke... | A signed graph is a graph whose edges are labeled either positive or negative.Corresponding to the two signed distance matrices defined for signed graphs, we define two signed distance laplacian matrices.We characterize balance in signed graphs using these matrices and find signed distance laplacian spectra of some cla... | [
{
"title": "Introduction",
"content": "Throughout this article, unless otherwise mentioned, by a graph we mean a finite, connected, simple graph.For any terms which are not mentioned here, the reader may refer to [4].\nA signed graph Σ = (G, σ) is an underlying graph G = (V, E) with a signature function σ :... | [
{
"id": "",
"caption": "are negative, and +1 otherwise.(S2) σ min (u, v) = +1 if all shortest uv -paths are positive, and −1 otherwise.",
"content": "are negative, and +1 otherwise.(S2) σ min (u, v) = +1 if all shortest uv -paths are positive, and −1 otherwise."
},
{
"id": "",
"caption": "[3... | [
{
"text": "Two Laplacians for the distance matrix of a graph. M Aouchiche, P Hansen, Linear Algebra Appl. 4392013",
"doi": ""
},
{
"text": "Comprehensive Introduction to Linear Algebra. J G Broida, S G Williamson, 1989Addison WesleyRedwood City, Cal",
"doi": ""
},
{
"text": "Signed dista... |
NONVANISHING OF HYPERELLIPTIC ZETA FUNCTIONS OVER FINITE FIELDS
6 Oct 2021
Jordan S Ellenberg
Wanlin Li
Mark Shusterman
NONVANISHING OF HYPERELLIPTIC ZETA FUNCTIONS OVER FINITE FIELDS
6 Oct 2021
Fixing t ∈ R and a finite field F q of odd characteristic, we give an explicit upper bound on the proportion of genus g ... | 119,735,177 | {
"arxiv": "1901.08202",
"mag": 3079316661,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.2140/ant.2020.14.1895"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1901.08202v3.pdf"
],
"pdf_sha": "eef679288cacab49eaa1110f697171735c9613c0",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1901.08202",
"status": "GREEN"
}
} | NONVANISHING OF HYPERELLIPTIC ZETA FUNCTIONS OVER FINITE FIELDS | [
"Jordan S Ellenberg",
"Wanlin Li",
"Mark Shusterman"
] | Fixing t ∈ R and a finite field F q of odd characteristic, we give an explicit upper bound on the proportion of genus g hyperelliptic curves over F q whose zeta function vanishes at 1 2 + it. Our upper bound is independent of g and tends to 0 as q grows. | [
{
"title": "INTRODUCTION",
"content": "Let p be an odd prime, set q = p k for some positive integer k, and denote by F q the finite field with q elements. To (the smooth completion of) any hyperelliptic curve C over F q one associates a zeta function Z C (s). Weil has shown that Z C (s) = 0 implies that s =... | [
{
"id": "",
"caption": "Corollary 3.3). Let A be a constant abelian variety defined overF q (x). For each f ∈ F q m [x], denote by A f the quadratic twist of A ⊗ F q m (x) by f . Let R n,m be the set { f ∈ F q m [x],squarefree, of deg n : A f has positive rank}. Then, lim m→∞ lim sup n→∞ |R n,m | q m(n+1) =... | [
{
"text": "Small zeros of Dirichlet L-functions of quadratic characters of prime modulus. J Andrade, S Baluyot, arXiv preprintJ. Andrade and S. Baluyot, Small zeros of Dirichlet L-functions of quadratic characters of prime modulus, (2018), arXiv preprint.",
"doi": ""
},
{
"text": "Average values of ... |
26 Oct 2007
26 Oct 2007arXiv:math/0605701v2 [math.AG] VANISHING RESULTS FOR TORIC VARIETIES ASSOCIATED TO GL n AND G 2 QËNDRIM R. GASHI Dedicated to Scarlett MccGwire and Dr. Christian Duhamel
Toric varieties associated with root systems appeared very naturally in the theory of group compactifications. Here they are ... | 16,153,264 | {
"arxiv": "math/0605701",
"mag": 2950849610,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1007/s00031-008-9002-4"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/math/0605701v2.pdf"
],
"pdf_sha": "35921a21b512a70e88b8c8c408e3e50f901da599",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/math/0605701",
"status": "GREEN"
}
} | [] | Toric varieties associated with root systems appeared very naturally in the theory of group compactifications. Here they are considered in a very different context.We prove the vanishing of higher cohomology groups for certain line bundles on toric varieties associated to GLn and G 2 . This can be considered of general... | [
{
"title": "Introduction",
"content": "Fix an isocrystal and the corresponding Newton vector. Mazur's inequality (see [10], pg.658, and the group-theoretic generalization in [13], Theorem 4.2) states that, given a lattice in our isocrystal, the Hodge vector associated to it is greater than or equal to the N... | [
{
"id": "",
"caption": "cf. [4], pg. 68) Let D be a Cartier divisor on V . Then O(D) is generated by its sections if and only if ψ D is (lower) convex. b) (cf. [4], pg.74) For every i ≥ 0 there exist canonical isomorphisms H i (V, O(D)) ∼ = u∈X * (T ) H i (V, O(D)) u , and H i (V, O(D)) u ∼ = H i (|∆| , |∆|... | [
{
"text": "The characters of discrete series as orbital integrals. J Arthur, Invent. Math. 323J. ARTHUR, The characters of discrete series as orbital integrals, Invent. Math. 32 1976, no.3, 205-261.",
"doi": ""
},
{
"text": "Frobenius splitting methods in geometry and representation theory. M Brion,... | |
Some Hierarchies of QCB 0 -Spaces
5 Apr 2013
Matthias Schröder
Victor Selivanov
A P Ershov
University of Bundeswehr-Munich
NeubiebergGermany
Institute of Informatics Systems SB RAS Novisibirsk
Russia
Some Hierarchies of QCB 0 -Spaces
5 Apr 2013A6A1372DF7738BD473BE804CEAD5F652arXiv:1304.1647v1[cs.LO]
We define ... | 15,509,555 | {
"arxiv": "1304.1647",
"mag": 2963651973,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/mscs/SchroderS15",
"doi": "10.1017/s0960129513000376"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1304.1647v1.pdf"
],
"pdf_sha": "5085c86d9fc3ec91522cb1a42f26919dacad5597",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1304.1647",
"status": "GREEN"
}
} | Some Hierarchies of QCB 0 -Spaces | [
"Matthias Schröder",
"Victor Selivanov",
"A P Ershov",
"University of Bundeswehr-Munich\nNeubiebergGermany",
"Institute of Informatics Systems SB RAS Novisibirsk\nRussia"
] | We define and study hierarchies of topological spaces induced by the classical Borel and Luzin hierarchies of sets.Our hierarchies are divided into two classes: hierarchies of countably based spaces induced by their embeddings into Pω, and hierarchies of spaces (not necessarily countably based) induced by their admissi... | [
{
"title": "Introduction",
"content": "A basic notion of Computable Analysis (CA) [Wei00] is the notion of an admissible representation of a topological space X.This is a partial continuous surjection δ from the Baire space N onto X satisfying a certain universality property (see Subsection 2.6 for some mor... | [
{
"id": "",
"caption": "by Proposition 2.8.Define the function g : Y → Pω by g(y) = {n | y ∈ A n }.Then for any y ∈ Y we have",
"content": "by Proposition 2.8.Define the function g : Y → Pω by g(y) = {n | y ∈ A n }.Then for any y ∈ Y we have"
},
{
"id": "",
"caption": "by Proposition 2.10.Si... | [
{
"text": "Topological properties of real number representations. V Brattka, P Hertling, Theoretical Computer Science. 2842002",
"doi": ""
},
{
"text": "Quasi-Polish spaces. M De Brecht, arXiv:1108.1445v1Annals of Pure and Applied Logic. 16432013",
"doi": ""
},
{
"text": "M De Brecht, A ... |
Hidden AGNs in Early-Type Galaxies
version April 8, 2016
Alessandro Paggi apaggi@cfa.harvard.edu
Harvard-Smithsonian Center for Astrophysics
60 Garden St02138CambridgeMAUSA
Giuseppina Fabbiano
Harvard-Smithsonian Center for Astrophysics
60 Garden St02138CambridgeMAUSA
Francesca Civano
Harvard-Smithsonian Center ... | 118,373,611 | {
"arxiv": "1507.03170",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.3847/0004-637x/823/2/112"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1507.03170v2.pdf"
],
"pdf_sha": "841b18653dc96f857f4f5621a35dc255a2c6bd18",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1507.03170",
"status": "GREEN"
}
} | Hidden AGNs in Early-Type Galaxies | [
"Alessandro Paggi apaggi@cfa.harvard.edu \nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA",
"Giuseppina Fabbiano \nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA",
"Francesca Civano \nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138Cambri... | We present a stacking analysis of the complete sample of Early Type Galaxies (ETGs) in the Chandra COSMOS (C-COSMOS) survey, to explore the nature of the X-ray luminosity in the redshift and stellar luminosity ranges 0 < z < 1.5 and 10 9 < L K /L < 10 13 . Using established scaling relations, we subtract the contributi... | [
{
"title": "Introduction",
"content": "Diffuse X-ray emission from normal early type galaxies (ETG) was first detected in the Virgo cluster with the Einstein X-ray Observatory (Forman et al. 1979). Although in a few cases (e.g., M86) the displacement of the X-ray emission from the stellar body of the galaxy... | [
{
"id": "",
"caption": "shows the results of the L K -first stacking, with L X−LMXB plotted against stellar mass M * and the mass of the nuclear black hole, M BH , obtained from the M BH − M * relation of Kormendy & Ho (2013). Evaluating M BH from the M BH − M * relation McConnell & Ma (2013) would yield bl... | [
{
"text": ". R D Blandford, M C Begelman, MNRAS. 3031Blandford, R. D., & Begelman, M. C. 1999, MNRAS, 303, L1",
"doi": ""
},
{
"text": ". B Boroson, D.-W Kim, G Fabbiano, ApJ. 729BKF1112Boroson, B., Kim, D.-W., & Fabbiano, G. 2011, ApJ, 729, 12 (BKF11)",
"doi": ""
},
{
"text": ". G Bruzu... |
Precise radial velocities of giant stars. IV. ⋆ A correlation between surface gravity and radial velocity variation and a statistical investigation of companion properties
4 Jan 2008 February 2, 2008
S Hekker
Leiden Observatory
Leiden University
P.O. Box 95132300 RALeidenThe Netherlands
Royal Observatory of Belgium... | 33,442,610 | {
"arxiv": "0801.0741",
"mag": 2103449991,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1051/0004-6361:20078321"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/0801.0741v1.pdf"
],
"pdf_sha": "11c7588046924474bb2d92e471addbf94e00406e",
"open_access": {
"license": null,
"url": "https://www.aanda.org/articles/aa/pdf/2008/10/aa8321-07.pdf",
"status": "BRONZE"
}
} | Precise radial velocities of giant stars. IV. ⋆ A correlation between surface gravity and radial velocity variation and a statistical investigation of companion properties | [
"S Hekker \nLeiden Observatory\nLeiden University\nP.O. Box 95132300 RALeidenThe Netherlands\n\nRoyal Observatory of Belgium\nRinglaan 31180BrusselsBelgium",
"I A G Snellen \nLeiden Observatory\nLeiden University\nP.O. Box 95132300 RALeidenThe Netherlands",
"C Aerts \nInstituut voor Sterrenkunde\nKatholieke Uni... | Context. Since 1999, we have been conducting a radial velocity survey of 179 K giants using the Coudé Auxiliary Telescope at UCO/Lick observatory. At present ∼20−100 measurements have been collected per star with a precision of 5 to 8 m s −1 . Of the stars monitored, 145 (80%) show radial velocity (RV) variations at a ... | [
{
"title": "Introduction",
"content": "For more than a decade, radial velocity observations with accuracies of order m s −1 have been within reach (see for instance Marcy & Butler (2000) and Queloz et al. (2001)). Even accuracies of less than 1 m s −1 (Pepe et al. 2003) are possible now. With these observat... | [
{
"id": "",
"caption": "Radial velocity variations as a function of phase for a star (HIP34693) with a highly significant period (top), with its periodogram (bottom). The dashed line in the periodogram indicates the significance threshold.",
"content": "Fig. 1 .\n1Radial velocity variations as a functio... | [
{
"text": ". P J Armitage, I A Bonnell, MNRAS. 33011Armitage, P. J. & Bonnell, I. A. 2002, MNRAS, 330, L11",
"doi": ""
},
{
"text": ". T R Bedding, R P Butler, F Carrier, ApJ. 647558Bedding, T. R., Butler, R. P., Carrier, F., et al. 2006, ApJ, 647, 558",
"doi": ""
},
{
"text": ". R P But... |
Multi-wavelength analysis of the dust emission in the Small Magellanic Cloud ⋆
4 May 2004 November 7, 2018
C Bot
Centre de Données Astronomiques de Strasbourg (CDS)
UMR 7550
Université Louis Pasteur
F-67000STRASBOURGFrance
Institut d'Astrophysique Spatiale
Université Paris-Sud
F-91405OrsayFrance
F Boulanger
Insti... | 1,432,248 | {
"arxiv": "astro-ph/0405045",
"mag": 2127714165,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1051/0004-6361:20035918"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/astro-ph/0405045v1.pdf"
],
"pdf_sha": "ccd3d843f2b79c5920f682d7ba1ff316c4e0d221",
"open_access": {
"license": null,
"url": "https://www.aanda.org/articles/aa/pdf/2004/32/aa0918.pdf",
"status": "BRONZE"
}
} | Multi-wavelength analysis of the dust emission in the Small Magellanic Cloud ⋆ | [
"C Bot \nCentre de Données Astronomiques de Strasbourg (CDS)\nUMR 7550\nUniversité Louis Pasteur\nF-67000STRASBOURGFrance\n\nInstitut d'Astrophysique Spatiale\nUniversité Paris-Sud\nF-91405OrsayFrance",
"F Boulanger \nInstitut d'Astrophysique Spatiale\nUniversité Paris-Sud\nF-91405OrsayFrance",
"G Lagache \nIns... | We present an analysis of dust grain emission in the diffuse interstellar medium of the Small Magellanic Cloud (SMC). This study is motivated by the availability of 170 µm ISOPHOT data covering a large part of the SMC, with a resolution enabling to disentangle the diffuse medium from the star forming regions. After dat... | [
{
"title": "Introduction",
"content": "The Small Magellanic Cloud (SMC) is a dwarf irregular galaxy at a distance of approximately 60 kpc (Westerlund 1991). Its relative proximity enables to have a global view of the galaxy, as well as a sufficiently good resolution to disentangle the diffuse medium from th... | [
{
"id": "",
"caption": "Histogram of the 170 micron flux to H column density ratio (in MJy.cm 2 /sr). The gaussian fit and the residuals are presented as continuous and dotted lines respectively",
"content": "Fig. 5 .\n5Histogram of the 170 micron flux to H column density ratio (in MJy.cm 2 /sr). Th... | [
{
"text": ". J E Aguirre, J J Bezaire, E S Cheng, ApJ. 596273Aguirre, J. E., Bezaire, J. J., Cheng, E. S., et al. 2003, ApJ, 596, 273",
"doi": ""
},
{
"text": ". H H Aumann, J W Fowler, M Melnyk, AJ. 991674Aumann, H. H., Fowler, J. W., & Melnyk, M. 1990, AJ, 99, 1674",
"doi": ""
},
{
"te... |
Learning Features that Predict Cue Usage
22 Oct 1997
Barbara Di
Johanna D Moore jmoore@cs.pitt.edu
Computer Science Department
and Learning Re-search & Development Center ‡ Intelligent Systems Program
Massimo Paolucci paolucci@cs.pitt.edu
University of Pittsburgh Pittsburgh
15260PAUSA
Learning Features that Pr... | 979 | {
"arxiv": "cmp-lg/9710006",
"mag": 2951950125,
"acl": "P97-1011",
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/acl/EugenioMP97",
"doi": "10.3115/976909.979628"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/cmp-lg/9710006v1.pdf"
],
"pdf_sha": "c5b2e949e7ad88c7d2922b8e5c5f016e178db493",
"open_access": {
"license": null,
"url": "http://dl.acm.org/ft_gateway.cfm?id=979628&type=pdf",
"status": "BRONZE"
}
} | Learning Features that Predict Cue Usage | [
"Barbara Di",
"Johanna D Moore jmoore@cs.pitt.edu \nComputer Science Department\nand Learning Re-search & Development Center ‡ Intelligent Systems Program",
"Massimo Paolucci paolucci@cs.pitt.edu",
"University of Pittsburgh Pittsburgh\n15260PAUSA"
] | Our goal is to identify the features that predict the occurrence and placement of discourse cues in tutorial explanations in order to aid in the automatic generation of explanations.Previous attempts to devise rules for text generation were based on intuition or small numbers of constructed examples.We apply a machine ... | [
{
"title": "Introduction",
"content": "Discourse cues are words or phrases, such as because, first, and although, that mark structural and semantic relationships between discourse entities.They play a crucial role in many discourse processing tasks, including plan recognition (Litman and Allen, 1987), text ... | [
{
"id": "",
"caption": "Figure 1: The RDA analysis of (1)",
"content": "Figure 1 :\n1\nFigure 1: The RDA analysis of (1)"
},
{
"id": "",
"caption": "Figure 2: Decision tree for Core2 -occurrence",
"content": "Figure 2 :\n2\nFigure 2: Decision tree for Core2 -occurrence"
},
{
"id"... | [
{
"text": "A computational theory of the function of clue words in argument understanding. Robin ; Cohen, Michael Elhadad, Kathleen Mckeown, Proceedings of COLING84. COLING84Stanford, CA; Helsinki, Finland1984. 1990Proceedings of COLING90",
"doi": ""
},
{
"text": "The role of sequence markers in rea... |
M-Theory and the Light Cone
Jun 1999
Joseph Polchinski
Institute for Theoretical Physics
University of California
93106-4030Santa BarbaraCA
M-Theory and the Light Cone
Jun 1999arXiv:hep-th/9903165v2 14 1
I discuss D0-brane quantum mechanics as a nonperturbative formulation of string theory, in particular the relati... | 119,417,724 | {
"arxiv": "hep-th/9903165",
"mag": 2057905855,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1143/ptps.134.158"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/hep-th/9903165v2.pdf"
],
"pdf_sha": "981ed619ad6cc1752e7dfadba808277c6bddf0cb",
"open_access": {
"license": null,
"url": "https://academic.oup.com/ptps/article-pdf/doi/10.1143/PTPS.134.158/5160623/134-158.pdf",
"status": "BRONZE"
}
} | M-Theory and the Light Cone | [
"Joseph Polchinski \nInstitute for Theoretical Physics\nUniversity of California\n93106-4030Santa BarbaraCA"
] | I discuss D0-brane quantum mechanics as a nonperturbative formulation of string theory, in particular the relation between the Banks-Fischler-Shenker-Susskind matrix model, the Maldacena conjecture for D0-branes, and type IIA/M-theory duality. Some features of the quantum mechanics of D0-branes are also discussed. §1. ... | [
{
"title": "§1. Introduction",
"content": "The main focus of this lecture is D0-brane quantum mechanics as a nonperturbative formulation of string theory. In particular I would like to discuss the relation between the Banks-Fischler-Shenker-Susskind matrix model, 1) the Maldacena conjecture for D0-branes 2)... | [
{
"id": "",
"caption": "Space of string vacua. The cusps are limits in which a weakly coupled string description is possible (except for the M-theory limit). §2. The Douglas-Kabat-Pouliot-Shenker (DKPS) Limit Consider a state in the IIA string theory containing N D0-branes plus any collection of open and cl... | [
{
"text": ". T Banks, W Fischler, S H Shenker, L Susskind, hep-th/9610043Phys. Rev. 55T. Banks, W. Fischler, S. H. Shenker, and L. Susskind, Phys. Rev. D55 (1961) 5112, hep-th/9610043.",
"doi": "hep-th/9610043"
},
{
"text": ". J Maldacena, hep-th/9711200Adv. Theor. Math. Phys. 2J. Maldacena, Adv. Th... |
Made-to-measure N -body systems
18 September 2018
D Syer
Canadian Institute for Theoretical Astrophysics
McLennan Labs
60 St. George StreetM5S 1A7TorontoOntarioCanada
Max-Planck-Institut für Astrophysik
Karl-Schwarzschild-Straße 185748Garching-bei-MünchenGermany
S Tremaine
Canadian Institute for Theoretical Astro... | 3,256,508 | {
"arxiv": "astro-ph/9605061",
"mag": 2112706506,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1093/mnras/282.1.223"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/astro-ph/9605061v1.pdf"
],
"pdf_sha": "612a335a9ddf0fb303994b5eabf821cc41d89fca",
"open_access": {
"license": null,
"url": "https://academic.oup.com/mnras/article-pdf/282/1/223/18199992/282-1-223.pdf",
"status": "BRONZE"
}
} | Made-to-measure N -body systems | [
"D Syer \nCanadian Institute for Theoretical Astrophysics\nMcLennan Labs\n60 St. George StreetM5S 1A7TorontoOntarioCanada\n\nMax-Planck-Institut für Astrophysik\nKarl-Schwarzschild-Straße 185748Garching-bei-MünchenGermany",
"S Tremaine \nCanadian Institute for Theoretical Astrophysics\nMcLennan Labs\n60 St. Georg... | We describe an algorithm for constructing N -body realisations of equilibrium stellar systems. The algorithm complements existing orbit-based modelling techniques using linear programming or other optimization algorithms. The equilibria are constructed by integrating an N -body system while slowly adjusting the masses ... | [
{
"title": "INTRODUCTION",
"content": "One of the central problems in stellar dynamics is to construct made-to-measure stellar systems. For example, (i) when modelling observations of an elliptical galaxy we wish to find a phase-space distribution function f (r, v) (hereafter DF) that solves Poisson's equat... | [
{
"id": "",
"caption": "The weights {w i } as a function of time for simulation A.",
"content": "Figure 1 .\n1The weights {w i } as a function of time for simulation A."
},
{
"id": "",
"caption": "The density error χ for simulations A (top curve), C (middle curve) and D (bottom curve).",
... | [
{
"text": ". J Binney, G A Mamon, MNRAS. 361Binney, J., & Mamon, G. A. 1982,MNRAS, 200, 361",
"doi": ""
},
{
"text": ". J J Binney, R L Davies, G D Illingworth, ApJ. 36178Binney, J. J., Davies, R. L., & Illingworth, G. D. (1990), ApJ, 361, 78",
"doi": ""
},
{
"text": ". H Dejonghe, ApJ. ... |
From Arguments to Key Points: Towards Automatic Argument Summarization
Association for Computational LinguisticsCopyright Association for Computational LinguisticsJuly 5 -10, 2020. 2020
Roy Bar roybar@il.ibm.com
IBM Research
Haim Lilach lilache@il.ibm.com
IBM Research
Eden Roni Friedman
IBM Research
Yoav Kan... | 218,487,506 | {
"arxiv": "2005.01619",
"mag": 3035554416,
"acl": "2020.acl-main.371",
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/acl/Bar-HaimEFKLS20",
"doi": "10.18653/v1/2020.acl-main.371"
} | {
"pdf_urls": [
"https://www.aclweb.org/anthology/2020.acl-main.371.pdf"
],
"pdf_sha": "eb3718caf089629c90b58775019e74b41b4028ca",
"open_access": {
"license": "CCBY",
"url": "https://www.aclweb.org/anthology/2020.acl-main.371.pdf",
"status": "HYBRID"
}
} | From Arguments to Key Points: Towards Automatic Argument Summarization | [
"Roy Bar roybar@il.ibm.com \nIBM Research",
"Haim Lilach lilache@il.ibm.com \nIBM Research",
"Eden Roni Friedman \nIBM Research",
"Yoav Kantor yoavka@il.ibm.com \nIBM Research",
"Dan Lahav dan.lahav@ibm.com \nIBM Research",
"Noam Slonim noams@il.ibm.com \nIBM Research"
] | Generating a concise summary from a large collection of arguments on a given topic is an intriguing yet understudied problem. We propose to represent such summaries as a small set of talking points, termed key points, each scored according to its salience. We show, by analyzing a large dataset of crowd-contributed argu... | [
{
"title": "Introduction",
"content": "Governments, businesses and individuals, all need to make decisions on a daily basis: \"Should cannabis be legalized?\",\"Should we develop this product?\", \"Should I become a vegetarian?\". When making an important decision, the process typically comprises several st... | [
{
"id": "",
"caption": "Argument coverage per number of key points.",
"content": "Figure 1 :\n1Argument coverage per number of key points."
},
{
"id": "",
"caption": "Precision/Recall trade-off for different key point selection policies. For each method, the highest F1 score, as well as the ... | [
{
"text": "Association for Computational Linguistics. Yamen Ajjour, Milad Alshomary, Henning Wachsmuth, Benno Stein, 10.18653/v1/D19-1290Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)... |
Nonlocal scalar field in an external potential: WKB approximation
Valeri P Frolov
Theoretical Physics Institute
University of Alberta
T6G 2E1EdmontonAlbertaCanada
Nonlocal scalar field in an external potential: WKB approximation
We consider a nonlocal theory of a scalar massive field in a flat spacetime backgroun... | 227,152,308 | {
"arxiv": "2011.11700",
"mag": 3124882961,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevd.103.025020"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2011.11700v1.pdf"
],
"pdf_sha": "cc3e7ff94da27a10534aa0979b31d6b6f12fefa3",
"open_access": {
"license": "CCBY",
"url": "http://link.aps.org/pdf/10.1103/PhysRevD.103.025020",
"status": "HYBRID"
}
} | Nonlocal scalar field in an external potential: WKB approximation | [
"Valeri P Frolov \nTheoretical Physics Institute\nUniversity of Alberta\nT6G 2E1EdmontonAlbertaCanada"
] | We consider a nonlocal theory of a scalar massive field in a flat spacetime background in the presence of an external potential and construct WKB solutions for this theory. We use a model in which the kinetic part of the scalar field action is modified by changing to f ( ) operator. We discuss conditions when the corre... | [
{
"title": "I. INTRODUCTION",
"content": "The idea of nonlocality is quite old in the theoretical physics. Nonlocal modifications of the field theory were discussed already in the publications [1][2][3]. Even if one works with a local quantum field theory, already at the one loop level its effective action ... | [
{
"id": "",
"caption": "Function X(x).",
"content": "FIG. 1 .\n1Function X(x)."
},
{
"id": "",
"caption": "defines a closed curve on (q, η) phase plane (see figure 2).",
"content": "defines a closed curve on (q, η) phase plane (see figure 2)."
},
{
"id": "",
"caption": "Phase... | [
{
"text": ". G Wataghin, 10.1007/BF01352311Z. Phys. 8892G. Wataghin, Z. Phys. 88, 92 (1934).",
"doi": ""
},
{
"text": ". H Yukawa, 10.1103/PhysRev.77.219Phys. Rev. 77219H. Yukawa, Phys. Rev. 77, 219 (1950).",
"doi": ""
},
{
"text": ". H Yukawa, 10.1103/PhysRev.80.1047Phys. Rev. 801047H. ... |
Polynomial equations for additive functions I. The inner parameter case
August 31, 2023
Eszter Gselmann gselmann@science.unideb.hu
University of Debrecen
Gergely Kiss
University of Debrecen
Polynomial equations for additive functions I. The inner parameter case
August 31, 2023EA85966ED7279EBC9B8F373053D85867arX... | 253,384,750 | {
"arxiv": "2211.03605",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1007/s00025-023-02087-y"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2211.03605v2.pdf"
],
"pdf_sha": "64840f61a1a64b4b36a0646a7121a2e7ae8b7088",
"open_access": {
"license": "CCBY",
"url": "https://link.springer.com/content/pdf/10.1007/s00025-023-02087-y.pdf",
"status": "HYBRID"
}
} | Polynomial equations for additive functions I. The inner parameter case | [
"Eszter Gselmann gselmann@science.unideb.hu \nUniversity of Debrecen",
"Gergely Kiss \nUniversity of Debrecen"
] | The aim of this sequence of work is to investigate polynomial equations satisfied by additive functions.As a result of this, new characterization theorems for homomorphisms and derivations can be given.More exactly, in this paper the following type of equation is consideredwhere n is a positive integer, F Ă C is a fiel... | [
{
"title": "Introduction and preliminaries",
"content": "Equations satisfied by additive functions play an important role not only in the theory of commutative algebra, but also in the theory of functional equations.It is an important and challenging question how special morphisms (such as homomorphisms and... | [
{
"id": "",
"caption": "˝¨¨¨˝d β r r pxq ¨dα 1 ´β1 1 ˝¨¨¨˝d α r ´βr r pyq \" ÿ βďα ˆα β ˙ϕβ pxqϕ α´β pyq `x, y P F ˆ˘.",
"content": "α 1 times ˝¨¨¨˝d r ˝¨¨¨˝d r looooomooooon α r times pxyq \" d 1 ˝¨¨¨˝d 1 looooomooooon α 1 times ˝¨¨¨˝d r´1 ˝¨¨¨˝d r´1 loooooooomoooooooon α r´1 times pd α r r pxyqq \" d ... | [
{
"text": "Characterizing ring derivations of all orders via functional equations: results and open problems. Bruce Ebanks, Aequationes Math. 8932015",
"doi": ""
},
{
"text": "Polynomially linked additive functions. Bruce Ebanks, Aequationes Math. 9122017",
"doi": ""
},
{
"text": "On the... |